Article CSSPACE, MML version 4.99.1005
:: CSSPACE:funcnot 1 => CSSPACE:func 1
definition
func the_set_of_ComplexSequences -> non empty set means
for b1 being set holds
b1 in it
iff
b1 is Function-like quasi_total Relation of NAT,COMPLEX;
end;
:: CSSPACE:def 1
theorem
for b1 being non empty set holds
b1 = the_set_of_ComplexSequences
iff
for b2 being set holds
b2 in b1
iff
b2 is Function-like quasi_total Relation of NAT,COMPLEX;
:: CSSPACE:funcnot 2 => CSSPACE:func 2
definition
let a1 be set;
assume a1 in the_set_of_ComplexSequences;
func seq_id A1 -> Function-like quasi_total Relation of NAT,COMPLEX equals
a1;
end;
:: CSSPACE:def 2
theorem
for b1 being set
st b1 in the_set_of_ComplexSequences
holds seq_id b1 = b1;
:: CSSPACE:funcnot 3 => CSSPACE:func 3
definition
let a1 be set;
assume a1 in COMPLEX;
func C_id A1 -> Element of COMPLEX equals
a1;
end;
:: CSSPACE:def 3
theorem
for b1 being set
st b1 in COMPLEX
holds C_id b1 = b1;
:: CSSPACE:th 1
theorem
ex b1 being Function-like quasi_total Relation of [:the_set_of_ComplexSequences,the_set_of_ComplexSequences:],the_set_of_ComplexSequences st
(for b2, b3 being Element of the_set_of_ComplexSequences holds
b1 .(b2,b3) = (seq_id b2) + seq_id b3) &
b1 is commutative(the_set_of_ComplexSequences) &
b1 is associative(the_set_of_ComplexSequences);
:: CSSPACE:th 2
theorem
ex b1 being Function-like quasi_total Relation of [:COMPLEX,the_set_of_ComplexSequences:],the_set_of_ComplexSequences st
for b2, b3 being set
st b2 in COMPLEX & b3 in the_set_of_ComplexSequences
holds b1 .(b2,b3) = (C_id b2) (#) seq_id b3;
:: CSSPACE:funcnot 4 => CSSPACE:func 4
definition
func l_add -> Function-like quasi_total Relation of [:the_set_of_ComplexSequences,the_set_of_ComplexSequences:],the_set_of_ComplexSequences means
for b1, b2 being Element of the_set_of_ComplexSequences holds
it .(b1,b2) = (seq_id b1) + seq_id b2;
end;
:: CSSPACE:def 4
theorem
for b1 being Function-like quasi_total Relation of [:the_set_of_ComplexSequences,the_set_of_ComplexSequences:],the_set_of_ComplexSequences holds
b1 = l_add
iff
for b2, b3 being Element of the_set_of_ComplexSequences holds
b1 .(b2,b3) = (seq_id b2) + seq_id b3;
:: CSSPACE:funcnot 5 => CSSPACE:func 5
definition
func l_mult -> Function-like quasi_total Relation of [:COMPLEX,the_set_of_ComplexSequences:],the_set_of_ComplexSequences means
for b1, b2 being set
st b1 in COMPLEX & b2 in the_set_of_ComplexSequences
holds it .(b1,b2) = (C_id b1) (#) seq_id b2;
end;
:: CSSPACE:def 5
theorem
for b1 being Function-like quasi_total Relation of [:COMPLEX,the_set_of_ComplexSequences:],the_set_of_ComplexSequences holds
b1 = l_mult
iff
for b2, b3 being set
st b2 in COMPLEX & b3 in the_set_of_ComplexSequences
holds b1 .(b2,b3) = (C_id b2) (#) seq_id b3;
:: CSSPACE:funcnot 6 => CSSPACE:func 6
definition
func CZeroseq -> Element of the_set_of_ComplexSequences means
for b1 being Element of NAT holds
(seq_id it) . b1 = 0c;
end;
:: CSSPACE:def 6
theorem
for b1 being Element of the_set_of_ComplexSequences holds
b1 = CZeroseq
iff
for b2 being Element of NAT holds
(seq_id b1) . b2 = 0c;
:: CSSPACE:th 3
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
seq_id b1 = b1;
:: CSSPACE:th 4
theorem
for b1, b2 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
b1 + b2 = (seq_id b1) + seq_id b2;
:: CSSPACE:th 5
theorem
for b1 being Element of COMPLEX
for b2 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
b1 * b2 = b1 (#) seq_id b2;
:: CSSPACE:funcreg 1
registration
cluster CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) -> Abelian strict;
end;
:: CSSPACE:th 6
theorem
for b1, b2, b3 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
(b1 + b2) + b3 = b1 + (b2 + b3);
:: CSSPACE:th 7
theorem
for b1 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
b1 + 0. CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) = b1;
:: CSSPACE:th 8
theorem
for b1 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
ex b2 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) st
b1 + b2 = 0. CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#);
:: CSSPACE:th 9
theorem
for b1 being Element of COMPLEX
for b2, b3 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
b1 * (b2 + b3) = (b1 * b2) + (b1 * b3);
:: CSSPACE:th 10
theorem
for b1, b2 being Element of COMPLEX
for b3 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
(b1 + b2) * b3 = (b1 * b3) + (b2 * b3);
:: CSSPACE:th 11
theorem
for b1, b2 being Element of COMPLEX
for b3 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
(b1 * b2) * b3 = b1 * (b2 * b3);
:: CSSPACE:th 12
theorem
for b1 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
1r * b1 = b1;
:: CSSPACE:funcnot 7 => CSSPACE:func 7
definition
func Linear_Space_of_ComplexSequences -> non empty strict CLSStruct equals
CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#);
end;
:: CSSPACE:def 7
theorem
Linear_Space_of_ComplexSequences = CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#);
:: CSSPACE:funcreg 2
registration
cluster Linear_Space_of_ComplexSequences -> non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like;
end;
:: CSSPACE:funcnot 8 => CSSPACE:func 8
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;
assume a2 is linearly-closed(a1) & a2 is not empty;
func Add_(A2,A1) -> Function-like quasi_total Relation of [:a2,a2:],a2 equals
(the addF of a1) || a2;
end;
:: CSSPACE:def 8
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) & b2 is not empty
holds Add_(b2,b1) = (the addF of b1) || b2;
:: CSSPACE:funcnot 9 => CSSPACE:func 9
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;
assume a2 is linearly-closed(a1) & a2 is not empty;
func Mult_(A2,A1) -> Function-like quasi_total Relation of [:COMPLEX,a2:],a2 equals
(the Mult of a1) | [:COMPLEX,a2:];
end;
:: CSSPACE:def 9
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) & b2 is not empty
holds Mult_(b2,b1) = (the Mult of b1) | [:COMPLEX,b2:];
:: CSSPACE:funcnot 10 => CSSPACE:func 10
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;
assume a2 is linearly-closed(a1) & a2 is not empty;
func Zero_(A2,A1) -> Element of a2 equals
0. a1;
end;
:: CSSPACE:def 10
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) & b2 is not empty
holds Zero_(b2,b1) = 0. b1;
:: CSSPACE:th 13
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) & b2 is not empty
holds CLSStruct(#b2,Zero_(b2,b1),Add_(b2,b1),Mult_(b2,b1)#) is Subspace of b1;
:: CSSPACE:funcnot 11 => CSSPACE:func 11
definition
func the_set_of_l2ComplexSequences -> Element of bool the carrier of Linear_Space_of_ComplexSequences means
it is not empty &
(for b1 being set holds
b1 in it
iff
b1 in the_set_of_ComplexSequences &
|.seq_id b1.| (#) |.seq_id b1.| is summable);
end;
:: CSSPACE:def 11
theorem
for b1 being Element of bool the carrier of Linear_Space_of_ComplexSequences holds
b1 = the_set_of_l2ComplexSequences
iff
b1 is not empty &
(for b2 being set holds
b2 in b1
iff
b2 in the_set_of_ComplexSequences &
|.seq_id b2.| (#) |.seq_id b2.| is summable);
:: CSSPACE:th 14
theorem
the_set_of_l2ComplexSequences is linearly-closed(Linear_Space_of_ComplexSequences) & the_set_of_l2ComplexSequences is not empty;
:: CSSPACE:th 15
theorem
CLSStruct(#the_set_of_l2ComplexSequences,Zero_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences)#) is Subspace of Linear_Space_of_ComplexSequences;
:: CSSPACE:th 16
theorem
CLSStruct(#the_set_of_l2ComplexSequences,Zero_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences)#) is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
:: CSSPACE:th 17
theorem
the carrier of Linear_Space_of_ComplexSequences = the_set_of_ComplexSequences &
(for b1 being set holds
b1 is Element of the carrier of Linear_Space_of_ComplexSequences
iff
b1 is Function-like quasi_total Relation of NAT,COMPLEX) &
(for b1 being set holds
b1 is Element of the carrier of Linear_Space_of_ComplexSequences
iff
b1 is Function-like quasi_total Relation of NAT,COMPLEX) &
(for b1 being Element of the carrier of Linear_Space_of_ComplexSequences holds
b1 = seq_id b1) &
(for b1, b2 being Element of the carrier of Linear_Space_of_ComplexSequences holds
b1 + b2 = (seq_id b1) + seq_id b2) &
(for b1 being Element of COMPLEX
for b2 being Element of the carrier of Linear_Space_of_ComplexSequences holds
b1 * b2 = b1 (#) seq_id b2);
:: CSSPACE:structnot 1 => CSSPACE:struct 1
definition
struct(CLSStruct) CUNITSTR(#
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,
scalar -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],COMPLEX
#);
end;
:: CSSPACE:attrnot 1 => CSSPACE:attr 1
definition
let a1 be CUNITSTR;
attr a1 is strict;
end;
:: CSSPACE:exreg 1
registration
cluster strict CUNITSTR;
end;
:: CSSPACE:aggrnot 1 => CSSPACE: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;
let a5 be Function-like quasi_total Relation of [:a1,a1:],COMPLEX;
aggr CUNITSTR(#a1,a2,a3,a4,a5#) -> strict CUNITSTR;
end;
:: CSSPACE:selnot 1 => CSSPACE:sel 1
definition
let a1 be CUNITSTR;
sel the scalar of a1 -> Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],COMPLEX;
end;
:: CSSPACE:exreg 2
registration
cluster non empty strict CUNITSTR;
end;
:: CSSPACE:funcreg 3
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;
let a5 be Function-like quasi_total Relation of [:a1,a1:],COMPLEX;
cluster CUNITSTR(#a1,a2,a3,a4,a5#) -> non empty strict;
end;
:: CSSPACE:funcnot 12 => CSSPACE:func 12
definition
let a1 be non empty CUNITSTR;
let a2, a3 be Element of the carrier of a1;
func A2 .|. A3 -> Element of COMPLEX equals
(the scalar of a1) .(a2,a3);
end;
:: CSSPACE:def 12
theorem
for b1 being non empty CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
b2 .|. b3 = (the scalar of b1) .(b2,b3);
:: CSSPACE:attrnot 2 => CSSPACE:attr 2
definition
let a1 be non empty CUNITSTR;
attr a1 is ComplexUnitarySpace-like means
for b1, b2, b3 being Element of the carrier of a1
for b4 being Element of COMPLEX holds
(b1 .|. b1 = 0 implies b1 = 0. a1) &
(b1 = 0. a1 implies b1 .|. b1 = 0) &
0 <= Re (b1 .|. b1) &
0 = Im (b1 .|. b1) &
b1 .|. b2 = (b2 .|. b1) *' &
(b1 + b2) .|. b3 = (b1 .|. b3) + (b2 .|. b3) &
(b4 * b1) .|. b2 = b4 * (b1 .|. b2);
end;
:: CSSPACE:dfs 13
definiens
let a1 be non empty CUNITSTR;
To prove
a1 is ComplexUnitarySpace-like
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1
for b4 being Element of COMPLEX holds
(b1 .|. b1 = 0 implies b1 = 0. a1) &
(b1 = 0. a1 implies b1 .|. b1 = 0) &
0 <= Re (b1 .|. b1) &
0 = Im (b1 .|. b1) &
b1 .|. b2 = (b2 .|. b1) *' &
(b1 + b2) .|. b3 = (b1 .|. b3) + (b2 .|. b3) &
(b4 * b1) .|. b2 = b4 * (b1 .|. b2);
:: CSSPACE:def 13
theorem
for b1 being non empty CUNITSTR holds
b1 is ComplexUnitarySpace-like
iff
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of COMPLEX holds
(b2 .|. b2 = 0 implies b2 = 0. b1) &
(b2 = 0. b1 implies b2 .|. b2 = 0) &
0 <= Re (b2 .|. b2) &
0 = Im (b2 .|. b2) &
b2 .|. b3 = (b3 .|. b2) *' &
(b2 + b3) .|. b4 = (b2 .|. b4) + (b3 .|. b4) &
(b5 * b2) .|. b3 = b5 * (b2 .|. b3);
:: CSSPACE:exreg 3
registration
cluster non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like strict ComplexUnitarySpace-like CUNITSTR;
end;
:: CSSPACE:modenot 1
definition
mode ComplexUnitarySpace is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
end;
:: CSSPACE:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR holds
(0. b1) .|. 0. b1 = 0;
:: CSSPACE:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
b2 .|. (b3 + b4) = (b2 .|. b3) + (b2 .|. b4);
:: CSSPACE:th 20
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b3, b4 being Element of the carrier of b2 holds
b3 .|. (b1 * b4) = b1 *' * (b3 .|. b4);
:: CSSPACE:th 21
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b3, b4 being Element of the carrier of b2 holds
(b1 * b3) .|. b4 = b3 .|. (b1 *' * b4);
:: CSSPACE:th 22
theorem
for b1, b2 being Element of COMPLEX
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b4, b5, b6 being Element of the carrier of b3 holds
((b1 * b4) + (b2 * b5)) .|. b6 = (b1 * (b4 .|. b6)) + (b2 * (b5 .|. b6));
:: CSSPACE:th 23
theorem
for b1, b2 being Element of COMPLEX
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b4, b5, b6 being Element of the carrier of b3 holds
b4 .|. ((b1 * b5) + (b2 * b6)) = (b1 *' * (b4 .|. b5)) + (b2 *' * (b4 .|. b6));
:: CSSPACE:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
(- b2) .|. b3 = b2 .|. - b3;
:: CSSPACE:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
(- b2) .|. b3 = - (b2 .|. b3);
:: CSSPACE:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
b2 .|. - b3 = - (b2 .|. b3);
:: CSSPACE:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
(- b2) .|. - b3 = b2 .|. b3;
:: CSSPACE:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 - b3) .|. b4 = (b2 .|. b4) - (b3 .|. b4);
:: CSSPACE:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
b2 .|. (b3 - b4) = (b2 .|. b3) - (b2 .|. b4);
:: CSSPACE:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(b2 - b3) .|. (b4 - b5) = (((b2 .|. b4) - (b2 .|. b5)) - (b3 .|. b4)) + (b3 .|. b5);
:: CSSPACE:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
(0. b1) .|. b2 = 0;
:: CSSPACE:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
b2 .|. 0. b1 = 0;
:: CSSPACE:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
(b2 + b3) .|. (b2 + b3) = (((b2 .|. b2) + (b2 .|. b3)) + (b3 .|. b2)) + (b3 .|. b3);
:: CSSPACE:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
(b2 + b3) .|. (b2 - b3) = (((b2 .|. b2) - (b2 .|. b3)) + (b3 .|. b2)) - (b3 .|. b3);
:: CSSPACE:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
(b2 - b3) .|. (b2 - b3) = (((b2 .|. b2) - (b2 .|. b3)) - (b3 .|. b2)) + (b3 .|. b3);
:: CSSPACE:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
|.b2 .|. b2.| = Re (b2 .|. b2);
:: CSSPACE:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
|.b2 .|. b3.| <= (sqrt |.b2 .|. b2.|) * sqrt |.b3 .|. b3.|;
:: CSSPACE:prednot 1 => CSSPACE:pred 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2, a3 be Element of the carrier of a1;
pred A2,A3 are_orthogonal means
a2 .|. a3 = 0;
symmetry;
:: for a1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
:: for a2, a3 being Element of the carrier of a1
:: st a2,a3 are_orthogonal
:: holds a3,a2 are_orthogonal;
end;
:: CSSPACE:dfs 14
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2, a3 be Element of the carrier of a1;
To prove
a2,a3 are_orthogonal
it is sufficient to prove
thus a2 .|. a3 = 0;
:: CSSPACE:def 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
b2,b3 are_orthogonal
iff
b2 .|. b3 = 0;
:: CSSPACE:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1
st b2,b3 are_orthogonal
holds b2,- b3 are_orthogonal;
:: CSSPACE:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1
st b2,b3 are_orthogonal
holds - b2,b3 are_orthogonal;
:: CSSPACE:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1
st b2,b3 are_orthogonal
holds - b2,- b3 are_orthogonal;
:: CSSPACE:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
b2,0. b1 are_orthogonal;
:: CSSPACE:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1
st b2,b3 are_orthogonal
holds (b2 + b3) .|. (b2 + b3) = (b2 .|. b2) + (b3 .|. b3);
:: CSSPACE:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1
st b2,b3 are_orthogonal
holds (b2 - b3) .|. (b2 - b3) = (b2 .|. b2) + (b3 .|. b3);
:: CSSPACE:funcnot 13 => CSSPACE:func 13
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2 be Element of the carrier of a1;
func ||.A2.|| -> Element of REAL equals
sqrt |.a2 .|. a2.|;
end;
:: CSSPACE:def 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
||.b2.|| = sqrt |.b2 .|. b2.|;
:: CSSPACE:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
||.b2.|| = 0
iff
b2 = 0. b1;
:: CSSPACE:th 45
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b3 being Element of the carrier of b2 holds
||.b1 * b3.|| = |.b1.| * ||.b3.||;
:: CSSPACE:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
0 <= ||.b2.||;
:: CSSPACE:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
|.b2 .|. b3.| <= ||.b2.|| * ||.b3.||;
:: CSSPACE:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 + b3.|| <= ||.b2.|| + ||.b3.||;
:: CSSPACE:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
||.- b2.|| = ||.b2.||;
:: CSSPACE:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2.|| - ||.b3.|| <= ||.b2 - b3.||;
:: CSSPACE:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
abs (||.b2.|| - ||.b3.||) <= ||.b2 - b3.||;
:: CSSPACE:funcnot 14 => CSSPACE:func 14
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2, a3 be Element of the carrier of a1;
func dist(A2,A3) -> Element of REAL equals
||.a2 - a3.||;
end;
:: CSSPACE:def 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
dist(b2,b3) = ||.b2 - b3.||;
:: CSSPACE:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
dist(b2,b3) = dist(b3,b2);
:: CSSPACE:funcnot 15 => CSSPACE:func 15
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2, a3 be Element of the carrier of a1;
redefine func dist(a2,a3) -> Element of REAL;
commutativity;
:: for a1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
:: for a2, a3 being Element of the carrier of a1 holds
:: dist(a2,a3) = dist(a3,a2);
end;
:: CSSPACE:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
dist(b2,b2) = 0;
:: CSSPACE:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
dist(b2,b3) <= (dist(b2,b4)) + dist(b4,b3);
:: CSSPACE:th 55
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
b2 <> b3
iff
dist(b2,b3) <> 0;
:: CSSPACE:th 56
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
0 <= dist(b2,b3);
:: CSSPACE:th 57
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
b2 <> b3
iff
0 < dist(b2,b3);
:: CSSPACE:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
dist(b2,b3) = sqrt |.(b2 - b3) .|. (b2 - b3).|;
:: CSSPACE:th 59
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4, b5 being Element of the carrier of b1 holds
dist(b2 + b3,b4 + b5) <= (dist(b2,b4)) + dist(b3,b5);
:: CSSPACE:th 60
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4, b5 being Element of the carrier of b1 holds
dist(b2 - b3,b4 - b5) <= (dist(b2,b4)) + dist(b3,b5);
:: CSSPACE:th 61
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
dist(b2 - b3,b4 - b3) = dist(b2,b4);
:: CSSPACE:th 62
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
dist(b2 - b3,b4 - b3) <= (dist(b3,b2)) + dist(b3,b4);
:: CSSPACE:th 63
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 + b3 = b3 + b2;
:: CSSPACE:funcnot 16 => CSSPACE:func 16
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
redefine func a2 + a3 -> Function-like quasi_total Relation of NAT,the carrier of a1;
commutativity;
:: for a1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
:: for a2, a3 being Function-like quasi_total Relation of NAT,the carrier of a1 holds
:: a2 + a3 = a3 + a2;
end;
:: CSSPACE:th 64
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 + (b3 + b4) = (b2 + b3) + b4;
:: CSSPACE:th 65
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is constant & b3 is constant & b4 = b2 + b3
holds b4 is constant;
:: CSSPACE:th 66
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is constant & b3 is constant & b4 = b2 - b3
holds b4 is constant;
:: CSSPACE:th 67
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b2
st b3 is constant & b4 = b1 * b3
holds b4 is constant;
:: CSSPACE:th 68
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is constant
iff
for b3 being Element of NAT holds
b2 . b3 = b2 . (b3 + 1);
:: CSSPACE:th 69
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is constant
iff
for b3, b4 being Element of NAT holds
b2 . b3 = b2 . (b3 + b4);
:: CSSPACE:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is constant
iff
for b3, b4 being Element of NAT holds
b2 . b3 = b2 . b4;
:: CSSPACE:th 71
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 - b3 = b2 + - b3;
:: CSSPACE:th 72
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 = b2 + 0. b1;
:: CSSPACE:th 73
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
b1 * (b3 + b4) = (b1 * b3) + (b1 * b4);
:: CSSPACE:th 74
theorem
for b1, b2 being Element of COMPLEX
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b3 holds
(b1 + b2) * b4 = (b1 * b4) + (b2 * b4);
:: CSSPACE:th 75
theorem
for b1, b2 being Element of COMPLEX
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b3 holds
(b1 * b2) * b4 = b1 * (b2 * b4);
:: CSSPACE:th 76
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
1r * b2 = b2;
:: CSSPACE:th 77
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
(- 1r) * b2 = - b2;
:: CSSPACE:th 78
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b3 - b2 = b3 + - b2;
:: CSSPACE:th 79
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 - b3 = - (b3 - b2);
:: CSSPACE:th 80
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 = b2 - 0. b1;
:: CSSPACE:th 81
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 = - - b2;
:: CSSPACE:th 82
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 - (b3 + b4) = (b2 - b3) - b4;
:: CSSPACE:th 83
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
(b2 + b3) - b4 = b2 + (b3 - b4);
:: CSSPACE:th 84
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 - (b3 - b4) = (b2 - b3) + b4;
:: CSSPACE:th 85
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
b1 * (b3 - b4) = (b1 * b3) - (b1 * b4);
:: CSSPACE:th 86
theorem
ex b1 being Function-like quasi_total Relation of [:the_set_of_l2ComplexSequences,the_set_of_l2ComplexSequences:],COMPLEX st
for b2, b3 being set
st b2 in the_set_of_l2ComplexSequences & b3 in the_set_of_l2ComplexSequences
holds b1 .(b2,b3) = Sum ((seq_id b2) (#) ((seq_id b3) *'));
:: CSSPACE:funcnot 17 => CSSPACE:func 17
definition
func cl_scalar -> Function-like quasi_total Relation of [:the_set_of_l2ComplexSequences,the_set_of_l2ComplexSequences:],COMPLEX means
for b1, b2 being set
st b1 in the_set_of_l2ComplexSequences & b2 in the_set_of_l2ComplexSequences
holds it .(b1,b2) = Sum ((seq_id b1) (#) ((seq_id b2) *'));
end;
:: CSSPACE:def 19
theorem
for b1 being Function-like quasi_total Relation of [:the_set_of_l2ComplexSequences,the_set_of_l2ComplexSequences:],COMPLEX holds
b1 = cl_scalar
iff
for b2, b3 being set
st b2 in the_set_of_l2ComplexSequences & b3 in the_set_of_l2ComplexSequences
holds b1 .(b2,b3) = Sum ((seq_id b2) (#) ((seq_id b3) *'));
:: CSSPACE:funcreg 4
registration
cluster CUNITSTR(#the_set_of_l2ComplexSequences,Zero_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),cl_scalar#) -> non empty strict;
end;
:: CSSPACE:funcnot 18 => CSSPACE:func 18
definition
func Complex_l2_Space -> non empty CUNITSTR equals
CUNITSTR(#the_set_of_l2ComplexSequences,Zero_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),cl_scalar#);
end;
:: CSSPACE:def 20
theorem
Complex_l2_Space = CUNITSTR(#the_set_of_l2ComplexSequences,Zero_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),cl_scalar#);
:: CSSPACE:th 87
theorem
for b1 being CLSStruct
st CLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#) is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
holds b1 is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
:: CSSPACE:th 88
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
st for b2 being Element of NAT holds
b1 . b2 = 0c
holds b1 is summable & Sum b1 = 0c;
:: CSSPACE:funcreg 5
registration
cluster Complex_l2_Space -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like;
end;