Article RUSUB_3, MML version 4.99.1005
:: RUSUB_3:funcnot 1 => RUSUB_3:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of bool the carrier of a1;
func Lin A2 -> strict Subspace of a1 means
the carrier of it = {Sum b1 where b1 is Linear_Combination of a2: TRUE};
end;
:: RUSUB_3:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being strict Subspace of b1 holds
b3 = Lin b2
iff
the carrier of b3 = {Sum b4 where b4 is Linear_Combination of b2: TRUE};
:: RUSUB_3:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being set holds
b3 in Lin b2
iff
ex b4 being Linear_Combination of b2 st
b3 = Sum b4;
:: RUSUB_3:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being set
st b3 in b2
holds b3 in Lin b2;
:: RUSUB_3:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
Lin {} the carrier of b1 = (0). b1;
:: RUSUB_3:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
st Lin b2 = (0). b1 & b2 <> {}
holds b2 = {0. b1};
:: RUSUB_3:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being strict Subspace of b1
for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds Lin b3 = b2;
:: RUSUB_3:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
st b2 = the carrier of b1
holds Lin b2 = b1;
:: RUSUB_3:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of bool the carrier of b1
st b2 c= b3
holds Lin b2 is Subspace of Lin b3;
:: RUSUB_3:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like UNITSTR
for b2, b3 being Element of bool the carrier of b1
st Lin b2 = b1 & b2 c= b3
holds Lin b3 = b1;
:: RUSUB_3:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of bool the carrier of b1 holds
Lin (b2 \/ b3) = (Lin b2) + Lin b3;
:: RUSUB_3:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of bool the carrier of b1 holds
Lin (b2 /\ b3) is Subspace of (Lin b2) /\ Lin b3;
:: RUSUB_3:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
st b2 is linearly-independent(b1)
holds ex b3 being Element of bool the carrier of b1 st
b2 c= b3 &
b3 is linearly-independent(b1) &
Lin b3 = UNITSTR(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1,the scalar of b1#);
:: RUSUB_3:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
st Lin b2 = b1
holds ex b3 being Element of bool the carrier of b1 st
b3 c= b2 & b3 is linearly-independent(b1) & Lin b3 = b1;
:: RUSUB_3:modenot 1 => RUSUB_3:mode 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
mode Basis of A1 -> Element of bool the carrier of a1 means
it is linearly-independent(a1) &
Lin it = UNITSTR(#the carrier of a1,the ZeroF of a1,the addF of a1,the Mult of a1,the scalar of a1#);
end;
:: RUSUB_3:dfs 2
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Element of bool the carrier of a1;
To prove
a2 is Basis of a1
it is sufficient to prove
thus a2 is linearly-independent(a1) &
Lin a2 = UNITSTR(#the carrier of a1,the ZeroF of a1,the addF of a1,the Mult of a1,the scalar of a1#);
:: RUSUB_3:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1 holds
b2 is Basis of b1
iff
b2 is linearly-independent(b1) &
Lin b2 = UNITSTR(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1,the scalar of b1#);
:: RUSUB_3:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like UNITSTR
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;
:: RUSUB_3:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
st Lin b2 = b1
holds ex b3 being Basis of b1 st
b3 c= b2;
:: RUSUB_3:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;
:: RUSUB_3:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;
:: RUSUB_3:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;
:: RUSUB_3:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;
:: RUSUB_3:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;
:: RUSUB_3:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;
:: RUSUB_3:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Basis of b1
for b3 being Element of the carrier of b1 holds
b3 in Lin b2;
:: RUSUB_3:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;
:: RUSUB_3:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;
:: RUSUB_3:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Basis of b2 holds
ex b4 being Basis of b1 st
b3 c= b4;
:: RUSUB_3:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;
:: RUSUB_3:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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);
:: RUSUB_3:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;
:: RUSUB_3:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;
:: RUSUB_3:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being set holds
b3 in Lin {b2}
iff
ex b4 being Element of REAL st
b3 = b4 * b2;
:: RUSUB_3:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
b2 in Lin {b2};
:: RUSUB_3:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being set holds
b4 in b2 + Lin {b3}
iff
ex b5 being Element of REAL st
b4 = b2 + (b5 * b3);
:: RUSUB_3:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being set holds
b4 in Lin {b2,b3}
iff
ex b5, b6 being Element of REAL st
b4 = (b5 * b2) + (b6 * b3);
:: RUSUB_3:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
b2 in Lin {b2,b3} & b3 in Lin {b2,b3};
:: RUSUB_3:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1
for b5 being set holds
b5 in b2 + Lin {b3,b4}
iff
ex b6, b7 being Element of REAL st
b5 = (b2 + (b6 * b3)) + (b7 * b4);
:: RUSUB_3:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1
for b5 being set holds
b5 in Lin {b2,b3,b4}
iff
ex b6, b7, b8 being Element of REAL st
b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4);
:: RUSUB_3:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
b2 in Lin {b2,b3,b4} & b3 in Lin {b2,b3,b4} & b4 in Lin {b2,b3,b4};
:: RUSUB_3:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being set holds
b6 in b2 + Lin {b3,b4,b5}
iff
ex b7, b8, b9 being Element of REAL st
b6 = ((b2 + (b7 * b3)) + (b8 * b4)) + (b9 * b5);
:: RUSUB_3:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
st b2 in Lin {b3} & b2 <> 0. b1
holds Lin {b2} = Lin {b3};
:: RUSUB_3:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4, b5 being Element of the carrier of b1
st b2 <> b3 & {b2,b3} is linearly-independent(b1) & b2 in Lin {b4,b5} & b3 in Lin {b4,b5}
holds Lin {b4,b5} = Lin {b2,b3} &
{b4,b5} is linearly-independent(b1) &
b4 <> b5;
:: RUSUB_3:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being set holds
b2 in (0). b1
iff
b2 = 0. b1;
:: RUSUB_3:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Subspace of b1
st b2 is Subspace of b4
holds b2 /\ b3 is Subspace of b4;
:: RUSUB_3:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Subspace of b1
st b2 is Subspace of b3 & b2 is Subspace of b4
holds b2 is Subspace of b3 /\ b4;
:: RUSUB_3:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Subspace of b1
st b2 is Subspace of b4 & b3 is Subspace of b4
holds b2 + b3 is Subspace of b4;
:: RUSUB_3:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Subspace of b1
st b2 is Subspace of b3
holds b2 is Subspace of b3 + b4;
:: RUSUB_3:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Subspace of b1
for b4 being Element of the carrier of b1
st b2 is Subspace of b3
holds b4 + b2 c= b4 + b3;