Article CLVECT_1, MML version 4.99.1005
:: CLVECT_1:structnot 1 => CLVECT_1:struct 1
definition
struct(addLoopStr) CLSStruct(#
carrier -> set,
ZeroF -> Element of the carrier of it,
addF -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],the carrier of it,
Mult -> Function-like quasi_total Relation of [:COMPLEX,the carrier of it:],the carrier of it
#);
end;
:: CLVECT_1:attrnot 1 => CLVECT_1:attr 1
definition
let a1 be CLSStruct;
attr a1 is strict;
end;
:: CLVECT_1:exreg 1
registration
cluster strict CLSStruct;
end;
:: CLVECT_1:aggrnot 1 => CLVECT_1:aggr 1
definition
let a1 be set;
let a2 be Element of a1;
let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
let a4 be Function-like quasi_total Relation of [:COMPLEX,a1:],a1;
aggr CLSStruct(#a1,a2,a3,a4#) -> strict CLSStruct;
end;
:: CLVECT_1:selnot 1 => CLVECT_1:sel 1
definition
let a1 be CLSStruct;
sel the Mult of a1 -> Function-like quasi_total Relation of [:COMPLEX,the carrier of a1:],the carrier of a1;
end;
:: CLVECT_1:exreg 2
registration
cluster non empty CLSStruct;
end;
:: CLVECT_1:modenot 1
definition
let a1 be CLSStruct;
mode VECTOR of a1 is Element of the carrier of a1;
end;
:: CLVECT_1:funcnot 1 => CLVECT_1:func 1
definition
let a1 be non empty CLSStruct;
let a2 be Element of the carrier of a1;
let a3 be Element of COMPLEX;
func A3 * A2 -> Element of the carrier of a1 equals
(the Mult of a1) . [a3,a2];
end;
:: CLVECT_1:def 1
theorem
for b1 being non empty CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX holds
b3 * b2 = (the Mult of b1) . [b3,b2];
:: CLVECT_1:funcreg 1
registration
let a1 be non empty set;
let a2 be Element of a1;
let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
let a4 be Function-like quasi_total Relation of [:COMPLEX,a1:],a1;
cluster CLSStruct(#a1,a2,a3,a4#) -> non empty strict;
end;
:: CLVECT_1:attrnot 2 => CLVECT_1:attr 2
definition
let a1 be non empty CLSStruct;
attr a1 is ComplexLinearSpace-like means
(for b1 being Element of COMPLEX
for b2, b3 being Element of the carrier of a1 holds
b1 * (b2 + b3) = (b1 * b2) + (b1 * b3)) &
(for b1, b2 being Element of COMPLEX
for b3 being Element of the carrier of a1 holds
(b1 + b2) * b3 = (b1 * b3) + (b2 * b3)) &
(for b1, b2 being Element of COMPLEX
for b3 being Element of the carrier of a1 holds
(b1 * b2) * b3 = b1 * (b2 * b3)) &
(for b1 being Element of the carrier of a1 holds
1r * b1 = b1);
end;
:: CLVECT_1:dfs 2
definiens
let a1 be non empty CLSStruct;
To prove
a1 is ComplexLinearSpace-like
it is sufficient to prove
thus (for b1 being Element of COMPLEX
for b2, b3 being Element of the carrier of a1 holds
b1 * (b2 + b3) = (b1 * b2) + (b1 * b3)) &
(for b1, b2 being Element of COMPLEX
for b3 being Element of the carrier of a1 holds
(b1 + b2) * b3 = (b1 * b3) + (b2 * b3)) &
(for b1, b2 being Element of COMPLEX
for b3 being Element of the carrier of a1 holds
(b1 * b2) * b3 = b1 * (b2 * b3)) &
(for b1 being Element of the carrier of a1 holds
1r * b1 = b1);
:: CLVECT_1:def 2
theorem
for b1 being non empty CLSStruct holds
b1 is ComplexLinearSpace-like
iff
(for b2 being Element of COMPLEX
for b3, b4 being Element of the carrier of b1 holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4)) &
(for b2, b3 being Element of COMPLEX
for b4 being Element of the carrier of b1 holds
(b2 + b3) * b4 = (b2 * b4) + (b3 * b4)) &
(for b2, b3 being Element of COMPLEX
for b4 being Element of the carrier of b1 holds
(b2 * b3) * b4 = b2 * (b3 * b4)) &
(for b2 being Element of the carrier of b1 holds
1r * b2 = b2);
:: CLVECT_1:funcnot 2 => CLVECT_1:func 2
definition
func Trivial-CLSStruct -> strict CLSStruct equals
CLSStruct(#1,op0,op2,pr2(COMPLEX,1)#);
end;
:: CLVECT_1:def 3
theorem
Trivial-CLSStruct = CLSStruct(#1,op0,op2,pr2(COMPLEX,1)#);
:: CLVECT_1:funcreg 2
registration
cluster Trivial-CLSStruct -> non empty trivial strict;
end;
:: CLVECT_1:exreg 3
registration
cluster non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like CLSStruct;
end;
:: CLVECT_1:modenot 2
definition
mode ComplexLinearSpace is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
end;
:: CLVECT_1:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
st (b3 = 0 or b2 = 0. b1)
holds b3 * b2 = 0. b1;
:: CLVECT_1:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
st b3 * b2 = 0. b1 & b3 <> 0
holds b2 = 0. b1;
:: CLVECT_1:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
- b2 = (- 1r) * b2;
:: CLVECT_1:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
st b2 = - b2
holds b2 = 0. b1;
:: CLVECT_1:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
st b2 + b2 = 0. b1
holds b2 = 0. b1;
:: CLVECT_1:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX holds
b3 * - b2 = (- b3) * b2;
:: CLVECT_1:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX holds
b3 * - b2 = - (b3 * b2);
:: CLVECT_1:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX holds
(- b3) * - b2 = b3 * b2;
:: CLVECT_1:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of COMPLEX holds
b4 * (b2 - b3) = (b4 * b2) - (b4 * b3);
:: CLVECT_1:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of COMPLEX holds
(b3 - b4) * b2 = (b3 * b2) - (b4 * b2);
:: CLVECT_1:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of COMPLEX
st b4 <> 0 & b4 * b2 = b4 * b3
holds b2 = b3;
:: CLVECT_1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of COMPLEX
st b2 <> 0. b1 & b3 * b2 = b4 * b2
holds b3 = b4;
:: CLVECT_1:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of COMPLEX
for b3, b4 being FinSequence of the carrier of b1
st len b3 = len b4 &
(for b5 being Element of NAT
for b6 being Element of the carrier of b1
st b5 in dom b3 & b6 = b4 . b5
holds b3 . b5 = b2 * b6)
holds Sum b3 = b2 * Sum b4;
:: CLVECT_1:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of COMPLEX holds
b2 * Sum <*> the carrier of b1 = 0. b1;
:: CLVECT_1:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of COMPLEX holds
b4 * Sum <*b2,b3*> = (b4 * b2) + (b4 * b3);
:: CLVECT_1:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of COMPLEX holds
b5 * Sum <*b2,b3,b4*> = ((b5 * b2) + (b5 * b3)) + (b5 * b4);
:: CLVECT_1:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
Sum <*b2,b2*> = [*2,0*] * b2;
:: CLVECT_1:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
Sum <*- b2,- b2*> = [*- 2,0*] * b2;
:: CLVECT_1:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
Sum <*b2,b2,b2*> = [*3,0*] * b2;
:: CLVECT_1:attrnot 3 => CLVECT_1:attr 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is linearly-closed means
(for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2
holds b1 + b2 in a2) &
(for b1 being Element of COMPLEX
for b2 being Element of the carrier of a1
st b2 in a2
holds b1 * b2 in a2);
end;
:: CLVECT_1:dfs 4
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is linearly-closed
it is sufficient to prove
thus (for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2
holds b1 + b2 in a2) &
(for b1 being Element of COMPLEX
for b2 being Element of the carrier of a1
st b2 in a2
holds b1 * b2 in a2);
:: CLVECT_1:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1 holds
b2 is linearly-closed(b1)
iff
(for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2
holds b3 + b4 in b2) &
(for b3 being Element of COMPLEX
for b4 being Element of the carrier of b1
st b4 in b2
holds b3 * b4 in b2);
:: CLVECT_1:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
st b2 <> {} & b2 is linearly-closed(b1)
holds 0. b1 in b2;
:: CLVECT_1:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
st b2 is linearly-closed(b1)
for b3 being Element of the carrier of b1
st b3 in b2
holds - b3 in b2;
:: CLVECT_1:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
st b2 is linearly-closed(b1)
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2
holds b3 - b4 in b2;
:: CLVECT_1:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
{0. b1} is linearly-closed(b1);
:: CLVECT_1:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
st the carrier of b1 = b2
holds b2 is linearly-closed(b1);
:: CLVECT_1:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b2 is linearly-closed(b1) &
b3 is linearly-closed(b1) &
b4 = {b5 + b6 where b5 is Element of the carrier of b1, b6 is Element of the carrier of b1: b5 in b2 & b6 in b3}
holds b4 is linearly-closed(b1);
:: CLVECT_1:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is linearly-closed(b1) & b3 is linearly-closed(b1)
holds b2 /\ b3 is linearly-closed(b1);
:: CLVECT_1:modenot 3 => CLVECT_1:mode 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
mode Subspace of A1 -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct means
the carrier of it c= the carrier of a1 &
0. it = 0. a1 &
the addF of it = (the addF of a1) || the carrier of it &
the Mult of it = (the Mult of a1) | [:COMPLEX,the carrier of it:];
end;
:: CLVECT_1:dfs 5
definiens
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
To prove
a2 is Subspace of a1
it is sufficient to prove
thus the carrier of a2 c= the carrier of a1 &
0. a2 = 0. a1 &
the addF of a2 = (the addF of a1) || the carrier of a2 &
the Mult of a2 = (the Mult of a1) | [:COMPLEX,the carrier of a2:];
:: CLVECT_1:def 5
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
b2 is Subspace of b1
iff
the carrier of b2 c= the carrier of b1 &
0. b2 = 0. b1 &
the addF of b2 = (the addF of b1) || the carrier of b2 &
the Mult of b2 = (the Mult of b1) | [:COMPLEX,the carrier of b2:];
:: CLVECT_1:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1
for b4 being set
st b4 in b2 & b2 is Subspace of b3
holds b4 in b3;
:: CLVECT_1:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1
for b3 being set
st b3 in b2
holds b3 in b1;
:: CLVECT_1:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1
for b3 being Element of the carrier of b2 holds
b3 is Element of the carrier of b1;
:: CLVECT_1:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
0. b2 = 0. b1;
:: CLVECT_1:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1 holds
0. b2 = 0. b3;
:: CLVECT_1:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
for b5, b6 being Element of the carrier of b4
st b5 = b2 & b6 = b3
holds b5 + b6 = b2 + b3;
:: CLVECT_1:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
for b4 being Subspace of b1
for b5 being Element of the carrier of b4
st b5 = b2
holds b3 * b5 = b3 * b2;
:: CLVECT_1:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
for b4 being Element of the carrier of b3
st b4 = b2
holds - b2 = - b4;
:: CLVECT_1:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
for b5, b6 being Element of the carrier of b4
st b5 = b2 & b6 = b3
holds b5 - b6 = b2 - b3;
:: CLVECT_1:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
0. b1 in b2;
:: CLVECT_1:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1 holds
0. b2 in b3;
:: CLVECT_1:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
0. b2 in b1;
:: CLVECT_1:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
st b2 in b4 & b3 in b4
holds b2 + b3 in b4;
:: CLVECT_1:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
for b4 being Subspace of b1
st b2 in b4
holds b3 * b2 in b4;
:: CLVECT_1:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
st b2 in b3
holds - b2 in b3;
:: CLVECT_1:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
st b2 in b4 & b3 in b4
holds b2 - b3 in b4;
:: CLVECT_1:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being non empty set
for b4 being Element of b3
for b5 being Function-like quasi_total Relation of [:b3,b3:],b3
for b6 being Function-like quasi_total Relation of [:COMPLEX,b3:],b3
st b2 = b3 &
b4 = 0. b1 &
b5 = (the addF of b1) || b2 &
b6 = (the Mult of b1) | [:COMPLEX,b2:]
holds CLSStruct(#b3,b4,b5,b6#) is Subspace of b1;
:: CLVECT_1:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
b1 is Subspace of b1;
:: CLVECT_1:th 46
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like CLSStruct
st b1 is Subspace of b2 & b2 is Subspace of b1
holds b1 = b2;
:: CLVECT_1:th 47
theorem
for b1, b2, b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
st b1 is Subspace of b2 & b2 is Subspace of b3
holds b1 is Subspace of b3;
:: CLVECT_1:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1
st the carrier of b2 c= the carrier of b3
holds b2 is Subspace of b3;
:: CLVECT_1:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1
st for b4 being Element of the carrier of b1
st b4 in b2
holds b4 in b3
holds b2 is Subspace of b3;
:: CLVECT_1:exreg 4
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
cluster non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like Subspace of a1;
end;
:: CLVECT_1:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being strict Subspace of b1
st the carrier of b2 = the carrier of b3
holds b2 = b3;
:: CLVECT_1:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being strict Subspace of b1
st for b4 being Element of the carrier of b1 holds
b4 in b2
iff
b4 in b3
holds b2 = b3;
:: CLVECT_1:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like CLSStruct
for b2 being strict Subspace of b1
st the carrier of b2 = the carrier of b1
holds b2 = b1;
:: CLVECT_1:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like CLSStruct
for b2 being strict Subspace of b1
st for b3 being Element of the carrier of b1 holds
b3 in b2
iff
b3 in b1
holds b2 = b1;
:: CLVECT_1:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Subspace of b1
st the carrier of b3 = b2
holds b2 is linearly-closed(b1);
:: CLVECT_1:th 55
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
st b2 <> {} & b2 is linearly-closed(b1)
holds ex b3 being strict Subspace of b1 st
b2 = the carrier of b3;
:: CLVECT_1:funcnot 3 => CLVECT_1:func 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
func (0). A1 -> strict Subspace of a1 means
the carrier of it = {0. a1};
end;
:: CLVECT_1:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being strict Subspace of b1 holds
b2 = (0). b1
iff
the carrier of b2 = {0. b1};
:: CLVECT_1:funcnot 4 => CLVECT_1:func 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
func (Omega). A1 -> strict Subspace of a1 equals
CLSStruct(#the carrier of a1,the ZeroF of a1,the addF of a1,the Mult of a1#);
end;
:: CLVECT_1:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
(Omega). b1 = CLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#);
:: CLVECT_1:th 56
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
(0). b2 = (0). b1;
:: CLVECT_1:th 57
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1 holds
(0). b2 = (0). b3;
:: CLVECT_1:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
(0). b2 is Subspace of b1;
:: CLVECT_1:th 59
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
(0). b1 is Subspace of b2;
:: CLVECT_1:th 60
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1 holds
(0). b2 is Subspace of b3;
:: CLVECT_1:th 61
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like CLSStruct holds
b1 is Subspace of (Omega). b1;
:: CLVECT_1:funcnot 5 => CLVECT_1:func 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
let a2 be Element of the carrier of a1;
let a3 be Subspace of a1;
func A2 + A3 -> Element of bool the carrier of a1 equals
{a2 + b1 where b1 is Element of the carrier of a1: b1 in a3};
end;
:: CLVECT_1:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
b2 + b3 = {b2 + b4 where b4 is Element of the carrier of b1: b4 in b3};
:: CLVECT_1:modenot 4 => CLVECT_1:mode 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
let a2 be Subspace of a1;
mode Coset of A2 -> Element of bool the carrier of a1 means
ex b1 being Element of the carrier of a1 st
it = b1 + a2;
end;
:: CLVECT_1:dfs 9
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
let a2 be Subspace of a1;
let a3 be Element of bool the carrier of a1;
To prove
a3 is Coset of a2
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
a3 = b1 + a2;
:: CLVECT_1:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1
for b3 being Element of bool the carrier of b1 holds
b3 is Coset of b2
iff
ex b4 being Element of the carrier of b1 st
b3 = b4 + b2;
:: CLVECT_1:th 62
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
0. b1 in b2 + b3
iff
b2 in b3;
:: CLVECT_1:th 63
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
b2 in b2 + b3;
:: CLVECT_1:th 64
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
(0. b1) + b2 = the carrier of b2;
:: CLVECT_1:th 65
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
b2 + (0). b1 = {b2};
:: CLVECT_1:th 66
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
b2 + (Omega). b1 = the carrier of b1;
:: CLVECT_1:th 67
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
0. b1 in b2 + b3
iff
b2 + b3 = the carrier of b3;
:: CLVECT_1:th 68
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
b2 in b3
iff
b2 + b3 = the carrier of b3;
:: CLVECT_1:th 69
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
for b4 being Subspace of b1
st b2 in b4
holds (b3 * b2) + b4 = the carrier of b4;
:: CLVECT_1:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
for b4 being Subspace of b1
st b3 <> 0 & (b3 * b2) + b4 = the carrier of b4
holds b2 in b4;
:: CLVECT_1:th 71
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
b2 in b3
iff
(- b2) + b3 = the carrier of b3;
:: CLVECT_1:th 72
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
b2 in b4
iff
b3 + b4 = (b3 + b2) + b4;
:: CLVECT_1:th 73
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
b2 in b4
iff
b3 + b4 = (b3 - b2) + b4;
:: CLVECT_1:th 74
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
b2 in b3 + b4
iff
b3 + b4 = b2 + b4;
:: CLVECT_1:th 75
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
b2 + b3 = (- b2) + b3
iff
b2 in b3;
:: CLVECT_1:th 76
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Subspace of b1
st b2 in b3 + b5 & b2 in b4 + b5
holds b3 + b5 = b4 + b5;
:: CLVECT_1:th 77
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
st b2 in b3 + b4 & b2 in (- b3) + b4
holds b3 in b4;
:: CLVECT_1:th 78
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
for b4 being Subspace of b1
st b3 <> 1r & b3 * b2 in b2 + b4
holds b2 in b4;
:: CLVECT_1:th 79
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
for b4 being Subspace of b1
st b2 in b4
holds b3 * b2 in b2 + b4;
:: CLVECT_1:th 80
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
- b2 in b2 + b3
iff
b2 in b3;
:: CLVECT_1:th 81
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
b2 + b3 in b3 + b4
iff
b2 in b4;
:: CLVECT_1:th 82
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
b2 - b3 in b2 + b4
iff
b3 in b4;
:: CLVECT_1:th 83
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
b2 in b3 + b4
iff
ex b5 being Element of the carrier of b1 st
b5 in b4 & b2 = b3 + b5;
:: CLVECT_1:th 84
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
b2 in b3 + b4
iff
ex b5 being Element of the carrier of b1 st
b5 in b4 & b2 = b3 - b5;
:: CLVECT_1:th 85
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
ex b5 being Element of the carrier of b1 st
b2 in b5 + b4 & b3 in b5 + b4
iff
b2 - b3 in b4;
:: CLVECT_1:th 86
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
st b2 + b4 = b3 + b4
holds ex b5 being Element of the carrier of b1 st
b5 in b4 & b2 + b5 = b3;
:: CLVECT_1:th 87
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
st b2 + b4 = b3 + b4
holds ex b5 being Element of the carrier of b1 st
b5 in b4 & b2 - b5 = b3;
:: CLVECT_1:th 88
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3, b4 being strict Subspace of b1 holds
b2 + b3 = b2 + b4
iff
b3 = b4;
:: CLVECT_1:th 89
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being strict Subspace of b1
st b2 + b4 = b3 + b5
holds b4 = b5;
:: CLVECT_1:th 90
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1
for b3 being Coset of b2 holds
b3 is linearly-closed(b1)
iff
b3 = the carrier of b2;
:: CLVECT_1:th 91
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being strict Subspace of b1
for b4 being Coset of b2
for b5 being Coset of b3
st b4 = b5
holds b2 = b3;
:: CLVECT_1:th 92
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
{b2} is Coset of (0). b1;
:: CLVECT_1:th 93
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
st b2 is Coset of (0). b1
holds ex b3 being Element of the carrier of b1 st
b2 = {b3};
:: CLVECT_1:th 94
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
the carrier of b2 is Coset of b2;
:: CLVECT_1:th 95
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
the carrier of b1 is Coset of (Omega). b1;
:: CLVECT_1:th 96
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
st b2 is Coset of (Omega). b1
holds b2 = the carrier of b1;
:: CLVECT_1:th 97
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1
for b3 being Coset of b2 holds
0. b1 in b3
iff
b3 = the carrier of b2;
:: CLVECT_1:th 98
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
for b4 being Coset of b3 holds
b2 in b4
iff
b4 = b2 + b3;
:: CLVECT_1:th 99
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
for b5 being Coset of b4
st b2 in b5 & b3 in b5
holds ex b6 being Element of the carrier of b1 st
b6 in b4 & b2 + b6 = b3;
:: CLVECT_1:th 100
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
for b5 being Coset of b4
st b2 in b5 & b3 in b5
holds ex b6 being Element of the carrier of b1 st
b6 in b4 & b2 - b6 = b3;
:: CLVECT_1:th 101
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
ex b5 being Coset of b4 st
b2 in b5 & b3 in b5
iff
b2 - b3 in b4;
:: CLVECT_1:th 102
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
for b4, b5 being Coset of b3
st b2 in b4 & b2 in b5
holds b4 = b5;
:: CLVECT_1:structnot 2 => CLVECT_1:struct 2
definition
struct(CLSStruct) CNORMSTR(#
carrier -> set,
ZeroF -> Element of the carrier of it,
addF -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],the carrier of it,
Mult -> Function-like quasi_total Relation of [:COMPLEX,the carrier of it:],the carrier of it,
norm -> Function-like quasi_total Relation of the carrier of it,REAL
#);
end;
:: CLVECT_1:attrnot 4 => CLVECT_1:attr 4
definition
let a1 be CNORMSTR;
attr a1 is strict;
end;
:: CLVECT_1:exreg 5
registration
cluster strict CNORMSTR;
end;
:: CLVECT_1:aggrnot 2 => CLVECT_1:aggr 2
definition
let a1 be set;
let a2 be Element of a1;
let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
let a4 be Function-like quasi_total Relation of [:COMPLEX,a1:],a1;
let a5 be Function-like quasi_total Relation of a1,REAL;
aggr CNORMSTR(#a1,a2,a3,a4,a5#) -> strict CNORMSTR;
end;
:: CLVECT_1:selnot 2 => CLVECT_1:sel 2
definition
let a1 be CNORMSTR;
sel the norm of a1 -> Function-like quasi_total Relation of the carrier of a1,REAL;
end;
:: CLVECT_1:exreg 6
registration
cluster non empty CNORMSTR;
end;
:: CLVECT_1:funcnot 6 => CLVECT_1:func 6
definition
let a1 be non empty CNORMSTR;
let a2 be Element of the carrier of a1;
func ||.A2.|| -> Element of REAL equals
(the norm of a1) . a2;
end;
:: CLVECT_1:def 10
theorem
for b1 being non empty CNORMSTR
for b2 being Element of the carrier of b1 holds
||.b2.|| = (the norm of b1) . b2;
:: CLVECT_1:attrnot 5 => CLVECT_1:attr 5
definition
let a1 be non empty CNORMSTR;
attr a1 is ComplexNormSpace-like means
for b1, b2 being Element of the carrier of a1
for b3 being Element of COMPLEX holds
(||.b1.|| = 0 implies b1 = 0. a1) &
(b1 = 0. a1 implies ||.b1.|| = 0) &
||.b3 * b1.|| = |.b3.| * ||.b1.|| &
||.b1 + b2.|| <= ||.b1.|| + ||.b2.||;
end;
:: CLVECT_1:dfs 11
definiens
let a1 be non empty CNORMSTR;
To prove
a1 is ComplexNormSpace-like
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
for b3 being Element of COMPLEX holds
(||.b1.|| = 0 implies b1 = 0. a1) &
(b1 = 0. a1 implies ||.b1.|| = 0) &
||.b3 * b1.|| = |.b3.| * ||.b1.|| &
||.b1 + b2.|| <= ||.b1.|| + ||.b2.||;
:: CLVECT_1:def 11
theorem
for b1 being non empty CNORMSTR holds
b1 is ComplexNormSpace-like
iff
for b2, b3 being Element of the carrier of b1
for b4 being Element of COMPLEX holds
(||.b2.|| = 0 implies b2 = 0. b1) &
(b2 = 0. b1 implies ||.b2.|| = 0) &
||.b4 * b2.|| = |.b4.| * ||.b2.|| &
||.b2 + b3.|| <= ||.b2.|| + ||.b3.||;
:: CLVECT_1:exreg 7
registration
cluster non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like strict ComplexNormSpace-like CNORMSTR;
end;
:: CLVECT_1:modenot 5
definition
mode ComplexNormSpace is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
end;
:: CLVECT_1:th 103
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
||.0. b1.|| = 0;
:: CLVECT_1:th 104
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1 holds
||.- b2.|| = ||.b2.||;
:: CLVECT_1:th 105
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 - b3.|| <= ||.b2.|| + ||.b3.||;
:: CLVECT_1:th 106
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1 holds
0 <= ||.b2.||;
:: CLVECT_1:th 107
theorem
for b1, b2 being Element of COMPLEX
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4, b5 being Element of the carrier of b3 holds
||.(b1 * b4) + (b2 * b5).|| <= (|.b1.| * ||.b4.||) + (|.b2.| * ||.b5.||);
:: CLVECT_1:th 108
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 - b3.|| = 0
iff
b2 = b3;
:: CLVECT_1:th 109
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 - b3.|| = ||.b3 - b2.||;
:: CLVECT_1:th 110
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2.|| - ||.b3.|| <= ||.b2 - b3.||;
:: CLVECT_1:th 111
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1 holds
abs (||.b2.|| - ||.b3.||) <= ||.b2 - b3.||;
:: CLVECT_1:th 112
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3, b4 being Element of the carrier of b1 holds
||.b2 - b3.|| <= ||.b2 - b4.|| + ||.b4 - b3.||;
:: CLVECT_1:th 113
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1
st b2 <> b3
holds ||.b2 - b3.|| <> 0;
:: CLVECT_1:funcnot 7 => CLVECT_1:func 7
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of COMPLEX;
func A3 * A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
for b1 being Element of NAT holds
it . b1 = a3 * (a2 . b1);
end;
:: CLVECT_1:def 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of COMPLEX
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b4 = b3 * b2
iff
for b5 being Element of NAT holds
b4 . b5 = b3 * (b2 . b5);
:: CLVECT_1:attrnot 6 => CLVECT_1:attr 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is convergent means
ex b1 being Element of the carrier of a1 st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds ||.(a2 . b4) - b1.|| < b2;
end;
:: CLVECT_1:dfs 13
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 is convergent
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds ||.(a2 . b4) - b1.|| < b2;
:: CLVECT_1:def 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is convergent(b1)
iff
ex b3 being Element of the carrier of b1 st
for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of NAT st
for b6 being Element of NAT
st b5 <= b6
holds ||.(b2 . b6) - b3.|| < b4;
:: CLVECT_1:th 115
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds b2 + b3 is convergent(b1);
:: CLVECT_1:th 116
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds b2 - b3 is convergent(b1);
:: CLVECT_1:th 117
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is convergent(b1)
holds b3 - b2 is convergent(b1);
:: CLVECT_1:th 118
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
st b3 is convergent(b2)
holds b1 * b3 is convergent(b2);
:: CLVECT_1:funcnot 8 => CLVECT_1:func 8
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
func ||.A2.|| -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = ||.a2 . b1.||;
end;
:: CLVECT_1:def 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL holds
b3 = ||.b2.||
iff
for b4 being Element of NAT holds
b3 . b4 = ||.b2 . b4.||;
:: CLVECT_1:th 119
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1)
holds ||.b2.|| is convergent;
:: CLVECT_1:funcnot 9 => CLVECT_1:func 9
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
assume a2 is convergent(a1);
func lim A2 -> Element of the carrier of a1 means
for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds ||.(a2 . b3) - it.|| < b1;
end;
:: CLVECT_1:def 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1)
for b3 being Element of the carrier of b1 holds
b3 = lim b2
iff
for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of NAT st
for b6 being Element of NAT
st b5 <= b6
holds ||.(b2 . b6) - b3.|| < b4;
:: CLVECT_1:th 120
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is convergent(b1) & lim b3 = b2
holds ||.b3 - b2.|| is convergent & lim ||.b3 - b2.|| = 0;
:: CLVECT_1:th 121
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds lim (b2 + b3) = (lim b2) + lim b3;
:: CLVECT_1:th 122
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds lim (b2 - b3) = (lim b2) - lim b3;
:: CLVECT_1:th 123
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is convergent(b1)
holds lim (b3 - b2) = (lim b3) - b2;
:: CLVECT_1:th 124
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
st b3 is convergent(b2)
holds lim (b1 * b3) = b1 * lim b3;