Article MOD_3, MML version 4.99.1005
:: MOD_3:th 2
theorem
for b1 being non empty non degenerated right_complementable add-associative right_zeroed doubleLoopStr holds
0. b1 <> - 1. b1;
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Linear_Combination of b2
for b4 being finite Element of bool the carrier of b2
st Carrier b3 c= b4
holds ex b5 being FinSequence of the carrier of b2 st
b5 is one-to-one & proj2 b5 = b4 & Sum b3 = Sum (b3 (#) b5);
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of the carrier of b1 holds
Sum (b4 * b3) = b4 * Sum b3;
:: MOD_3:funcnot 1 => MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3 be Element of bool the carrier of a2;
func Lin A3 -> strict Subspace of a2 means
the carrier of it = {Sum b1 where b1 is Linear_Combination of a3: TRUE};
end;
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
for b4 being strict Subspace of b2 holds
b4 = Lin b3
iff
the carrier of b4 = {Sum b5 where b5 is Linear_Combination of b3: TRUE};
:: MOD_3:th 11
theorem
for b1 being set
for b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2
for b4 being Element of bool the carrier of b3 holds
b1 in Lin b4
iff
ex b5 being Linear_Combination of b4 st
b1 = Sum b5;
:: MOD_3:th 12
theorem
for b1 being set
for b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2
for b4 being Element of bool the carrier of b3
st b1 in b4
holds b1 in Lin b4;
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
Lin {} the carrier of b2 = (0). b2;
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
st Lin b3 = (0). b2 & b3 <> {}
holds b3 = {0. b2};
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
for b4 being strict Subspace of b2
st 0. b1 <> 1. b1 & b3 = the carrier of b4
holds Lin b3 = b4;
:: MOD_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 strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
st 0. b1 <> 1. b1 & b3 = the carrier of b2
holds Lin b3 = b2;
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
st b3 c= b4
holds Lin b3 is Subspace of Lin b4;
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
st Lin b3 = b2 & b3 c= b4
holds Lin b4 = b2;
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2 holds
Lin (b3 \/ b4) = (Lin b3) + Lin b4;
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2 holds
Lin (b3 /\ b4) is Subspace of (Lin b3) /\ Lin b4;
:: MOD_3:attrnot 1 => MOD_3:attr 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 be Element of bool the carrier of a2;
attr a3 is base means
a3 is linearly-independent(a1, a2) &
Lin a3 = VectSpStr(#the carrier of a2,the addF of a2,the ZeroF of a2,the lmult of a2#);
end;
:: MOD_3:dfs 2
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 Element of bool the carrier of a2;
To prove
a3 is base
it is sufficient to prove
thus a3 is linearly-independent(a1, a2) &
Lin a3 = VectSpStr(#the carrier of a2,the addF of a2,the ZeroF of a2,the lmult of a2#);
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2 holds
b3 is base(b1, b2)
iff
b3 is linearly-independent(b1, b2) &
Lin b3 = VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#);
:: MOD_3:attrnot 2 => MOD_3:attr 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;
attr a2 is free means
ex b1 being Element of bool the carrier of a2 st
b1 is base(a1, a2);
end;
:: MOD_3:dfs 3
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;
To prove
a2 is free
it is sufficient to prove
thus ex b1 being Element of bool the carrier of a2 st
b1 is base(a1, a2);
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
b2 is free(b1)
iff
ex b3 being Element of bool the carrier of b2 st
b3 is base(b1, b2);
:: MOD_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 VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
(0). b2 is free(b1);
:: MOD_3:exreg 1
registration
let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
cluster non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed free VectSpStr over a1;
end;
:: MOD_3:th 23
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible 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 holds
{b3} is linearly-independent(b1, b2)
iff
b3 <> 0. b2;
:: MOD_3:th 24
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible 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 Element of the carrier of b2 holds
b3 <> b4 & {b3,b4} is linearly-independent(b1, b2)
iff
b4 <> 0. b2 &
(for b5 being Element of the carrier of b1 holds
b3 <> b5 * b4);
:: MOD_3:th 25
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible 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 Element of the carrier of b2 holds
b3 <> b4 & {b3,b4} is linearly-independent(b1, b2)
iff
for b5, b6 being Element of the carrier of b1
st (b5 * b3) + (b6 * b4) = 0. b2
holds b5 = 0. b1 & b6 = 0. b1;
:: MOD_3:th 26
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible 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 bool the carrier of b2
st b3 is linearly-independent(b1, b2)
holds ex b4 being Element of bool the carrier of b2 st
b3 c= b4 & b4 is base(b1, b2);
:: MOD_3:th 27
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible 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 bool the carrier of b2
st Lin b3 = b2
holds ex b4 being Element of bool the carrier of b2 st
b4 c= b3 & b4 is base(b1, b2);
:: MOD_3:th 28
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible 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 free(b1);
:: MOD_3:modenot 1 => MOD_3:mode 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible 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 Basis of A2 -> Element of bool the carrier of a2 means
it is base(a1, a2);
end;
:: MOD_3:dfs 4
definiens
let a1 be non empty non degenerated right_complementable almost_left_invertible 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 bool the carrier of a2;
To prove
a3 is Basis of a2
it is sufficient to prove
thus a3 is base(a1, a2);
:: MOD_3:def 5
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible 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 bool the carrier of b2 holds
b3 is Basis of b2
iff
b3 is base(b1, b2);
:: MOD_3:th 29
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible 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 bool the carrier of b2
st b3 is linearly-independent(b1, b2)
holds ex b4 being Basis of b2 st
b3 c= b4;
:: MOD_3:th 30
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible 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 bool the carrier of b2
st Lin b3 = b2
holds ex b4 being Basis of b2 st
b4 c= b3;