Theorem efiatan2 20710

Description: Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 3-Apr-2015.)

Assertion

Ref	Expression
efiatan2	|- ( A e. dom arctan -> ( exp ` ( _i x. (arctan ` A ) ) ) = ( ( 1 + ( _i x. A ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )

Proof of Theorem efiatan2

Step	Hyp	Ref	Expression
1		ax-icn 9005	. . . . 5 |- _i e. CC
2		atancl 20674	. . . . 5 |- ( A e. dom arctan -> (arctan ` A ) e. CC )
3		mulcl 9030	. . . . 5 |- ( ( _i e. CC /\ (arctan ` A ) e. CC ) -> ( _i x. (arctan ` A ) ) e. CC )
4	1, 2, 3	sylancr 645	. . . 4 |- ( A e. dom arctan -> ( _i x. (arctan ` A ) ) e. CC )
5		efcl 12640	. . . 4 |- ( ( _i x. (arctan ` A ) ) e. CC -> ( exp ` ( _i x. (arctan ` A ) ) ) e. CC )
6	4, 5	syl 16	. . 3 |- ( A e. dom arctan -> ( exp ` ( _i x. (arctan ` A ) ) ) e. CC )
7		ax-1cn 9004	. . . . 5 |- 1 e. CC
8		atandm2 20670	. . . . . . 7 |- ( A e. dom arctan <-> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) )
9	8	simp1bi 972	. . . . . 6 |- ( A e. dom arctan -> A e. CC )
10	9	sqcld 11476	. . . . 5 |- ( A e. dom arctan -> ( A ^ 2 ) e. CC )
11		addcl 9028	. . . . 5 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 + ( A ^ 2 ) ) e. CC )
12	7, 10, 11	sylancr 645	. . . 4 |- ( A e. dom arctan -> ( 1 + ( A ^ 2 ) ) e. CC )
13	12	sqrcld 12194	. . 3 |- ( A e. dom arctan -> ( sqr ` ( 1 + ( A ^ 2 ) ) ) e. CC )
14	12	sqsqrd 12196	. . . . 5 |- ( A e. dom arctan -> ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) = ( 1 + ( A ^ 2 ) ) )
15		atandm4 20672	. . . . . 6 |- ( A e. dom arctan <-> ( A e. CC /\ ( 1 + ( A ^ 2 ) ) =/= 0 ) )
16	15	simprbi 451	. . . . 5 |- ( A e. dom arctan -> ( 1 + ( A ^ 2 ) ) =/= 0 )
17	14, 16	eqnetrd 2585	. . . 4 |- ( A e. dom arctan -> ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) =/= 0 )
18		sqne0 11403	. . . . 5 |- ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) e. CC -> ( ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) =/= 0 <-> ( sqr ` ( 1 + ( A ^ 2 ) ) ) =/= 0 ) )
19	13, 18	syl 16	. . . 4 |- ( A e. dom arctan -> ( ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) =/= 0 <-> ( sqr ` ( 1 + ( A ^ 2 ) ) ) =/= 0 ) )
20	17, 19	mpbid 202	. . 3 |- ( A e. dom arctan -> ( sqr ` ( 1 + ( A ^ 2 ) ) ) =/= 0 )
21	6, 13, 20	divcan4d 9752	. 2 |- ( A e. dom arctan -> ( ( ( exp ` ( _i x. (arctan ` A ) ) ) x. ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) = ( exp ` ( _i x. (arctan ` A ) ) ) )
22		2cn 10026	. . . . . . . 8 |- 2 e. CC
23		2ne0 10039	. . . . . . . 8 |- 2 =/= 0
24	22, 23	reccli 9700	. . . . . . 7 |- ( 1 / 2 ) e. CC
25	12, 16	logcld 20421	. . . . . . 7 |- ( A e. dom arctan -> ( log ` ( 1 + ( A ^ 2 ) ) ) e. CC )
26		mulcl 9030	. . . . . . 7 |- ( ( ( 1 / 2 ) e. CC /\ ( log ` ( 1 + ( A ^ 2 ) ) ) e. CC ) -> ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) e. CC )
27	24, 25, 26	sylancr 645	. . . . . 6 |- ( A e. dom arctan -> ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) e. CC )
28		efadd 12651	. . . . . 6 |- ( ( ( _i x. (arctan ` A ) ) e. CC /\ ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) e. CC ) -> ( exp ` ( ( _i x. (arctan ` A ) ) + ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) ) = ( ( exp ` ( _i x. (arctan ` A ) ) ) x. ( exp ` ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) ) )
29	4, 27, 28	syl2anc 643	. . . . 5 |- ( A e. dom arctan -> ( exp ` ( ( _i x. (arctan ` A ) ) + ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) ) = ( ( exp ` ( _i x. (arctan ` A ) ) ) x. ( exp ` ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) ) )
30	22	a1i 11	. . . . . . . . . . 11 |- ( A e. dom arctan -> 2 e. CC )
31		mulcl 9030	. . . . . . . . . . . . . 14 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
32	1, 9, 31	sylancr 645	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( _i x. A ) e. CC )
33		addcl 9028	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC )
34	7, 32, 33	sylancr 645	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) e. CC )
35	8	simp3bi 974	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) =/= 0 )
36	34, 35	logcld 20421	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( log ` ( 1 + ( _i x. A ) ) ) e. CC )
37	30, 36, 4	subdid 9445	. . . . . . . . . 10 |- ( A e. dom arctan -> ( 2 x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) ) = ( ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) - ( 2 x. ( _i x. (arctan ` A ) ) ) ) )
38		atanval 20677	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> (arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
39	38	oveq2d 6056	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( ( 2 x. _i ) x. (arctan ` A ) ) = ( ( 2 x. _i ) x. ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) )
40	1	a1i 11	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> _i e. CC )
41	30, 40, 2	mulassd 9067	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( ( 2 x. _i ) x. (arctan ` A ) ) = ( 2 x. ( _i x. (arctan ` A ) ) ) )
42		halfcl 10149	. . . . . . . . . . . . . . . . . 18 |- ( _i e. CC -> ( _i / 2 ) e. CC )
43	1, 42	ax-mp 8	. . . . . . . . . . . . . . . . 17 |- ( _i / 2 ) e. CC
44	22, 1, 43	mulassi 9055	. . . . . . . . . . . . . . . 16 |- ( ( 2 x. _i ) x. ( _i / 2 ) ) = ( 2 x. ( _i x. ( _i / 2 ) ) )
45	22, 1, 43	mul12i 9217	. . . . . . . . . . . . . . . 16 |- ( 2 x. ( _i x. ( _i / 2 ) ) ) = ( _i x. ( 2 x. ( _i / 2 ) ) )
46	1, 22, 23	divcan2i 9713	. . . . . . . . . . . . . . . . . 18 |- ( 2 x. ( _i / 2 ) ) = _i
47	46	oveq2i 6051	. . . . . . . . . . . . . . . . 17 |- ( _i x. ( 2 x. ( _i / 2 ) ) ) = ( _i x. _i )
48		ixi 9607	. . . . . . . . . . . . . . . . 17 |- ( _i x. _i ) = -u 1
49	47, 48	eqtri 2424	. . . . . . . . . . . . . . . 16 |- ( _i x. ( 2 x. ( _i / 2 ) ) ) = -u 1
50	44, 45, 49	3eqtri 2428	. . . . . . . . . . . . . . 15 |- ( ( 2 x. _i ) x. ( _i / 2 ) ) = -u 1
51	50	oveq1i 6050	. . . . . . . . . . . . . 14 |- ( ( ( 2 x. _i ) x. ( _i / 2 ) ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( -u 1 x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) )
52		subcl 9261	. . . . . . . . . . . . . . . . . 18 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 - ( _i x. A ) ) e. CC )
53	7, 32, 52	sylancr 645	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) e. CC )
54	8	simp2bi 973	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) =/= 0 )
55	53, 54	logcld 20421	. . . . . . . . . . . . . . . 16 |- ( A e. dom arctan -> ( log ` ( 1 - ( _i x. A ) ) ) e. CC )
56	55, 36	subcld 9367	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC )
57	56	mulm1d 9441	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> ( -u 1 x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = -u ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) )
58	51, 57	syl5eq 2448	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( ( ( 2 x. _i ) x. ( _i / 2 ) ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = -u ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) )
59	22, 1	mulcli 9051	. . . . . . . . . . . . . . 15 |- ( 2 x. _i ) e. CC
60	59	a1i 11	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> ( 2 x. _i ) e. CC )
61	43	a1i 11	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> ( _i / 2 ) e. CC )
62	60, 61, 56	mulassd 9067	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( ( ( 2 x. _i ) x. ( _i / 2 ) ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( ( 2 x. _i ) x. ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) )
63	55, 36	negsubdi2d 9383	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> -u ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) )
64	58, 62, 63	3eqtr3d 2444	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( ( 2 x. _i ) x. ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) )
65	39, 41, 64	3eqtr3d 2444	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( 2 x. ( _i x. (arctan ` A ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) )
66	65	oveq2d 6056	. . . . . . . . . 10 |- ( A e. dom arctan -> ( ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) - ( 2 x. ( _i x. (arctan ` A ) ) ) ) = ( ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) - ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) )
67		mulcl 9030	. . . . . . . . . . . . 13 |- ( ( 2 e. CC /\ ( log ` ( 1 + ( _i x. A ) ) ) e. CC ) -> ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC )
68	22, 36, 67	sylancr 645	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC )
69	68, 36, 55	subsubd 9395	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) - ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( ( ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) )
70	36	2timesd 10166	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 + ( _i x. A ) ) ) ) )
71	70	oveq1d 6055	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) = ( ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 + ( _i x. A ) ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) )
72	36, 36	pncand 9368	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 + ( _i x. A ) ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) = ( log ` ( 1 + ( _i x. A ) ) ) )
73	71, 72	eqtrd 2436	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) = ( log ` ( 1 + ( _i x. A ) ) ) )
74	73	oveq1d 6055	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( ( ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) )
75		atanlogadd 20707	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log )
76		logef 20429	. . . . . . . . . . . . 13 |- ( ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log -> ( log ` ( exp ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) )
77	75, 76	syl 16	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( log ` ( exp ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) )
78		efadd 12651	. . . . . . . . . . . . . . 15 |- ( ( ( log ` ( 1 + ( _i x. A ) ) ) e. CC /\ ( log ` ( 1 - ( _i x. A ) ) ) e. CC ) -> ( exp ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) x. ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) )
79	36, 55, 78	syl2anc 643	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> ( exp ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) x. ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) )
80		eflog 20427	. . . . . . . . . . . . . . . 16 |- ( ( ( 1 + ( _i x. A ) ) e. CC /\ ( 1 + ( _i x. A ) ) =/= 0 ) -> ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) = ( 1 + ( _i x. A ) ) )
81	34, 35, 80	syl2anc 643	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) = ( 1 + ( _i x. A ) ) )
82		eflog 20427	. . . . . . . . . . . . . . . 16 |- ( ( ( 1 - ( _i x. A ) ) e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 ) -> ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) = ( 1 - ( _i x. A ) ) )
83	53, 54, 82	syl2anc 643	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) = ( 1 - ( _i x. A ) ) )
84	81, 83	oveq12d 6058	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> ( ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) x. ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( ( 1 + ( _i x. A ) ) x. ( 1 - ( _i x. A ) ) ) )
85		sq1 11431	. . . . . . . . . . . . . . . . 17 |- ( 1 ^ 2 ) = 1
86	85	a1i 11	. . . . . . . . . . . . . . . 16 |- ( A e. dom arctan -> ( 1 ^ 2 ) = 1 )
87		sqmul 11400	. . . . . . . . . . . . . . . . . 18 |- ( ( _i e. CC /\ A e. CC ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
88	1, 9, 87	sylancr 645	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
89		i2 11436	. . . . . . . . . . . . . . . . . . 19 |- ( _i ^ 2 ) = -u 1
90	89	oveq1i 6050	. . . . . . . . . . . . . . . . . 18 |- ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) )
91	10	mulm1d 9441	. . . . . . . . . . . . . . . . . 18 |- ( A e. dom arctan -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
92	90, 91	syl5eq 2448	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
93	88, 92	eqtrd 2436	. . . . . . . . . . . . . . . 16 |- ( A e. dom arctan -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
94	86, 93	oveq12d 6058	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( ( 1 ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( 1 - -u ( A ^ 2 ) ) )
95		subsq 11443	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( ( 1 ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( 1 + ( _i x. A ) ) x. ( 1 - ( _i x. A ) ) ) )
96	7, 32, 95	sylancr 645	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( ( 1 ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( 1 + ( _i x. A ) ) x. ( 1 - ( _i x. A ) ) ) )
97		subneg 9306	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 - -u ( A ^ 2 ) ) = ( 1 + ( A ^ 2 ) ) )
98	7, 10, 97	sylancr 645	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( 1 - -u ( A ^ 2 ) ) = ( 1 + ( A ^ 2 ) ) )
99	94, 96, 98	3eqtr3d 2444	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> ( ( 1 + ( _i x. A ) ) x. ( 1 - ( _i x. A ) ) ) = ( 1 + ( A ^ 2 ) ) )
100	79, 84, 99	3eqtrd 2440	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( exp ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( 1 + ( A ^ 2 ) ) )
101	100	fveq2d 5691	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( log ` ( exp ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) ) = ( log ` ( 1 + ( A ^ 2 ) ) ) )
102	77, 101	eqtr3d 2438	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) = ( log ` ( 1 + ( A ^ 2 ) ) ) )
103	69, 74, 102	3eqtrd 2440	. . . . . . . . . 10 |- ( A e. dom arctan -> ( ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) - ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( log ` ( 1 + ( A ^ 2 ) ) ) )
104	37, 66, 103	3eqtrd 2440	. . . . . . . . 9 |- ( A e. dom arctan -> ( 2 x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) ) = ( log ` ( 1 + ( A ^ 2 ) ) ) )
105	104	oveq1d 6055	. . . . . . . 8 |- ( A e. dom arctan -> ( ( 2 x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) ) / 2 ) = ( ( log ` ( 1 + ( A ^ 2 ) ) ) / 2 ) )
106	36, 4	subcld 9367	. . . . . . . . 9 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) e. CC )
107	23	a1i 11	. . . . . . . . 9 |- ( A e. dom arctan -> 2 =/= 0 )
108	106, 30, 107	divcan3d 9751	. . . . . . . 8 |- ( A e. dom arctan -> ( ( 2 x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) ) / 2 ) = ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) )
109	25, 30, 107	divrec2d 9750	. . . . . . . 8 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( A ^ 2 ) ) ) / 2 ) = ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) )
110	105, 108, 109	3eqtr3d 2444	. . . . . . 7 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) = ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) )
111	36, 4, 27	subaddd 9385	. . . . . . 7 |- ( A e. dom arctan -> ( ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) = ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) <-> ( ( _i x. (arctan ` A ) ) + ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) = ( log ` ( 1 + ( _i x. A ) ) ) ) )
112	110, 111	mpbid 202	. . . . . 6 |- ( A e. dom arctan -> ( ( _i x. (arctan ` A ) ) + ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) = ( log ` ( 1 + ( _i x. A ) ) ) )
113	112	fveq2d 5691	. . . . 5 |- ( A e. dom arctan -> ( exp ` ( ( _i x. (arctan ` A ) ) + ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) ) = ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) )
114	29, 113	eqtr3d 2438	. . . 4 |- ( A e. dom arctan -> ( ( exp ` ( _i x. (arctan ` A ) ) ) x. ( exp ` ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) ) = ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) )
115	24	a1i 11	. . . . . . 7 |- ( A e. dom arctan -> ( 1 / 2 ) e. CC )
116	12, 16, 115	cxpefd 20556	. . . . . 6 |- ( A e. dom arctan -> ( ( 1 + ( A ^ 2 ) ) ^ c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) )
117		cxpsqr 20547	. . . . . . 7 |- ( ( 1 + ( A ^ 2 ) ) e. CC -> ( ( 1 + ( A ^ 2 ) ) ^ c ( 1 / 2 ) ) = ( sqr ` ( 1 + ( A ^ 2 ) ) ) )
118	12, 117	syl 16	. . . . . 6 |- ( A e. dom arctan -> ( ( 1 + ( A ^ 2 ) ) ^ c ( 1 / 2 ) ) = ( sqr ` ( 1 + ( A ^ 2 ) ) ) )
119	116, 118	eqtr3d 2438	. . . . 5 |- ( A e. dom arctan -> ( exp ` ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) = ( sqr ` ( 1 + ( A ^ 2 ) ) ) )
120	119	oveq2d 6056	. . . 4 |- ( A e. dom arctan -> ( ( exp ` ( _i x. (arctan ` A ) ) ) x. ( exp ` ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) ) = ( ( exp ` ( _i x. (arctan ` A ) ) ) x. ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )
121	114, 120, 81	3eqtr3d 2444	. . 3 |- ( A e. dom arctan -> ( ( exp ` ( _i x. (arctan ` A ) ) ) x. ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) = ( 1 + ( _i x. A ) ) )
122	121	oveq1d 6055	. 2 |- ( A e. dom arctan -> ( ( ( exp ` ( _i x. (arctan ` A ) ) ) x. ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) = ( ( 1 + ( _i x. A ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )
123	21, 122	eqtr3d 2438	1 |- ( A e. dom arctan -> ( exp ` ( _i x. (arctan ` A ) ) ) = ( ( 1 + ( _i x. A ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 177   = wceq 1649   e. wcel 1721   =/= wne 2567   dom cdm 4837   ran crn 4838   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947   _ici 8948   + caddc 8949   x. cmul 8951   - cmin 9247   -ucneg 9248   / cdiv 9633   2c2 10005   ^cexp 11337   sqrcsqr 11993   expce 12619   logclog 20405   ^ c ccxp 20406  arctancatan 20657

This theorem is referenced by:  cosatan  20714

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-atan 20660