Article CONVEX4, MML version 4.99.1005
:: CONVEX4:modenot 1 => CONVEX4:mode 1
definition
let a1 be non empty ZeroStr;
mode C_Linear_Combination of A1 -> Element of Funcs(the carrier of a1,COMPLEX) means
ex b1 being finite Element of bool the carrier of a1 st
for b2 being Element of the carrier of a1
st not b2 in b1
holds it . b2 = {};
end;
:: CONVEX4:dfs 1
definiens
let a1 be non empty ZeroStr;
let a2 be Element of Funcs(the carrier of a1,COMPLEX);
To prove
a2 is C_Linear_Combination of a1
it is sufficient to prove
thus ex b1 being finite Element of bool the carrier of a1 st
for b2 being Element of the carrier of a1
st not b2 in b1
holds a2 . b2 = {};
:: CONVEX4:def 1
theorem
for b1 being non empty ZeroStr
for b2 being Element of Funcs(the carrier of b1,COMPLEX) holds
b2 is C_Linear_Combination of b1
iff
ex b3 being finite Element of bool the carrier of b1 st
for b4 being Element of the carrier of b1
st not b4 in b3
holds b2 . b4 = {};
:: CONVEX4:funcnot 1 => CONVEX4:func 1
definition
let a1 be non empty addLoopStr;
let a2 be Element of Funcs(the carrier of a1,COMPLEX);
func Carrier A2 -> Element of bool the carrier of a1 equals
{b1 where b1 is Element of the carrier of a1: a2 . b1 <> 0c};
end;
:: CONVEX4:def 2
theorem
for b1 being non empty addLoopStr
for b2 being Element of Funcs(the carrier of b1,COMPLEX) holds
Carrier b2 = {b3 where b3 is Element of the carrier of b1: b2 . b3 <> 0c};
:: CONVEX4:funcreg 1
registration
let a1 be non empty addLoopStr;
let a2 be C_Linear_Combination of a1;
cluster Carrier a2 -> finite;
end;
:: CONVEX4:th 1
theorem
for b1 being non empty addLoopStr
for b2 being C_Linear_Combination of b1
for b3 being Element of the carrier of b1 holds
b2 . b3 = 0c
iff
not b3 in Carrier b2;
:: CONVEX4:funcnot 2 => CONVEX4:func 2
definition
let a1 be non empty addLoopStr;
func ZeroCLC A1 -> C_Linear_Combination of a1 means
Carrier it = {};
end;
:: CONVEX4:def 3
theorem
for b1 being non empty addLoopStr
for b2 being C_Linear_Combination of b1 holds
b2 = ZeroCLC b1
iff
Carrier b2 = {};
:: CONVEX4:funcreg 2
registration
let a1 be non empty addLoopStr;
cluster Carrier ZeroCLC a1 -> empty;
end;
:: CONVEX4:th 2
theorem
for b1 being non empty addLoopStr
for b2 being Element of the carrier of b1 holds
(ZeroCLC b1) . b2 = 0c;
:: CONVEX4:modenot 2 => CONVEX4:mode 2
definition
let a1 be non empty addLoopStr;
let a2 be Element of bool the carrier of a1;
mode C_Linear_Combination of A2 -> C_Linear_Combination of a1 means
Carrier it c= a2;
end;
:: CONVEX4:dfs 4
definiens
let a1 be non empty addLoopStr;
let a2 be Element of bool the carrier of a1;
let a3 be C_Linear_Combination of a1;
To prove
a3 is C_Linear_Combination of a2
it is sufficient to prove
thus Carrier a3 c= a2;
:: CONVEX4:def 4
theorem
for b1 being non empty addLoopStr
for b2 being Element of bool the carrier of b1
for b3 being C_Linear_Combination of b1 holds
b3 is C_Linear_Combination of b2
iff
Carrier b3 c= b2;
:: CONVEX4:th 3
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of bool the carrier of b1
for b4 being C_Linear_Combination of b2
st b2 c= b3
holds b4 is C_Linear_Combination of b3;
:: CONVEX4:th 4
theorem
for b1 being non empty addLoopStr
for b2 being Element of bool the carrier of b1 holds
ZeroCLC b1 is C_Linear_Combination of b2;
:: CONVEX4:th 5
theorem
for b1 being non empty addLoopStr
for b2 being C_Linear_Combination of {} the carrier of b1 holds
b2 = ZeroCLC b1;
:: CONVEX4:funcnot 3 => CONVEX4:func 3
definition
let a1 be non empty CLSStruct;
let a2 be FinSequence of the carrier of a1;
let a3 be Function-like quasi_total Relation of the carrier of a1,COMPLEX;
func A3 (#) A2 -> FinSequence of the carrier of a1 means
len it = len a2 &
(for b1 being natural set
st b1 in dom it
holds it . b1 = (a3 . (a2 /. b1)) * (a2 /. b1));
end;
:: CONVEX4:def 5
theorem
for b1 being non empty CLSStruct
for b2 being FinSequence of the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,COMPLEX
for b4 being FinSequence of the carrier of b1 holds
b4 = b3 (#) b2
iff
len b4 = len b2 &
(for b5 being natural set
st b5 in dom b4
holds b4 . b5 = (b3 . (b2 /. b5)) * (b2 /. b5));
:: CONVEX4:th 6
theorem
for b1 being non empty CLSStruct
for b2 being Element of the carrier of b1
for b3 being set
for b4 being FinSequence of the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,COMPLEX
st b3 in dom b4 & b2 = b4 . b3
holds (b5 (#) b4) . b3 = (b5 . b2) * b2;
:: CONVEX4:th 7
theorem
for b1 being non empty CLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,COMPLEX holds
b2 (#) <*> the carrier of b1 = <*> the carrier of b1;
:: CONVEX4:th 8
theorem
for b1 being non empty CLSStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,COMPLEX holds
b3 (#) <*b2*> = <*(b3 . b2) * b2*>;
:: CONVEX4:th 9
theorem
for b1 being non empty CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,COMPLEX holds
b4 (#) <*b2,b3*> = <*(b4 . b2) * b2,(b4 . b3) * b3*>;
:: CONVEX4:th 10
theorem
for b1 being non empty CLSStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,COMPLEX holds
b5 (#) <*b2,b3,b4*> = <*(b5 . b2) * b2,(b5 . b3) * b3,(b5 . b4) * b4*>;
:: CONVEX4:funcnot 4 => CONVEX4:func 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed CLSStruct;
let a2 be C_Linear_Combination of a1;
func Sum A2 -> Element of the carrier of a1 means
ex b1 being FinSequence of the carrier of a1 st
b1 is one-to-one & rng b1 = Carrier a2 & it = Sum (a2 (#) b1);
end;
:: CONVEX4:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed CLSStruct
for b2 being C_Linear_Combination of b1
for b3 being Element of the carrier of b1 holds
b3 = Sum b2
iff
ex b4 being FinSequence of the carrier of b1 st
b4 is one-to-one & rng b4 = Carrier b2 & b3 = Sum (b2 (#) b4);
:: CONVEX4:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed CLSStruct holds
Sum ZeroCLC b1 = 0. b1;
:: CONVEX4:th 12
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 <> {}
holds b2 is linearly-closed(b1)
iff
for b3 being C_Linear_Combination of b2 holds
Sum b3 in b2;
:: CONVEX4:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed CLSStruct
for b2 being C_Linear_Combination of {} the carrier of b1 holds
Sum b2 = 0. b1;
:: CONVEX4:th 14
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 C_Linear_Combination of {b2} holds
Sum b3 = (b3 . b2) * b2;
:: CONVEX4:th 15
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
st b2 <> b3
for b4 being C_Linear_Combination of {b2,b3} holds
Sum b4 = ((b4 . b2) * b2) + ((b4 . b3) * b3);
:: CONVEX4:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed CLSStruct
for b2 being C_Linear_Combination of b1
st Carrier b2 = {}
holds Sum b2 = 0. b1;
:: CONVEX4:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being C_Linear_Combination of b1
for b3 being Element of the carrier of b1
st Carrier b2 = {b3}
holds Sum b2 = (b2 . b3) * b3;
:: CONVEX4:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being C_Linear_Combination of b1
for b3, b4 being Element of the carrier of b1
st Carrier b2 = {b3,b4} & b3 <> b4
holds Sum b2 = ((b2 . b3) * b3) + ((b2 . b4) * b4);
:: CONVEX4:prednot 1 => CONVEX4:pred 1
definition
let a1 be non empty addLoopStr;
let a2, a3 be C_Linear_Combination of a1;
redefine pred A2 = A3 means
for b1 being Element of the carrier of a1 holds
a2 . b1 = a3 . b1;
symmetry;
:: for a1 being non empty addLoopStr
:: for a2, a3 being C_Linear_Combination of a1
:: st a2 = a3
:: holds a3 = a2;
reflexivity;
:: for a1 being non empty addLoopStr
:: for a2 being C_Linear_Combination of a1 holds
:: a2 = a2;
end;
:: CONVEX4:dfs 7
definiens
let a1 be non empty addLoopStr;
let a2, a3 be C_Linear_Combination of a1;
To prove
a2 = a3
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
a2 . b1 = a3 . b1;
:: CONVEX4:def 7
theorem
for b1 being non empty addLoopStr
for b2, b3 being C_Linear_Combination of b1 holds
b2 = b3
iff
for b4 being Element of the carrier of b1 holds
b2 . b4 = b3 . b4;
:: CONVEX4:funcnot 5 => CONVEX4:func 5
definition
let a1 be non empty addLoopStr;
let a2, a3 be C_Linear_Combination of a1;
redefine func A2 + A3 -> C_Linear_Combination of a1 means
for b1 being Element of the carrier of a1 holds
it . b1 = (a2 . b1) + (a3 . b1);
commutativity;
:: for a1 being non empty addLoopStr
:: for a2, a3 being C_Linear_Combination of a1 holds
:: a2 + a3 = a3 + a2;
end;
:: CONVEX4:def 8
theorem
for b1 being non empty addLoopStr
for b2, b3, b4 being C_Linear_Combination of b1 holds
b4 = b2 + b3
iff
for b5 being Element of the carrier of b1 holds
b4 . b5 = (b2 . b5) + (b3 . b5);
:: CONVEX4:th 19
theorem
for b1 being non empty CLSStruct
for b2, b3 being C_Linear_Combination of b1 holds
Carrier (b2 + b3) c= (Carrier b2) \/ Carrier b3;
:: CONVEX4:th 20
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being C_Linear_Combination of b1
st b3 is C_Linear_Combination of b2 & b4 is C_Linear_Combination of b2
holds b3 + b4 is C_Linear_Combination of b2;
:: CONVEX4:funcnot 6 => CONVEX4:func 6
definition
let a1 be non empty CLSStruct;
let a2 be Element of bool the carrier of a1;
let a3, a4 be C_Linear_Combination of a2;
redefine func a3 + a4 -> C_Linear_Combination of a2;
commutativity;
:: for a1 being non empty CLSStruct
:: for a2 being Element of bool the carrier of a1
:: for a3, a4 being C_Linear_Combination of a2 holds
:: a3 + a4 = a4 + a3;
end;
:: CONVEX4:th 21
theorem
for b1 being non empty addLoopStr
for b2, b3 being C_Linear_Combination of b1 holds
b2 + b3 = b3 + b2;
:: CONVEX4:th 22
theorem
for b1 being non empty CLSStruct
for b2, b3, b4 being C_Linear_Combination of b1 holds
b2 + (b3 + b4) = (b2 + b3) + b4;
:: CONVEX4:th 23
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
b2 + ZeroCLC b1 = b2;
:: CONVEX4:funcnot 7 => CONVEX4:func 7
definition
let a1 be non empty CLSStruct;
let a2 be Element of COMPLEX;
let a3 be C_Linear_Combination of a1;
func A2 * A3 -> C_Linear_Combination of a1 means
for b1 being Element of the carrier of a1 holds
it . b1 = a2 * (a3 . b1);
end;
:: CONVEX4:def 9
theorem
for b1 being non empty CLSStruct
for b2 being Element of COMPLEX
for b3, b4 being C_Linear_Combination of b1 holds
b4 = b2 * b3
iff
for b5 being Element of the carrier of b1 holds
b4 . b5 = b2 * (b3 . b5);
:: CONVEX4:th 24
theorem
for b1 being non empty CLSStruct
for b2 being Element of COMPLEX
for b3 being C_Linear_Combination of b1
st b2 <> 0c
holds Carrier (b2 * b3) = Carrier b3;
:: CONVEX4:th 25
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
0c * b2 = ZeroCLC b1;
:: CONVEX4:th 26
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of COMPLEX
for b4 being C_Linear_Combination of b1
st b4 is C_Linear_Combination of b2
holds b3 * b4 is C_Linear_Combination of b2;
:: CONVEX4:th 27
theorem
for b1 being non empty CLSStruct
for b2, b3 being Element of COMPLEX
for b4 being C_Linear_Combination of b1 holds
(b2 + b3) * b4 = (b2 * b4) + (b3 * b4);
:: CONVEX4:th 28
theorem
for b1 being non empty CLSStruct
for b2 being Element of COMPLEX
for b3, b4 being C_Linear_Combination of b1 holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4);
:: CONVEX4:th 29
theorem
for b1 being non empty CLSStruct
for b2, b3 being Element of COMPLEX
for b4 being C_Linear_Combination of b1 holds
b2 * (b3 * b4) = (b2 * b3) * b4;
:: CONVEX4:th 30
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
1r * b2 = b2;
:: CONVEX4:funcnot 8 => CONVEX4:func 8
definition
let a1 be non empty CLSStruct;
let a2 be C_Linear_Combination of a1;
func - A2 -> C_Linear_Combination of a1 equals
(- 1r) * a2;
end;
:: CONVEX4:def 10
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
- b2 = (- 1r) * b2;
:: CONVEX4:th 31
theorem
for b1 being non empty CLSStruct
for b2 being Element of the carrier of b1
for b3 being C_Linear_Combination of b1 holds
(- b3) . b2 = - (b3 . b2);
:: CONVEX4:th 32
theorem
for b1 being non empty CLSStruct
for b2, b3 being C_Linear_Combination of b1
st b2 + b3 = ZeroCLC b1
holds b3 = - b2;
:: CONVEX4:th 33
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
- - b2 = b2;
:: CONVEX4:funcnot 9 => CONVEX4:func 9
definition
let a1 be non empty CLSStruct;
let a2, a3 be C_Linear_Combination of a1;
func A2 - A3 -> C_Linear_Combination of a1 equals
a2 + - a3;
end;
:: CONVEX4:def 11
theorem
for b1 being non empty CLSStruct
for b2, b3 being C_Linear_Combination of b1 holds
b2 - b3 = b2 + - b3;
:: CONVEX4:th 34
theorem
for b1 being non empty CLSStruct
for b2 being Element of the carrier of b1
for b3, b4 being C_Linear_Combination of b1 holds
(b3 - b4) . b2 = (b3 . b2) - (b4 . b2);
:: CONVEX4:th 35
theorem
for b1 being non empty CLSStruct
for b2, b3 being C_Linear_Combination of b1 holds
Carrier (b2 - b3) c= (Carrier b2) \/ Carrier b3;
:: CONVEX4:th 36
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being C_Linear_Combination of b1
st b3 is C_Linear_Combination of b2 & b4 is C_Linear_Combination of b2
holds b3 - b4 is C_Linear_Combination of b2;
:: CONVEX4:th 37
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
b2 - b2 = ZeroCLC b1;
:: CONVEX4:funcnot 10 => CONVEX4:func 10
definition
let a1 be non empty CLSStruct;
func C_LinComb A1 -> set means
for b1 being set holds
b1 in it
iff
b1 is C_Linear_Combination of a1;
end;
:: CONVEX4:def 12
theorem
for b1 being non empty CLSStruct
for b2 being set holds
b2 = C_LinComb b1
iff
for b3 being set holds
b3 in b2
iff
b3 is C_Linear_Combination of b1;
:: CONVEX4:funcreg 3
registration
let a1 be non empty CLSStruct;
cluster C_LinComb a1 -> non empty;
end;
:: CONVEX4:funcnot 11 => CONVEX4:func 11
definition
let a1 be non empty CLSStruct;
let a2 be Element of C_LinComb a1;
func @ A2 -> C_Linear_Combination of a1 equals
a2;
end;
:: CONVEX4:def 13
theorem
for b1 being non empty CLSStruct
for b2 being Element of C_LinComb b1 holds
@ b2 = b2;
:: CONVEX4:funcnot 12 => CONVEX4:func 12
definition
let a1 be non empty CLSStruct;
let a2 be C_Linear_Combination of a1;
func @ A2 -> Element of C_LinComb a1 equals
a2;
end;
:: CONVEX4:def 14
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
@ b2 = b2;
:: CONVEX4:funcnot 13 => CONVEX4:func 13
definition
let a1 be non empty CLSStruct;
func C_LCAdd A1 -> Function-like quasi_total Relation of [:C_LinComb a1,C_LinComb a1:],C_LinComb a1 means
for b1, b2 being Element of C_LinComb a1 holds
it .(b1,b2) = (@ b1) + @ b2;
end;
:: CONVEX4:def 15
theorem
for b1 being non empty CLSStruct
for b2 being Function-like quasi_total Relation of [:C_LinComb b1,C_LinComb b1:],C_LinComb b1 holds
b2 = C_LCAdd b1
iff
for b3, b4 being Element of C_LinComb b1 holds
b2 .(b3,b4) = (@ b3) + @ b4;
:: CONVEX4:funcnot 14 => CONVEX4:func 14
definition
let a1 be non empty CLSStruct;
func C_LCMult A1 -> Function-like quasi_total Relation of [:COMPLEX,C_LinComb a1:],C_LinComb a1 means
for b1 being Element of COMPLEX
for b2 being Element of C_LinComb a1 holds
it . [b1,b2] = b1 * @ b2;
end;
:: CONVEX4:def 16
theorem
for b1 being non empty CLSStruct
for b2 being Function-like quasi_total Relation of [:COMPLEX,C_LinComb b1:],C_LinComb b1 holds
b2 = C_LCMult b1
iff
for b3 being Element of COMPLEX
for b4 being Element of C_LinComb b1 holds
b2 . [b3,b4] = b3 * @ b4;
:: CONVEX4:funcnot 15 => CONVEX4:func 15
definition
let a1 be non empty CLSStruct;
func LC_CLSpace A1 -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct equals
CLSStruct(#C_LinComb a1,@ ZeroCLC a1,C_LCAdd a1,C_LCMult a1#);
end;
:: CONVEX4:def 17
theorem
for b1 being non empty CLSStruct holds
LC_CLSpace b1 = CLSStruct(#C_LinComb b1,@ ZeroCLC b1,C_LCAdd b1,C_LCMult b1#);
:: CONVEX4:funcreg 4
registration
let a1 be non empty CLSStruct;
cluster LC_CLSpace a1 -> non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like;
end;
:: CONVEX4:th 38
theorem
for b1 being non empty CLSStruct
for b2, b3 being C_Linear_Combination of b1 holds
(vector(LC_CLSpace b1,b2)) + vector(LC_CLSpace b1,b3) = b2 + b3;
:: CONVEX4:th 39
theorem
for b1 being non empty CLSStruct
for b2 being Element of COMPLEX
for b3 being C_Linear_Combination of b1 holds
b2 * vector(LC_CLSpace b1,b3) = b2 * b3;
:: CONVEX4:th 40
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
- vector(LC_CLSpace b1,b2) = - b2;
:: CONVEX4:th 41
theorem
for b1 being non empty CLSStruct
for b2, b3 being C_Linear_Combination of b1 holds
(vector(LC_CLSpace b1,b2)) - vector(LC_CLSpace b1,b3) = b2 - b3;
:: CONVEX4:funcnot 16 => CONVEX4:func 16
definition
let a1 be non empty CLSStruct;
let a2 be Element of bool the carrier of a1;
func LC_CLSpace A2 -> strict Subspace of LC_CLSpace a1 means
the carrier of it = {b1 where b1 is C_Linear_Combination of a2: TRUE};
end;
:: CONVEX4:def 18
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being strict Subspace of LC_CLSpace b1 holds
b3 = LC_CLSpace b2
iff
the carrier of b3 = {b4 where b4 is C_Linear_Combination of b2: TRUE};
:: CONVEX4:funcnot 17 => CONVEX4:func 17
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
let a2 be Subspace of a1;
func Up A2 -> Element of bool the carrier of a1 equals
the carrier of a2;
end;
:: CONVEX4:def 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
Up b2 = the carrier of b2;
:: CONVEX4:funcreg 5
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
let a2 be Subspace of a1;
cluster Up a2 -> non empty;
end;
:: CONVEX4:attrnot 1 => CONVEX4:attr 1
definition
let a1 be non empty CLSStruct;
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 COMPLEX
st (ex b4 being Element of REAL st
b4 = b3) &
b1 in a2 &
b2 in a2
holds ((1r - b3) * b1) + (b3 * b2) in a2;
end;
:: CONVEX4:dfs 20
definiens
let a1 be non empty CLSStruct;
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 COMPLEX
st (ex b4 being Element of REAL st
b4 = b3) &
b1 in a2 &
b2 in a2
holds ((1r - b3) * b1) + (b3 * b2) in a2;
:: CONVEX4:def 20
theorem
for b1 being non empty CLSStruct
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 COMPLEX
st (ex b6 being Element of REAL st
b6 = b5) &
b3 in b2 &
b4 in b2
holds ((1r - b5) * b3) + (b5 * b4) in b2;
:: CONVEX4:funcnot 18 => CONVEX4:func 18
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;
:: CONVEX4:def 21
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#);
:: CONVEX4:funcreg 6
registration
let a1 be non empty CLSStruct;
cluster [#] a1 -> Affine;
end;
:: CONVEX4:funcreg 7
registration
let a1 be non empty CLSStruct;
cluster {} a1 -> Affine;
end;
:: CONVEX4:exreg 1
registration
let a1 be non empty CLSStruct;
cluster non empty Affine Element of bool the carrier of a1;
end;
:: CONVEX4:exreg 2
registration
let a1 be non empty CLSStruct;
cluster empty Affine Element of bool the carrier of a1;
end;
:: CONVEX4:th 42
theorem
for b1 being real set
for b2 being complex set holds
Re (b1 * b2) = b1 * Re b2;
:: CONVEX4:th 43
theorem
for b1 being real set
for b2 being complex set holds
Im (b1 * b2) = b1 * Im b2;
:: CONVEX4:th 44
theorem
for b1 being real set
for b2 being complex set
st {} <= b1 & b1 <= 1
holds |.b1 * b2.| = b1 * |.b2.| &
|.(1r - b1) * b2.| = (1r - b1) * |.b2.|;
:: CONVEX4:funcnot 19 => CONVEX4:func 19
definition
let a1 be non empty CLSStruct;
let a2 be Element of bool the carrier of a1;
let a3 be Element of COMPLEX;
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;
:: CONVEX4:def 22
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of COMPLEX holds
b3 * b2 = {b3 * b4 where b4 is Element of the carrier of b1: b4 in b2};
:: CONVEX4:attrnot 2 => CONVEX4:attr 2
definition
let a1 be non empty CLSStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is convex means
for b1, b2 being Element of the carrier of a1
for b3 being Element of COMPLEX
st (ex b4 being Element of REAL st
b3 = b4 & {} < b4 & b4 < 1) &
b1 in a2 &
b2 in a2
holds (b3 * b1) + ((1r - b3) * b2) in a2;
end;
:: CONVEX4:dfs 23
definiens
let a1 be non empty CLSStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is convex
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
for b3 being Element of COMPLEX
st (ex b4 being Element of REAL st
b3 = b4 & {} < b4 & b4 < 1) &
b1 in a2 &
b2 in a2
holds (b3 * b1) + ((1r - b3) * b2) in a2;
:: CONVEX4:def 23
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1 holds
b2 is convex(b1)
iff
for b3, b4 being Element of the carrier of b1
for b5 being Element of COMPLEX
st (ex b6 being Element of REAL st
b5 = b6 & {} < b6 & b6 < 1) &
b3 in b2 &
b4 in b2
holds (b5 * b3) + ((1r - b5) * b4) in b2;
:: CONVEX4:th 45
theorem
for b1 being non empty ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of COMPLEX
st b2 is convex(b1)
holds b3 * b2 is convex(b1);
:: CONVEX4:th 46
theorem
for b1 being non empty Abelian add-associative ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is convex(b1) & b3 is convex(b1)
holds b2 + b3 is convex(b1);
:: CONVEX4:th 47
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 convex(b1) & b3 is convex(b1)
holds b2 - b3 is convex(b1);
:: CONVEX4:th 48
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1 holds
b2 is convex(b1)
iff
for b3 being Element of COMPLEX
st ex b4 being Element of REAL st
b3 = b4 & {} < b4 & b4 < 1
holds (b3 * b2) + ((1r - b3) * b2) c= b2;
:: CONVEX4:th 49
theorem
for b1 being non empty Abelian CLSStruct
for b2 being Element of bool the carrier of b1
st b2 is convex(b1)
for b3 being Element of COMPLEX
st ex b4 being Element of REAL st
b3 = b4 & {} < b4 & b4 < 1
holds ((1r - b3) * b2) + (b3 * b2) c= b2;
:: CONVEX4:th 50
theorem
for b1 being non empty Abelian add-associative ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is convex(b1) & b3 is convex(b1)
for b4 being Element of COMPLEX
st ex b5 being Element of REAL st
b4 = b5
holds (b4 * b2) + ((1r - b4) * b3) is convex(b1);
:: CONVEX4:th 51
theorem
for b1 being non empty ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1 holds
1r * b2 = b2;
:: CONVEX4:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being non empty Element of bool the carrier of b1 holds
0c * b2 = {0. b1};
:: CONVEX4:th 53
theorem
for b1 being non empty add-associative addLoopStr
for b2, b3, b4 being Element of bool the carrier of b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4);
:: CONVEX4:th 54
theorem
for b1 being non empty ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of COMPLEX holds
b3 * (b4 * b2) = (b3 * b4) * b2;
:: CONVEX4:th 55
theorem
for b1 being non empty ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of COMPLEX holds
b4 * (b2 + b3) = (b4 * b2) + (b4 * b3);
:: CONVEX4:th 56
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 Element of the carrier of b1 holds
b2 is convex(b1)
iff
b3 + b2 is convex(b1);
:: CONVEX4:th 57
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
Up (0). b1 is convex(b1);
:: CONVEX4:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
Up (Omega). b1 is convex(b1);
:: CONVEX4:th 59
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
st b2 = {}
holds b2 is convex(b1);
:: CONVEX4:th 60
theorem
for b1 being non empty Abelian add-associative ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of bool the carrier of b1
for b4, b5 being Element of COMPLEX
st b2 is convex(b1) & b3 is convex(b1)
holds (b4 * b2) + (b5 * b3) is convex(b1);
:: CONVEX4:th 61
theorem
for b1 being non empty ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of COMPLEX holds
(b3 + b4) * b2 c= (b3 * b2) + (b4 * b2);
:: CONVEX4:th 62
theorem
for b1 being non empty CLSStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of COMPLEX
st b2 c= b3
holds b4 * b2 c= b4 * b3;
:: CONVEX4:th 63
theorem
for b1 being non empty CLSStruct
for b2 being empty Element of bool the carrier of b1
for b3 being Element of COMPLEX holds
b3 * b2 = {};
:: CONVEX4:th 64
theorem
for b1 being non empty addLoopStr
for b2 being empty Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b2 + b3 = {};
:: CONVEX4:th 65
theorem
for b1 being non empty right_zeroed addLoopStr
for b2 being Element of bool the carrier of b1 holds
b2 + {0. b1} = b2;
:: CONVEX4:th 66
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, b4 being Element of COMPLEX
st (ex b5, b6 being Element of REAL st
b3 = b5 & b4 = b6 & {} <= b5 & {} <= b6) &
b2 is convex(b1)
holds (b3 * b2) + (b4 * b2) = (b3 + b4) * b2;
:: CONVEX4:th 67
theorem
for b1 being non empty Abelian add-associative ComplexLinearSpace-like CLSStruct
for b2, b3, b4 being Element of bool the carrier of b1
for b5, b6, b7 being Element of COMPLEX
st b2 is convex(b1) & b3 is convex(b1) & b4 is convex(b1)
holds ((b5 * b2) + (b6 * b3)) + (b7 * b4) is convex(b1);
:: CONVEX4:th 68
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool bool the carrier of b1
st for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is convex(b1)
holds meet b2 is convex(b1);
:: CONVEX4:th 69
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
st b2 is Affine(b1)
holds b2 is convex(b1);
:: CONVEX4:exreg 3
registration
let a1 be non empty CLSStruct;
cluster non empty convex Element of bool the carrier of a1;
end;
:: CONVEX4:exreg 4
registration
let a1 be non empty CLSStruct;
cluster empty convex Element of bool the carrier of a1;
end;
:: CONVEX4:th 70
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 = {b5 where b5 is Element of the carrier of b1: b4 <= Re (b5 .|. b3)}
holds b2 is convex(b1);
:: CONVEX4:th 71
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 = {b5 where b5 is Element of the carrier of b1: b4 < Re (b5 .|. b3)}
holds b2 is convex(b1);
:: CONVEX4:th 72
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 = {b5 where b5 is Element of the carrier of b1: Re (b5 .|. b3) <= b4}
holds b2 is convex(b1);
:: CONVEX4:th 73
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 = {b5 where b5 is Element of the carrier of b1: Re (b5 .|. b3) < b4}
holds b2 is convex(b1);
:: CONVEX4:th 74
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 = {b5 where b5 is Element of the carrier of b1: b4 <= Im (b5 .|. b3)}
holds b2 is convex(b1);
:: CONVEX4:th 75
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 = {b5 where b5 is Element of the carrier of b1: b4 < Im (b5 .|. b3)}
holds b2 is convex(b1);
:: CONVEX4:th 76
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 = {b5 where b5 is Element of the carrier of b1: Im (b5 .|. b3) <= b4}
holds b2 is convex(b1);
:: CONVEX4:th 77
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 = {b5 where b5 is Element of the carrier of b1: Im (b5 .|. b3) < b4}
holds b2 is convex(b1);
:: CONVEX4:th 78
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 = {b5 where b5 is Element of the carrier of b1: |.b5 .|. b3.| <= b4}
holds b2 is convex(b1);
:: CONVEX4:th 79
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
st b2 = {b5 where b5 is Element of the carrier of b1: |.b5 .|. b3.| < b4}
holds b2 is convex(b1);
:: CONVEX4:attrnot 3 => CONVEX4:attr 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
let a2 be C_Linear_Combination of a1;
attr a2 is convex means
ex b1 being FinSequence of the carrier of a1 st
b1 is one-to-one &
rng b1 = Carrier a2 &
(ex b2 being FinSequence of REAL st
len b2 = len b1 &
Sum b2 = 1 &
(for b3 being natural set
st b3 in dom b2
holds b2 . b3 = a2 . (b1 . b3) & {} <= b2 . b3));
end;
:: CONVEX4:dfs 24
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
let a2 be C_Linear_Combination of a1;
To prove
a2 is convex
it is sufficient to prove
thus ex b1 being FinSequence of the carrier of a1 st
b1 is one-to-one &
rng b1 = Carrier a2 &
(ex b2 being FinSequence of REAL st
len b2 = len b1 &
Sum b2 = 1 &
(for b3 being natural set
st b3 in dom b2
holds b2 . b3 = a2 . (b1 . b3) & {} <= b2 . b3));
:: CONVEX4:def 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being C_Linear_Combination of b1 holds
b2 is convex(b1)
iff
ex b3 being FinSequence of the carrier of b1 st
b3 is one-to-one &
rng b3 = Carrier b2 &
(ex b4 being FinSequence of REAL st
len b4 = len b3 &
Sum b4 = 1 &
(for b5 being natural set
st b5 in dom b4
holds b4 . b5 = b2 . (b3 . b5) & {} <= b4 . b5));
:: CONVEX4:th 80
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being C_Linear_Combination of b1
st b2 is convex(b1)
holds Carrier b2 <> {};
:: CONVEX4:th 81
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being C_Linear_Combination of b1
for b3 being Element of the carrier of b1
st b2 is convex(b1) &
(ex b4 being Element of REAL st
b4 = b2 . b3 & b4 <= {})
holds not b3 in Carrier b2;
:: CONVEX4:th 82
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being C_Linear_Combination of b1
st b2 is convex(b1)
holds b2 <> ZeroCLC b1;
:: CONVEX4:th 83
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 C_Linear_Combination of b1
st b3 is convex(b1) & Carrier b3 = {b2}
holds (ex b4 being Element of REAL st
b4 = b3 . b2 & b4 = 1) &
Sum b3 = (b3 . b2) * b2;
:: CONVEX4: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 C_Linear_Combination of b1
st b4 is convex(b1) & Carrier b4 = {b2,b3} & b2 <> b3
holds (ex b5, b6 being Element of REAL st
b5 = b4 . b2 & b6 = b4 . b3 & b5 + b6 = 1 & {} <= b5 & {} <= b6) &
Sum b4 = ((b4 . b2) * b2) + ((b4 . b3) * b3);
:: CONVEX4:th 85
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 C_Linear_Combination of b1
st b5 is convex(b1) & Carrier b5 = {b2,b3,b4} & b2 <> b3 & b3 <> b4 & b4 <> b2
holds (ex b6, b7, b8 being real set st
b6 = b5 . b2 & b7 = b5 . b3 & b8 = b5 . b4 & (b6 + b7) + b8 = 1 & {} <= b6 & {} <= b7 & {} <= b8) &
Sum b5 = (((b5 . b2) * b2) + ((b5 . b3) * b3)) + ((b5 . b4) * b4);
:: CONVEX4:th 86
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 C_Linear_Combination of {b2}
st b3 is convex(b1)
holds (ex b4 being Element of REAL st
b4 = b3 . b2 & b4 = 1) &
Sum b3 = (b3 . b2) * b2;
:: CONVEX4: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 C_Linear_Combination of {b2,b3}
st b2 <> b3 & b4 is convex(b1)
holds (ex b5, b6 being real set st
b5 = b4 . b2 & b6 = b4 . b3 & {} <= b5 & {} <= b6) &
Sum b4 = ((b4 . b2) * b2) + ((b4 . b3) * b3);
:: CONVEX4:th 88
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 C_Linear_Combination of {b2,b3,b4}
st b2 <> b3 & b3 <> b4 & b4 <> b2 & b5 is convex(b1)
holds (ex b6, b7, b8 being real set st
b6 = b5 . b2 & b7 = b5 . b3 & b8 = b5 . b4 & (b6 + b7) + b8 = 1 & {} <= b6 & {} <= b7 & {} <= b8) &
Sum b5 = (((b5 . b2) * b2) + ((b5 . b3) * b3)) + ((b5 . b4) * b4);
:: CONVEX4:funcnot 20 => CONVEX4:func 20
definition
let a1 be non empty CLSStruct;
let a2 be Element of bool the carrier of a1;
func Convex-Family A2 -> Element of bool bool the carrier of a1 means
for b1 being Element of bool the carrier of a1 holds
b1 in it
iff
b1 is convex(a1) & a2 c= b1;
end;
:: CONVEX4:def 25
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1 holds
b3 = Convex-Family b2
iff
for b4 being Element of bool the carrier of b1 holds
b4 in b3
iff
b4 is convex(b1) & b2 c= b4;
:: CONVEX4:funcnot 21 => CONVEX4:func 21
definition
let a1 be non empty CLSStruct;
let a2 be Element of bool the carrier of a1;
func conv A2 -> convex Element of bool the carrier of a1 equals
meet Convex-Family a2;
end;
:: CONVEX4:def 26
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1 holds
conv b2 = meet Convex-Family b2;
:: CONVEX4:th 89
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being convex Element of bool the carrier of b1
st b2 c= b3
holds conv b2 c= b3;