Article VECTSP_5, MML version 4.99.1005
:: VECTSP_5:funcnot 1 => VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3, a4 be Subspace of a2;
func A3 + A4 -> strict Subspace 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;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2
for b5 being strict Subspace 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};
:: VECTSP_5:funcnot 2 => VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3, a4 be Subspace of a2;
func A3 /\ A4 -> strict Subspace of a2 means
the carrier of it = (the carrier of a3) /\ the carrier of a4;
commutativity;
:: for a1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
:: for a2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1
:: for a3, a4 being Subspace of a2 holds
:: a3 /\ a4 = a4 /\ a3;
end;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2
for b5 being strict Subspace of b2 holds
b5 = b3 /\ b4
iff
the carrier of b5 = (the carrier of b3) /\ the carrier of b4;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace 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;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2
for b5 being Element of the carrier of b2
st (b5 in b3 or b5 in b4)
holds b5 in b3 + b4;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2
for b5 being set holds
b5 in b3 /\ b4
iff
b5 in b3 & b5 in b4;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being strict Subspace of b2 holds
b3 + b3 = b3;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2 holds
b3 + b4 = b4 + b3;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Subspace of b2 holds
b3 + (b4 + b5) = (b3 + b4) + b5;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2 holds
b3 is Subspace of b3 + b4 & b4 is Subspace of b3 + b4;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being strict Subspace of b2 holds
b3 is Subspace of b4
iff
b3 + b4 = b4;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being strict Subspace of b2 holds
((0). b2) + b3 = b3 & b3 + (0). b2 = b3;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
((0). b2) + (Omega). b2 = VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#) &
((Omega). b2) + (0). b2 = VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#);
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
((Omega). b2) + b3 = VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#) &
b3 + (Omega). b2 = VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#);
:: VECTSP_5: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 strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
((Omega). b2) + (Omega). b2 = b2;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being strict Subspace of b2 holds
b3 /\ b3 = b3;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Subspace of b2 holds
b3 /\ (b4 /\ b5) = (b3 /\ b4) /\ b5;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2 holds
b3 /\ b4 is Subspace of b3 & b3 /\ b4 is Subspace of b4;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
(for b4 being strict Subspace of b2
st b4 is Subspace of b3
holds b4 /\ b3 = b4) &
(for b4 being Subspace of b2
st b4 /\ b3 = b4
holds b4 is Subspace of b3);
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Subspace of b2
st b3 is Subspace of b4
holds b3 /\ b5 is Subspace of b4 /\ b5;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Subspace of b2
st b3 is Subspace of b4
holds b3 /\ b5 is Subspace of b4;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Subspace of b2
st b3 is Subspace of b4 & b3 is Subspace of b5
holds b3 is Subspace of b4 /\ b5;
:: VECTSP_5: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
for b3 being Subspace of b2 holds
((0). b2) /\ b3 = (0). b2 & b3 /\ (0). b2 = (0). b2;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being strict Subspace of b2 holds
((Omega). b2) /\ b3 = b3 & b3 /\ (Omega). b2 = b3;
:: VECTSP_5: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 strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
((Omega). b2) /\ (Omega). b2 = b2;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2 holds
b3 /\ b4 is Subspace of b3 + b4;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being strict Subspace of b2 holds
(b3 /\ b4) + b4 = b4;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being strict Subspace of b2 holds
b4 /\ (b4 + b3) = b4;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Subspace of b2 holds
(b3 /\ b4) + (b4 /\ b5) is Subspace of b4 /\ (b3 + b5);
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Subspace of b2
st b3 is Subspace of b4
holds b4 /\ (b3 + b5) = (b3 /\ b4) + (b4 /\ b5);
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Subspace of b2 holds
b3 + (b4 /\ b5) is Subspace of (b4 + b3) /\ (b3 + b5);
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Subspace of b2
st b3 is Subspace of b4
holds b4 + (b3 /\ b5) = (b3 + b4) /\ (b4 + b5);
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2
for b5 being strict Subspace of b2
st b5 is Subspace of b3
holds b5 + (b4 /\ b3) = (b5 + b4) /\ b3;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being strict Subspace of b2 holds
b3 + b4 = b4
iff
b3 /\ b4 = b3;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4, b5 being strict Subspace of b2
st b3 is Subspace of b4
holds b3 + b5 is Subspace of b4 + b5;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Subspace of b2
st b3 is Subspace of b4
holds b3 is Subspace of b4 + b5;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Subspace of b2
st b3 is Subspace of b4 & b5 is Subspace of b4
holds b3 + b5 is Subspace of b4;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2 holds
ex b5 being Subspace of b2 st
the carrier of b5 = (the carrier of b3) \/ the carrier of b4
iff
(b3 is Subspace of b4 or b4 is Subspace of b3);
:: VECTSP_5:funcnot 3 => VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
func Subspaces A2 -> set means
for b1 being set holds
b1 in it
iff
ex b2 being strict Subspace of a2 st
b2 = b1;
end;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being set holds
b3 = Subspaces b2
iff
for b4 being set holds
b4 in b3
iff
ex b5 being strict Subspace of b2 st
b5 = b4;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
cluster Subspaces a2 -> non empty;
end;
:: VECTSP_5: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 strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
b2 in Subspaces b2;
:: VECTSP_5:prednot 1 => VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3, a4 be Subspace of a2;
pred A2 is_the_direct_sum_of A3,A4 means
VectSpStr(#the carrier of a2,the addF of a2,the ZeroF of a2,the lmult of a2#) = a3 + a4 &
a3 /\ a4 = (0). a2;
end;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3, a4 be Subspace of a2;
To prove
a2 is_the_direct_sum_of a3,a4
it is sufficient to prove
thus VectSpStr(#the carrier of a2,the addF of a2,the ZeroF of a2,the lmult of a2#) = a3 + a4 &
a3 /\ a4 = (0). a2;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2 holds
b2 is_the_direct_sum_of b3,b4
iff
VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#) = b3 + b4 &
b3 /\ b4 = (0). b2;
:: VECTSP_5:modenot 1 => VECTSP_5:mode 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative 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 Subspace of a2;
mode Linear_Compl of A3 -> Subspace of a2 means
a2 is_the_direct_sum_of it,a3;
end;
:: VECTSP_5:dfs 5
definiens
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative 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, a4 be Subspace of a2;
To prove
a4 is Linear_Compl of a3
it is sufficient to prove
thus a2 is_the_direct_sum_of a4,a3;
:: VECTSP_5:def 5
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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, b4 being Subspace of b2 holds
b4 is Linear_Compl of b3
iff
b2 is_the_direct_sum_of b4,b3;
:: VECTSP_5:th 47
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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, b4 being Subspace of b2
st b2 is_the_direct_sum_of b3,b4
holds b4 is Linear_Compl of b3;
:: VECTSP_5:th 48
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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 Subspace of b2
for b4 being Linear_Compl of b3 holds
b2 is_the_direct_sum_of b4,b3 & b2 is_the_direct_sum_of b3,b4;
:: VECTSP_5:th 49
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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 Subspace of b2
for b4 being Linear_Compl of b3 holds
b3 + b4 = VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#) &
b4 + b3 = VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#);
:: VECTSP_5:th 50
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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 Subspace of b2
for b4 being Linear_Compl of b3 holds
b3 /\ b4 = (0). b2 & b4 /\ b3 = (0). b2;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2
st b2 is_the_direct_sum_of b3,b4
holds b2 is_the_direct_sum_of b4,b3;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
b2 is_the_direct_sum_of (0). b2,(Omega). b2 & b2 is_the_direct_sum_of (Omega). b2,(0). b2;
:: VECTSP_5:th 53
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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 Subspace of b2
for b4 being Linear_Compl of b3 holds
b3 is Linear_Compl of b4;
:: VECTSP_5:th 54
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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
(0). b2 is Linear_Compl of (Omega). b2 & (Omega). b2 is Linear_Compl of (0). b2;
:: VECTSP_5:th 55
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, b4 being Subspace 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;
:: VECTSP_5:th 56
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, b4 being Subspace 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};
:: VECTSP_5: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 strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace 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;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace 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;
:: VECTSP_5:th 59
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, b4 being Subspace 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;
:: VECTSP_5:funcnot 4 => VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Element of the carrier of a2;
let a4, a5 be Subspace 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;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4, b5 being Subspace 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;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace 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;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace 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;
:: VECTSP_5:th 66
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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 Subspace of b2
for b4 being Linear_Compl of b3
for b5 being Element of the carrier of b2
for b6 being Element of [:the carrier of b2,the carrier of b2:]
st b6 `1 + (b6 `2) = b5 & b6 `1 in b3 & b6 `2 in b4
holds b6 = b5 |--(b3,b4);
:: VECTSP_5:th 67
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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 Subspace of b2
for b4 being Linear_Compl of b3
for b5 being Element of the carrier of b2 holds
(b5 |--(b3,b4)) `1 + ((b5 |--(b3,b4)) `2) = b5;
:: VECTSP_5:th 68
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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 Subspace of b2
for b4 being Linear_Compl of b3
for b5 being Element of the carrier of b2 holds
(b5 |--(b3,b4)) `1 in b3 & (b5 |--(b3,b4)) `2 in b4;
:: VECTSP_5:th 69
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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 Subspace of b2
for b4 being Linear_Compl of b3
for b5 being Element of the carrier of b2 holds
(b5 |--(b3,b4)) `1 = (b5 |--(b4,b3)) `2;
:: VECTSP_5:th 70
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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 Subspace of b2
for b4 being Linear_Compl of b3
for b5 being Element of the carrier of b2 holds
(b5 |--(b3,b4)) `2 = (b5 |--(b4,b3)) `1;
:: VECTSP_5:funcnot 5 => VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
func SubJoin A2 -> Function-like quasi_total Relation of [:Subspaces a2,Subspaces a2:],Subspaces a2 means
for b1, b2 being Element of Subspaces a2
for b3, b4 being Subspace of a2
st b1 = b3 & b2 = b4
holds it .(b1,b2) = b3 + b4;
end;
:: VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Function-like quasi_total Relation of [:Subspaces b2,Subspaces b2:],Subspaces b2 holds
b3 = SubJoin b2
iff
for b4, b5 being Element of Subspaces b2
for b6, b7 being Subspace of b2
st b4 = b6 & b5 = b7
holds b3 .(b4,b5) = b6 + b7;
:: VECTSP_5:funcnot 6 => VECTSP_5: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 VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
func SubMeet A2 -> Function-like quasi_total Relation of [:Subspaces a2,Subspaces a2:],Subspaces a2 means
for b1, b2 being Element of Subspaces a2
for b3, b4 being Subspace of a2
st b1 = b3 & b2 = b4
holds it .(b1,b2) = b3 /\ b4;
end;
:: VECTSP_5:def 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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Function-like quasi_total Relation of [:Subspaces b2,Subspaces b2:],Subspaces b2 holds
b3 = SubMeet b2
iff
for b4, b5 being Element of Subspaces b2
for b6, b7 being Subspace of b2
st b4 = b6 & b5 = b7
holds b3 .(b4,b5) = b6 /\ b7;
:: VECTSP_5:th 75
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
LattStr(#Subspaces b2,SubJoin b2,SubMeet b2#) is non empty Lattice-like LattStr;
:: VECTSP_5:th 76
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
LattStr(#Subspaces b2,SubJoin b2,SubMeet b2#) is non empty Lattice-like lower-bounded LattStr;
:: VECTSP_5:th 77
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
LattStr(#Subspaces b2,SubJoin b2,SubMeet b2#) is non empty Lattice-like upper-bounded LattStr;
:: VECTSP_5:th 78
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
LattStr(#Subspaces b2,SubJoin b2,SubMeet b2#) is non empty Lattice-like bounded LattStr;
:: VECTSP_5:th 79
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
LattStr(#Subspaces b2,SubJoin b2,SubMeet b2#) is non empty Lattice-like modular LattStr;
:: VECTSP_5:th 80
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative 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
LattStr(#Subspaces b2,SubJoin b2,SubMeet b2#) is non empty Lattice-like bounded complemented LattStr;