atancj - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > atancj		Structured version Visualization version GIF version

Theorem atancj 26874

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 26871 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 482	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → 𝐴 ∈ ℂ)
2		simpr 484	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) ≠ 0)
3		fveq2 6840	. . . . . 6 ⊢ (𝐴 = -i → (ℜ‘𝐴) = (ℜ‘-i))
4		ax-icn 11097	. . . . . . . 8 ⊢ i ∈ ℂ
5	4	renegi 15142	. . . . . . 7 ⊢ (ℜ‘-i) = -(ℜ‘i)
6		rei 15118	. . . . . . . 8 ⊢ (ℜ‘i) = 0
7	6	negeqi 11386	. . . . . . 7 ⊢ -(ℜ‘i) = -0
8		neg0 11440	. . . . . . 7 ⊢ -0 = 0
9	5, 7, 8	3eqtri 2763	. . . . . 6 ⊢ (ℜ‘-i) = 0
10	3, 9	eqtrdi 2787	. . . . 5 ⊢ (𝐴 = -i → (ℜ‘𝐴) = 0)
11	10	necon3i 2964	. . . 4 ⊢ ((ℜ‘𝐴) ≠ 0 → 𝐴 ≠ -i)
12	2, 11	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → 𝐴 ≠ -i)
13		fveq2 6840	. . . . . 6 ⊢ (𝐴 = i → (ℜ‘𝐴) = (ℜ‘i))
14	13, 6	eqtrdi 2787	. . . . 5 ⊢ (𝐴 = i → (ℜ‘𝐴) = 0)
15	14	necon3i 2964	. . . 4 ⊢ ((ℜ‘𝐴) ≠ 0 → 𝐴 ≠ i)
16	2, 15	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → 𝐴 ≠ i)
17		atandm 26840	. . 3 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i))
18	1, 12, 16, 17	syl3anbrc 1345	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → 𝐴 ∈ dom arctan)
19		halfcl 12403	. . . . . 6 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
20	4, 19	ax-mp 5	. . . . 5 ⊢ (i / 2) ∈ ℂ
21		ax-1cn 11096	. . . . . . . 8 ⊢ 1 ∈ ℂ
22		mulcl 11122	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
23	4, 1, 22	sylancr 588	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (i · 𝐴) ∈ ℂ)
24		subcl 11392	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
25	21, 23, 24	sylancr 588	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − (i · 𝐴)) ∈ ℂ)
26		atandm2 26841	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
27	18, 26	sylib 218	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
28	27	simp2d 1144	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − (i · 𝐴)) ≠ 0)
29	25, 28	logcld 26534	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(1 − (i · 𝐴))) ∈ ℂ)
30		addcl 11120	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
31	21, 23, 30	sylancr 588	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + (i · 𝐴)) ∈ ℂ)
32	27	simp3d 1145	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + (i · 𝐴)) ≠ 0)
33	31, 32	logcld 26534	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(1 + (i · 𝐴))) ∈ ℂ)
34	29, 33	subcld 11505	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
35		cjmul 15104	. . . . 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 588	. . . 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 12285	. . . . . . . 8 ⊢ 2 ≠ 0
38		2cn 12256	. . . . . . . . 9 ⊢ 2 ∈ ℂ
39	4, 38	cjdivi 15153	. . . . . . . 8 ⊢ (2 ≠ 0 → (∗‘(i / 2)) = ((∗‘i) / (∗‘2)))
40	37, 39	ax-mp 5	. . . . . . 7 ⊢ (∗‘(i / 2)) = ((∗‘i) / (∗‘2))
41		divneg 11846	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(i / 2) = (-i / 2))
42	4, 38, 37, 41	mp3an 1464	. . . . . . . 8 ⊢ -(i / 2) = (-i / 2)
43		cji 15121	. . . . . . . . 9 ⊢ (∗‘i) = -i
44		2re 12255	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
45		cjre 15101	. . . . . . . . . 10 ⊢ (2 ∈ ℝ → (∗‘2) = 2)
46	44, 45	ax-mp 5	. . . . . . . . 9 ⊢ (∗‘2) = 2
47	43, 46	oveq12i 7379	. . . . . . . 8 ⊢ ((∗‘i) / (∗‘2)) = (-i / 2)
48	42, 47	eqtr4i 2762	. . . . . . 7 ⊢ -(i / 2) = ((∗‘i) / (∗‘2))
49	40, 48	eqtr4i 2762	. . . . . 6 ⊢ (∗‘(i / 2)) = -(i / 2)
50	49	oveq1i 7377	. . . . 5 ⊢ ((∗‘(i / 2)) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = (-(i / 2) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
51	34	cjcld 15158	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) ∈ ℂ)
52		mulneg12 11588	. . . . . 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 588	. . . . 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	eqtrid 2783	. . . 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 15111	. . . . . . . . 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 585	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = ((∗‘(log‘(1 − (i · 𝐴)))) − (∗‘(log‘(1 + (i · 𝐴))))))
57		imsub 15097	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (ℑ‘(1 − (i · 𝐴))) = ((ℑ‘1) − (ℑ‘(i · 𝐴))))
58	21, 23, 57	sylancr 588	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 − (i · 𝐴))) = ((ℑ‘1) − (ℑ‘(i · 𝐴))))
59		reim 15071	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = (ℑ‘(i · 𝐴)))
60	59	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) = (ℑ‘(i · 𝐴)))
61	60	oveq2d 7383	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((ℑ‘1) − (ℜ‘𝐴)) = ((ℑ‘1) − (ℑ‘(i · 𝐴))))
62	58, 61	eqtr4d 2774	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 − (i · 𝐴))) = ((ℑ‘1) − (ℜ‘𝐴)))
63		df-neg 11380	. . . . . . . . . . . . . 14 ⊢ -(ℜ‘𝐴) = (0 − (ℜ‘𝐴))
64		im1 15117	. . . . . . . . . . . . . . 15 ⊢ (ℑ‘1) = 0
65	64	oveq1i 7377	. . . . . . . . . . . . . 14 ⊢ ((ℑ‘1) − (ℜ‘𝐴)) = (0 − (ℜ‘𝐴))
66	63, 65	eqtr4i 2762	. . . . . . . . . . . . 13 ⊢ -(ℜ‘𝐴) = ((ℑ‘1) − (ℜ‘𝐴))
67	62, 66	eqtr4di 2789	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 − (i · 𝐴))) = -(ℜ‘𝐴))
68		recl 15072	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ)
69	68	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) ∈ ℝ)
70	69	recnd 11173	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) ∈ ℂ)
71	70, 2	negne0d 11503	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → -(ℜ‘𝐴) ≠ 0)
72	67, 71	eqnetrd 2999	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 − (i · 𝐴))) ≠ 0)
73		logcj 26570	. . . . . . . . . . 11 ⊢ (((1 − (i · 𝐴)) ∈ ℂ ∧ (ℑ‘(1 − (i · 𝐴))) ≠ 0) → (log‘(∗‘(1 − (i · 𝐴)))) = (∗‘(log‘(1 − (i · 𝐴)))))
74	25, 72, 73	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(∗‘(1 − (i · 𝐴)))) = (∗‘(log‘(1 − (i · 𝐴)))))
75		cjsub 15111	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (∗‘(1 − (i · 𝐴))) = ((∗‘1) − (∗‘(i · 𝐴))))
76	21, 23, 75	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(1 − (i · 𝐴))) = ((∗‘1) − (∗‘(i · 𝐴))))
77		1re 11144	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
78		cjre 15101	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℝ → (∗‘1) = 1)
79	77, 78	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘1) = 1)
80		cjmul 15104	. . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (∗‘(i · 𝐴)) = ((∗‘i) · (∗‘𝐴)))
81	4, 1, 80	sylancr 588	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(i · 𝐴)) = ((∗‘i) · (∗‘𝐴)))
82	43	oveq1i 7377	. . . . . . . . . . . . . . 15 ⊢ ((∗‘i) · (∗‘𝐴)) = (-i · (∗‘𝐴))
83		cjcl 15067	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)
84	83	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘𝐴) ∈ ℂ)
85		mulneg1 11586	. . . . . . . . . . . . . . . 16 ⊢ ((i ∈ ℂ ∧ (∗‘𝐴) ∈ ℂ) → (-i · (∗‘𝐴)) = -(i · (∗‘𝐴)))
86	4, 84, 85	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (-i · (∗‘𝐴)) = -(i · (∗‘𝐴)))
87	82, 86	eqtrid 2783	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘i) · (∗‘𝐴)) = -(i · (∗‘𝐴)))
88	81, 87	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(i · 𝐴)) = -(i · (∗‘𝐴)))
89	79, 88	oveq12d 7385	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘1) − (∗‘(i · 𝐴))) = (1 − -(i · (∗‘𝐴))))
90		mulcl 11122	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ (∗‘𝐴) ∈ ℂ) → (i · (∗‘𝐴)) ∈ ℂ)
91	4, 84, 90	sylancr 588	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (i · (∗‘𝐴)) ∈ ℂ)
92		subneg 11443	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · (∗‘𝐴)) ∈ ℂ) → (1 − -(i · (∗‘𝐴))) = (1 + (i · (∗‘𝐴))))
93	21, 91, 92	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − -(i · (∗‘𝐴))) = (1 + (i · (∗‘𝐴))))
94	76, 89, 93	3eqtrd 2775	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(1 − (i · 𝐴))) = (1 + (i · (∗‘𝐴))))
95	94	fveq2d 6844	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(∗‘(1 − (i · 𝐴)))) = (log‘(1 + (i · (∗‘𝐴)))))
96	74, 95	eqtr3d 2773	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(log‘(1 − (i · 𝐴)))) = (log‘(1 + (i · (∗‘𝐴)))))
97		imadd 15096	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (ℑ‘(1 + (i · 𝐴))) = ((ℑ‘1) + (ℑ‘(i · 𝐴))))
98	21, 23, 97	sylancr 588	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 + (i · 𝐴))) = ((ℑ‘1) + (ℑ‘(i · 𝐴))))
99	60	oveq2d 7383	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (0 + (ℜ‘𝐴)) = (0 + (ℑ‘(i · 𝐴))))
100	64	oveq1i 7377	. . . . . . . . . . . . . 14 ⊢ ((ℑ‘1) + (ℑ‘(i · 𝐴))) = (0 + (ℑ‘(i · 𝐴)))
101	99, 100	eqtr4di 2789	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (0 + (ℜ‘𝐴)) = ((ℑ‘1) + (ℑ‘(i · 𝐴))))
102	70	addlidd 11347	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (0 + (ℜ‘𝐴)) = (ℜ‘𝐴))
103	98, 101, 102	3eqtr2d 2777	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 + (i · 𝐴))) = (ℜ‘𝐴))
104	103, 2	eqnetrd 2999	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 + (i · 𝐴))) ≠ 0)
105		logcj 26570	. . . . . . . . . . 11 ⊢ (((1 + (i · 𝐴)) ∈ ℂ ∧ (ℑ‘(1 + (i · 𝐴))) ≠ 0) → (log‘(∗‘(1 + (i · 𝐴)))) = (∗‘(log‘(1 + (i · 𝐴)))))
106	31, 104, 105	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(∗‘(1 + (i · 𝐴)))) = (∗‘(log‘(1 + (i · 𝐴)))))
107		cjadd 15103	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (∗‘(1 + (i · 𝐴))) = ((∗‘1) + (∗‘(i · 𝐴))))
108	21, 23, 107	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(1 + (i · 𝐴))) = ((∗‘1) + (∗‘(i · 𝐴))))
109	79, 88	oveq12d 7385	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘1) + (∗‘(i · 𝐴))) = (1 + -(i · (∗‘𝐴))))
110		negsub 11442	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · (∗‘𝐴)) ∈ ℂ) → (1 + -(i · (∗‘𝐴))) = (1 − (i · (∗‘𝐴))))
111	21, 91, 110	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + -(i · (∗‘𝐴))) = (1 − (i · (∗‘𝐴))))
112	108, 109, 111	3eqtrd 2775	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(1 + (i · 𝐴))) = (1 − (i · (∗‘𝐴))))
113	112	fveq2d 6844	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(∗‘(1 + (i · 𝐴)))) = (log‘(1 − (i · (∗‘𝐴)))))
114	106, 113	eqtr3d 2773	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(log‘(1 + (i · 𝐴)))) = (log‘(1 − (i · (∗‘𝐴)))))
115	96, 114	oveq12d 7385	. . . . . . . 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 2771	. . . . . . 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 11387	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = -((log‘(1 + (i · (∗‘𝐴)))) − (log‘(1 − (i · (∗‘𝐴))))))
118		addcl 11120	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (i · (∗‘𝐴)) ∈ ℂ) → (1 + (i · (∗‘𝐴))) ∈ ℂ)
119	21, 91, 118	sylancr 588	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + (i · (∗‘𝐴))) ∈ ℂ)
120		atandmcj 26873	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → (∗‘𝐴) ∈ dom arctan)
121	18, 120	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘𝐴) ∈ dom arctan)
122		atandm2 26841	. . . . . . . . . 10 ⊢ ((∗‘𝐴) ∈ dom arctan ↔ ((∗‘𝐴) ∈ ℂ ∧ (1 − (i · (∗‘𝐴))) ≠ 0 ∧ (1 + (i · (∗‘𝐴))) ≠ 0))
123	122	simp3bi 1148	. . . . . . . . 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 26534	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(1 + (i · (∗‘𝐴)))) ∈ ℂ)
126		subcl 11392	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (i · (∗‘𝐴)) ∈ ℂ) → (1 − (i · (∗‘𝐴))) ∈ ℂ)
127	21, 91, 126	sylancr 588	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − (i · (∗‘𝐴))) ∈ ℂ)
128	122	simp2bi 1147	. . . . . . . . 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 26534	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(1 − (i · (∗‘𝐴)))) ∈ ℂ)
131	125, 130	negsubdi2d 11521	. . . . . 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 2771	. . . . 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 7383	. . . 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 2775	. . 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 26848	. . . . 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 6844	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(arctan‘𝐴)) = (∗‘((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
138		atanval 26848	. . . 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 2781	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(arctan‘𝐴)) = (arctan‘(∗‘𝐴)))
141	18, 140	jca 511	1 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (𝐴 ∈ dom arctan ∧ (∗‘(arctan‘𝐴)) = (arctan‘(∗‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 dom cdm 5631 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 ici 11040 + caddc 11041 · cmul 11043 − cmin 11377 -cneg 11378 / cdiv 11807 2c2 12236 ∗ccj 15058 ℜcre 15059 ℑcim 15060 logclog 26518 arctancatan 26828

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520 df-atan 26831

This theorem is referenced by: atanrecl 26875

W3C validator