:: CFUNCDOM semantic presentation
definition
let c
1 be non
empty set ;
func ComplexFuncAdd c
1 -> BinOp of
Funcs a
1,
COMPLEX means :
Def1:
:: CFUNCDOM:def 1
for b
1, b
2 being
Element of
Funcs a
1,
COMPLEX holds a
2 . b
1,b
2 = addcomplex .: b
1,b
2;
existence
ex b1 being BinOp of Funcs c1,COMPLEX st
for b2, b3 being Element of Funcs c1,COMPLEX holds b1 . b2,b3 = addcomplex .: b2,b3
uniqueness
for b1, b2 being BinOp of Funcs c1,COMPLEX holds
( ( for b3, b4 being Element of Funcs c1,COMPLEX holds b1 . b3,b4 = addcomplex .: b3,b4 ) & ( for b3, b4 being Element of Funcs c1,COMPLEX holds b2 . b3,b4 = addcomplex .: b3,b4 ) implies b1 = b2 )
end;
:: deftheorem Def1 defines ComplexFuncAdd CFUNCDOM:def 1 :
definition
let c
1 be non
empty set ;
func ComplexFuncMult c
1 -> BinOp of
Funcs a
1,
COMPLEX means :
Def2:
:: CFUNCDOM:def 2
for b
1, b
2 being
Element of
Funcs a
1,
COMPLEX holds a
2 . b
1,b
2 = multcomplex .: b
1,b
2;
existence
ex b1 being BinOp of Funcs c1,COMPLEX st
for b2, b3 being Element of Funcs c1,COMPLEX holds b1 . b2,b3 = multcomplex .: b2,b3
uniqueness
for b1, b2 being BinOp of Funcs c1,COMPLEX holds
( ( for b3, b4 being Element of Funcs c1,COMPLEX holds b1 . b3,b4 = multcomplex .: b3,b4 ) & ( for b3, b4 being Element of Funcs c1,COMPLEX holds b2 . b3,b4 = multcomplex .: b3,b4 ) implies b1 = b2 )
end;
:: deftheorem Def2 defines ComplexFuncMult CFUNCDOM:def 2 :
definition
let c
1 be non
empty set ;
func ComplexFuncExtMult c
1 -> Function of
[:COMPLEX ,(Funcs a1,COMPLEX ):],
Funcs a
1,
COMPLEX means :
Def3:
:: CFUNCDOM:def 3
for b
1 being
Complexfor b
2 being
Element of
Funcs a
1,
COMPLEX for b
3 being
Element of a
1 holds
(a2 . [b1,b2]) . b
3 = b
1 * (b2 . b3);
existence
ex b1 being Function of [:COMPLEX ,(Funcs c1,COMPLEX ):], Funcs c1,COMPLEX st
for b2 being Complex
for b3 being Element of Funcs c1,COMPLEX
for b4 being Element of c1 holds (b1 . [b2,b3]) . b4 = b2 * (b3 . b4)
uniqueness
for b1, b2 being Function of [:COMPLEX ,(Funcs c1,COMPLEX ):], Funcs c1,COMPLEX holds
( ( for b3 being Complex
for b4 being Element of Funcs c1,COMPLEX
for b5 being Element of c1 holds (b1 . [b3,b4]) . b5 = b3 * (b4 . b5) ) & ( for b3 being Complex
for b4 being Element of Funcs c1,COMPLEX
for b5 being Element of c1 holds (b2 . [b3,b4]) . b5 = b3 * (b4 . b5) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines ComplexFuncExtMult CFUNCDOM:def 3 :
:: deftheorem Def4 defines ComplexFuncZero CFUNCDOM:def 4 :
:: deftheorem Def5 defines ComplexFuncUnit CFUNCDOM:def 5 :
Lemma4:
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being Function of b1,b2 holds b3 in dom b4
theorem Th1: :: CFUNCDOM:1
theorem Th2: :: CFUNCDOM:2
theorem Th3: :: CFUNCDOM:3
theorem Th4: :: CFUNCDOM:4
theorem Th5: :: CFUNCDOM:5
theorem Th6: :: CFUNCDOM:6
theorem Th7: :: CFUNCDOM:7
theorem Th8: :: CFUNCDOM:8
theorem Th9: :: CFUNCDOM:9
theorem Th10: :: CFUNCDOM:10
theorem Th11: :: CFUNCDOM:11
theorem Th12: :: CFUNCDOM:12
theorem Th13: :: CFUNCDOM:13
theorem Th14: :: CFUNCDOM:14
theorem Th15: :: CFUNCDOM:15
theorem Th16: :: CFUNCDOM:16
Lemma20:
for b1 being non empty set
for b2, b3 being Element of Funcs b1,COMPLEX
for b4 being Complex holds (ComplexFuncAdd b1) . ((ComplexFuncExtMult b1) . [b4,b2]),((ComplexFuncExtMult b1) . [b4,b3]) = (ComplexFuncExtMult b1) . [b4,((ComplexFuncAdd b1) . b2,b3)]
theorem Th17: :: CFUNCDOM:17
theorem Th18: :: CFUNCDOM:18
theorem Th19: :: CFUNCDOM:19
for b
1 being
set for b
2 being non
empty set holds
ex b
3, b
4 being
Element of
Funcs b
2,
COMPLEX st
( ( for b
5 being
set holds
( b
5 in b
2 implies ( ( b
5 = b
1 implies b
3 . b
5 = 1r ) & ( b
5 <> b
1 implies b
3 . b
5 = 0 ) ) ) ) & ( for b
5 being
set holds
( b
5 in b
2 implies ( ( b
5 = b
1 implies b
4 . b
5 = 0 ) & ( b
5 <> b
1 implies b
4 . b
5 = 1r ) ) ) ) )
theorem Th20: :: CFUNCDOM:20
theorem Th21: :: CFUNCDOM:21
theorem Th22: :: CFUNCDOM:22
theorem Th23: :: CFUNCDOM:23
theorem Th24: :: CFUNCDOM:24
for b
1, b
2 being
set for b
3 being non
empty set holds
not ( b
3 = {b1,b2} & b
1 <> b
2 & ( for b
4, b
5 being
Element of
Funcs b
3,
COMPLEX holds
not ( ( for b
6, b
7 being
Complex holds
(
(ComplexFuncAdd b3) . ((ComplexFuncExtMult b3) . [b6,b4]),
((ComplexFuncExtMult b3) . [b7,b5]) = ComplexFuncZero b
3 implies ( b
6 = 0 & b
7 = 0 ) ) ) & ( for b
6 being
Element of
Funcs b
3,
COMPLEX holds
ex b
7, b
8 being
Complex st b
6 = (ComplexFuncAdd b3) . ((ComplexFuncExtMult b3) . [b7,b4]),
((ComplexFuncExtMult b3) . [b8,b5]) ) ) ) )
theorem Th25: :: CFUNCDOM:25
:: deftheorem Def6 defines ComplexVectSpace CFUNCDOM:def 6 :
Lemma28:
ex b1 being non empty set ex b2, b3 being set st
( b1 = {b2,b3} & b2 <> b3 )
theorem Th26: :: CFUNCDOM:26
definition
let c
1 be non
empty set ;
func CRing c
1 -> strict doubleLoopStr equals :: CFUNCDOM:def 7
doubleLoopStr(#
(Funcs a1,COMPLEX ),
(ComplexFuncAdd a1),
(ComplexFuncMult a1),
(ComplexFuncUnit a1),
(ComplexFuncZero a1) #);
correctness
coherence
doubleLoopStr(# (Funcs c1,COMPLEX ),(ComplexFuncAdd c1),(ComplexFuncMult c1),(ComplexFuncUnit c1),(ComplexFuncZero c1) #) is strict doubleLoopStr ;
;
end;
:: deftheorem Def7 defines CRing CFUNCDOM:def 7 :
Lemma29:
for b1 being non empty set
for b2, b3 being Element of (CRing b1) holds b2 * b3 = b3 * b2
by Th9;
theorem Th27: :: CFUNCDOM:27
theorem Th28: :: CFUNCDOM:28
definition
let c
1 be non
empty set ;
func CAlgebra c
1 -> strict ComplexAlgebraStr equals :: CFUNCDOM:def 8
ComplexAlgebraStr(#
(Funcs a1,COMPLEX ),
(ComplexFuncMult a1),
(ComplexFuncAdd a1),
(ComplexFuncExtMult a1),
(ComplexFuncUnit a1),
(ComplexFuncZero a1) #);
correctness
coherence
ComplexAlgebraStr(# (Funcs c1,COMPLEX ),(ComplexFuncMult c1),(ComplexFuncAdd c1),(ComplexFuncExtMult c1),(ComplexFuncUnit c1),(ComplexFuncZero c1) #) is strict ComplexAlgebraStr ;
;
end;
:: deftheorem Def8 defines CAlgebra CFUNCDOM:def 8 :
Lemma33:
for b1 being non empty set
for b2, b3 being Element of (CAlgebra b1) holds b2 * b3 = b3 * b2
by Th9;
theorem Th29: :: CFUNCDOM:29
:: deftheorem Def9 defines ComplexAlgebra-like CFUNCDOM:def 9 :
theorem Th30: :: CFUNCDOM:30
theorem Th31: :: CFUNCDOM:31
theorem Th32: :: CFUNCDOM:32