Article VECTSP_9, MML version 4.99.1005
:: VECTSP_9:exreg 1
registration
let a1 be non empty 1-sorted;
cluster non empty Element of bool the carrier of a1;
end;
:: VECTSP_9:th 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 being Linear_Combination of b2
for b4, b5 being FinSequence of the carrier of b2
for b6 being Function-like quasi_total bijective Relation of dom b4,dom b4
st b5 = b4 * b6
holds Sum (b3 (#) b4) = Sum (b3 (#) b5);
:: VECTSP_9:th 6
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 Linear_Combination of b2
for b4 being FinSequence of the carrier of b2
st Carrier b3 misses proj2 b4
holds Sum (b3 (#) b4) = 0. b2;
:: VECTSP_9:th 7
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 FinSequence of the carrier of b2
st b3 is one-to-one
for b4 being Linear_Combination of b2
st Carrier b4 c= proj2 b3
holds Sum (b4 (#) b3) = Sum b4;
:: VECTSP_9: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
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 FinSequence of the carrier of b2 holds
ex b5 being Linear_Combination of b2 st
Carrier b5 = (proj2 b4) /\ Carrier b3 & b3 (#) b4 = b5 (#) b4;
:: VECTSP_9:th 9
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 Linear_Combination of b2
for b4 being Element of bool the carrier of b2
for b5 being FinSequence of the carrier of b2
st proj2 b5 c= the carrier of Lin b4
holds ex b6 being Linear_Combination of b4 st
Sum (b3 (#) b5) = Sum b6;
:: VECTSP_9:th 10
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 Linear_Combination of b2
for b4 being Element of bool the carrier of b2
st Carrier b3 c= the carrier of Lin b4
holds ex b5 being Linear_Combination of b4 st
Sum b3 = Sum b5;
:: VECTSP_9:th 11
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_Combination of b2
st Carrier b4 c= the carrier of b3
for b5 being Linear_Combination of b3
st b5 = b4 | the carrier of b3
holds Carrier b4 = Carrier b5 & Sum b4 = Sum b5;
:: VECTSP_9:th 12
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_Combination of b3 holds
ex b5 being Linear_Combination of b2 st
Carrier b4 = Carrier b5 & Sum b4 = Sum b5;
:: VECTSP_9:th 13
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_Combination of b2
st Carrier b4 c= the carrier of b3
holds ex b5 being Linear_Combination of b3 st
Carrier b5 = Carrier b4 & Sum b5 = Sum b4;
:: VECTSP_9:th 14
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 Basis of b2
for b4 being Element of the carrier of b2 holds
b4 in Lin b3;
:: VECTSP_9:th 15
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 Element of bool the carrier of b3
st b4 is linearly-independent(b1, b3)
holds b4 is linearly-independent Element of bool the carrier of b2;
:: VECTSP_9:th 16
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 Element of bool the carrier of b2
st b4 is linearly-independent(b1, b2) & b4 c= the carrier of b3
holds b4 is linearly-independent Element of bool the carrier of b3;
:: VECTSP_9:th 17
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 Basis of b3 holds
ex b5 being Basis of b2 st
b4 c= b5;
:: VECTSP_9:th 18
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 Element of bool the carrier of b2
st b3 is linearly-independent(b1, b2)
for b4 being Element of the carrier of b2
st b4 in b3
for b5 being Element of bool the carrier of b2
st b5 = b3 \ {b4}
holds not b4 in Lin b5;
:: VECTSP_9:th 19
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 Basis of b2
for b4 being non empty Element of bool the carrier of b2
st b4 misses b3
for b5 being Element of bool the carrier of b2
st b5 = b3 \/ b4
holds b5 is linearly-dependent(b1, b2);
:: VECTSP_9:th 20
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 Element of bool the carrier of b2
st b4 c= the carrier of b3
holds Lin b4 is Subspace of b3;
:: VECTSP_9:th 21
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 Element of bool the carrier of b2
for b5 being Element of bool the carrier of b3
st b4 = b5
holds Lin b4 = Lin b5;
:: VECTSP_9:th 22
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 finite Element of bool the carrier of b2
for b5 being Element of the carrier of b2
st b5 in Lin (b3 \/ b4) & not b5 in Lin b4
holds ex b6 being Element of the carrier of b2 st
b6 in b3 &
b6 in Lin (((b3 \/ b4) \ {b6}) \/ {b5});
:: VECTSP_9:th 23
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 finite Element of bool the carrier of b2
st VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#) = Lin b3 &
b4 is linearly-independent(b1, b2)
holds card b4 <= card b3 &
(ex b5 being finite Element of bool the carrier of b2 st
b5 c= b3 &
card b5 = (card b3) - card b4 &
VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#) = Lin (b4 \/ b5));
:: VECTSP_9:attrnot 1 => MATRLIN:attr 2
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;
attr a2 is finite-dimensional means
ex b1 being finite Element of bool the carrier of a2 st
b1 is Basis of a2;
end;
:: VECTSP_9:dfs 1
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;
To prove
a2 is finite-dimensional
it is sufficient to prove
thus ex b1 being finite Element of bool the carrier of a2 st
b1 is Basis of a2;
:: VECTSP_9:def 1
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
b2 is finite-dimensional(b1)
iff
ex b3 being finite Element of bool the carrier of b2 st
b3 is Basis of b2;
:: VECTSP_9:th 24
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
st b2 is finite-dimensional(b1)
for b3 being Basis of b2 holds
b3 is finite;
:: VECTSP_9:th 25
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
st b2 is finite-dimensional(b1)
for b3 being Element of bool the carrier of b2
st b3 is linearly-independent(b1, b2)
holds b3 is finite;
:: VECTSP_9:th 26
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
st b2 is finite-dimensional(b1)
for b3, b4 being Basis of b2 holds
Card b3 = Card b4;
:: VECTSP_9:th 27
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 finite-dimensional(b1);
:: VECTSP_9:th 28
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
st b2 is finite-dimensional(b1)
holds b3 is finite-dimensional(b1);
:: VECTSP_9:exreg 2
registration
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;
cluster non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed finite-dimensional Subspace of a2;
end;
:: VECTSP_9:condreg 1
registration
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 finite-dimensional VectSpStr over a1;
cluster -> finite-dimensional (Subspace of a2);
end;
:: VECTSP_9:exreg 3
registration
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 finite-dimensional VectSpStr over a1;
cluster non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed finite-dimensional Subspace of a2;
end;
:: VECTSP_9:funcnot 1 => VECTSP_9:func 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;
assume a2 is finite-dimensional(a1);
func dim A2 -> natural set means
for b1 being Basis of a2 holds
it = Card b1;
end;
:: VECTSP_9:def 2
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
st b2 is finite-dimensional(b1)
for b3 being natural set holds
b3 = dim b2
iff
for b4 being Basis of b2 holds
b3 = Card b4;
:: VECTSP_9:th 29
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 finite-dimensional VectSpStr over b1
for b3 being Subspace of b2 holds
dim b3 <= dim b2;
:: VECTSP_9:th 30
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 finite-dimensional VectSpStr over b1
for b3 being Element of bool the carrier of b2
st b3 is linearly-independent(b1, b2)
holds Card b3 = dim Lin b3;
:: VECTSP_9:th 31
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 finite-dimensional VectSpStr over b1 holds
dim b2 = dim (Omega). b2;
:: VECTSP_9:th 32
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 finite-dimensional VectSpStr over b1
for b3 being Subspace of b2 holds
dim b2 = dim b3
iff
(Omega). b2 = (Omega). b3;
:: VECTSP_9:th 33
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 finite-dimensional VectSpStr over b1 holds
dim b2 = 0
iff
(Omega). b2 = (0). b2;
:: VECTSP_9:th 34
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 finite-dimensional VectSpStr over b1 holds
dim b2 = 1
iff
ex b3 being Element of the carrier of b2 st
b3 <> 0. b2 & (Omega). b2 = Lin {b3};
:: VECTSP_9:th 35
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 finite-dimensional VectSpStr over b1 holds
dim b2 = 2
iff
ex b3, b4 being Element of the carrier of b2 st
b3 <> b4 & {b3,b4} is linearly-independent(b1, b2) & (Omega). b2 = Lin {b3,b4};
:: VECTSP_9:th 36
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 finite-dimensional VectSpStr over b1
for b3, b4 being Subspace of b2 holds
(dim (b3 + b4)) + dim (b3 /\ b4) = (dim b3) + dim b4;
:: VECTSP_9:th 37
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 finite-dimensional VectSpStr over b1
for b3, b4 being Subspace of b2 holds
((dim b3) + dim b4) - dim b2 <= dim (b3 /\ b4);
:: VECTSP_9:th 38
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 finite-dimensional VectSpStr over b1
for b3, b4 being Subspace of b2
st b2 is_the_direct_sum_of b3,b4
holds dim b2 = (dim b3) + dim b4;
:: VECTSP_9:th 39
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 natural set
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1 holds
b2 <= dim b3
iff
ex b4 being strict Subspace of b3 st
dim b4 = b2;
:: VECTSP_9:funcnot 2 => VECTSP_9:func 2
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 finite-dimensional VectSpStr over a1;
let a3 be natural set;
func A3 Subspaces_of A2 -> set means
for b1 being set holds
b1 in it
iff
ex b2 being strict Subspace of a2 st
b2 = b1 & dim b2 = a3;
end;
:: VECTSP_9:def 3
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 finite-dimensional VectSpStr over b1
for b3 being natural set
for b4 being set holds
b4 = b3 Subspaces_of b2
iff
for b5 being set holds
b5 in b4
iff
ex b6 being strict Subspace of b2 st
b6 = b5 & dim b6 = b3;
:: VECTSP_9:th 40
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 natural set
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
st b2 <= dim b3
holds b2 Subspaces_of b3 is not empty;
:: VECTSP_9:th 41
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 natural set
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
st dim b3 < b2
holds b2 Subspaces_of b3 = {};
:: VECTSP_9:th 42
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 natural set
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b4 being Subspace of b3 holds
b2 Subspaces_of b4 c= b2 Subspaces_of b3;