Article RUSUB_4, MML version 4.99.1005
:: RUSUB_4:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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});
:: RUSUB_4:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being finite Element of bool the carrier of b1
st UNITSTR(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1,the scalar 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 &
UNITSTR(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1,the scalar of b1#) = Lin (b3 \/ b4));
:: RUSUB_4:attrnot 1 => RUSUB_4:attr 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
attr a1 is finite-dimensional means
ex b1 being finite Element of bool the carrier of a1 st
b1 is Basis of a1;
end;
:: RUSUB_4:dfs 1
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
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;
:: RUSUB_4:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
b1 is finite-dimensional
iff
ex b2 being finite Element of bool the carrier of b1 st
b2 is Basis of b1;
:: RUSUB_4:exreg 1
registration
cluster non empty left_complementable right_complementable Abelian add-associative right_zeroed RealLinearSpace-like zeroed strict RealUnitarySpace-like finite-dimensional UNITSTR;
end;
:: RUSUB_4:attrnot 2 => RUSUB_4:attr 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
attr a1 is finite-dimensional means
ex b1 being finite Element of bool the carrier of a1 st
b1 is Basis of a1;
end;
:: RUSUB_4:dfs 2
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
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;
:: RUSUB_4:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
b1 is finite-dimensional
iff
ex b2 being finite Element of bool the carrier of b1 st
b2 is Basis of b1;
:: RUSUB_4:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
st b1 is finite-dimensional
for b2 being Basis of b1 holds
b2 is finite;
:: RUSUB_4: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 b1 is finite-dimensional & b2 is linearly-independent(b1)
holds b2 is finite;
:: RUSUB_4:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Basis of b1
st b1 is finite-dimensional
holds Card b2 = Card b3;
:: RUSUB_4:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
(0). b1 is finite-dimensional;
:: RUSUB_4:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
st b1 is finite-dimensional
holds b2 is finite-dimensional;
:: RUSUB_4:exreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
cluster non empty left_complementable right_complementable Abelian add-associative right_zeroed RealLinearSpace-like zeroed strict RealUnitarySpace-like finite-dimensional Subspace of a1;
end;
:: RUSUB_4:condreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR;
cluster -> finite-dimensional (Subspace of a1);
end;
:: RUSUB_4:exreg 3
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR;
cluster non empty left_complementable right_complementable Abelian add-associative right_zeroed RealLinearSpace-like zeroed strict RealUnitarySpace-like finite-dimensional Subspace of a1;
end;
:: RUSUB_4:funcnot 1 => RUSUB_4:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
assume a1 is finite-dimensional;
func dim A1 -> Element of NAT means
for b1 being Basis of a1 holds
it = Card b1;
end;
:: RUSUB_4:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;
:: RUSUB_4:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Subspace of b1 holds
dim b2 <= dim b1;
:: RUSUB_4:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Element of bool the carrier of b1
st b2 is linearly-independent(b1)
holds Card b2 = dim Lin b2;
:: RUSUB_4:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR holds
dim b1 = dim (Omega). b1;
:: RUSUB_4:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Subspace of b1 holds
dim b1 = dim b2
iff
(Omega). b1 = (Omega). b2;
:: RUSUB_4:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR holds
dim b1 = 0
iff
(Omega). b1 = (0). b1;
:: RUSUB_4:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR holds
dim b1 = 1
iff
ex b2 being Element of the carrier of b1 st
b2 <> 0. b1 & (Omega). b1 = Lin {b2};
:: RUSUB_4:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR 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};
:: RUSUB_4:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2, b3 being Subspace of b1 holds
(dim (b2 + b3)) + dim (b2 /\ b3) = (dim b2) + dim b3;
:: RUSUB_4:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2, b3 being Subspace of b1 holds
((dim b2) + dim b3) - dim b1 <= dim (b2 /\ b3);
:: RUSUB_4:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2, b3 being Subspace of b1
st b1 is_the_direct_sum_of b2,b3
holds dim b1 = (dim b2) + dim b3;
:: RUSUB_4:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Subspace of b1
for b3 being Element of NAT holds
b3 <= dim b1
iff
ex b4 being strict Subspace of b1 st
dim b4 = b3;
:: RUSUB_4:funcnot 2 => RUSUB_4:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR;
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;
:: RUSUB_4:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
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;
:: RUSUB_4:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Element of NAT
st b2 <= dim b1
holds b2 Subspaces_of b1 is not empty;
:: RUSUB_4:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Element of NAT
st dim b1 < b2
holds b2 Subspaces_of b1 = {};
:: RUSUB_4:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like finite-dimensional UNITSTR
for b2 being Subspace of b1
for b3 being Element of NAT holds
b3 Subspaces_of b2 c= b3 Subspaces_of b1;
:: RUSUB_4:attrnot 3 => RUSUB_4:attr 2
definition
let a1 be non empty RLSStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is Affine means
for b1, b2 being Element of the carrier of a1
for b3 being Element of REAL
st b1 in a2 & b2 in a2
holds ((1 - b3) * b1) + (b3 * b2) in a2;
end;
:: RUSUB_4:dfs 5
definiens
let a1 be non empty RLSStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is Affine
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
for b3 being Element of REAL
st b1 in a2 & b2 in a2
holds ((1 - b3) * b1) + (b3 * b2) in a2;
:: RUSUB_4:def 5
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1 holds
b2 is Affine(b1)
iff
for b3, b4 being Element of the carrier of b1
for b5 being Element of REAL
st b3 in b2 & b4 in b2
holds ((1 - b5) * b3) + (b5 * b4) in b2;
:: RUSUB_4:th 22
theorem
for b1 being non empty RLSStruct holds
[#] b1 is Affine(b1) & {} b1 is Affine(b1);
:: RUSUB_4:th 23
theorem
for b1 being non empty RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
{b2} is Affine(b1);
:: RUSUB_4:exreg 4
registration
let a1 be non empty RLSStruct;
cluster non empty Affine Element of bool the carrier of a1;
end;
:: RUSUB_4:exreg 5
registration
let a1 be non empty RLSStruct;
cluster empty Affine Element of bool the carrier of a1;
end;
:: RUSUB_4:funcnot 3 => RUSUB_4:func 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Subspace of a1;
func Up A2 -> non empty Element of bool the carrier of a1 equals
the carrier of a2;
end;
:: RUSUB_4:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
Up b2 = the carrier of b2;
:: RUSUB_4:funcnot 4 => RUSUB_4:func 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
let a2 be Subspace of a1;
func Up A2 -> non empty Element of bool the carrier of a1 equals
the carrier of a2;
end;
:: RUSUB_4:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
Up b2 = the carrier of b2;
:: RUSUB_4:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
Up b2 is Affine(b1) & 0. b1 in the carrier of b2;
:: RUSUB_4:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Affine Element of bool the carrier of b1
st 0. b1 in b2
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b3 in b2
holds b4 * b3 in b2;
:: RUSUB_4:attrnot 4 => RUSUB_4:attr 3
definition
let a1 be non empty RLSStruct;
let a2 be non empty Element of bool the carrier of a1;
attr a2 is Subspace-like means
0. a1 in a2 &
(for b1, b2 being Element of the carrier of a1
for b3 being Element of REAL
st b1 in a2 & b2 in a2
holds b1 + b2 in a2 & b3 * b1 in a2);
end;
:: RUSUB_4:dfs 8
definiens
let a1 be non empty RLSStruct;
let a2 be non empty Element of bool the carrier of a1;
To prove
a2 is Subspace-like
it is sufficient to prove
thus 0. a1 in a2 &
(for b1, b2 being Element of the carrier of a1
for b3 being Element of REAL
st b1 in a2 & b2 in a2
holds b1 + b2 in a2 & b3 * b1 in a2);
:: RUSUB_4:def 8
theorem
for b1 being non empty RLSStruct
for b2 being non empty Element of bool the carrier of b1 holds
b2 is Subspace-like(b1)
iff
0. b1 in b2 &
(for b3, b4 being Element of the carrier of b1
for b5 being Element of REAL
st b3 in b2 & b4 in b2
holds b3 + b4 in b2 & b5 * b3 in b2);
:: RUSUB_4:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Affine Element of bool the carrier of b1
st 0. b1 in b2
holds b2 is Subspace-like(b1) & b2 = the carrier of Lin b2;
:: RUSUB_4:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
Up b2 is Subspace-like(b1);
:: RUSUB_4:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being non empty Affine Element of bool the carrier of b1
st 0. b1 in b2
holds b2 = the carrier of Lin b2;
:: RUSUB_4:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1 holds
Up b2 is Subspace-like(b1);
:: RUSUB_4:funcnot 5 => RUSUB_4:func 5
definition
let a1 be non empty addLoopStr;
let a2 be Element of bool the carrier of a1;
let a3 be Element of the carrier of a1;
func A3 + A2 -> Element of bool the carrier of a1 equals
{a3 + b1 where b1 is Element of the carrier of a1: b1 in a2};
end;
:: RUSUB_4:def 9
theorem
for b1 being non empty addLoopStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 + b2 = {b3 + b4 where b4 is Element of the carrier of b1: b4 in b2};
:: RUSUB_4:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being strict Subspace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
st Up b2 = b3
holds b4 + b2 = b4 + b3;
:: RUSUB_4:th 33
theorem
for b1 being non empty Abelian add-associative RealLinearSpace-like RLSStruct
for b2 being Affine Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 + b2 is Affine(b1);
:: RUSUB_4:th 34
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
for b4 being Element of the carrier of b1
st Up b2 = b3
holds b4 + b2 = b4 + b3;
:: RUSUB_4:funcnot 6 => RUSUB_4:func 6
definition
let a1 be non empty addLoopStr;
let a2, a3 be Element of bool the carrier of a1;
func A2 + A3 -> Element of bool the carrier of a1 equals
{b1 + b2 where b1 is Element of the carrier of a1, b2 is Element of the carrier of a1: b1 in a2 & b2 in a3};
end;
:: RUSUB_4:def 10
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of bool the carrier of b1 holds
b2 + b3 = {b4 + b5 where b4 is Element of the carrier of b1, b5 is Element of the carrier of b1: b4 in b2 & b5 in b3};
:: RUSUB_4:th 35
theorem
for b1 being non empty Abelian addLoopStr
for b2, b3 being Element of bool the carrier of b1 holds
b3 + b2 = b2 + b3;
:: RUSUB_4:funcnot 7 => RUSUB_4:func 7
definition
let a1 be non empty Abelian addLoopStr;
let a2, a3 be Element of bool the carrier of a1;
redefine func a2 + a3 -> Element of bool the carrier of a1;
commutativity;
:: for a1 being non empty Abelian addLoopStr
:: for a2, a3 being Element of bool the carrier of a1 holds
:: a2 + a3 = a3 + a2;
end;
:: RUSUB_4:th 36
theorem
for b1 being non empty addLoopStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
{b3} + b2 = b3 + b2;
:: RUSUB_4:th 37
theorem
for b1 being non empty Abelian add-associative RealLinearSpace-like RLSStruct
for b2 being Affine Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
{b3} + b2 is Affine(b1);
:: RUSUB_4:th 38
theorem
for b1 being non empty RLSStruct
for b2, b3 being Affine Element of bool the carrier of b1 holds
b2 /\ b3 is Affine(b1);
:: RUSUB_4:th 39
theorem
for b1 being non empty Abelian add-associative RealLinearSpace-like RLSStruct
for b2, b3 being Affine Element of bool the carrier of b1 holds
b2 + b3 is Affine(b1);