Theorem efiatan2 24644

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 9995	. . . . 5 |- _i e. CC
2		atancl 24608	. . . . 5 |- ( A e. dom arctan -> (arctan ` A ) e. CC )
3		mulcl 10020	. . . . 5 |- ( ( _i e. CC /\ (arctan ` A ) e. CC ) -> ( _i x. (arctan ` A ) ) e. CC )
4	1, 2, 3	sylancr 695	. . . 4 |- ( A e. dom arctan -> ( _i x. (arctan ` A ) ) e. CC )
5		efcl 14813	. . . 4 |- ( ( _i x. (arctan ` A ) ) e. CC -> ( exp ` ( _i x. (arctan ` A ) ) ) e. CC )
6	4, 5	syl 17	. . 3 |- ( A e. dom arctan -> ( exp ` ( _i x. (arctan ` A ) ) ) e. CC )
7		ax-1cn 9994	. . . . 5 |- 1 e. CC
8		atandm2 24604	. . . . . . 7 |- ( A e. dom arctan <-> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) )
9	8	simp1bi 1076	. . . . . 6 |- ( A e. dom arctan -> A e. CC )
10	9	sqcld 13006	. . . . 5 |- ( A e. dom arctan -> ( A ^ 2 ) e. CC )
11		addcl 10018	. . . . 5 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 + ( A ^ 2 ) ) e. CC )
12	7, 10, 11	sylancr 695	. . . 4 |- ( A e. dom arctan -> ( 1 + ( A ^ 2 ) ) e. CC )
13	12	sqrtcld 14176	. . 3 |- ( A e. dom arctan -> ( sqr ` ( 1 + ( A ^ 2 ) ) ) e. CC )
14	12	sqsqrtd 14178	. . . . 5 |- ( A e. dom arctan -> ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) = ( 1 + ( A ^ 2 ) ) )
15		atandm4 24606	. . . . . 6 |- ( A e. dom arctan <-> ( A e. CC /\ ( 1 + ( A ^ 2 ) ) =/= 0 ) )
16	15	simprbi 480	. . . . 5 |- ( A e. dom arctan -> ( 1 + ( A ^ 2 ) ) =/= 0 )
17	14, 16	eqnetrd 2861	. . . 4 |- ( A e. dom arctan -> ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) =/= 0 )
18		sqne0 12930	. . . . 5 |- ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) e. CC -> ( ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) =/= 0 <-> ( sqr ` ( 1 + ( A ^ 2 ) ) ) =/= 0 ) )
19	13, 18	syl 17	. . . 4 |- ( A e. dom arctan -> ( ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) =/= 0 <-> ( sqr ` ( 1 + ( A ^ 2 ) ) ) =/= 0 ) )
20	17, 19	mpbid 222	. . 3 |- ( A e. dom arctan -> ( sqr ` ( 1 + ( A ^ 2 ) ) ) =/= 0 )
21	6, 13, 20	divcan4d 10807	. 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 11247	. . . . . . 7 |- ( 1 / 2 ) e. CC
23	12, 16	logcld 24317	. . . . . . 7 |- ( A e. dom arctan -> ( log ` ( 1 + ( A ^ 2 ) ) ) e. CC )
24		mulcl 10020	. . . . . . 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 695	. . . . . 6 |- ( A e. dom arctan -> ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) e. CC )
26		efadd 14824	. . . . . 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 693	. . . . 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 11091	. . . . . . . . . . . 12 |- 2 e. CC
29	28	a1i 11	. . . . . . . . . . 11 |- ( A e. dom arctan -> 2 e. CC )
30		mulcl 10020	. . . . . . . . . . . . . 14 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
31	1, 9, 30	sylancr 695	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( _i x. A ) e. CC )
32		addcl 10018	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC )
33	7, 31, 32	sylancr 695	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) e. CC )
34	8	simp3bi 1078	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) =/= 0 )
35	33, 34	logcld 24317	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( log ` ( 1 + ( _i x. A ) ) ) e. CC )
36	29, 35, 4	subdid 10486	. . . . . . . . . 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 24611	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> (arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
38	37	oveq2d 6666	. . . . . . . . . . . 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 10063	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( ( 2 x. _i ) x. (arctan ` A ) ) = ( 2 x. ( _i x. (arctan ` A ) ) ) )
41		halfcl 11257	. . . . . . . . . . . . . . . . . 18 |- ( _i e. CC -> ( _i / 2 ) e. CC )
42	1, 41	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- ( _i / 2 ) e. CC
43	28, 1, 42	mulassi 10049	. . . . . . . . . . . . . . . 16 |- ( ( 2 x. _i ) x. ( _i / 2 ) ) = ( 2 x. ( _i x. ( _i / 2 ) ) )
44	28, 1, 42	mul12i 10231	. . . . . . . . . . . . . . . 16 |- ( 2 x. ( _i x. ( _i / 2 ) ) ) = ( _i x. ( 2 x. ( _i / 2 ) ) )
45		2ne0 11113	. . . . . . . . . . . . . . . . . . 19 |- 2 =/= 0
46	1, 28, 45	divcan2i 10768	. . . . . . . . . . . . . . . . . 18 |- ( 2 x. ( _i / 2 ) ) = _i
47	46	oveq2i 6661	. . . . . . . . . . . . . . . . 17 |- ( _i x. ( 2 x. ( _i / 2 ) ) ) = ( _i x. _i )
48		ixi 10656	. . . . . . . . . . . . . . . . 17 |- ( _i x. _i ) = -u 1
49	47, 48	eqtri 2644	. . . . . . . . . . . . . . . 16 |- ( _i x. ( 2 x. ( _i / 2 ) ) ) = -u 1
50	43, 44, 49	3eqtri 2648	. . . . . . . . . . . . . . 15 |- ( ( 2 x. _i ) x. ( _i / 2 ) ) = -u 1
51	50	oveq1i 6660	. . . . . . . . . . . . . 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 10280	. . . . . . . . . . . . . . . . . 18 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 - ( _i x. A ) ) e. CC )
53	7, 31, 52	sylancr 695	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) e. CC )
54	8	simp2bi 1077	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) =/= 0 )
55	53, 54	logcld 24317	. . . . . . . . . . . . . . . 16 |- ( A e. dom arctan -> ( log ` ( 1 - ( _i x. A ) ) ) e. CC )
56	55, 35	subcld 10392	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC )
57	56	mulm1d 10482	. . . . . . . . . . . . . 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 2668	. . . . . . . . . . . . 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 11255	. . . . . . . . . . . . . . 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 10063	. . . . . . . . . . . . 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 10408	. . . . . . . . . . . . 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 2664	. . . . . . . . . . . 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 2664	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( 2 x. ( _i x. (arctan ` A ) ) ) = ( ( log ` ( 1 + ( _i x. A ) ) ) - ( log ` ( 1 - ( _i x. A ) ) ) ) )
66	65	oveq2d 6666	. . . . . . . . . 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 10020	. . . . . . . . . . . . 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 695	. . . . . . . . . . . 12 |- ( A e. dom arctan -> ( 2 x. ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC )
69	68, 35, 55	subsubd 10420	. . . . . . . . . . 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 11275	. . . . . . . . . . . . . 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 6665	. . . . . . . . . . . . 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 10393	. . . . . . . . . . . . 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 2656	. . . . . . . . . . . 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 6665	. . . . . . . . . . 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 24641	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) e. ran log )
76		logef 24328	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 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 14824	. . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . 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 24323	. . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( exp ` ( log ` ( 1 + ( _i x. A ) ) ) ) = ( 1 + ( _i x. A ) ) )
82		eflog 24323	. . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( exp ` ( log ` ( 1 - ( _i x. A ) ) ) ) = ( 1 - ( _i x. A ) ) )
84	81, 83	oveq12d 6668	. . . . . . . . . . . . . 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 12958	. . . . . . . . . . . . . . . . 17 |- ( 1 ^ 2 ) = 1
86	85	a1i 11	. . . . . . . . . . . . . . . 16 |- ( A e. dom arctan -> ( 1 ^ 2 ) = 1 )
87		sqmul 12926	. . . . . . . . . . . . . . . . . 18 |- ( ( _i e. CC /\ A e. CC ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
88	1, 9, 87	sylancr 695	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
89		i2 12965	. . . . . . . . . . . . . . . . . . 19 |- ( _i ^ 2 ) = -u 1
90	89	oveq1i 6660	. . . . . . . . . . . . . . . . . 18 |- ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) )
91	10	mulm1d 10482	. . . . . . . . . . . . . . . . . 18 |- ( A e. dom arctan -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
92	90, 91	syl5eq 2668	. . . . . . . . . . . . . . . . 17 |- ( A e. dom arctan -> ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
93	88, 92	eqtrd 2656	. . . . . . . . . . . . . . . 16 |- ( A e. dom arctan -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
94	86, 93	oveq12d 6668	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( ( 1 ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( 1 - -u ( A ^ 2 ) ) )
95		subsq 12972	. . . . . . . . . . . . . . . 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 695	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( ( 1 ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( 1 + ( _i x. A ) ) x. ( 1 - ( _i x. A ) ) ) )
97		subneg 10330	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 - -u ( A ^ 2 ) ) = ( 1 + ( A ^ 2 ) ) )
98	7, 10, 97	sylancr 695	. . . . . . . . . . . . . . 15 |- ( A e. dom arctan -> ( 1 - -u ( A ^ 2 ) ) = ( 1 + ( A ^ 2 ) ) )
99	94, 96, 98	3eqtr3d 2664	. . . . . . . . . . . . . 14 |- ( A e. dom arctan -> ( ( 1 + ( _i x. A ) ) x. ( 1 - ( _i x. A ) ) ) = ( 1 + ( A ^ 2 ) ) )
100	79, 84, 99	3eqtrd 2660	. . . . . . . . . . . . 13 |- ( A e. dom arctan -> ( exp ` ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) ) = ( 1 + ( A ^ 2 ) ) )
101	100	fveq2d 6195	. . . . . . . . . . . 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 2658	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( _i x. A ) ) ) + ( log ` ( 1 - ( _i x. A ) ) ) ) = ( log ` ( 1 + ( A ^ 2 ) ) ) )
103	69, 74, 102	3eqtrd 2660	. . . . . . . . . 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 2660	. . . . . . . . 9 |- ( A e. dom arctan -> ( 2 x. ( ( log ` ( 1 + ( _i x. A ) ) ) - ( _i x. (arctan ` A ) ) ) ) = ( log ` ( 1 + ( A ^ 2 ) ) ) )
105	104	oveq1d 6665	. . . . . . . 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 10392	. . . . . . . . 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 10806	. . . . . . . 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 10805	. . . . . . . 8 |- ( A e. dom arctan -> ( ( log ` ( 1 + ( A ^ 2 ) ) ) / 2 ) = ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) )
110	105, 108, 109	3eqtr3d 2664	. . . . . . 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 10410	. . . . . . 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 222	. . . . . 6 |- ( A e. dom arctan -> ( ( _i x. (arctan ` A ) ) + ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) = ( log ` ( 1 + ( _i x. A ) ) ) )
113	112	fveq2d 6195	. . . . 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 2658	. . . 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 24458	. . . . . 6 |- ( A e. dom arctan -> ( ( 1 + ( A ^ 2 ) ) ^c ( 1 / 2 ) ) = ( exp ` ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) )
117		cxpsqrt 24449	. . . . . . 7 |- ( ( 1 + ( A ^ 2 ) ) e. CC -> ( ( 1 + ( A ^ 2 ) ) ^c ( 1 / 2 ) ) = ( sqr ` ( 1 + ( A ^ 2 ) ) ) )
118	12, 117	syl 17	. . . . . 6 |- ( A e. dom arctan -> ( ( 1 + ( A ^ 2 ) ) ^c ( 1 / 2 ) ) = ( sqr ` ( 1 + ( A ^ 2 ) ) ) )
119	116, 118	eqtr3d 2658	. . . . 5 |- ( A e. dom arctan -> ( exp ` ( ( 1 / 2 ) x. ( log ` ( 1 + ( A ^ 2 ) ) ) ) ) = ( sqr ` ( 1 + ( A ^ 2 ) ) ) )
120	119	oveq2d 6666	. . . 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 2664	. . 3 |- ( A e. dom arctan -> ( ( exp ` ( _i x. (arctan ` A ) ) ) x. ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) = ( 1 + ( _i x. A ) ) )
122	121	oveq1d 6665	. 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 2658	1 |- ( A e. dom arctan -> ( exp ` ( _i x. (arctan ` A ) ) ) = ( ( 1 + ( _i x. A ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   ^cexp 12860   sqrcsqrt 13973   expce 14792   logclog 24301   ^c ccxp 24302  arctancatan 24591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-atan 24594

This theorem is referenced by:  cosatan  24648