Article RMOD_3, MML version 4.99.1005
:: RMOD_3:funcnot 1 => RMOD_3:func 1
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 Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
let a3, a4 be Submodule of a2;
func A3 + A4 -> strict Submodule of a2 means
the carrier of it = {b1 + b2 where b1 is Element of the carrier of a2, b2 is Element of the carrier of a2: b1 in a3 & b2 in a4};
end;
:: RMOD_3:def 1
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
for b3, b4 being Submodule of b2
for b5 being strict Submodule of b2 holds
b5 = b3 + b4
iff
the carrier of b5 = {b6 + b7 where b6 is Element of the carrier of b2, b7 is Element of the carrier of b2: b6 in b3 & b7 in b4};
:: RMOD_3:funcnot 2 => RMOD_3:func 2
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 Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
let a3, a4 be Submodule of a2;
func A3 /\ A4 -> strict Submodule of a2 means
the carrier of it = (the carrier of a3) /\ the carrier of a4;
end;
:: RMOD_3:def 2
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
for b3, b4 being Submodule of b2
for b5 being strict Submodule of b2 holds
b5 = b3 /\ b4
iff
the carrier of b5 = (the carrier of b3) /\ the carrier of b4;
:: RMOD_3:th 5
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
for b3, b4 being Submodule of b2
for b5 being set holds
b5 in b3 + b4
iff
ex b6, b7 being Element of the carrier of b2 st
b6 in b3 & b7 in b4 & b5 = b6 + b7;
:: RMOD_3:th 6
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
for b3, b4 being Submodule of b2
for b5 being Element of the carrier of b2
st (b5 in b3 or b5 in b4)
holds b5 in b3 + b4;
:: RMOD_3:th 7
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
for b3, b4 being Submodule of b2
for b5 being set holds
b5 in b3 /\ b4
iff
b5 in b3 & b5 in b4;
:: RMOD_3:th 8
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
for b3 being strict Submodule of b2 holds
b3 + b3 = b3;
:: RMOD_3:th 9
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
for b3, b4 being Submodule of b2 holds
b3 + b4 = b4 + b3;
:: RMOD_3:th 10
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
for b3, b4, b5 being Submodule of b2 holds
b3 + (b4 + b5) = (b3 + b4) + b5;
:: RMOD_3:th 11
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
for b3, b4 being Submodule of b2 holds
b3 is Submodule of b3 + b4 & b4 is Submodule of b3 + b4;
:: RMOD_3:th 12
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
for b3 being Submodule of b2
for b4 being strict Submodule of b2 holds
b3 is Submodule of b4
iff
b3 + b4 = b4;
:: RMOD_3:th 13
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
for b3 being strict Submodule of b2 holds
((0). b2) + b3 = b3 & b3 + (0). b2 = b3;
:: RMOD_3:th 14
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 strict RightMod-like RightModStr over b1 holds
((0). b2) + (Omega). b2 = b2 & ((Omega). b2) + (0). b2 = b2;
:: RMOD_3:th 15
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
for b3 being Submodule of b2 holds
((Omega). b2) + b3 = RightModStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the rmult of b2#) &
b3 + (Omega). b2 = RightModStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the rmult of b2#);
:: RMOD_3:th 16
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 strict RightMod-like RightModStr over b1 holds
((Omega). b2) + (Omega). b2 = b2;
:: RMOD_3:th 17
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
for b3 being strict Submodule of b2 holds
b3 /\ b3 = b3;
:: RMOD_3:th 18
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
for b3, b4 being Submodule of b2 holds
b3 /\ b4 = b4 /\ b3;
:: RMOD_3:th 19
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
for b3, b4, b5 being Submodule of b2 holds
b3 /\ (b4 /\ b5) = (b3 /\ b4) /\ b5;
:: RMOD_3:th 20
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
for b3, b4 being Submodule of b2 holds
b3 /\ b4 is Submodule of b3 & b3 /\ b4 is Submodule of b4;
:: RMOD_3:th 21
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
for b3 being Submodule of b2 holds
(for b4 being strict Submodule of b2
st b4 is Submodule of b3
holds b4 /\ b3 = b4) &
(for b4 being Submodule of b2
st b4 /\ b3 = b4
holds b4 is Submodule of b3);
:: RMOD_3:th 22
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
for b3, b4, b5 being Submodule of b2
st b3 is Submodule of b4
holds b3 /\ b5 is Submodule of b4 /\ b5;
:: RMOD_3:th 23
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
for b3, b4, b5 being Submodule of b2
st b3 is Submodule of b4
holds b3 /\ b5 is Submodule of b4;
:: RMOD_3:th 24
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
for b3, b4, b5 being Submodule of b2
st b3 is Submodule of b4 & b3 is Submodule of b5
holds b3 is Submodule of b4 /\ b5;
:: RMOD_3: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 Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Submodule of b2 holds
((0). b2) /\ b3 = (0). b2 & b3 /\ (0). b2 = (0). b2;
:: RMOD_3:th 27
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
for b3 being strict Submodule of b2 holds
((Omega). b2) /\ b3 = b3 & b3 /\ (Omega). b2 = b3;
:: RMOD_3:th 28
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 strict RightMod-like RightModStr over b1 holds
((Omega). b2) /\ (Omega). b2 = b2;
:: RMOD_3:th 29
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
for b3, b4 being Submodule of b2 holds
b3 /\ b4 is Submodule of b3 + b4;
:: RMOD_3:th 30
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
for b3 being Submodule of b2
for b4 being strict Submodule of b2 holds
(b3 /\ b4) + b4 = b4;
:: RMOD_3:th 31
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
for b3 being Submodule of b2
for b4 being strict Submodule of b2 holds
b4 /\ (b4 + b3) = b4;
:: RMOD_3:th 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 Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4, b5 being Submodule of b2 holds
(b3 /\ b4) + (b4 /\ b5) is Submodule of b4 /\ (b3 + b5);
:: RMOD_3:th 33
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
for b3, b4, b5 being Submodule of b2
st b3 is Submodule of b4
holds b4 /\ (b3 + b5) = (b3 /\ b4) + (b4 /\ b5);
:: RMOD_3:th 34
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
for b3, b4, b5 being Submodule of b2 holds
b3 + (b4 /\ b5) is Submodule of (b4 + b3) /\ (b3 + b5);
:: RMOD_3:th 35
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
for b3, b4, b5 being Submodule of b2
st b3 is Submodule of b4
holds b4 + (b3 /\ b5) = (b3 + b4) /\ (b4 + b5);
:: RMOD_3:th 36
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
for b3, b4 being Submodule of b2
for b5 being strict Submodule of b2
st b5 is Submodule of b3
holds b5 + (b4 /\ b3) = (b5 + b4) /\ b3;
:: RMOD_3:th 37
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
for b3, b4 being strict Submodule of b2 holds
b3 + b4 = b4
iff
b3 /\ b4 = b3;
:: RMOD_3:th 38
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
for b3 being Submodule of b2
for b4, b5 being strict Submodule of b2
st b3 is Submodule of b4
holds b3 + b5 is Submodule of b4 + b5;
:: RMOD_3:th 39
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
for b3, b4, b5 being Submodule of b2
st b3 is Submodule of b4
holds b3 is Submodule of b4 + b5;
:: RMOD_3:th 40
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
for b3, b4, b5 being Submodule of b2
st b3 is Submodule of b4 & b5 is Submodule of b4
holds b3 + b5 is Submodule of b4;
:: RMOD_3:th 41
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
for b3, b4 being Submodule of b2 holds
ex b5 being Submodule of b2 st
the carrier of b5 = (the carrier of b3) \/ the carrier of b4
iff
(b3 is Submodule of b4 or b4 is Submodule of b3);
:: RMOD_3:funcnot 3 => RMOD_3:func 3
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 Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
func Submodules A2 -> set means
for b1 being set holds
b1 in it
iff
ex b2 being strict Submodule of a2 st
b2 = b1;
end;
:: RMOD_3:def 3
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
for b3 being set holds
b3 = Submodules b2
iff
for b4 being set holds
b4 in b3
iff
ex b5 being strict Submodule of b2 st
b5 = b4;
:: RMOD_3:funcreg 1
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 Submodules a2 -> non empty;
end;
:: RMOD_3:th 44
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 strict RightMod-like RightModStr over b1 holds
b2 in Submodules b2;
:: RMOD_3:prednot 1 => RMOD_3:pred 1
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 Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
let a3, a4 be Submodule of a2;
pred A2 is_the_direct_sum_of A3,A4 means
RightModStr(#the carrier of a2,the addF of a2,the ZeroF of a2,the rmult of a2#) = a3 + a4 &
a3 /\ a4 = (0). a2;
end;
:: RMOD_3:dfs 4
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 Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
let a3, a4 be Submodule of a2;
To prove
a2 is_the_direct_sum_of a3,a4
it is sufficient to prove
thus RightModStr(#the carrier of a2,the addF of a2,the ZeroF of a2,the rmult of a2#) = a3 + a4 &
a3 /\ a4 = (0). a2;
:: RMOD_3:def 4
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
for b3, b4 being Submodule of b2 holds
b2 is_the_direct_sum_of b3,b4
iff
RightModStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the rmult of b2#) = b3 + b4 &
b3 /\ b4 = (0). b2;
:: RMOD_3:th 46
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
for b3, b4 being Submodule of b2
st b2 is_the_direct_sum_of b3,b4
holds b2 is_the_direct_sum_of b4,b3;
:: RMOD_3:th 47
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 strict RightMod-like RightModStr over b1 holds
b2 is_the_direct_sum_of (0). b2,(Omega). b2 & b2 is_the_direct_sum_of (Omega). b2,(0). b2;
:: RMOD_3:th 48
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
for b3, b4 being Submodule of b2
for b5 being Coset of b3
for b6 being Coset of b4
st b5 meets b6
holds b5 /\ b6 is Coset of b3 /\ b4;
:: RMOD_3:th 49
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
for b3, b4 being Submodule of b2 holds
b2 is_the_direct_sum_of b3,b4
iff
for b5 being Coset of b3
for b6 being Coset of b4 holds
ex b7 being Element of the carrier of b2 st
b5 /\ b6 = {b7};
:: RMOD_3:th 50
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 strict RightMod-like RightModStr over b1
for b3, b4 being Submodule of b2 holds
b3 + b4 = b2
iff
for b5 being Element of the carrier of b2 holds
ex b6, b7 being Element of the carrier of b2 st
b6 in b3 & b7 in b4 & b5 = b6 + b7;
:: RMOD_3:th 51
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
for b3, b4 being Submodule of b2
for b5, b6, b7, b8, b9 being Element of the carrier of b2
st b2 is_the_direct_sum_of b3,b4 & b5 = b6 + b7 & b5 = b8 + b9 & b6 in b3 & b8 in b3 & b7 in b4 & b9 in b4
holds b6 = b8 & b7 = b9;
:: RMOD_3:th 52
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
for b3, b4 being Submodule of b2
st b2 = b3 + b4 &
(ex b5 being Element of the carrier of b2 st
for b6, b7, b8, b9 being Element of the carrier of b2
st b5 = b6 + b7 & b5 = b8 + b9 & b6 in b3 & b8 in b3 & b7 in b4 & b9 in b4
holds b6 = b8 & b7 = b9)
holds b2 is_the_direct_sum_of b3,b4;
:: RMOD_3:funcnot 4 => RMOD_3:func 4
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 Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
let a3 be Element of the carrier of a2;
let a4, a5 be Submodule of a2;
assume a2 is_the_direct_sum_of a4,a5;
func A3 |--(A4,A5) -> Element of [:the carrier of a2,the carrier of a2:] means
a3 = it `1 + (it `2) & it `1 in a4 & it `2 in a5;
end;
:: RMOD_3:def 5
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
for b3 being Element of the carrier of b2
for b4, b5 being Submodule of b2
st b2 is_the_direct_sum_of b4,b5
for b6 being Element of [:the carrier of b2,the carrier of b2:] holds
b6 = b3 |--(b4,b5)
iff
b3 = b6 `1 + (b6 `2) & b6 `1 in b4 & b6 `2 in b5;
:: RMOD_3:th 57
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
for b3, b4 being Submodule of b2
for b5 being Element of the carrier of b2
st b2 is_the_direct_sum_of b3,b4
holds (b5 |--(b3,b4)) `1 = (b5 |--(b4,b3)) `2;
:: RMOD_3:th 58
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
for b3, b4 being Submodule of b2
for b5 being Element of the carrier of b2
st b2 is_the_direct_sum_of b3,b4
holds (b5 |--(b3,b4)) `2 = (b5 |--(b4,b3)) `1;
:: RMOD_3:funcnot 5 => RMOD_3:func 5
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 Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
func SubJoin A2 -> Function-like quasi_total Relation of [:Submodules a2,Submodules a2:],Submodules a2 means
for b1, b2 being Element of Submodules a2
for b3, b4 being Submodule of a2
st b1 = b3 & b2 = b4
holds it .(b1,b2) = b3 + b4;
end;
:: RMOD_3:def 6
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
for b3 being Function-like quasi_total Relation of [:Submodules b2,Submodules b2:],Submodules b2 holds
b3 = SubJoin b2
iff
for b4, b5 being Element of Submodules b2
for b6, b7 being Submodule of b2
st b4 = b6 & b5 = b7
holds b3 .(b4,b5) = b6 + b7;
:: RMOD_3:funcnot 6 => RMOD_3:func 6
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 Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
func SubMeet A2 -> Function-like quasi_total Relation of [:Submodules a2,Submodules a2:],Submodules a2 means
for b1, b2 being Element of Submodules a2
for b3, b4 being Submodule of a2
st b1 = b3 & b2 = b4
holds it .(b1,b2) = b3 /\ b4;
end;
:: RMOD_3:def 7
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
for b3 being Function-like quasi_total Relation of [:Submodules b2,Submodules b2:],Submodules b2 holds
b3 = SubMeet b2
iff
for b4, b5 being Element of Submodules b2
for b6, b7 being Submodule of b2
st b4 = b6 & b5 = b7
holds b3 .(b4,b5) = b6 /\ b7;
:: RMOD_3:th 63
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
LattStr(#Submodules b2,SubJoin b2,SubMeet b2#) is non empty Lattice-like LattStr;
:: RMOD_3:th 64
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
LattStr(#Submodules b2,SubJoin b2,SubMeet b2#) is non empty Lattice-like lower-bounded LattStr;
:: RMOD_3:th 65
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
LattStr(#Submodules b2,SubJoin b2,SubMeet b2#) is non empty Lattice-like upper-bounded LattStr;
:: RMOD_3:th 66
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
LattStr(#Submodules b2,SubJoin b2,SubMeet b2#) is non empty Lattice-like bounded LattStr;
:: RMOD_3:th 67
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
LattStr(#Submodules b2,SubJoin b2,SubMeet b2#) is non empty Lattice-like modular LattStr;