Article MOD_4, MML version 4.99.1005
:: MOD_4:funcnot 1 => MOD_4:func 1
definition
let a1, a2, a3 be non empty set;
let a4 be Function-like quasi_total Relation of [:a1,a2:],a3;
redefine func ~ a4 -> Function-like quasi_total Relation of [:a2,a1:],a3;
end;
:: MOD_4:th 1
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of [:b2,b3:],b1
for b5 being Element of b2
for b6 being Element of b3 holds
b4 .(b5,b6) = (~ b4) .(b6,b5);
:: MOD_4:funcnot 2 => MOD_4:func 2
definition
let a1 be non empty doubleLoopStr;
func opp A1 -> strict doubleLoopStr equals
doubleLoopStr(#the carrier of a1,the addF of a1,~ the multF of a1,1. a1,0. a1#);
end;
:: MOD_4:def 1
theorem
for b1 being non empty doubleLoopStr holds
opp b1 = doubleLoopStr(#the carrier of b1,the addF of b1,~ the multF of b1,1. b1,0. b1#);
:: MOD_4:funcreg 1
registration
let a1 be non empty doubleLoopStr;
cluster opp a1 -> non empty strict;
end;
:: MOD_4:funcreg 2
registration
let a1 be non empty well-unital doubleLoopStr;
cluster opp a1 -> strict well-unital;
end;
:: MOD_4:funcreg 3
registration
let a1 be non empty right_complementable add-associative right_zeroed doubleLoopStr;
cluster opp a1 -> right_complementable strict add-associative right_zeroed;
end;
:: MOD_4:th 2
theorem
for b1 being non empty doubleLoopStr holds
addLoopStr(#the carrier of opp b1,the addF of opp b1,the ZeroF of opp b1#) = addLoopStr(#the carrier of b1,the addF of b1,the ZeroF of b1#) &
(b1 is add-associative & b1 is right_zeroed & b1 is right_complementable implies comp opp b1 = comp b1) &
(for b2 being set holds
b2 is Element of the carrier of opp b1
iff
b2 is Element of the carrier of b1);
:: MOD_4:th 4
theorem
(for b1 being non empty unital doubleLoopStr holds
1. b1 = 1. opp b1) &
(for b1 being non empty right_complementable add-associative right_zeroed doubleLoopStr holds
0. b1 = 0. opp b1 &
(for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8, b9 being Element of the carrier of opp b1
st b2 = b6 & b3 = b7 & b4 = b8 & b5 = b9
holds b2 + b3 = b6 + b7 &
b2 * b3 = b7 * b6 &
- b2 = - b6 &
(b2 + b3) + b4 = (b6 + b7) + b8 &
b2 + (b3 + b4) = b6 + (b7 + b8) &
(b2 * b3) * b4 = b8 * (b7 * b6) &
b2 * (b3 * b4) = (b8 * b7) * b6 &
b2 * (b3 + b4) = (b7 + b8) * b6 &
(b3 + b4) * b2 = b6 * (b7 + b8) &
(b2 * b3) + (b4 * b5) = (b7 * b6) + (b9 * b8)));
:: MOD_4:funcreg 4
registration
let a1 be non empty Abelian doubleLoopStr;
cluster opp a1 -> strict Abelian;
end;
:: MOD_4:funcreg 5
registration
let a1 be non empty add-associative doubleLoopStr;
cluster opp a1 -> strict add-associative;
end;
:: MOD_4:funcreg 6
registration
let a1 be non empty right_zeroed doubleLoopStr;
cluster opp a1 -> strict right_zeroed;
end;
:: MOD_4:funcreg 7
registration
let a1 be non empty right_complementable doubleLoopStr;
cluster opp a1 -> right_complementable strict;
end;
:: MOD_4:funcreg 8
registration
let a1 be non empty associative doubleLoopStr;
cluster opp a1 -> strict associative;
end;
:: MOD_4:funcreg 9
registration
let a1 be non empty well-unital doubleLoopStr;
cluster opp a1 -> strict well-unital;
end;
:: MOD_4:funcreg 10
registration
let a1 be non empty distributive doubleLoopStr;
cluster opp a1 -> strict distributive;
end;
:: MOD_4:th 5
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
opp b1 is non empty right_complementable strict associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
:: MOD_4:funcreg 11
registration
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
cluster opp a1 -> right_complementable strict unital distributive Abelian add-associative right_zeroed;
end;
:: MOD_4:th 6
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
opp b1 is non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
:: MOD_4:funcreg 12
registration
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
cluster opp a1 -> strict associative;
end;
:: MOD_4:th 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
opp b1 is non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
:: MOD_4:funcreg 13
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
cluster opp a1 -> non degenerated right_complementable almost_left_invertible strict unital associative distributive Abelian add-associative right_zeroed;
end;
:: MOD_4:th 8
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
opp b1 is non empty non degenerated right_complementable almost_left_invertible strict associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
:: MOD_4:funcreg 14
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
cluster opp a1 -> almost_left_invertible strict;
end;
:: MOD_4:funcnot 3 => MOD_4:func 3
definition
let a1 be non empty doubleLoopStr;
let a2 be non empty VectSpStr over a1;
func opp A2 -> strict RightModStr over opp a1 means
for b1 being Function-like quasi_total Relation of [:the carrier of a2,the carrier of opp a1:],the carrier of a2
st b1 = ~ the lmult of a2
holds it = RightModStr(#the carrier of a2,the addF of a2,0. a2,b1#);
end;
:: MOD_4:def 2
theorem
for b1 being non empty doubleLoopStr
for b2 being non empty VectSpStr over b1
for b3 being strict RightModStr over opp b1 holds
b3 = opp b2
iff
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of opp b1:],the carrier of b2
st b4 = ~ the lmult of b2
holds b3 = RightModStr(#the carrier of b2,the addF of b2,0. b2,b4#);
:: MOD_4:funcreg 15
registration
let a1 be non empty doubleLoopStr;
let a2 be non empty VectSpStr over a1;
cluster opp a2 -> non empty strict;
end;
:: MOD_4:th 9
theorem
for b1 being non empty doubleLoopStr
for b2 being non empty VectSpStr over b1 holds
addLoopStr(#the carrier of opp b2,the addF of opp b2,the ZeroF of opp b2#) = addLoopStr(#the carrier of b2,the addF of b2,the ZeroF of b2#) &
(for b3 being set holds
b3 is Element of the carrier of b2
iff
b3 is Element of the carrier of opp b2);
:: MOD_4:funcnot 4 => MOD_4:func 4
definition
let a1 be non empty doubleLoopStr;
let a2 be non empty VectSpStr over a1;
let a3 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a2:],the carrier of a2;
func opp A3 -> Function-like quasi_total Relation of [:the carrier of opp a2,the carrier of opp a1:],the carrier of opp a2 equals
~ a3;
end;
:: MOD_4:def 3
theorem
for b1 being non empty doubleLoopStr
for b2 being non empty VectSpStr over b1
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b2:],the carrier of b2 holds
opp b3 = ~ b3;
:: MOD_4:th 10
theorem
for b1 being non empty doubleLoopStr
for b2 being non empty VectSpStr over b1 holds
the rmult of opp b2 = opp the lmult of b2;
:: MOD_4:funcnot 5 => MOD_4:func 5
definition
let a1 be non empty doubleLoopStr;
let a2 be non empty RightModStr over a1;
func opp A2 -> strict VectSpStr over opp a1 means
for b1 being Function-like quasi_total Relation of [:the carrier of opp a1,the carrier of a2:],the carrier of a2
st b1 = ~ the rmult of a2
holds it = VectSpStr(#the carrier of a2,the addF of a2,0. a2,b1#);
end;
:: MOD_4:def 4
theorem
for b1 being non empty doubleLoopStr
for b2 being non empty RightModStr over b1
for b3 being strict VectSpStr over opp b1 holds
b3 = opp b2
iff
for b4 being Function-like quasi_total Relation of [:the carrier of opp b1,the carrier of b2:],the carrier of b2
st b4 = ~ the rmult of b2
holds b3 = VectSpStr(#the carrier of b2,the addF of b2,0. b2,b4#);
:: MOD_4:funcreg 16
registration
let a1 be non empty doubleLoopStr;
let a2 be non empty RightModStr over a1;
cluster opp a2 -> non empty strict;
end;
:: MOD_4:th 12
theorem
for b1 being non empty doubleLoopStr
for b2 being non empty RightModStr over b1 holds
addLoopStr(#the carrier of opp b2,the addF of opp b2,the ZeroF of opp b2#) = addLoopStr(#the carrier of b2,the addF of b2,the ZeroF of b2#) &
(for b3 being set holds
b3 is Element of the carrier of b2
iff
b3 is Element of the carrier of opp b2);
:: MOD_4:funcnot 6 => MOD_4:func 6
definition
let a1 be non empty doubleLoopStr;
let a2 be non empty RightModStr over a1;
let a3 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a1:],the carrier of a2;
func opp A3 -> Function-like quasi_total Relation of [:the carrier of opp a1,the carrier of opp a2:],the carrier of opp a2 equals
~ a3;
end;
:: MOD_4:def 5
theorem
for b1 being non empty doubleLoopStr
for b2 being non empty RightModStr over b1
for b3 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b1:],the carrier of b2 holds
opp b3 = ~ b3;
:: MOD_4:th 13
theorem
for b1 being non empty doubleLoopStr
for b2 being non empty RightModStr over b1 holds
the lmult of opp b2 = opp the rmult of b2;
:: MOD_4:th 16
theorem
for b1 being non empty doubleLoopStr
for b2 being non empty VectSpStr over b1
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b2:],the carrier of b2
for b4 being Element of the carrier of b1
for b5 being Element of the carrier of opp b1
for b6 being Element of the carrier of b2
for b7 being Element of the carrier of opp b2
st b4 = b5 & b6 = b7
holds (opp b3) .(b7,b5) = b3 .(b4,b6);
:: MOD_4:th 17
theorem
for b1, b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty VectSpStr over b1
for b4 being non empty RightModStr over b2
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2
for b7 being Element of the carrier of b3
for b8 being Element of the carrier of b4
st b2 = opp b1 & b4 = opp b3 & b5 = b6 & b7 = b8
holds b8 * b6 = b5 * b7;
:: MOD_4:th 18
theorem
for b1, b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty VectSpStr over b1
for b4 being non empty RightModStr over b2
for b5, b6 being Element of the carrier of b3
for b7, b8 being Element of the carrier of b4
st b2 = opp b1 & b4 = opp b3 & b5 = b7 & b6 = b8
holds b7 + b8 = b5 + b6;
:: MOD_4:th 20
theorem
for b1 being non empty doubleLoopStr
for b2 being non empty RightModStr over b1
for b3 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b1:],the carrier of b2
for b4 being Element of the carrier of b1
for b5 being Element of the carrier of opp b1
for b6 being Element of the carrier of b2
for b7 being Element of the carrier of opp b2
st b4 = b5 & b6 = b7
holds (opp b3) .(b5,b7) = b3 .(b6,b4);
:: MOD_4:th 21
theorem
for b1, b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty VectSpStr over b1
for b4 being non empty RightModStr over b2
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2
for b7 being Element of the carrier of b3
for b8 being Element of the carrier of b4
st b1 = opp b2 & b3 = opp b4 & b5 = b6 & b7 = b8
holds b8 * b6 = b5 * b7;
:: MOD_4:th 22
theorem
for b1, b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty VectSpStr over b1
for b4 being non empty RightModStr over b2
for b5, b6 being Element of the carrier of b3
for b7, b8 being Element of the carrier of b4
st b1 = opp b2 & b3 = opp b4 & b5 = b7 & b6 = b8
holds b7 + b8 = b5 + b6;
:: MOD_4:th 23
theorem
for b1 being non empty strict doubleLoopStr
for b2 being non empty VectSpStr over b1 holds
opp opp b2 = VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#);
:: MOD_4:th 24
theorem
for b1 being non empty strict doubleLoopStr
for b2 being non empty RightModStr over b1 holds
opp opp b2 = RightModStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the rmult of b2#);
:: MOD_4:th 25
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
opp b2 is non empty right_complementable Abelian add-associative right_zeroed strict RightMod-like RightModStr over opp b1;
:: MOD_4:funcreg 17
registration
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
cluster opp a2 -> right_complementable Abelian add-associative right_zeroed strict RightMod-like;
end;
:: MOD_4:th 26
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1 holds
opp b2 is non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over opp b1;
:: MOD_4:funcreg 18
registration
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
cluster opp a2 -> right_complementable strict VectSp-like Abelian add-associative right_zeroed;
end;
:: MOD_4:attrnot 1 => MOD_4:attr 1
definition
let a1, a2 be non empty multMagma;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is antimultiplicative means
for b1, b2 being Element of the carrier of a1 holds
a3 . (b1 * b2) = (a3 . b2) * (a3 . b1);
end;
:: MOD_4:dfs 6
definiens
let a1, a2 be non empty multMagma;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is antimultiplicative
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
a3 . (b1 * b2) = (a3 . b2) * (a3 . b1);
:: MOD_4:def 6
theorem
for b1, b2 being non empty multMagma
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is antimultiplicative(b1, b2)
iff
for b4, b5 being Element of the carrier of b1 holds
b3 . (b4 * b5) = (b3 . b5) * (b3 . b4);
:: MOD_4:attrnot 2 => MOD_4:attr 2
definition
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is antilinear means
a3 is additive(a1, a2) & a3 is antimultiplicative(a1, a2) & a3 is unity-preserving(a1, a2);
end;
:: MOD_4:dfs 7
definiens
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is antilinear
it is sufficient to prove
thus a3 is additive(a1, a2) & a3 is antimultiplicative(a1, a2) & a3 is unity-preserving(a1, a2);
:: MOD_4:def 7
theorem
for b1, b2 being non empty doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is antilinear(b1, b2)
iff
b3 is additive(b1, b2) & b3 is antimultiplicative(b1, b2) & b3 is unity-preserving(b1, b2);
:: MOD_4:condreg 1
registration
let a1, a2 be non empty doubleLoopStr;
cluster Function-like quasi_total antilinear -> unity-preserving additive antimultiplicative (Relation of the carrier of a1,the carrier of a2);
end;
:: MOD_4:condreg 2
registration
let a1, a2 be non empty doubleLoopStr;
cluster Function-like quasi_total unity-preserving additive antimultiplicative -> antilinear (Relation of the carrier of a1,the carrier of a2);
end;
:: MOD_4:attrnot 3 => MOD_4:attr 3
definition
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is monomorphism means
a3 is linear(a1, a2) & a3 is one-to-one;
end;
:: MOD_4:dfs 8
definiens
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is monomorphism
it is sufficient to prove
thus a3 is linear(a1, a2) & a3 is one-to-one;
:: MOD_4:def 8
theorem
for b1, b2 being non empty doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is monomorphism(b1, b2)
iff
b3 is linear(b1, b2) & b3 is one-to-one;
:: MOD_4:attrnot 4 => MOD_4:attr 4
definition
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is antimonomorphism means
a3 is antilinear(a1, a2) & a3 is one-to-one;
end;
:: MOD_4:dfs 9
definiens
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is antimonomorphism
it is sufficient to prove
thus a3 is antilinear(a1, a2) & a3 is one-to-one;
:: MOD_4:def 9
theorem
for b1, b2 being non empty doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is antimonomorphism(b1, b2)
iff
b3 is antilinear(b1, b2) & b3 is one-to-one;
:: MOD_4:attrnot 5 => MOD_4:attr 5
definition
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is epimorphism means
a3 is linear(a1, a2) & proj2 a3 = the carrier of a2;
end;
:: MOD_4:dfs 10
definiens
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is epimorphism
it is sufficient to prove
thus a3 is linear(a1, a2) & proj2 a3 = the carrier of a2;
:: MOD_4:def 10
theorem
for b1, b2 being non empty doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is epimorphism(b1, b2)
iff
b3 is linear(b1, b2) & proj2 b3 = the carrier of b2;
:: MOD_4:attrnot 6 => MOD_4:attr 6
definition
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is antiepimorphism means
a3 is antilinear(a1, a2) & proj2 a3 = the carrier of a2;
end;
:: MOD_4:dfs 11
definiens
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is antiepimorphism
it is sufficient to prove
thus a3 is antilinear(a1, a2) & proj2 a3 = the carrier of a2;
:: MOD_4:def 11
theorem
for b1, b2 being non empty doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is antiepimorphism(b1, b2)
iff
b3 is antilinear(b1, b2) & proj2 b3 = the carrier of b2;
:: MOD_4:attrnot 7 => MOD_4:attr 7
definition
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is isomorphism means
a3 is monomorphism(a1, a2) & proj2 a3 = the carrier of a2;
end;
:: MOD_4:dfs 12
definiens
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is isomorphism
it is sufficient to prove
thus a3 is monomorphism(a1, a2) & proj2 a3 = the carrier of a2;
:: MOD_4:def 12
theorem
for b1, b2 being non empty doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is isomorphism(b1, b2)
iff
b3 is monomorphism(b1, b2) & proj2 b3 = the carrier of b2;
:: MOD_4:attrnot 8 => MOD_4:attr 8
definition
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is antiisomorphism means
a3 is antimonomorphism(a1, a2) & proj2 a3 = the carrier of a2;
end;
:: MOD_4:dfs 13
definiens
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is antiisomorphism
it is sufficient to prove
thus a3 is antimonomorphism(a1, a2) & proj2 a3 = the carrier of a2;
:: MOD_4:def 13
theorem
for b1, b2 being non empty doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is antiisomorphism(b1, b2)
iff
b3 is antimonomorphism(b1, b2) & proj2 b3 = the carrier of b2;
:: MOD_4:attrnot 9 => MOD_4:attr 9
definition
let a1 be non empty doubleLoopStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
attr a2 is endomorphism means
a2 is linear(a1, a1);
end;
:: MOD_4:dfs 14
definiens
let a1 be non empty doubleLoopStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
To prove
a2 is endomorphism
it is sufficient to prove
thus a2 is linear(a1, a1);
:: MOD_4:def 14
theorem
for b1 being non empty doubleLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 is endomorphism(b1)
iff
b2 is linear(b1, b1);
:: MOD_4:attrnot 10 => MOD_4:attr 10
definition
let a1 be non empty doubleLoopStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
attr a2 is antiendomorphism means
a2 is antilinear(a1, a1);
end;
:: MOD_4:dfs 15
definiens
let a1 be non empty doubleLoopStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
To prove
a2 is antiendomorphism
it is sufficient to prove
thus a2 is antilinear(a1, a1);
:: MOD_4:def 15
theorem
for b1 being non empty doubleLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 is antiendomorphism(b1)
iff
b2 is antilinear(b1, b1);
:: MOD_4:attrnot 11 => MOD_4:attr 11
definition
let a1 be non empty doubleLoopStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
attr a2 is automorphism means
a2 is isomorphism(a1, a1);
end;
:: MOD_4:dfs 16
definiens
let a1 be non empty doubleLoopStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
To prove
a2 is automorphism
it is sufficient to prove
thus a2 is isomorphism(a1, a1);
:: MOD_4:def 16
theorem
for b1 being non empty doubleLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 is automorphism(b1)
iff
b2 is isomorphism(b1, b1);
:: MOD_4:attrnot 12 => MOD_4:attr 12
definition
let a1 be non empty doubleLoopStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
attr a2 is antiautomorphism means
a2 is antiisomorphism(a1, a1);
end;
:: MOD_4:dfs 17
definiens
let a1 be non empty doubleLoopStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
To prove
a2 is antiautomorphism
it is sufficient to prove
thus a2 is antiisomorphism(a1, a1);
:: MOD_4:def 17
theorem
for b1 being non empty doubleLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 is antiautomorphism(b1)
iff
b2 is antiisomorphism(b1, b1);
:: MOD_4:th 27
theorem
for b1 being non empty doubleLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 is automorphism(b1)
iff
(for b3, b4 being Element of the carrier of b1 holds
b2 . (b3 + b4) = (b2 . b3) + (b2 . b4)) &
(for b3, b4 being Element of the carrier of b1 holds
b2 . (b3 * b4) = (b2 . b3) * (b2 . b4)) &
b2 . 1_ b1 = 1_ b1 &
b2 is one-to-one &
proj2 b2 = the carrier of b1;
:: MOD_4:th 28
theorem
for b1 being non empty doubleLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 is antiautomorphism(b1)
iff
(for b3, b4 being Element of the carrier of b1 holds
b2 . (b3 + b4) = (b2 . b3) + (b2 . b4)) &
(for b3, b4 being Element of the carrier of b1 holds
b2 . (b3 * b4) = (b2 . b4) * (b2 . b3)) &
b2 . 1_ b1 = 1_ b1 &
b2 is one-to-one &
proj2 b2 = the carrier of b1;
:: MOD_4:th 29
theorem
for b1 being non empty doubleLoopStr holds
id b1 is automorphism(b1);
:: MOD_4:th 30
theorem
for b1, b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4, b5 being Element of the carrier of b1
st b3 is linear(b1, b2)
holds b3 . 0. b1 = 0. b2 &
b3 . - b4 = - (b3 . b4) &
b3 . (b4 - b5) = (b3 . b4) - (b3 . b5);
:: MOD_4:th 31
theorem
for b1, b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4, b5 being Element of the carrier of b1
st b3 is antilinear(b1, b2)
holds b3 . 0. b1 = 0. b2 &
b3 . - b4 = - (b3 . b4) &
b3 . (b4 - b5) = (b3 . b4) - (b3 . b5);
:: MOD_4:th 32
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
id b1 is antiautomorphism(b1)
iff
b1 is non empty right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
:: MOD_4:th 33
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
id b1 is antiautomorphism(b1)
iff
b1 is non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
:: MOD_4:funcnot 7 => MOD_4:func 7
definition
let a1, a2 be non empty doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
func opp A3 -> Function-like quasi_total Relation of the carrier of a1,the carrier of opp a2 equals
a3;
end;
:: MOD_4:def 18
theorem
for b1, b2 being non empty doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
opp b3 = b3;
:: MOD_4:th 34
theorem
for b1, b2 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
opp opp b3 = b3;
:: MOD_4:th 35
theorem
for b1 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is linear(b1, b2)
iff
opp b3 is antilinear(b1, opp b2);
:: MOD_4:th 36
theorem
for b1 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is antilinear(b1, b2)
iff
opp b3 is linear(b1, opp b2);
:: MOD_4:th 37
theorem
for b1 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is monomorphism(b1, b2)
iff
opp b3 is antimonomorphism(b1, opp b2);
:: MOD_4:th 38
theorem
for b1 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is antimonomorphism(b1, b2)
iff
opp b3 is monomorphism(b1, opp b2);
:: MOD_4:th 39
theorem
for b1 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is epimorphism(b1, b2)
iff
opp b3 is antiepimorphism(b1, opp b2);
:: MOD_4:th 40
theorem
for b1 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is antiepimorphism(b1, b2)
iff
opp b3 is epimorphism(b1, opp b2);
:: MOD_4:th 41
theorem
for b1 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is isomorphism(b1, b2)
iff
opp b3 is antiisomorphism(b1, opp b2);
:: MOD_4:th 42
theorem
for b1 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is antiisomorphism(b1, b2)
iff
opp b3 is isomorphism(b1, opp b2);
:: MOD_4:th 43
theorem
for b1 being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 is endomorphism(b1)
iff
opp b2 is antilinear(b1, opp b1);
:: MOD_4:th 44
theorem
for b1 being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 is antiendomorphism(b1)
iff
opp b2 is linear(b1, opp b1);
:: MOD_4:th 45
theorem
for b1 being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 is automorphism(b1)
iff
opp b2 is antiisomorphism(b1, opp b1);
:: MOD_4:th 46
theorem
for b1 being non empty right_complementable well-unital add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 is antiautomorphism(b1)
iff
opp b2 is isomorphism(b1, opp b1);
:: MOD_4:modenot 1 => MOD_4:mode 1
definition
let a1, a2 be non empty right_complementable add-associative right_zeroed addLoopStr;
mode Homomorphism of A1,A2 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a2 means
for b1, b2 being Element of the carrier of a1 holds
it . (b1 + b2) = (it . b1) + (it . b2);
end;
:: MOD_4:dfs 19
definiens
let a1, a2 be non empty right_complementable add-associative right_zeroed addLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is Homomorphism of a1,a2
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
a3 . (b1 + b2) = (a3 . b1) + (a3 . b2);
:: MOD_4:def 19
theorem
for b1, b2 being non empty right_complementable add-associative right_zeroed addLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is Homomorphism of b1,b2
iff
for b4, b5 being Element of the carrier of b1 holds
b3 . (b4 + b5) = (b3 . b4) + (b3 . b5);
:: MOD_4:funcnot 8 => MOD_4:func 8
definition
let a1, a2 be non empty right_complementable add-associative right_zeroed addLoopStr;
redefine func ZeroMap(a1,a2) -> Homomorphism of a1,a2;
end;
:: MOD_4:attrnot 13 => MOD_4:attr 13
definition
let a1, a2 be non empty right_complementable add-associative right_zeroed addLoopStr;
let a3 be Homomorphism of a1,a2;
attr a3 is monomorphism means
a3 is one-to-one;
end;
:: MOD_4:dfs 20
definiens
let a1, a2 be non empty right_complementable add-associative right_zeroed addLoopStr;
let a3 be Homomorphism of a1,a2;
To prove
a3 is monomorphism
it is sufficient to prove
thus a3 is one-to-one;
:: MOD_4:def 20
theorem
for b1, b2 being non empty right_complementable add-associative right_zeroed addLoopStr
for b3 being Homomorphism of b1,b2 holds
b3 is monomorphism(b1, b2)
iff
b3 is one-to-one;
:: MOD_4:attrnot 14 => MOD_4:attr 14
definition
let a1, a2 be non empty right_complementable add-associative right_zeroed addLoopStr;
let a3 be Homomorphism of a1,a2;
attr a3 is epimorphism means
proj2 a3 = the carrier of a2;
end;
:: MOD_4:dfs 21
definiens
let a1, a2 be non empty right_complementable add-associative right_zeroed addLoopStr;
let a3 be Homomorphism of a1,a2;
To prove
a3 is epimorphism
it is sufficient to prove
thus proj2 a3 = the carrier of a2;
:: MOD_4:def 21
theorem
for b1, b2 being non empty right_complementable add-associative right_zeroed addLoopStr
for b3 being Homomorphism of b1,b2 holds
b3 is epimorphism(b1, b2)
iff
proj2 b3 = the carrier of b2;
:: MOD_4:attrnot 15 => MOD_4:attr 15
definition
let a1, a2 be non empty right_complementable add-associative right_zeroed addLoopStr;
let a3 be Homomorphism of a1,a2;
attr a3 is isomorphism means
a3 is one-to-one & proj2 a3 = the carrier of a2;
end;
:: MOD_4:dfs 22
definiens
let a1, a2 be non empty right_complementable add-associative right_zeroed addLoopStr;
let a3 be Homomorphism of a1,a2;
To prove
a3 is isomorphism
it is sufficient to prove
thus a3 is one-to-one & proj2 a3 = the carrier of a2;
:: MOD_4:def 22
theorem
for b1, b2 being non empty right_complementable add-associative right_zeroed addLoopStr
for b3 being Homomorphism of b1,b2 holds
b3 is isomorphism(b1, b2)
iff
b3 is one-to-one & proj2 b3 = the carrier of b2;
:: MOD_4:modenot 2
definition
let a1 be non empty right_complementable add-associative right_zeroed addLoopStr;
mode Endomorphism of a1 is Homomorphism of a1,a1;
end;
:: MOD_4:exreg 1
registration
let a1 be non empty right_complementable add-associative right_zeroed addLoopStr;
cluster Relation-like Function-like quasi_total isomorphism Homomorphism of a1,a1;
end;
:: MOD_4:modenot 3
definition
let a1 be non empty right_complementable add-associative right_zeroed addLoopStr;
mode Automorphism of a1 is isomorphism Homomorphism of a1,a1;
end;
:: MOD_4:funcnot 9 => MOD_4:func 9
definition
let a1 be non empty right_complementable add-associative right_zeroed addLoopStr;
redefine func id a1 -> isomorphism Homomorphism of a1,a1;
end;
:: MOD_4:th 48
theorem
for b1, b2 being non empty right_complementable add-associative right_zeroed addLoopStr
for b3 being Homomorphism of b1,b2
for b4, b5 being Element of the carrier of b1 holds
b3 . 0. b1 = 0. b2 &
b3 . - b4 = - (b3 . b4) &
b3 . (b4 - b5) = (b3 . b4) - (b3 . b5);
:: MOD_4:th 49
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3 being Homomorphism of b2,b1
for b4, b5 being Element of the carrier of b2 holds
b3 . (b4 - b5) = (b3 . b4) - (b3 . b5);
:: MOD_4:modenot 4 => MOD_4:mode 2
definition
let a1, a2 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a5 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a2;
mode Homomorphism of A3,A4,A5 -> Function-like quasi_total Relation of the carrier of a4,the carrier of a5 means
(for b1, b2 being Element of the carrier of a4 holds
it . (b1 + b2) = (it . b1) + (it . b2)) &
(for b1 being Element of the carrier of a1
for b2 being Element of the carrier of a4 holds
it . (b1 * b2) = (a3 . b1) * (it . b2));
end;
:: MOD_4:dfs 23
definiens
let a1, a2 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a5 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a2;
let a6 be Function-like quasi_total Relation of the carrier of a4,the carrier of a5;
To prove
a6 is Homomorphism of a3,a4,a5
it is sufficient to prove
thus (for b1, b2 being Element of the carrier of a4 holds
a6 . (b1 + b2) = (a6 . b1) + (a6 . b2)) &
(for b1 being Element of the carrier of a1
for b2 being Element of the carrier of a4 holds
a6 . (b1 * b2) = (a3 . b1) * (a6 . b2));
:: MOD_4:def 24
theorem
for b1, b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b5 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b5 holds
b6 is Homomorphism of b3,b4,b5
iff
(for b7, b8 being Element of the carrier of b4 holds
b6 . (b7 + b8) = (b6 . b7) + (b6 . b8)) &
(for b7 being Element of the carrier of b1
for b8 being Element of the carrier of b4 holds
b6 . (b7 * b8) = (b3 . b7) * (b6 . b8));
:: MOD_4:th 50
theorem
for b1, b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b5 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2 holds
ZeroMap(b4,b5) is Homomorphism of b3,b4,b5;
:: MOD_4:prednot 1 => MOD_4:pred 1
definition
let a1, a2 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a5 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a2;
let a6 be Homomorphism of a3,a4,a5;
pred A6 is_monomorphism_wrp A3 means
a6 is one-to-one;
end;
:: MOD_4:dfs 24
definiens
let a1, a2 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a5 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a2;
let a6 be Homomorphism of a3,a4,a5;
To prove
a6 is_monomorphism_wrp a3
it is sufficient to prove
thus a6 is one-to-one;
:: MOD_4:def 25
theorem
for b1, b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b5 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2
for b6 being Homomorphism of b3,b4,b5 holds
b6 is_monomorphism_wrp b3
iff
b6 is one-to-one;
:: MOD_4:prednot 2 => MOD_4:pred 2
definition
let a1, a2 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a5 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a2;
let a6 be Homomorphism of a3,a4,a5;
pred A6 is_epimorphism_wrp A3 means
proj2 a6 = the carrier of a5;
end;
:: MOD_4:dfs 25
definiens
let a1, a2 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a5 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a2;
let a6 be Homomorphism of a3,a4,a5;
To prove
a6 is_epimorphism_wrp a3
it is sufficient to prove
thus proj2 a6 = the carrier of a5;
:: MOD_4:def 26
theorem
for b1, b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b5 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2
for b6 being Homomorphism of b3,b4,b5 holds
b6 is_epimorphism_wrp b3
iff
proj2 b6 = the carrier of b5;
:: MOD_4:prednot 3 => MOD_4:pred 3
definition
let a1, a2 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a5 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a2;
let a6 be Homomorphism of a3,a4,a5;
pred A6 is_isomorphism_wrp A3 means
a6 is one-to-one & proj2 a6 = the carrier of a5;
end;
:: MOD_4:dfs 26
definiens
let a1, a2 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a5 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a2;
let a6 be Homomorphism of a3,a4,a5;
To prove
a6 is_isomorphism_wrp a3
it is sufficient to prove
thus a6 is one-to-one & proj2 a6 = the carrier of a5;
:: MOD_4:def 27
theorem
for b1, b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b5 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2
for b6 being Homomorphism of b3,b4,b5 holds
b6 is_isomorphism_wrp b3
iff
b6 is one-to-one & proj2 b6 = the carrier of b5;
:: MOD_4:modenot 5
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
let a3 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
mode Endomorphism of a2,a3 is Homomorphism of a2,a3,a3;
end;
:: MOD_4:prednot 4 => MOD_4:pred 4
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
let a3 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a4 be Homomorphism of a2,a3,a3;
pred A4 is_automorphism_wrp A2 means
a4 is one-to-one & proj2 a4 = the carrier of a3;
end;
:: MOD_4:dfs 27
definiens
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
let a3 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a4 be Homomorphism of a2,a3,a3;
To prove
a4 is_automorphism_wrp a2
it is sufficient to prove
thus a4 is one-to-one & proj2 a4 = the carrier of a3;
:: MOD_4:def 28
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b4 being Homomorphism of b2,b3,b3 holds
b4 is_automorphism_wrp b2
iff
b4 is one-to-one & proj2 b4 = the carrier of b3;
:: MOD_4:modenot 6
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2, a3 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
mode Homomorphism of a2,a3 is Homomorphism of id a1,a2,a3;
end;
:: MOD_4:th 51
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3 holds
b4 is Homomorphism of id b1,b2,b3
iff
(for b5, b6 being Element of the carrier of b2 holds
b4 . (b5 + b6) = (b4 . b5) + (b4 . b6)) &
(for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2 holds
b4 . (b5 * b6) = b5 * (b4 . b6));
:: MOD_4:attrnot 16 => MOD_4:attr 16
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2, a3 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a4 be Homomorphism of id a1,a2,a3;
attr a4 is monomorphism means
a4 is one-to-one;
end;
:: MOD_4:dfs 28
definiens
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2, a3 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a4 be Homomorphism of id a1,a2,a3;
To prove
a4 is monomorphism
it is sufficient to prove
thus a4 is one-to-one;
:: MOD_4:def 29
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b4 being Homomorphism of id b1,b2,b3 holds
b4 is monomorphism(b1, b2, b3)
iff
b4 is one-to-one;
:: MOD_4:attrnot 17 => MOD_4:attr 17
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2, a3 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a4 be Homomorphism of id a1,a2,a3;
attr a4 is epimorphism means
proj2 a4 = the carrier of a3;
end;
:: MOD_4:dfs 29
definiens
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2, a3 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a4 be Homomorphism of id a1,a2,a3;
To prove
a4 is epimorphism
it is sufficient to prove
thus proj2 a4 = the carrier of a3;
:: MOD_4:def 30
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b4 being Homomorphism of id b1,b2,b3 holds
b4 is epimorphism(b1, b2, b3)
iff
proj2 b4 = the carrier of b3;
:: MOD_4:attrnot 18 => MOD_4:attr 18
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2, a3 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a4 be Homomorphism of id a1,a2,a3;
attr a4 is isomorphism means
a4 is one-to-one & proj2 a4 = the carrier of a3;
end;
:: MOD_4:dfs 30
definiens
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2, a3 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a4 be Homomorphism of id a1,a2,a3;
To prove
a4 is isomorphism
it is sufficient to prove
thus a4 is one-to-one & proj2 a4 = the carrier of a3;
:: MOD_4:def 31
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b4 being Homomorphism of id b1,b2,b3 holds
b4 is isomorphism(b1, b2, b3)
iff
b4 is one-to-one & proj2 b4 = the carrier of b3;
:: MOD_4:modenot 7
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
mode Endomorphism of a2 is Homomorphism of id a1,a2,a2;
end;
:: MOD_4:attrnot 19 => MOD_4:attr 19
definition
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Homomorphism of id a1,a2,a2;
attr a3 is automorphism means
a3 is one-to-one & proj2 a3 = the carrier of a2;
end;
:: MOD_4:dfs 31
definiens
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Homomorphism of id a1,a2,a2;
To prove
a3 is automorphism
it is sufficient to prove
thus a3 is one-to-one & proj2 a3 = the carrier of a2;
:: MOD_4:def 32
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Homomorphism of id b1,b2,b2 holds
b3 is automorphism(b1, b2)
iff
b3 is one-to-one & proj2 b3 = the carrier of b2;