Theorem atancj 24637

Description: The arctangent function distributes under conjugation. (The condition that Re ( A ) =/= 0 is necessary because the branch cuts are chosen so that the negative imaginary line "agrees with" neighboring values with negative real part, while the positive imaginary line agrees with values with positive real part. This makes atanneg 24634 true unconditionally but messes up conjugation symmetry, and it is impossible to have both in a single-valued function. The claim is true on the imaginary line between -u 1 and 1, though.) (Contributed by Mario Carneiro, 31-Mar-2015.)

Assertion

Ref	Expression
atancj	|- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( A e. dom arctan /\ ( * ` (arctan ` A ) ) = (arctan ` ( * ` A ) ) ) )

Proof of Theorem atancj

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> A e. CC )
2		simpr 477	. . . 4 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( Re ` A ) =/= 0 )
3		fveq2 6191	. . . . . 6 |- ( A = -u _i -> ( Re ` A ) = ( Re ` -u _i ) )
4		ax-icn 9995	. . . . . . . 8 |- _i e. CC
5	4	renegi 13920	. . . . . . 7 |- ( Re ` -u _i ) = -u ( Re ` _i )
6		rei 13896	. . . . . . . 8 |- ( Re ` _i ) = 0
7	6	negeqi 10274	. . . . . . 7 |- -u ( Re ` _i ) = -u 0
8		neg0 10327	. . . . . . 7 |- -u 0 = 0
9	5, 7, 8	3eqtri 2648	. . . . . 6 |- ( Re ` -u _i ) = 0
10	3, 9	syl6eq 2672	. . . . 5 |- ( A = -u _i -> ( Re ` A ) = 0 )
11	10	necon3i 2826	. . . 4 |- ( ( Re ` A ) =/= 0 -> A =/= -u _i )
12	2, 11	syl 17	. . 3 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> A =/= -u _i )
13		fveq2 6191	. . . . . 6 |- ( A = _i -> ( Re ` A ) = ( Re ` _i ) )
14	13, 6	syl6eq 2672	. . . . 5 |- ( A = _i -> ( Re ` A ) = 0 )
15	14	necon3i 2826	. . . 4 |- ( ( Re ` A ) =/= 0 -> A =/= _i )
16	2, 15	syl 17	. . 3 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> A =/= _i )
17		atandm 24603	. . 3 |- ( A e. dom arctan <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) )
18	1, 12, 16, 17	syl3anbrc 1246	. 2 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> A e. dom arctan )
19		halfcl 11257	. . . . . 6 |- ( _i e. CC -> ( _i / 2 ) e. CC )
20	4, 19	ax-mp 5	. . . . 5 |- ( _i / 2 ) e. CC
21		ax-1cn 9994	. . . . . . . 8 |- 1 e. CC
22		mulcl 10020	. . . . . . . . 9 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
23	4, 1, 22	sylancr 695	. . . . . . . 8 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( _i x. A ) e. CC )
24		subcl 10280	. . . . . . . 8 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 - ( _i x. A ) ) e. CC )
25	21, 23, 24	sylancr 695	. . . . . . 7 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 1 - ( _i x. A ) ) e. CC )
26		atandm2 24604	. . . . . . . . 9 |- ( A e. dom arctan <-> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) )
27	18, 26	sylib 208	. . . . . . . 8 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( A e. CC /\ ( 1 - ( _i x. A ) ) =/= 0 /\ ( 1 + ( _i x. A ) ) =/= 0 ) )
28	27	simp2d 1074	. . . . . . 7 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 1 - ( _i x. A ) ) =/= 0 )
29	25, 28	logcld 24317	. . . . . 6 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( log ` ( 1 - ( _i x. A ) ) ) e. CC )
30		addcl 10018	. . . . . . . 8 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC )
31	21, 23, 30	sylancr 695	. . . . . . 7 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 1 + ( _i x. A ) ) e. CC )
32	27	simp3d 1075	. . . . . . 7 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 1 + ( _i x. A ) ) =/= 0 )
33	31, 32	logcld 24317	. . . . . 6 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( log ` ( 1 + ( _i x. A ) ) ) e. CC )
34	29, 33	subcld 10392	. . . . 5 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC )
35		cjmul 13882	. . . . 5 |- ( ( ( _i / 2 ) e. CC /\ ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) e. CC ) -> ( * ` ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) = ( ( * ` ( _i / 2 ) ) x. ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) )
36	20, 34, 35	sylancr 695	. . . 4 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) = ( ( * ` ( _i / 2 ) ) x. ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) )
37		2ne0 11113	. . . . . . . 8 |- 2 =/= 0
38		2cn 11091	. . . . . . . . 9 |- 2 e. CC
39	4, 38	cjdivi 13931	. . . . . . . 8 |- ( 2 =/= 0 -> ( * ` ( _i / 2 ) ) = ( ( * ` _i ) / ( * ` 2 ) ) )
40	37, 39	ax-mp 5	. . . . . . 7 |- ( * ` ( _i / 2 ) ) = ( ( * ` _i ) / ( * ` 2 ) )
41		divneg 10719	. . . . . . . . 9 |- ( ( _i e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> -u ( _i / 2 ) = ( -u _i / 2 ) )
42	4, 38, 37, 41	mp3an 1424	. . . . . . . 8 |- -u ( _i / 2 ) = ( -u _i / 2 )
43		cji 13899	. . . . . . . . 9 |- ( * ` _i ) = -u _i
44		2re 11090	. . . . . . . . . 10 |- 2 e. RR
45		cjre 13879	. . . . . . . . . 10 |- ( 2 e. RR -> ( * ` 2 ) = 2 )
46	44, 45	ax-mp 5	. . . . . . . . 9 |- ( * ` 2 ) = 2
47	43, 46	oveq12i 6662	. . . . . . . 8 |- ( ( * ` _i ) / ( * ` 2 ) ) = ( -u _i / 2 )
48	42, 47	eqtr4i 2647	. . . . . . 7 |- -u ( _i / 2 ) = ( ( * ` _i ) / ( * ` 2 ) )
49	40, 48	eqtr4i 2647	. . . . . 6 |- ( * ` ( _i / 2 ) ) = -u ( _i / 2 )
50	49	oveq1i 6660	. . . . 5 |- ( ( * ` ( _i / 2 ) ) x. ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) = ( -u ( _i / 2 ) x. ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
51	34	cjcld 13936	. . . . . 6 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) e. CC )
52		mulneg12 10468	. . . . . 6 |- ( ( ( _i / 2 ) e. CC /\ ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) e. CC ) -> ( -u ( _i / 2 ) x. ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) = ( ( _i / 2 ) x. -u ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) )
53	20, 51, 52	sylancr 695	. . . . 5 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( -u ( _i / 2 ) x. ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) = ( ( _i / 2 ) x. -u ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) )
54	50, 53	syl5eq 2668	. . . 4 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( ( * ` ( _i / 2 ) ) x. ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) = ( ( _i / 2 ) x. -u ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) )
55		cjsub 13889	. . . . . . . . 9 |- ( ( ( log ` ( 1 - ( _i x. A ) ) ) e. CC /\ ( log ` ( 1 + ( _i x. A ) ) ) e. CC ) -> ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( ( * ` ( log ` ( 1 - ( _i x. A ) ) ) ) - ( * ` ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
56	29, 33, 55	syl2anc 693	. . . . . . . 8 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( ( * ` ( log ` ( 1 - ( _i x. A ) ) ) ) - ( * ` ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
57		imsub 13875	. . . . . . . . . . . . . . 15 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( Im ` ( 1 - ( _i x. A ) ) ) = ( ( Im ` 1 ) - ( Im ` ( _i x. A ) ) ) )
58	21, 23, 57	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( Im ` ( 1 - ( _i x. A ) ) ) = ( ( Im ` 1 ) - ( Im ` ( _i x. A ) ) ) )
59		reim 13849	. . . . . . . . . . . . . . . 16 |- ( A e. CC -> ( Re ` A ) = ( Im ` ( _i x. A ) ) )
60	59	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( Re ` A ) = ( Im ` ( _i x. A ) ) )
61	60	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( ( Im ` 1 ) - ( Re ` A ) ) = ( ( Im ` 1 ) - ( Im ` ( _i x. A ) ) ) )
62	58, 61	eqtr4d 2659	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( Im ` ( 1 - ( _i x. A ) ) ) = ( ( Im ` 1 ) - ( Re ` A ) ) )
63		df-neg 10269	. . . . . . . . . . . . . 14 |- -u ( Re ` A ) = ( 0 - ( Re ` A ) )
64		im1 13895	. . . . . . . . . . . . . . 15 |- ( Im ` 1 ) = 0
65	64	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( ( Im ` 1 ) - ( Re ` A ) ) = ( 0 - ( Re ` A ) )
66	63, 65	eqtr4i 2647	. . . . . . . . . . . . 13 |- -u ( Re ` A ) = ( ( Im ` 1 ) - ( Re ` A ) )
67	62, 66	syl6eqr 2674	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( Im ` ( 1 - ( _i x. A ) ) ) = -u ( Re ` A ) )
68		recl 13850	. . . . . . . . . . . . . . 15 |- ( A e. CC -> ( Re ` A ) e. RR )
69	68	adantr 481	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( Re ` A ) e. RR )
70	69	recnd 10068	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( Re ` A ) e. CC )
71	70, 2	negne0d 10390	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> -u ( Re ` A ) =/= 0 )
72	67, 71	eqnetrd 2861	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( Im ` ( 1 - ( _i x. A ) ) ) =/= 0 )
73		logcj 24352	. . . . . . . . . . 11 |- ( ( ( 1 - ( _i x. A ) ) e. CC /\ ( Im ` ( 1 - ( _i x. A ) ) ) =/= 0 ) -> ( log ` ( * ` ( 1 - ( _i x. A ) ) ) ) = ( * ` ( log ` ( 1 - ( _i x. A ) ) ) ) )
74	25, 72, 73	syl2anc 693	. . . . . . . . . 10 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( log ` ( * ` ( 1 - ( _i x. A ) ) ) ) = ( * ` ( log ` ( 1 - ( _i x. A ) ) ) ) )
75		cjsub 13889	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( * ` ( 1 - ( _i x. A ) ) ) = ( ( * ` 1 ) - ( * ` ( _i x. A ) ) ) )
76	21, 23, 75	sylancr 695	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( 1 - ( _i x. A ) ) ) = ( ( * ` 1 ) - ( * ` ( _i x. A ) ) ) )
77		1re 10039	. . . . . . . . . . . . . 14 |- 1 e. RR
78		cjre 13879	. . . . . . . . . . . . . 14 |- ( 1 e. RR -> ( * ` 1 ) = 1 )
79	77, 78	mp1i 13	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` 1 ) = 1 )
80		cjmul 13882	. . . . . . . . . . . . . . 15 |- ( ( _i e. CC /\ A e. CC ) -> ( * ` ( _i x. A ) ) = ( ( * ` _i ) x. ( * ` A ) ) )
81	4, 1, 80	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( _i x. A ) ) = ( ( * ` _i ) x. ( * ` A ) ) )
82	43	oveq1i 6660	. . . . . . . . . . . . . . 15 |- ( ( * ` _i ) x. ( * ` A ) ) = ( -u _i x. ( * ` A ) )
83		cjcl 13845	. . . . . . . . . . . . . . . . 17 |- ( A e. CC -> ( * ` A ) e. CC )
84	83	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` A ) e. CC )
85		mulneg1 10466	. . . . . . . . . . . . . . . 16 |- ( ( _i e. CC /\ ( * ` A ) e. CC ) -> ( -u _i x. ( * ` A ) ) = -u ( _i x. ( * ` A ) ) )
86	4, 84, 85	sylancr 695	. . . . . . . . . . . . . . 15 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( -u _i x. ( * ` A ) ) = -u ( _i x. ( * ` A ) ) )
87	82, 86	syl5eq 2668	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( ( * ` _i ) x. ( * ` A ) ) = -u ( _i x. ( * ` A ) ) )
88	81, 87	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( _i x. A ) ) = -u ( _i x. ( * ` A ) ) )
89	79, 88	oveq12d 6668	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( ( * ` 1 ) - ( * ` ( _i x. A ) ) ) = ( 1 - -u ( _i x. ( * ` A ) ) ) )
90		mulcl 10020	. . . . . . . . . . . . . 14 |- ( ( _i e. CC /\ ( * ` A ) e. CC ) -> ( _i x. ( * ` A ) ) e. CC )
91	4, 84, 90	sylancr 695	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( _i x. ( * ` A ) ) e. CC )
92		subneg 10330	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ ( _i x. ( * ` A ) ) e. CC ) -> ( 1 - -u ( _i x. ( * ` A ) ) ) = ( 1 + ( _i x. ( * ` A ) ) ) )
93	21, 91, 92	sylancr 695	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 1 - -u ( _i x. ( * ` A ) ) ) = ( 1 + ( _i x. ( * ` A ) ) ) )
94	76, 89, 93	3eqtrd 2660	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( 1 - ( _i x. A ) ) ) = ( 1 + ( _i x. ( * ` A ) ) ) )
95	94	fveq2d 6195	. . . . . . . . . 10 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( log ` ( * ` ( 1 - ( _i x. A ) ) ) ) = ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) )
96	74, 95	eqtr3d 2658	. . . . . . . . 9 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( log ` ( 1 - ( _i x. A ) ) ) ) = ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) )
97		imadd 13874	. . . . . . . . . . . . . 14 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( Im ` ( 1 + ( _i x. A ) ) ) = ( ( Im ` 1 ) + ( Im ` ( _i x. A ) ) ) )
98	21, 23, 97	sylancr 695	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( Im ` ( 1 + ( _i x. A ) ) ) = ( ( Im ` 1 ) + ( Im ` ( _i x. A ) ) ) )
99	60	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 0 + ( Re ` A ) ) = ( 0 + ( Im ` ( _i x. A ) ) ) )
100	64	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( ( Im ` 1 ) + ( Im ` ( _i x. A ) ) ) = ( 0 + ( Im ` ( _i x. A ) ) )
101	99, 100	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 0 + ( Re ` A ) ) = ( ( Im ` 1 ) + ( Im ` ( _i x. A ) ) ) )
102	70	addid2d 10237	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 0 + ( Re ` A ) ) = ( Re ` A ) )
103	98, 101, 102	3eqtr2d 2662	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( Im ` ( 1 + ( _i x. A ) ) ) = ( Re ` A ) )
104	103, 2	eqnetrd 2861	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( Im ` ( 1 + ( _i x. A ) ) ) =/= 0 )
105		logcj 24352	. . . . . . . . . . 11 |- ( ( ( 1 + ( _i x. A ) ) e. CC /\ ( Im ` ( 1 + ( _i x. A ) ) ) =/= 0 ) -> ( log ` ( * ` ( 1 + ( _i x. A ) ) ) ) = ( * ` ( log ` ( 1 + ( _i x. A ) ) ) ) )
106	31, 104, 105	syl2anc 693	. . . . . . . . . 10 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( log ` ( * ` ( 1 + ( _i x. A ) ) ) ) = ( * ` ( log ` ( 1 + ( _i x. A ) ) ) ) )
107		cjadd 13881	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( * ` ( 1 + ( _i x. A ) ) ) = ( ( * ` 1 ) + ( * ` ( _i x. A ) ) ) )
108	21, 23, 107	sylancr 695	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( 1 + ( _i x. A ) ) ) = ( ( * ` 1 ) + ( * ` ( _i x. A ) ) ) )
109	79, 88	oveq12d 6668	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( ( * ` 1 ) + ( * ` ( _i x. A ) ) ) = ( 1 + -u ( _i x. ( * ` A ) ) ) )
110		negsub 10329	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ ( _i x. ( * ` A ) ) e. CC ) -> ( 1 + -u ( _i x. ( * ` A ) ) ) = ( 1 - ( _i x. ( * ` A ) ) ) )
111	21, 91, 110	sylancr 695	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 1 + -u ( _i x. ( * ` A ) ) ) = ( 1 - ( _i x. ( * ` A ) ) ) )
112	108, 109, 111	3eqtrd 2660	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( 1 + ( _i x. A ) ) ) = ( 1 - ( _i x. ( * ` A ) ) ) )
113	112	fveq2d 6195	. . . . . . . . . 10 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( log ` ( * ` ( 1 + ( _i x. A ) ) ) ) = ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) )
114	106, 113	eqtr3d 2658	. . . . . . . . 9 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( log ` ( 1 + ( _i x. A ) ) ) ) = ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) )
115	96, 114	oveq12d 6668	. . . . . . . 8 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( ( * ` ( log ` ( 1 - ( _i x. A ) ) ) ) - ( * ` ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) - ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) ) )
116	56, 115	eqtrd 2656	. . . . . . 7 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) - ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) ) )
117	116	negeqd 10275	. . . . . 6 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> -u ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = -u ( ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) - ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) ) )
118		addcl 10018	. . . . . . . . 9 |- ( ( 1 e. CC /\ ( _i x. ( * ` A ) ) e. CC ) -> ( 1 + ( _i x. ( * ` A ) ) ) e. CC )
119	21, 91, 118	sylancr 695	. . . . . . . 8 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 1 + ( _i x. ( * ` A ) ) ) e. CC )
120		atandmcj 24636	. . . . . . . . . 10 |- ( A e. dom arctan -> ( * ` A ) e. dom arctan )
121	18, 120	syl 17	. . . . . . . . 9 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` A ) e. dom arctan )
122		atandm2 24604	. . . . . . . . . 10 |- ( ( * ` A ) e. dom arctan <-> ( ( * ` A ) e. CC /\ ( 1 - ( _i x. ( * ` A ) ) ) =/= 0 /\ ( 1 + ( _i x. ( * ` A ) ) ) =/= 0 ) )
123	122	simp3bi 1078	. . . . . . . . 9 |- ( ( * ` A ) e. dom arctan -> ( 1 + ( _i x. ( * ` A ) ) ) =/= 0 )
124	121, 123	syl 17	. . . . . . . 8 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 1 + ( _i x. ( * ` A ) ) ) =/= 0 )
125	119, 124	logcld 24317	. . . . . . 7 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) e. CC )
126		subcl 10280	. . . . . . . . 9 |- ( ( 1 e. CC /\ ( _i x. ( * ` A ) ) e. CC ) -> ( 1 - ( _i x. ( * ` A ) ) ) e. CC )
127	21, 91, 126	sylancr 695	. . . . . . . 8 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 1 - ( _i x. ( * ` A ) ) ) e. CC )
128	122	simp2bi 1077	. . . . . . . . 9 |- ( ( * ` A ) e. dom arctan -> ( 1 - ( _i x. ( * ` A ) ) ) =/= 0 )
129	121, 128	syl 17	. . . . . . . 8 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( 1 - ( _i x. ( * ` A ) ) ) =/= 0 )
130	127, 129	logcld 24317	. . . . . . 7 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) e. CC )
131	125, 130	negsubdi2d 10408	. . . . . 6 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> -u ( ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) - ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) ) = ( ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) - ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) ) )
132	117, 131	eqtrd 2656	. . . . 5 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> -u ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) = ( ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) - ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) ) )
133	132	oveq2d 6666	. . . 4 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( ( _i / 2 ) x. -u ( * ` ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) - ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) ) ) )
134	36, 54, 133	3eqtrd 2660	. . 3 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) - ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) ) ) )
135		atanval 24611	. . . . 5 |- ( A e. dom arctan -> (arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
136	18, 135	syl 17	. . . 4 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> (arctan ` A ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) )
137	136	fveq2d 6195	. . 3 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` (arctan ` A ) ) = ( * ` ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. A ) ) ) - ( log ` ( 1 + ( _i x. A ) ) ) ) ) ) )
138		atanval 24611	. . . 4 |- ( ( * ` A ) e. dom arctan -> (arctan ` ( * ` A ) ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) - ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) ) ) )
139	121, 138	syl 17	. . 3 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> (arctan ` ( * ` A ) ) = ( ( _i / 2 ) x. ( ( log ` ( 1 - ( _i x. ( * ` A ) ) ) ) - ( log ` ( 1 + ( _i x. ( * ` A ) ) ) ) ) ) )
140	134, 137, 139	3eqtr4d 2666	. 2 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( * ` (arctan ` A ) ) = (arctan ` ( * ` A ) ) )
141	18, 140	jca 554	1 |- ( ( A e. CC /\ ( Re ` A ) =/= 0 ) -> ( A e. dom arctan /\ ( * ` (arctan ` A ) ) = (arctan ` ( * ` A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   dom cdm 5114   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   *ccj 13836   Recre 13837   Imcim 13838   logclog 24301  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-atan 24594

This theorem is referenced by:  atanrecl  24638