Theorem efiatan2 23448

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 9540	. . . . 5 |- _i e. CC
2		atancl 23412	. . . . 5 |- ( A e. dom arctan -> (arctan ` A ) e. CC )
3		mulcl 9565	. . . . 5 |- ( ( _i e. CC /\ (arctan ` A ) e. CC ) -> ( _i x. (arctan ` A ) ) e. CC )
4	1, 2, 3	sylancr 661	. . . 4 |- ( A e. dom arctan -> ( _i x. (arctan ` A ) ) e. CC )
5		efcl 13903	. . . 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 9539	. . . . 5 |- 1 e. CC
8		atandm2 23408	. . . . . . 7 |- ( A e. dom arctan <-> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) )
9	8	simp1bi 1009	. . . . . 6 |- ( A e. dom arctan -> A e. CC )
10	9	sqcld 12293	. . . . 5 |- ( A e. dom arctan -> ( A ^ 2 ) e. CC )
11		addcl 9563	. . . . 5 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 + ( A ^ 2 ) ) e. CC )
12	7, 10, 11	sylancr 661	. . . 4 |- ( A e. dom arctan -> ( 1 + ( A ^ 2 ) ) e. CC )
13	12	sqrtcld 13353	. . 3 |- ( A e. dom arctan -> ( sqr ` ( 1 + ( A ^ 2 ) ) ) e. CC )
14	12	sqsqrtd 13355	. . . . 5 |- ( A e. dom arctan -> ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) = ( 1 + ( A ^ 2 ) ) )
15		atandm4 23410	. . . . . 6 |- ( A e. dom arctan <-> ( A e. CC /\ ( 1 + ( A ^ 2 ) ) =/= 0 ) )
16	15	simprbi 462	. . . . 5 |- ( A e. dom arctan -> ( 1 + ( A ^ 2 ) ) =/= 0 )
17	14, 16	eqnetrd 2747	. . . 4 |- ( A e. dom arctan -> ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) =/= 0 )
18		sqne0 12219	. . . . 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 210	. . 3 |- ( A e. dom arctan -> ( sqr ` ( 1 + ( A ^ 2 ) ) ) =/= 0 )
21	6, 13, 20	divcan4d 10322	. 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		halfcn 10751	. . . . . . 7 |- ( 1 / 2 ) e. CC
23	12, 16	logcld 23127	. . . . . . 7 |- ( A e. dom arctan -> ( log ` ( 1 + ( A ^ 2 ) ) ) e. CC )
24		mulcl 9565	. . . . . . 7 |- ( ( ( 1 / 2 ) e. CC /\ ( log ` ( 1 + ( A ^ 2 ) ) ) e. CC ) -> ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) e. CC )
25	22, 23, 24	sylancr 661	. . . . . 6 |- ( A e. dom arctan -> ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) e. CC )
26		efadd 13914	. . . . . 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 ) ) ) ) ) ) )
27	4, 25, 26	syl2anc 659	. . . . 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 ) ) ) ) ) ) )
28		2cn 10602	. . . . . . . . . . . 12 |- 2 e. CC
29	28	a1i 11	. . . . . . . . . . 11 |- ( A e. dom arctan -> 2 e. CC )
30		mulcl 9565	. . . . . . . . . . . . . 14 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
31	1, 9, 30	sylancr 661	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( _i x. A ) e. CC )
32		addcl 9563	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC )
33	7, 31, 32	sylancr 661	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) e. CC )
34	8	simp3bi 1011	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) =/= 0 )
35	33, 34	logcld 23127	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( log ` ( 1 + ( _i x. A ) ) ) e. CC )
36	29, 35, 4	subdid 10008	. . . . . . . . . 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 ) ) ) ) )
37		atanval 23415	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> (arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
38	37	oveq2d 6286	. . . . . . . . . . . 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 ) ) ) ) ) ) )
39	1	a1i 11	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> _i e. CC )
40	29, 39, 2	mulassd 9608	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( ( 2 x. _i ) x. (arctan ` A ) ) = ( 2 x. ( _i x. (arctan ` A ) ) ) )
41		halfcl 10760	. . . . . . . . . . . . . . . . . 18 |- ( _i e. CC -> ( _i / 2 ) e. CC )
42	1, 41	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- ( _i / 2 ) e. CC
43	28, 1, 42	mulassi 9594	. . . . . . . . . . . . . . . 16 |- ( ( 2 x. _i ) x. ( _i / 2 ) ) = ( 2 x. ( _i x. ( _i / 2 ) ) )
44	28, 1, 42	mul12i 9764	. . . . . . . . . . . . . . . 16 |- ( 2 x. ( _i x. ( _i / 2 ) ) ) = ( _i x. ( 2 x. ( _i / 2 ) ) )
45		2ne0 10624	. . . . . . . . . . . . . . . . . . 19 |- 2 =/= 0
46	1, 28, 45	divcan2i 10283	. . . . . . . . . . . . . . . . . 18 |- ( 2 x. ( _i / 2 ) ) = _i
47	46	oveq2i 6281	. . . . . . . . . . . . . . . . 17 |- ( _i x. ( 2 x. ( _i / 2 ) ) ) = ( _i x. _i )
48		ixi 10174	. . . . . . . . . . . . . . . . 17 |- ( _i x. _i ) = -u 1
49	47, 48	eqtri 2483	. . . . . . . . . . . . . . . 16 |- ( _i x. ( 2 x. ( _i / 2 ) ) ) = -u 1
50	43, 44, 49	3eqtri 2487	. . . . . . . . . . . . . . 15 |- ( ( 2 x. _i ) x. ( _i / 2 ) ) = -u 1
51	50	oveq1i 6280	. . . . . . . . . . . . . 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 9810	. . . . . . . . . . . . . . . . . 18 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 - ( _i x. A ) ) e. CC )
53	7, 31, 52	sylancr 661	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) e. CC )
54	8	simp2bi 1010	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) =/= 0 )
55	53, 54	logcld 23127	. . . . . . . . . . . . . . . 16 |- ( A e. dom arctan -> ( log ` ( 1 - ( _i x. A ) ) ) e. CC )
56	55, 35	subcld 9922	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC )
57	56	mulm1d 10004	. . . . . . . . . . . . . 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 2507	. . . . . . . . . . . . 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		2mulicn 10758	. . . . . . . . . . . . . . 15 |- ( 2 x. _i ) e. CC
60	59	a1i 11	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> ( 2 x. _i ) e. CC )
61	42	a1i 11	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> ( _i / 2 ) e. CC )
62	60, 61, 56	mulassd 9608	. . . . . . . . . . . . 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, 35	negsubdi2d 9938	. . . . . . . . . . . . 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 2503	. . . . . . . . . . . 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	38, 40, 64	3eqtr3d 2503	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( 2 x. ( _i x. (arctan ` A ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) )
66	65	oveq2d 6286	. . . . . . . . . 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 9565	. . . . . . . . . . . . 13 |- ( ( 2 e. CC /\ ( log ` ( 1 + ( _i x. A ) ) ) e. CC ) -> ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC )
68	28, 35, 67	sylancr 661	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC )
69	68, 35, 55	subsubd 9950	. . . . . . . . . . 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	35	2timesd 10777	. . . . . . . . . . . . . 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 6285	. . . . . . . . . . . . 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	35, 35	pncand 9923	. . . . . . . . . . . . 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 2495	. . . . . . . . . . . 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 6285	. . . . . . . . . . 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 23445	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log )
76		logef 23138	. . . . . . . . . . . . 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 13914	. . . . . . . . . . . . . . 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	35, 55, 78	syl2anc 659	. . . . . . . . . . . . . 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 23133	. . . . . . . . . . . . . . . 16 |- ( ( ( 1 + ( _i x. A ) ) e. CC /\ ( 1 + ( _i x. A ) ) =/= 0 ) -> ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) = ( 1 + ( _i x. A ) ) )
81	33, 34, 80	syl2anc 659	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) = ( 1 + ( _i x. A ) ) )
82		eflog 23133	. . . . . . . . . . . . . . . 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 659	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) = ( 1 - ( _i x. A ) ) )
84	81, 83	oveq12d 6288	. . . . . . . . . . . . . 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 12247	. . . . . . . . . . . . . . . . 17 |- ( 1 ^ 2 ) = 1
86	85	a1i 11	. . . . . . . . . . . . . . . 16 |- ( A e. dom arctan -> ( 1 ^ 2 ) = 1 )
87		sqmul 12216	. . . . . . . . . . . . . . . . . 18 |- ( ( _i e. CC /\ A e. CC ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
88	1, 9, 87	sylancr 661	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
89		i2 12253	. . . . . . . . . . . . . . . . . . 19 |- ( _i ^ 2 ) = -u 1
90	89	oveq1i 6280	. . . . . . . . . . . . . . . . . 18 |- ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) )
91	10	mulm1d 10004	. . . . . . . . . . . . . . . . . 18 |- ( A e. dom arctan -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
92	90, 91	syl5eq 2507	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
93	88, 92	eqtrd 2495	. . . . . . . . . . . . . . . 16 |- ( A e. dom arctan -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
94	86, 93	oveq12d 6288	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( ( 1 ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( 1 - -u ( A ^ 2 ) ) )
95		subsq 12260	. . . . . . . . . . . . . . . 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, 31, 95	sylancr 661	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( ( 1 ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( 1 + ( _i x. A ) ) x. ( 1 - ( _i x. A ) ) ) )
97		subneg 9859	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 - -u ( A ^ 2 ) ) = ( 1 + ( A ^ 2 ) ) )
98	7, 10, 97	sylancr 661	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( 1 - -u ( A ^ 2 ) ) = ( 1 + ( A ^ 2 ) ) )
99	94, 96, 98	3eqtr3d 2503	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> ( ( 1 + ( _i x. A ) ) x. ( 1 - ( _i x. A ) ) ) = ( 1 + ( A ^ 2 ) ) )
100	79, 84, 99	3eqtrd 2499	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( exp ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( 1 + ( A ^ 2 ) ) )
101	100	fveq2d 5852	. . . . . . . . . . . 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 2497	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) = ( log ` ( 1 + ( A ^ 2 ) ) ) )
103	69, 74, 102	3eqtrd 2499	. . . . . . . . . 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	36, 66, 103	3eqtrd 2499	. . . . . . . . 9 |- ( A e. dom arctan -> ( 2 x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) ) = ( log ` ( 1 + ( A ^ 2 ) ) ) )
105	104	oveq1d 6285	. . . . . . . 8 |- ( A e. dom arctan -> ( ( 2 x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) ) / 2 ) = ( ( log ` ( 1 + ( A ^ 2 ) ) ) / 2 ) )
106	35, 4	subcld 9922	. . . . . . . . 9 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) e. CC )
107	45	a1i 11	. . . . . . . . 9 |- ( A e. dom arctan -> 2 =/= 0 )
108	106, 29, 107	divcan3d 10321	. . . . . . . 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	23, 29, 107	divrec2d 10320	. . . . . . . 8 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( A ^ 2 ) ) ) / 2 ) = ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) )
110	105, 108, 109	3eqtr3d 2503	. . . . . . 7 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) = ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) )
111	35, 4, 25	subaddd 9940	. . . . . . 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 210	. . . . . 6 |- ( A e. dom arctan -> ( ( _i x. (arctan ` A ) ) + ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) = ( log ` ( 1 + ( _i x. A ) ) ) )
113	112	fveq2d 5852	. . . . 5 |- ( A e. dom arctan -> ( exp ` ( ( _i x. (arctan ` A ) ) + ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) ) = ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) )
114	27, 113	eqtr3d 2497	. . . 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	22	a1i 11	. . . . . . 7 |- ( A e. dom arctan -> ( 1 / 2 ) e. CC )
116	12, 16, 115	cxpefd 23264	. . . . . 6 |- ( A e. dom arctan -> ( ( 1 + ( A ^ 2 ) ) ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) )
117		cxpsqrt 23255	. . . . . . 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 2497	. . . . 5 |- ( A e. dom arctan -> ( exp ` ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) = ( sqr ` ( 1 + ( A ^ 2 ) ) ) )
120	119	oveq2d 6286	. . . 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 2503	. . 3 |- ( A e. dom arctan -> ( ( exp ` ( _i x. (arctan ` A ) ) ) x. ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) = ( 1 + ( _i x. A ) ) )
122	121	oveq1d 6285	. 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 2497	1 |- ( A e. dom arctan -> ( exp ` ( _i x. (arctan ` A ) ) ) = ( ( 1 + ( _i x. A ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1398   e. wcel 1823   =/= wne 2649   dom cdm 4988   ran crn 4989   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482   _ici 9483   + caddc 9484   x. cmul 9486   - cmin 9796   -ucneg 9797   / cdiv 10202   2c2 10581   ^cexp 12151   sqrcsqrt 13151   expce 13882   logclog 23111   ^c ccxp 23112  arctancatan 23395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-fac 12339  df-bc 12366  df-hash 12391  df-shft 12985  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-limsup 13379  df-clim 13396  df-rlim 13397  df-sum 13594  df-ef 13888  df-sin 13890  df-cos 13891  df-pi 13893  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-hom 14811  df-cco 14812  df-rest 14915  df-topn 14916  df-0g 14934  df-gsum 14935  df-topgen 14936  df-pt 14937  df-prds 14940  df-xrs 14994  df-qtop 14999  df-imas 15000  df-xps 15002  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-fbas 18614  df-fg 18615  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cld 19690  df-ntr 19691  df-cls 19692  df-nei 19769  df-lp 19807  df-perf 19808  df-cn 19898  df-cnp 19899  df-haus 19986  df-tx 20232  df-hmeo 20425  df-fil 20516  df-fm 20608  df-flim 20609  df-flf 20610  df-xms 20992  df-ms 20993  df-tms 20994  df-cncf 21551  df-limc 22439  df-dv 22440  df-log 23113  df-cxp 23114  df-atan 23398

This theorem is referenced by:  cosatan  23452