Theorem sincolem 8665

Description: Lemma for sinco 8667 and cosco 8668.

Hypotheses

Ref	Expression
sinco.1	|- F = {<.x, y>. | (x e. CC /\ y = (i x. x))}
sinco.2	|- G = {<.x, y>. | (x e. CC /\ y = (-ui x. x))}
sincolem.3	|- J = {<.x, y>. | (x e. CC /\ y = (x / A))}
sincolem.4	|- H = {<.w, v>. | (w e. CC /\ v = (((exp o. F)` w)O((exp o. G)` w)))}
sincolem.5	|- ((((exp o. F)` z) e. CC /\ ((exp o. G)` z) e. CC) -> (((exp o. F)` z)O((exp o. G)` z)) e. CC)
sincolem.6	|- A e. CC
sincolem.7	|- A =/= 0

Assertion

Ref	Expression
sincolem	|- ((J o. H) Fn CC /\ (z e. CC -> ((J o. H)` z) = (((exp` (i x. z))O(exp` (-ui x. z))) / A)))

Distinct variable groups:   v,F,w,z   v,G,w,z   v,O,w,x,y,z   x,A,y   z,H   z,J

Proof of Theorem sincolem

Step	Hyp	Ref	Expression
1		oprex 3983	. . . 4 |- (x / A) e. V
2		sincolem.3	. . . 4 |- J = {<.x, y>. | (x e. CC /\ y = (x / A))}
3	1, 2	fnopab2 3618	. . 3 |- J Fn CC
4		fveq2 3724	. . . . . . . 8 |- (w = z -> ((exp o. F)` w) = ((exp o. F)` z))
5		fveq2 3724	. . . . . . . 8 |- (w = z -> ((exp o. G)` w) = ((exp o. G)` z))
6	4, 5	opreq12d 3978	. . . . . . 7 |- (w = z -> (((exp o. F)` w)O((exp o. G)` w)) = (((exp o. F)` z)O((exp o. G)` z)))
7	6	eleq1d 1540	. . . . . 6 |- (w = z -> ((((exp o. F)` w)O((exp o. G)` w)) e. CC <-> (((exp o. F)` z)O((exp o. G)` z)) e. CC))
8	7	cbvralv 1800	. . . . 5 |- (A.w e. CC (((exp o. F)` w)O((exp o. G)` w)) e. CC <-> A.z e. CC (((exp o. F)` z)O((exp o. G)` z)) e. CC)
9		sincolem.5	. . . . . 6 |- ((((exp o. F)` z) e. CC /\ ((exp o. G)` z) e. CC) -> (((exp o. F)` z)O((exp o. G)` z)) e. CC)
10		eff 7313	. . . . . . . 8 |- exp:CC-->CC
11		sinco.1	. . . . . . . . 9 |- F = {<.x, y>. | (x e. CC /\ y = (i x. x))}
12		axicn 5270	. . . . . . . . . 10 |- i e. CC
13		axmulcl 5273	. . . . . . . . . 10 |- ((i e. CC /\ x e. CC) -> (i x. x) e. CC)
14	12, 13	mpan 695	. . . . . . . . 9 |- (x e. CC -> (i x. x) e. CC)
15	11, 14	fopab 3827	. . . . . . . 8 |- F:CC-->CC
16		fco 3636	. . . . . . . 8 |- ((exp:CC-->CC /\ F:CC-->CC) -> (exp o. F):CC-->CC)
17	10, 15, 16	mp2an 697	. . . . . . 7 |- (exp o. F):CC-->CC
18	17	ffvelrni 3815	. . . . . 6 |- (z e. CC -> ((exp o. F)` z) e. CC)
19		sinco.2	. . . . . . . . 9 |- G = {<.x, y>. | (x e. CC /\ y = (-ui x. x))}
20	12	negcl 5369	. . . . . . . . . 10 |- -ui e. CC
21		axmulcl 5273	. . . . . . . . . 10 |- ((-ui e. CC /\ x e. CC) -> (-ui x. x) e. CC)
22	20, 21	mpan 695	. . . . . . . . 9 |- (x e. CC -> (-ui x. x) e. CC)
23	19, 22	fopab 3827	. . . . . . . 8 |- G:CC-->CC
24		fco 3636	. . . . . . . 8 |- ((exp:CC-->CC /\ G:CC-->CC) -> (exp o. G):CC-->CC)
25	10, 23, 24	mp2an 697	. . . . . . 7 |- (exp o. G):CC-->CC
26	25	ffvelrni 3815	. . . . . 6 |- (z e. CC -> ((exp o. G)` z) e. CC)
27	9, 18, 26	sylanc 471	. . . . 5 |- (z e. CC -> (((exp o. F)` z)O((exp o. G)` z)) e. CC)
28	8, 27	mprgbir 1701	. . . 4 |- A.w e. CC (((exp o. F)` w)O((exp o. G)` w)) e. CC
29		sincolem.4	. . . . 5 |- H = {<.w, v>. | (w e. CC /\ v = (((exp o. F)` w)O((exp o. G)` w)))}
30	29	fopab2 3823	. . . 4 |- (A.w e. CC (((exp o. F)` w)O((exp o. G)` w)) e. CC <-> H:CC-->CC)
31	28, 30	mpbi 189	. . 3 |- H:CC-->CC
32		fnfco 3642	. . 3 |- ((J Fn CC /\ H:CC-->CC) -> (J o. H) Fn CC)
33	3, 31, 32	mp2an 697	. 2 |- (J o. H) Fn CC
34		fnfun 3585	. . . . 5 |- (J Fn CC -> Fun J)
35	3, 34	ax-mp 7	. . . 4 |- Fun J
36		fvco3 3776	. . . 4 |- ((Fun J /\ H:CC-->CC /\ z e. CC) -> ((J o. H)` z) = (J` (H` z)))
37	35, 31, 36	mp3an12 906	. . 3 |- (z e. CC -> ((J o. H)` z) = (J` (H` z)))
38	6, 29	fvopab4g 3779	. . . . . 6 |- ((z e. CC /\ (((exp o. F)` z)O((exp o. G)` z)) e. CC) -> (H` z) = (((exp o. F)` z)O((exp o. G)` z)))
39	27, 38	mpdan 704	. . . . 5 |- (z e. CC -> (H` z) = (((exp o. F)` z)O((exp o. G)` z)))
40		ffun 3629	. . . . . . . . 9 |- (exp:CC-->CC -> Fun exp)
41	10, 40	ax-mp 7	. . . . . . . 8 |- Fun exp
42		fvco3 3776	. . . . . . . 8 |- ((Fun exp /\ F:CC-->CC /\ z e. CC) -> ((exp o. F)` z) = (exp` (F` z)))
43	41, 15, 42	mp3an12 906	. . . . . . 7 |- (z e. CC -> ((exp o. F)` z) = (exp` (F` z)))
44		axmulcl 5273	. . . . . . . . . 10 |- ((i e. CC /\ z e. CC) -> (i x. z) e. CC)
45	12, 44	mpan 695	. . . . . . . . 9 |- (z e. CC -> (i x. z) e. CC)
46		opreq2 3969	. . . . . . . . . 10 |- (x = z -> (i x. x) = (i x. z))
47	46, 11	fvopab4g 3779	. . . . . . . . 9 |- ((z e. CC /\ (i x. z) e. CC) -> (F` z) = (i x. z))
48	45, 47	mpdan 704	. . . . . . . 8 |- (z e. CC -> (F` z) = (i x. z))
49	48	fveq2d 3728	. . . . . . 7 |- (z e. CC -> (exp` (F` z)) = (exp` (i x. z)))
50	43, 49	eqtrd 1507	. . . . . 6 |- (z e. CC -> ((exp o. F)` z) = (exp` (i x. z)))
51		fvco3 3776	. . . . . . . 8 |- ((Fun exp /\ G:CC-->CC /\ z e. CC) -> ((exp o. G)` z) = (exp` (G` z)))
52	41, 23, 51	mp3an12 906	. . . . . . 7 |- (z e. CC -> ((exp o. G)` z) = (exp` (G` z)))
53		axmulcl 5273	. . . . . . . . . 10 |- ((-ui e. CC /\ z e. CC) -> (-ui x. z) e. CC)
54	20, 53	mpan 695	. . . . . . . . 9 |- (z e. CC -> (-ui x. z) e. CC)
55		opreq2 3969	. . . . . . . . . 10 |- (x = z -> (-ui x. x) = (-ui x. z))
56	55, 19	fvopab4g 3779	. . . . . . . . 9 |- ((z e. CC /\ (-ui x. z) e. CC) -> (G` z) = (-ui x. z))
57	54, 56	mpdan 704	. . . . . . . 8 |- (z e. CC -> (G` z) = (-ui x. z))
58	57	fveq2d 3728	. . . . . . 7 |- (z e. CC -> (exp` (G` z)) = (exp` (-ui x. z)))
59	52, 58	eqtrd 1507	. . . . . 6 |- (z e. CC -> ((exp o. G)` z) = (exp` (-ui x. z)))
60	50, 59	opreq12d 3978	. . . . 5 |- (z e. CC -> (((exp o. F)` z)O((exp o. G)` z)) = ((exp` (i x. z))O(exp` (-ui x. z))))
61	39, 60	eqtrd 1507	. . . 4 |- (z e. CC -> (H` z) = ((exp` (i x. z))O(exp` (-ui x. z))))
62	61	fveq2d 3728	. . 3 |- (z e. CC -> (J` (H` z)) = (J` ((exp` (i x. z))O(exp` (-ui x. z)))))
63	60, 27	eqeltrrd 1549	. . . 4 |- (z e. CC -> ((exp` (i x. z))O(exp` (-ui x. z))) e. CC)
64		sincolem.6	. . . . . 6 |- A e. CC
65		sincolem.7	. . . . . 6 |- A =/= 0
66		divclt 5712	. . . . . 6 |- ((((exp` (i x. z))O(exp` (-ui x. z))) e. CC /\ A e. CC /\ A =/= 0) -> (((exp` (i x. z))O(exp` (-ui x. z))) / A) e. CC)
67	64, 65, 66	mp3an23 908	. . . . 5 |- (((exp` (i x. z))O(exp` (-ui x. z))) e. CC -> (((exp` (i x. z))O(exp` (-ui x. z))) / A) e. CC)
68		opreq1 3968	. . . . . 6 |- (x = ((exp` (i x. z))O(exp` (-ui x. z))) -> (x / A) = (((exp` (i x. z))O(exp` (-ui x. z))) / A))
69	68, 2	fvopab4g 3779	. . . . 5 |- ((((exp` (i x. z))O(exp` (-ui x. z))) e. CC /\ (((exp` (i x. z