Theorem efiatan2 22271

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 9337	. . . . 5 |- _i e. CC
2		atancl 22235	. . . . 5 |- ( A e. dom arctan -> (arctan ` A ) e. CC )
3		mulcl 9362	. . . . 5 |- ( ( _i e. CC /\ (arctan ` A ) e. CC ) -> ( _i x. (arctan ` A ) ) e. CC )
4	1, 2, 3	sylancr 658	. . . 4 |- ( A e. dom arctan -> ( _i x. (arctan ` A ) ) e. CC )
5		efcl 13364	. . . 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 9336	. . . . 5 |- 1 e. CC
8		atandm2 22231	. . . . . . 7 |- ( A e. dom arctan <-> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) )
9	8	simp1bi 998	. . . . . 6 |- ( A e. dom arctan -> A e. CC )
10	9	sqcld 12002	. . . . 5 |- ( A e. dom arctan -> ( A ^ 2 ) e. CC )
11		addcl 9360	. . . . 5 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 + ( A ^ 2 ) ) e. CC )
12	7, 10, 11	sylancr 658	. . . 4 |- ( A e. dom arctan -> ( 1 + ( A ^ 2 ) ) e. CC )
13	12	sqrcld 12919	. . 3 |- ( A e. dom arctan -> ( sqr ` ( 1 + ( A ^ 2 ) ) ) e. CC )
14	12	sqsqrd 12921	. . . . 5 |- ( A e. dom arctan -> ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) = ( 1 + ( A ^ 2 ) ) )
15		atandm4 22233	. . . . . 6 |- ( A e. dom arctan <-> ( A e. CC /\ ( 1 + ( A ^ 2 ) ) =/= 0 ) )
16	15	simprbi 461	. . . . 5 |- ( A e. dom arctan -> ( 1 + ( A ^ 2 ) ) =/= 0 )
17	14, 16	eqnetrd 2624	. . . 4 |- ( A e. dom arctan -> ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) =/= 0 )
18		sqne0 11928	. . . . 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 10109	. 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 10537	. . . . . . 7 |- ( 1 / 2 ) e. CC
23	12, 16	logcld 21981	. . . . . . 7 |- ( A e. dom arctan -> ( log ` ( 1 + ( A ^ 2 ) ) ) e. CC )
24		mulcl 9362	. . . . . . 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 658	. . . . . 6 |- ( A e. dom arctan -> ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) e. CC )
26		efadd 13375	. . . . . 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 656	. . . . 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 10388	. . . . . . . . . . . 12 |- 2 e. CC
29	28	a1i 11	. . . . . . . . . . 11 |- ( A e. dom arctan -> 2 e. CC )
30		mulcl 9362	. . . . . . . . . . . . . 14 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
31	1, 9, 30	sylancr 658	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( _i x. A ) e. CC )
32		addcl 9360	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC )
33	7, 31, 32	sylancr 658	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) e. CC )
34	8	simp3bi 1000	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) =/= 0 )
35	33, 34	logcld 21981	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( log ` ( 1 + ( _i x. A ) ) ) e. CC )
36	29, 35, 4	subdid 9796	. . . . . . . . . 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 22238	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> (arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
38	37	oveq2d 6106	. . . . . . . . . . . 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 9405	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( ( 2 x. _i ) x. (arctan ` A ) ) = ( 2 x. ( _i x. (arctan ` A ) ) ) )
41		halfcl 10546	. . . . . . . . . . . . . . . . . 18 |- ( _i e. CC -> ( _i / 2 ) e. CC )
42	1, 41	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- ( _i / 2 ) e. CC
43	28, 1, 42	mulassi 9391	. . . . . . . . . . . . . . . 16 |- ( ( 2 x. _i ) x. ( _i / 2 ) ) = ( 2 x. ( _i x. ( _i / 2 ) ) )
44	28, 1, 42	mul12i 9560	. . . . . . . . . . . . . . . 16 |- ( 2 x. ( _i x. ( _i / 2 ) ) ) = ( _i x. ( 2 x. ( _i / 2 ) ) )
45		2ne0 10410	. . . . . . . . . . . . . . . . . . 19 |- 2 =/= 0
46	1, 28, 45	divcan2i 10070	. . . . . . . . . . . . . . . . . 18 |- ( 2 x. ( _i / 2 ) ) = _i
47	46	oveq2i 6101	. . . . . . . . . . . . . . . . 17 |- ( _i x. ( 2 x. ( _i / 2 ) ) ) = ( _i x. _i )
48		ixi 9961	. . . . . . . . . . . . . . . . 17 |- ( _i x. _i ) = -u 1
49	47, 48	eqtri 2461	. . . . . . . . . . . . . . . 16 |- ( _i x. ( 2 x. ( _i / 2 ) ) ) = -u 1
50	43, 44, 49	3eqtri 2465	. . . . . . . . . . . . . . 15 |- ( ( 2 x. _i ) x. ( _i / 2 ) ) = -u 1
51	50	oveq1i 6100	. . . . . . . . . . . . . 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 9605	. . . . . . . . . . . . . . . . . 18 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 - ( _i x. A ) ) e. CC )
53	7, 31, 52	sylancr 658	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) e. CC )
54	8	simp2bi 999	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) =/= 0 )
55	53, 54	logcld 21981	. . . . . . . . . . . . . . . 16 |- ( A e. dom arctan -> ( log ` ( 1 - ( _i x. A ) ) ) e. CC )
56	55, 35	subcld 9715	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC )
57	56	mulm1d 9792	. . . . . . . . . . . . . 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 2485	. . . . . . . . . . . . 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 10544	. . . . . . . . . . . . . . 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 9405	. . . . . . . . . . . . 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 9731	. . . . . . . . . . . . 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 2481	. . . . . . . . . . . 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 2481	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( 2 x. ( _i x. (arctan ` A ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) )
66	65	oveq2d 6106	. . . . . . . . . 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 9362	. . . . . . . . . . . . 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 658	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC )
69	68, 35, 55	subsubd 9743	. . . . . . . . . . 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 10563	. . . . . . . . . . . . . 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 6105	. . . . . . . . . . . . 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 9716	. . . . . . . . . . . . 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 2473	. . . . . . . . . . . 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 6105	. . . . . . . . . . 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 22268	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log )
76		logef 21989	. . . . . . . . . . . . 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 13375	. . . . . . . . . . . . . . 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 656	. . . . . . . . . . . . . 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 21987	. . . . . . . . . . . . . . . 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 656	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) = ( 1 + ( _i x. A ) ) )
82		eflog 21987	. . . . . . . . . . . . . . . 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 656	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) = ( 1 - ( _i x. A ) ) )
84	81, 83	oveq12d 6108	. . . . . . . . . . . . . 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 11956	. . . . . . . . . . . . . . . . 17 |- ( 1 ^ 2 ) = 1
86	85	a1i 11	. . . . . . . . . . . . . . . 16 |- ( A e. dom arctan -> ( 1 ^ 2 ) = 1 )
87		sqmul 11925	. . . . . . . . . . . . . . . . . 18 |- ( ( _i e. CC /\ A e. CC ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
88	1, 9, 87	sylancr 658	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
89		i2 11962	. . . . . . . . . . . . . . . . . . 19 |- ( _i ^ 2 ) = -u 1
90	89	oveq1i 6100	. . . . . . . . . . . . . . . . . 18 |- ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) )
91	10	mulm1d 9792	. . . . . . . . . . . . . . . . . 18 |- ( A e. dom arctan -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
92	90, 91	syl5eq 2485	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
93	88, 92	eqtrd 2473	. . . . . . . . . . . . . . . 16 |- ( A e. dom arctan -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
94	86, 93	oveq12d 6108	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( ( 1 ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( 1 - -u ( A ^ 2 ) ) )
95		subsq 11969	. . . . . . . . . . . . . . . 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 658	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( ( 1 ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( 1 + ( _i x. A ) ) x. ( 1 - ( _i x. A ) ) ) )
97		subneg 9654	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 - -u ( A ^ 2 ) ) = ( 1 + ( A ^ 2 ) ) )
98	7, 10, 97	sylancr 658	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( 1 - -u ( A ^ 2 ) ) = ( 1 + ( A ^ 2 ) ) )
99	94, 96, 98	3eqtr3d 2481	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> ( ( 1 + ( _i x. A ) ) x. ( 1 - ( _i x. A ) ) ) = ( 1 + ( A ^ 2 ) ) )
100	79, 84, 99	3eqtrd 2477	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( exp ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( 1 + ( A ^ 2 ) ) )
101	100	fveq2d 5692	. . . . . . . . . . . 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 2475	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) = ( log ` ( 1 + ( A ^ 2 ) ) ) )
103	69, 74, 102	3eqtrd 2477	. . . . . . . . . 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 2477	. . . . . . . . 9 |- ( A e. dom arctan -> ( 2 x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) ) = ( log ` ( 1 + ( A ^ 2 ) ) ) )
105	104	oveq1d 6105	. . . . . . . 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 9715	. . . . . . . . 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 10108	. . . . . . . 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 10107	. . . . . . . 8 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( A ^ 2 ) ) ) / 2 ) = ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) )
110	105, 108, 109	3eqtr3d 2481	. . . . . . 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 9733	. . . . . . 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 5692	. . . . 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 2475	. . . 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 22116	. . . . . 6 |- ( A e. dom arctan -> ( ( 1 + ( A ^ 2 ) ) ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) )
117		cxpsqr 22107	. . . . . . 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 2475	. . . . 5 |- ( A e. dom arctan -> ( exp ` ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) = ( sqr ` ( 1 + ( A ^ 2 ) ) ) )
120	119	oveq2d 6106	. . . 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 2481	. . 3 |- ( A e. dom arctan -> ( ( exp ` ( _i x. (arctan ` A ) ) ) x. ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) = ( 1 + ( _i x. A ) ) )
122	121	oveq1d 6105	. 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 2475	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 1364   e. wcel 1761   =/= wne 2604   dom cdm 4836   ran crn 4837   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279   _ici 9280   + caddc 9281   x. cmul 9283   - cmin 9591   -ucneg 9592   / cdiv 9989   2c2 10367   ^cexp 11861   sqrcsqr 12718   expce 13343   logclog 21965   ^c ccxp 21966  arctancatan 22218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967  df-cxp 21968  df-atan 22221

This theorem is referenced by:  cosatan  22275