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Theorem atancj 25189

Description: The arctangent function distributes under conjugation. (The condition that ℜ(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 25186 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 -1 and 1, though.) (Contributed by Mario Carneiro, 31-Mar-2015.)

**Assertion**
Ref	Expression
atancj	⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (𝐴 ∈ dom arctan ∧ (∗‘(arctan‘𝐴)) = (arctan‘(∗‘𝐴))))

**Proof of Theorem atancj**
Step	Hyp	Ref	Expression
1		simpl 475	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → 𝐴 ∈ ℂ)
2		simpr 477	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) ≠ 0)
3		fveq2 6499	. . . . . 6 ⊢ (𝐴 = -i → (ℜ‘𝐴) = (ℜ‘-i))
4		ax-icn 10394	. . . . . . . 8 ⊢ i ∈ ℂ
5	4	renegi 14400	. . . . . . 7 ⊢ (ℜ‘-i) = -(ℜ‘i)
6		rei 14376	. . . . . . . 8 ⊢ (ℜ‘i) = 0
7	6	negeqi 10679	. . . . . . 7 ⊢ -(ℜ‘i) = -0
8		neg0 10733	. . . . . . 7 ⊢ -0 = 0
9	5, 7, 8	3eqtri 2807	. . . . . 6 ⊢ (ℜ‘-i) = 0
10	3, 9	syl6eq 2831	. . . . 5 ⊢ (𝐴 = -i → (ℜ‘𝐴) = 0)
11	10	necon3i 3000	. . . 4 ⊢ ((ℜ‘𝐴) ≠ 0 → 𝐴 ≠ -i)
12	2, 11	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → 𝐴 ≠ -i)
13		fveq2 6499	. . . . . 6 ⊢ (𝐴 = i → (ℜ‘𝐴) = (ℜ‘i))
14	13, 6	syl6eq 2831	. . . . 5 ⊢ (𝐴 = i → (ℜ‘𝐴) = 0)
15	14	necon3i 3000	. . . 4 ⊢ ((ℜ‘𝐴) ≠ 0 → 𝐴 ≠ i)
16	2, 15	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → 𝐴 ≠ i)
17		atandm 25155	. . 3 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i))
18	1, 12, 16, 17	syl3anbrc 1323	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → 𝐴 ∈ dom arctan)
19		halfcl 11672	. . . . . 6 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
20	4, 19	ax-mp 5	. . . . 5 ⊢ (i / 2) ∈ ℂ
21		ax-1cn 10393	. . . . . . . 8 ⊢ 1 ∈ ℂ
22		mulcl 10419	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
23	4, 1, 22	sylancr 578	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (i · 𝐴) ∈ ℂ)
24		subcl 10685	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
25	21, 23, 24	sylancr 578	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − (i · 𝐴)) ∈ ℂ)
26		atandm2 25156	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
27	18, 26	sylib 210	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
28	27	simp2d 1123	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − (i · 𝐴)) ≠ 0)
29	25, 28	logcld 24855	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(1 − (i · 𝐴))) ∈ ℂ)
30		addcl 10417	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
31	21, 23, 30	sylancr 578	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + (i · 𝐴)) ∈ ℂ)
32	27	simp3d 1124	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + (i · 𝐴)) ≠ 0)
33	31, 32	logcld 24855	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(1 + (i · 𝐴))) ∈ ℂ)
34	29, 33	subcld 10798	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
35		cjmul 14362	. . . . 5 ⊢ (((i / 2) ∈ ℂ ∧ ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ) → (∗‘((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((∗‘(i / 2)) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
36	20, 34, 35	sylancr 578	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((∗‘(i / 2)) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
37		2ne0 11551	. . . . . . . 8 ⊢ 2 ≠ 0
38		2cn 11515	. . . . . . . . 9 ⊢ 2 ∈ ℂ
39	4, 38	cjdivi 14411	. . . . . . . 8 ⊢ (2 ≠ 0 → (∗‘(i / 2)) = ((∗‘i) / (∗‘2)))
40	37, 39	ax-mp 5	. . . . . . 7 ⊢ (∗‘(i / 2)) = ((∗‘i) / (∗‘2))
41		divneg 11133	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(i / 2) = (-i / 2))
42	4, 38, 37, 41	mp3an 1440	. . . . . . . 8 ⊢ -(i / 2) = (-i / 2)
43		cji 14379	. . . . . . . . 9 ⊢ (∗‘i) = -i
44		2re 11514	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
45		cjre 14359	. . . . . . . . . 10 ⊢ (2 ∈ ℝ → (∗‘2) = 2)
46	44, 45	ax-mp 5	. . . . . . . . 9 ⊢ (∗‘2) = 2
47	43, 46	oveq12i 6988	. . . . . . . 8 ⊢ ((∗‘i) / (∗‘2)) = (-i / 2)
48	42, 47	eqtr4i 2806	. . . . . . 7 ⊢ -(i / 2) = ((∗‘i) / (∗‘2))
49	40, 48	eqtr4i 2806	. . . . . 6 ⊢ (∗‘(i / 2)) = -(i / 2)
50	49	oveq1i 6986	. . . . 5 ⊢ ((∗‘(i / 2)) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = (-(i / 2) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
51	34	cjcld 14416	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) ∈ ℂ)
52		mulneg12 10879	. . . . . 6 ⊢ (((i / 2) ∈ ℂ ∧ (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) ∈ ℂ) → (-(i / 2) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((i / 2) · -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
53	20, 51, 52	sylancr 578	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (-(i / 2) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((i / 2) · -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
54	50, 53	syl5eq 2827	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘(i / 2)) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((i / 2) · -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
55		cjsub 14369	. . . . . . . . 9 ⊢ (((log‘(1 − (i · 𝐴))) ∈ ℂ ∧ (log‘(1 + (i · 𝐴))) ∈ ℂ) → (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = ((∗‘(log‘(1 − (i · 𝐴)))) − (∗‘(log‘(1 + (i · 𝐴))))))
56	29, 33, 55	syl2anc 576	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = ((∗‘(log‘(1 − (i · 𝐴)))) − (∗‘(log‘(1 + (i · 𝐴))))))
57		imsub 14355	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (ℑ‘(1 − (i · 𝐴))) = ((ℑ‘1) − (ℑ‘(i · 𝐴))))
58	21, 23, 57	sylancr 578	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 − (i · 𝐴))) = ((ℑ‘1) − (ℑ‘(i · 𝐴))))
59		reim 14329	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = (ℑ‘(i · 𝐴)))
60	59	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) = (ℑ‘(i · 𝐴)))
61	60	oveq2d 6992	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((ℑ‘1) − (ℜ‘𝐴)) = ((ℑ‘1) − (ℑ‘(i · 𝐴))))
62	58, 61	eqtr4d 2818	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 − (i · 𝐴))) = ((ℑ‘1) − (ℜ‘𝐴)))
63		df-neg 10673	. . . . . . . . . . . . . 14 ⊢ -(ℜ‘𝐴) = (0 − (ℜ‘𝐴))
64		im1 14375	. . . . . . . . . . . . . . 15 ⊢ (ℑ‘1) = 0
65	64	oveq1i 6986	. . . . . . . . . . . . . 14 ⊢ ((ℑ‘1) − (ℜ‘𝐴)) = (0 − (ℜ‘𝐴))
66	63, 65	eqtr4i 2806	. . . . . . . . . . . . 13 ⊢ -(ℜ‘𝐴) = ((ℑ‘1) − (ℜ‘𝐴))
67	62, 66	syl6eqr 2833	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 − (i · 𝐴))) = -(ℜ‘𝐴))
68		recl 14330	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ)
69	68	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) ∈ ℝ)
70	69	recnd 10468	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) ∈ ℂ)
71	70, 2	negne0d 10796	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → -(ℜ‘𝐴) ≠ 0)
72	67, 71	eqnetrd 3035	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 − (i · 𝐴))) ≠ 0)
73		logcj 24890	. . . . . . . . . . 11 ⊢ (((1 − (i · 𝐴)) ∈ ℂ ∧ (ℑ‘(1 − (i · 𝐴))) ≠ 0) → (log‘(∗‘(1 − (i · 𝐴)))) = (∗‘(log‘(1 − (i · 𝐴)))))
74	25, 72, 73	syl2anc 576	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(∗‘(1 − (i · 𝐴)))) = (∗‘(log‘(1 − (i · 𝐴)))))
75		cjsub 14369	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (∗‘(1 − (i · 𝐴))) = ((∗‘1) − (∗‘(i · 𝐴))))
76	21, 23, 75	sylancr 578	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(1 − (i · 𝐴))) = ((∗‘1) − (∗‘(i · 𝐴))))
77		1re 10439	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
78		cjre 14359	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℝ → (∗‘1) = 1)
79	77, 78	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘1) = 1)
80		cjmul 14362	. . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (∗‘(i · 𝐴)) = ((∗‘i) · (∗‘𝐴)))
81	4, 1, 80	sylancr 578	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(i · 𝐴)) = ((∗‘i) · (∗‘𝐴)))
82	43	oveq1i 6986	. . . . . . . . . . . . . . 15 ⊢ ((∗‘i) · (∗‘𝐴)) = (-i · (∗‘𝐴))
83		cjcl 14325	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)
84	83	adantr 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘𝐴) ∈ ℂ)
85		mulneg1 10877	. . . . . . . . . . . . . . . 16 ⊢ ((i ∈ ℂ ∧ (∗‘𝐴) ∈ ℂ) → (-i · (∗‘𝐴)) = -(i · (∗‘𝐴)))
86	4, 84, 85	sylancr 578	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (-i · (∗‘𝐴)) = -(i · (∗‘𝐴)))
87	82, 86	syl5eq 2827	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘i) · (∗‘𝐴)) = -(i · (∗‘𝐴)))
88	81, 87	eqtrd 2815	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(i · 𝐴)) = -(i · (∗‘𝐴)))
89	79, 88	oveq12d 6994	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘1) − (∗‘(i · 𝐴))) = (1 − -(i · (∗‘𝐴))))
90		mulcl 10419	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ (∗‘𝐴) ∈ ℂ) → (i · (∗‘𝐴)) ∈ ℂ)
91	4, 84, 90	sylancr 578	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (i · (∗‘𝐴)) ∈ ℂ)
92		subneg 10736	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · (∗‘𝐴)) ∈ ℂ) → (1 − -(i · (∗‘𝐴))) = (1 + (i · (∗‘𝐴))))
93	21, 91, 92	sylancr 578	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − -(i · (∗‘𝐴))) = (1 + (i · (∗‘𝐴))))
94	76, 89, 93	3eqtrd 2819	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(1 − (i · 𝐴))) = (1 + (i · (∗‘𝐴))))
95	94	fveq2d 6503	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(∗‘(1 − (i · 𝐴)))) = (log‘(1 + (i · (∗‘𝐴)))))
96	74, 95	eqtr3d 2817	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(log‘(1 − (i · 𝐴)))) = (log‘(1 + (i · (∗‘𝐴)))))
97		imadd 14354	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (ℑ‘(1 + (i · 𝐴))) = ((ℑ‘1) + (ℑ‘(i · 𝐴))))
98	21, 23, 97	sylancr 578	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 + (i · 𝐴))) = ((ℑ‘1) + (ℑ‘(i · 𝐴))))
99	60	oveq2d 6992	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (0 + (ℜ‘𝐴)) = (0 + (ℑ‘(i · 𝐴))))
100	64	oveq1i 6986	. . . . . . . . . . . . . 14 ⊢ ((ℑ‘1) + (ℑ‘(i · 𝐴))) = (0 + (ℑ‘(i · 𝐴)))
101	99, 100	syl6eqr 2833	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (0 + (ℜ‘𝐴)) = ((ℑ‘1) + (ℑ‘(i · 𝐴))))
102	70	addid2d 10641	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (0 + (ℜ‘𝐴)) = (ℜ‘𝐴))
103	98, 101, 102	3eqtr2d 2821	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 + (i · 𝐴))) = (ℜ‘𝐴))
104	103, 2	eqnetrd 3035	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 + (i · 𝐴))) ≠ 0)
105		logcj 24890	. . . . . . . . . . 11 ⊢ (((1 + (i · 𝐴)) ∈ ℂ ∧ (ℑ‘(1 + (i · 𝐴))) ≠ 0) → (log‘(∗‘(1 + (i · 𝐴)))) = (∗‘(log‘(1 + (i · 𝐴)))))
106	31, 104, 105	syl2anc 576	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(∗‘(1 + (i · 𝐴)))) = (∗‘(log‘(1 + (i · 𝐴)))))
107		cjadd 14361	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (∗‘(1 + (i · 𝐴))) = ((∗‘1) + (∗‘(i · 𝐴))))
108	21, 23, 107	sylancr 578	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(1 + (i · 𝐴))) = ((∗‘1) + (∗‘(i · 𝐴))))
109	79, 88	oveq12d 6994	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘1) + (∗‘(i · 𝐴))) = (1 + -(i · (∗‘𝐴))))
110		negsub 10735	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · (∗‘𝐴)) ∈ ℂ) → (1 + -(i · (∗‘𝐴))) = (1 − (i · (∗‘𝐴))))
111	21, 91, 110	sylancr 578	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + -(i · (∗‘𝐴))) = (1 − (i · (∗‘𝐴))))
112	108, 109, 111	3eqtrd 2819	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(1 + (i · 𝐴))) = (1 − (i · (∗‘𝐴))))
113	112	fveq2d 6503	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(∗‘(1 + (i · 𝐴)))) = (log‘(1 − (i · (∗‘𝐴)))))
114	106, 113	eqtr3d 2817	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(log‘(1 + (i · 𝐴)))) = (log‘(1 − (i · (∗‘𝐴)))))
115	96, 114	oveq12d 6994	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘(log‘(1 − (i · 𝐴)))) − (∗‘(log‘(1 + (i · 𝐴))))) = ((log‘(1 + (i · (∗‘𝐴)))) − (log‘(1 − (i · (∗‘𝐴))))))
116	56, 115	eqtrd 2815	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = ((log‘(1 + (i · (∗‘𝐴)))) − (log‘(1 − (i · (∗‘𝐴))))))
117	116	negeqd 10680	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = -((log‘(1 + (i · (∗‘𝐴)))) − (log‘(1 − (i · (∗‘𝐴))))))
118		addcl 10417	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (i · (∗‘𝐴)) ∈ ℂ) → (1 + (i · (∗‘𝐴))) ∈ ℂ)
119	21, 91, 118	sylancr 578	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + (i · (∗‘𝐴))) ∈ ℂ)
120		atandmcj 25188	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → (∗‘𝐴) ∈ dom arctan)
121	18, 120	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘𝐴) ∈ dom arctan)
122		atandm2 25156	. . . . . . . . . 10 ⊢ ((∗‘𝐴) ∈ dom arctan ↔ ((∗‘𝐴) ∈ ℂ ∧ (1 − (i · (∗‘𝐴))) ≠ 0 ∧ (1 + (i · (∗‘𝐴))) ≠ 0))
123	122	simp3bi 1127	. . . . . . . . 9 ⊢ ((∗‘𝐴) ∈ dom arctan → (1 + (i · (∗‘𝐴))) ≠ 0)
124	121, 123	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + (i · (∗‘𝐴))) ≠ 0)
125	119, 124	logcld 24855	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(1 + (i · (∗‘𝐴)))) ∈ ℂ)
126		subcl 10685	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (i · (∗‘𝐴)) ∈ ℂ) → (1 − (i · (∗‘𝐴))) ∈ ℂ)
127	21, 91, 126	sylancr 578	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − (i · (∗‘𝐴))) ∈ ℂ)
128	122	simp2bi 1126	. . . . . . . . 9 ⊢ ((∗‘𝐴) ∈ dom arctan → (1 − (i · (∗‘𝐴))) ≠ 0)
129	121, 128	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − (i · (∗‘𝐴))) ≠ 0)
130	127, 129	logcld 24855	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(1 − (i · (∗‘𝐴)))) ∈ ℂ)
131	125, 130	negsubdi2d 10814	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → -((log‘(1 + (i · (∗‘𝐴)))) − (log‘(1 − (i · (∗‘𝐴))))) = ((log‘(1 − (i · (∗‘𝐴)))) − (log‘(1 + (i · (∗‘𝐴))))))
132	117, 131	eqtrd 2815	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = ((log‘(1 − (i · (∗‘𝐴)))) − (log‘(1 + (i · (∗‘𝐴))))))
133	132	oveq2d 6992	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((i / 2) · -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((i / 2) · ((log‘(1 − (i · (∗‘𝐴)))) − (log‘(1 + (i · (∗‘𝐴)))))))
134	36, 54, 133	3eqtrd 2819	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((i / 2) · ((log‘(1 − (i · (∗‘𝐴)))) − (log‘(1 + (i · (∗‘𝐴)))))))
135		atanval 25163	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
136	18, 135	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
137	136	fveq2d 6503	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(arctan‘𝐴)) = (∗‘((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
138		atanval 25163	. . . 4 ⊢ ((∗‘𝐴) ∈ dom arctan → (arctan‘(∗‘𝐴)) = ((i / 2) · ((log‘(1 − (i · (∗‘𝐴)))) − (log‘(1 + (i · (∗‘𝐴)))))))
139	121, 138	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (arctan‘(∗‘𝐴)) = ((i / 2) · ((log‘(1 − (i · (∗‘𝐴)))) − (log‘(1 + (i · (∗‘𝐴)))))))
140	134, 137, 139	3eqtr4d 2825	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(arctan‘𝐴)) = (arctan‘(∗‘𝐴)))
141	18, 140	jca 504	1 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (𝐴 ∈ dom arctan ∧ (∗‘(arctan‘𝐴)) = (arctan‘(∗‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ≠ wne 2968 dom cdm 5407 ‘cfv 6188 (class class class)co 6976 ℂcc 10333 ℝcr 10334 0cc0 10335 1c1 10336 ici 10337 + caddc 10338 · cmul 10340 − cmin 10670 -cneg 10671 / cdiv 11098 2c2 11495 ∗ccj 14316 ℜcre 14317 ℑcim 14318 logclog 24839 arctancatan 25143

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 ax-addf 10414 ax-mulf 10415

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-om 7397 df-1st 7501 df-2nd 7502 df-supp 7634 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fsupp 8629 df-fi 8670 df-sup 8701 df-inf 8702 df-oi 8769 df-card 9162 df-cda 9388 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-q 12163 df-rp 12205 df-xneg 12324 df-xadd 12325 df-xmul 12326 df-ioo 12558 df-ioc 12559 df-ico 12560 df-icc 12561 df-fz 12709 df-fzo 12850 df-fl 12977 df-mod 13053 df-seq 13185 df-exp 13245 df-fac 13449 df-bc 13478 df-hash 13506 df-shft 14287 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-limsup 14689 df-clim 14706 df-rlim 14707 df-sum 14904 df-ef 15281 df-sin 15283 df-cos 15284 df-pi 15286 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-starv 16436 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-unif 16444 df-hom 16445 df-cco 16446 df-rest 16552 df-topn 16553 df-0g 16571 df-gsum 16572 df-topgen 16573 df-pt 16574 df-prds 16577 df-xrs 16631 df-qtop 16636 df-imas 16637 df-xps 16639 df-mre 16715 df-mrc 16716 df-acs 16718 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-submnd 17804 df-mulg 18012 df-cntz 18218 df-cmn 18668 df-psmet 20239 df-xmet 20240 df-met 20241 df-bl 20242 df-mopn 20243 df-fbas 20244 df-fg 20245 df-cnfld 20248 df-top 21206 df-topon 21223 df-topsp 21245 df-bases 21258 df-cld 21331 df-ntr 21332 df-cls 21333 df-nei 21410 df-lp 21448 df-perf 21449 df-cn 21539 df-cnp 21540 df-haus 21627 df-tx 21874 df-hmeo 22067 df-fil 22158 df-fm 22250 df-flim 22251 df-flf 22252 df-xms 22633 df-ms 22634 df-tms 22635 df-cncf 23189 df-limc 24167 df-dv 24168 df-log 24841 df-atan 25146

This theorem is referenced by: atanrecl 25190

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