Article RLVECT_5, MML version 4.99.1005
:: RLVECT_5:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Linear_Combination of b1
for b4 being Element of bool the carrier of b1
st b4 is linearly-independent(b1) & Carrier b2 c= b4 & Carrier b3 c= b4 & Sum b2 = Sum b3
holds b2 = b3;
:: RLVECT_5:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st b2 is linearly-independent(b1)
holds ex b3 being Basis of b1 st
b2 c= b3;
:: RLVECT_5:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
for b3 being Element of the carrier of b1 holds
b3 in Carrier b2
iff
ex b4 being Element of the carrier of b1 st
b3 = b4 & b2 . b4 <> 0;
:: RLVECT_5:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
for b3, b4 being FinSequence of the carrier of b1
for b5 being Function-like quasi_total bijective Relation of dom b3,dom b3
st b4 = b3 * b5
holds Sum (b2 (#) b3) = Sum (b2 (#) b4);
:: RLVECT_5:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
for b3 being FinSequence of the carrier of b1
st Carrier b2 misses proj2 b3
holds Sum (b2 (#) b3) = 0. b1;
:: RLVECT_5:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being FinSequence of the carrier of b1
st b2 is one-to-one
for b3 being Linear_Combination of b1
st Carrier b3 c= proj2 b2
holds Sum (b3 (#) b2) = Sum b3;
:: RLVECT_5:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
for b3 being FinSequence of the carrier of b1 holds
ex b4 being Linear_Combination of b1 st
Carrier b4 = (proj2 b3) /\ Carrier b2 & b2 (#) b3 = b4 (#) b3;
:: RLVECT_5:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
for b3 being Element of bool the carrier of b1
for b4 being FinSequence of the carrier of b1
st proj2 b4 c= the carrier of Lin b3
holds ex b5 being Linear_Combination of b3 st
Sum (b2 (#) b4) = Sum b5;
:: RLVECT_5:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
for b3 being Element of bool the carrier of b1
st Carrier b2 c= the carrier of Lin b3
holds ex b4 being Linear_Combination of b3 st
Sum b2 = Sum b4;
:: RLVECT_5:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Linear_Combination of b1
st Carrier b3 c= the carrier of b2
for b4 being Linear_Combination of b2
st b4 = b3 | the carrier of b2
holds Carrier b3 = Carrier b4 & Sum b3 = Sum b4;
:: RLVECT_5:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Linear_Combination of b2 holds
ex b4 being Linear_Combination of b1 st
Carrier b3 = Carrier b4 & Sum b3 = Sum b4;
:: RLVECT_5:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Linear_Combination of b1
st Carrier b3 c= the carrier of b2
holds ex b4 being Linear_Combination of b2 st
Carrier b4 = Carrier b3 & Sum b4 = Sum b3;
:: RLVECT_5:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Basis of b1
for b3 being Element of the carrier of b1 holds
b3 in Lin b2;
:: RLVECT_5:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Element of bool the carrier of b2
st b3 is linearly-independent(b2)
holds b3 is linearly-independent Element of bool the carrier of b1;
:: RLVECT_5:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Element of bool the carrier of b1
st b3 is linearly-independent(b1) & b3 c= the carrier of b2
holds b3 is linearly-independent Element of bool the carrier of b2;
:: RLVECT_5:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Basis of b2 holds
ex b4 being Basis of b1 st
b3 c= b4;
:: RLVECT_5:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st b2 is linearly-independent(b1)
for b3 being Element of the carrier of b1
st b3 in b2
for b4 being Element of bool the carrier of b1
st b4 = b2 \ {b3}
holds not b3 in Lin b4;
:: RLVECT_5:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Basis of b1
for b3 being non empty Element of bool the carrier of b1
st b3 misses b2
for b4 being Element of bool the carrier of b1
st b4 = b2 \/ b3
holds b4 is linearly-dependent(b1);
:: RLVECT_5:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Element of bool the carrier of b1
st b3 c= the carrier of b2
holds Lin b3 is Subspace of b2;
:: RLVECT_5:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b3 = b4
holds Lin b3 = Lin b4;
:: RLVECT_5:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being finite Element of bool the carrier of b1
for b4 being Element of the carrier of b1
st b4 in Lin (b2 \/ b3) & not b4 in Lin b3
holds ex b5 being Element of the carrier of b1 st
b5 in b2 &
b5 in Lin (((b2 \/ b3) \ {b5}) \/ {b4});
:: RLVECT_5:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being finite Element of bool the carrier of b1
st RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#) = Lin b2 &
b3 is linearly-independent(b1)
holds card b3 <= card b2 &
(ex b4 being finite Element of bool the carrier of b1 st
b4 c= b2 &
card b4 = (card b2) - card b3 &
RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#) = Lin (b3 \/ b4));
:: RLVECT_5:attrnot 1 => RLVECT_5:attr 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
attr a1 is finite-dimensional means
ex b1 being finite Element of bool the carrier of a1 st
b1 is Basis of a1;
end;
:: RLVECT_5:dfs 1
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
To prove
a1 is finite-dimensional
it is sufficient to prove
thus ex b1 being finite Element of bool the carrier of a1 st
b1 is Basis of a1;
:: RLVECT_5:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
b1 is finite-dimensional
iff
ex b2 being finite Element of bool the carrier of b1 st
b2 is Basis of b1;
:: RLVECT_5:exreg 1
registration
cluster non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct;
end;
:: RLVECT_5:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st b1 is finite-dimensional
for b2 being Basis of b1 holds
b2 is finite;
:: RLVECT_5:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st b1 is finite-dimensional
for b2 being Element of bool the carrier of b1
st b2 is linearly-independent(b1)
holds b2 is finite;
:: RLVECT_5:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st b1 is finite-dimensional
for b2, b3 being Basis of b1 holds
Card b2 = Card b3;
:: RLVECT_5:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
(0). b1 is finite-dimensional;
:: RLVECT_5:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
st b1 is finite-dimensional
holds b2 is finite-dimensional;
:: RLVECT_5:exreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional Subspace of a1;
end;
:: RLVECT_5:condreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct;
cluster -> finite-dimensional (Subspace of a1);
end;
:: RLVECT_5:exreg 3
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct;
cluster non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional Subspace of a1;
end;
:: RLVECT_5:funcnot 1 => RLVECT_5:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
assume a1 is finite-dimensional;
func dim A1 -> Element of NAT means
for b1 being Basis of a1 holds
it = Card b1;
end;
:: RLVECT_5:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st b1 is finite-dimensional
for b2 being Element of NAT holds
b2 = dim b1
iff
for b3 being Basis of b1 holds
b2 = Card b3;
:: RLVECT_5:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct
for b2 being Subspace of b1 holds
dim b2 <= dim b1;
:: RLVECT_5:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct
for b2 being Element of bool the carrier of b1
st b2 is linearly-independent(b1)
holds Card b2 = dim Lin b2;
:: RLVECT_5:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct holds
dim b1 = dim (Omega). b1;
:: RLVECT_5:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct
for b2 being Subspace of b1 holds
dim b1 = dim b2
iff
(Omega). b1 = (Omega). b2;
:: RLVECT_5:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct holds
dim b1 = 0
iff
(Omega). b1 = (0). b1;
:: RLVECT_5:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct holds
dim b1 = 1
iff
ex b2 being Element of the carrier of b1 st
b2 <> 0. b1 & (Omega). b1 = Lin {b2};
:: RLVECT_5:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct holds
dim b1 = 2
iff
ex b2, b3 being Element of the carrier of b1 st
b2 <> b3 & {b2,b3} is linearly-independent(b1) & (Omega). b1 = Lin {b2,b3};
:: RLVECT_5:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct
for b2, b3 being Subspace of b1 holds
(dim (b2 + b3)) + dim (b2 /\ b3) = (dim b2) + dim b3;
:: RLVECT_5:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct
for b2, b3 being Subspace of b1 holds
((dim b2) + dim b3) - dim b1 <= dim (b2 /\ b3);
:: RLVECT_5:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct
for b2, b3 being Subspace of b1
st b1 is_the_direct_sum_of b2,b3
holds dim b1 = (dim b2) + dim b3;
:: RLVECT_5:th 39
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct holds
b1 <= dim b2
iff
ex b3 being strict Subspace of b2 st
dim b3 = b1;
:: RLVECT_5:funcnot 2 => RLVECT_5:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct;
let a2 be Element of NAT;
func A2 Subspaces_of A1 -> set means
for b1 being set holds
b1 in it
iff
ex b2 being strict Subspace of a1 st
b2 = b1 & dim b2 = a2;
end;
:: RLVECT_5:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct
for b2 being Element of NAT
for b3 being set holds
b3 = b2 Subspaces_of b1
iff
for b4 being set holds
b4 in b3
iff
ex b5 being strict Subspace of b1 st
b5 = b4 & dim b5 = b2;
:: RLVECT_5:th 40
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct
st b1 <= dim b2
holds b1 Subspaces_of b2 is not empty;
:: RLVECT_5:th 41
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct
st dim b2 < b1
holds b1 Subspaces_of b2 = {};
:: RLVECT_5:th 42
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like finite-dimensional RLSStruct
for b3 being Subspace of b2 holds
b1 Subspaces_of b3 c= b1 Subspaces_of b2;