Theorem isosctrlem2 24997

Description: Lemma for isosctr 24999. Corresponds to the case where one vertex is at 0, another at 1 and the third lies on the unit circle. (Contributed by Saveliy Skresanov, 31-Dec-2016.)

Assertion

Ref	Expression
isosctrlem2	⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))

Proof of Theorem isosctrlem2

Step	Hyp	Ref	Expression
1		1cnd 10371	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℂ)
2		simpl1 1199	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 𝐴 ∈ ℂ)
3	1, 2	negsubd 10740	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 + -𝐴) = (1 − 𝐴))
4		1rp 12141	. . . . . . . 8 ⊢ 1 ∈ ℝ+
5	4	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℝ+)
6		simpl3 1203	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ¬ 1 = 𝐴)
7		simpl2 1201	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (abs‘𝐴) = 1)
8	1, 2, 1	sub32d 10766	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) = ((1 − 1) − 𝐴))
9		1m1e0 11447	. . . . . . . . . . . . . . . . 17 ⊢ (1 − 1) = 0
10	9	oveq1i 6932	. . . . . . . . . . . . . . . 16 ⊢ ((1 − 1) − 𝐴) = (0 − 𝐴)
11		df-neg 10609	. . . . . . . . . . . . . . . 16 ⊢ -𝐴 = (0 − 𝐴)
12	10, 11	eqtr4i 2805	. . . . . . . . . . . . . . 15 ⊢ ((1 − 1) − 𝐴) = -𝐴
13	8, 12	syl6eq 2830	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) = -𝐴)
14		1cnd 10371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 1 ∈ ℂ)
15		simp1 1127	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 𝐴 ∈ ℂ)
16	14, 15	subcld 10734	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ)
17	16	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℂ)
18		ax-1cn 10330	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℂ
19		subeq0 10649	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
20	18, 19	mpan 680	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
21	20	biimpd 221	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 → 1 = 𝐴))
22	21	con3dimp 399	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → ¬ (1 − 𝐴) = 0)
23	22	neqned 2976	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
24	23	3adant2 1122	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
25	24	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ≠ 0)
26	17, 25	recrecd 11148	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 / (1 − 𝐴))) = (1 − 𝐴))
27	14, 16, 24	div2negd 11166	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / -(1 − 𝐴)) = (1 / (1 − 𝐴)))
28	27	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-1 / -(1 − 𝐴)) = (1 / (1 − 𝐴)))
29	15	negcld 10721	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -𝐴 ∈ ℂ)
30	29, 16, 24	cjdivd 14370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = ((∗‘-𝐴) / (∗‘(1 − 𝐴))))
31	15	cjnegd 14358	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘-𝐴) = -(∗‘𝐴))
32		fveq2 6446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
33		abs0 14432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (abs‘0) = 0
34	32, 33	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0)
35		eqtr2 2800	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((abs‘𝐴) = 1 ∧ (abs‘𝐴) = 0) → 1 = 0)
36	34, 35	sylan2 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((abs‘𝐴) = 1 ∧ 𝐴 = 0) → 1 = 0)
37		ax-1ne0 10341	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 1 ≠ 0
38		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (1 ≠ 0 → 1 ≠ 0)
39	38	neneqd 2974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (1 ≠ 0 → ¬ 1 = 0)
40	37, 39	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((abs‘𝐴) = 1 ∧ 𝐴 = 0) → ¬ 1 = 0)
41	36, 40	pm2.65da 807	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((abs‘𝐴) = 1 → ¬ 𝐴 = 0)
42	41	adantl 475	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ¬ 𝐴 = 0)
43		df-ne 2970	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
44		oveq1 6929	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴)↑2) = (1↑2))
45		sq1 13277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (1↑2) = 1
46	44, 45	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴)↑2) = 1)
47	46	adantl 475	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
48		absvalsq 14427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
49	48	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
50	47, 49	eqtr3d 2816	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 1 = (𝐴 · (∗‘𝐴)))
51	50	3adant3 1123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 1 = (𝐴 · (∗‘𝐴)))
52	51	oveq1d 6937	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (1 / 𝐴) = ((𝐴 · (∗‘𝐴)) / 𝐴))
53		simp1 1127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
54	53	cjcld 14343	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (∗‘𝐴) ∈ ℂ)
55		simp3 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
56	54, 53, 55	divcan3d 11156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
57	52, 56	eqtrd 2814	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (1 / 𝐴) = (∗‘𝐴))
58	43, 57	syl3an3br 1476	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 𝐴 = 0) → (1 / 𝐴) = (∗‘𝐴))
59	42, 58	mpd3an3 1535	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 / 𝐴) = (∗‘𝐴))
60	59	eqcomd 2784	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘𝐴) = (1 / 𝐴))
61	60	3adant3 1123	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘𝐴) = (1 / 𝐴))
62	61	negeqd 10616	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -(∗‘𝐴) = -(1 / 𝐴))
63	31, 62	eqtrd 2814	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘-𝐴) = -(1 / 𝐴))
64	63	oveq1d 6937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((∗‘-𝐴) / (∗‘(1 − 𝐴))) = (-(1 / 𝐴) / (∗‘(1 − 𝐴))))
65		cjsub 14296	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (∗‘(1 − 𝐴)) = ((∗‘1) − (∗‘𝐴)))
66	18, 65	mpan 680	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 ∈ ℂ → (∗‘(1 − 𝐴)) = ((∗‘1) − (∗‘𝐴)))
67		1red 10377	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℝ)
68	67	cjred 14373	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐴 ∈ ℂ → (∗‘1) = 1)
69	68	oveq1d 6937	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 ∈ ℂ → ((∗‘1) − (∗‘𝐴)) = (1 − (∗‘𝐴)))
70	66, 69	eqtrd 2814	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 ∈ ℂ → (∗‘(1 − 𝐴)) = (1 − (∗‘𝐴)))
71	70	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘(1 − 𝐴)) = (1 − (∗‘𝐴)))
72	60	oveq2d 6938	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 − (∗‘𝐴)) = (1 − (1 / 𝐴)))
73	71, 72	eqtrd 2814	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘(1 − 𝐴)) = (1 − (1 / 𝐴)))
74	73	3adant3 1123	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(1 − 𝐴)) = (1 − (1 / 𝐴)))
75	74	oveq2d 6938	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (∗‘(1 − 𝐴))) = (-(1 / 𝐴) / (1 − (1 / 𝐴))))
76	30, 64, 75	3eqtrd 2818	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = (-(1 / 𝐴) / (1 − (1 / 𝐴))))
77	41	3ad2ant2 1125	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ¬ 𝐴 = 0)
78	77	neqned 2976	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 𝐴 ≠ 0)
79		1cnd 10371	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 1 ∈ ℂ)
80		simpl 476	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
81		simpr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
82	79, 80, 81	divnegd 11164	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -(1 / 𝐴) = (-1 / 𝐴))
83	82	oveq1d 6937	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
84	15, 78, 83	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
85	14	negcld 10721	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -1 ∈ ℂ)
86	85, 15, 78	divcld 11151	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / 𝐴) ∈ ℂ)
87	15, 78	reccld 11144	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / 𝐴) ∈ ℂ)
88	14, 87	subcld 10734	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − (1 / 𝐴)) ∈ ℂ)
89	16, 24	cjne0d 14350	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(1 − 𝐴)) ≠ 0)
90	74, 89	eqnetrrd 3037	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − (1 / 𝐴)) ≠ 0)
91	86, 88, 15, 90, 78	divcan5d 11177	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · (-1 / 𝐴)) / (𝐴 · (1 − (1 / 𝐴)))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
92	85, 15, 78	divcan2d 11153	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (-1 / 𝐴)) = -1)
93	15, 14, 87	subdid 10831	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 − (1 / 𝐴))) = ((𝐴 · 1) − (𝐴 · (1 / 𝐴))))
94	15	mulid1d 10394	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · 1) = 𝐴)
95	15, 78	recidd 11146	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 / 𝐴)) = 1)
96	94, 95	oveq12d 6940	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · 1) − (𝐴 · (1 / 𝐴))) = (𝐴 − 1))
97	93, 96	eqtrd 2814	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 − (1 / 𝐴))) = (𝐴 − 1))
98	92, 97	oveq12d 6940	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · (-1 / 𝐴)) / (𝐴 · (1 − (1 / 𝐴)))) = (-1 / (𝐴 − 1)))
99	84, 91, 98	3eqtr2d 2820	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = (-1 / (𝐴 − 1)))
100		subcl 10621	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
101	100	negnegd 10725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → --(𝐴 − 1) = (𝐴 − 1))
102		negsubdi2 10682	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (1 − 𝐴))
103	102	negeqd 10616	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → --(𝐴 − 1) = -(1 − 𝐴))
104	101, 103	eqtr3d 2816	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) = -(1 − 𝐴))
105	15, 14, 104	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 − 1) = -(1 − 𝐴))
106	105	oveq2d 6938	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / (𝐴 − 1)) = (-1 / -(1 − 𝐴)))
107	76, 99, 106	3eqtrd 2818	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = (-1 / -(1 − 𝐴)))
108	107	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) = (-1 / -(1 − 𝐴)))
109	29, 16, 24	divcld 11151	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-𝐴 / (1 − 𝐴)) ∈ ℂ)
110	109	adantr 474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℂ)
111		simpr 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(-𝐴 / (1 − 𝐴))) = 0)
112	110, 111	reim0bd 14347	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℝ)
113	112	cjred 14373	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) = (-𝐴 / (1 − 𝐴)))
114	113, 112	eqeltrd 2859	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) ∈ ℝ)
115	108, 114	eqeltrrd 2860	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-1 / -(1 − 𝐴)) ∈ ℝ)
116	28, 115	eqeltrrd 2860	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 − 𝐴)) ∈ ℝ)
117	16, 24	recne0d 11145	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / (1 − 𝐴)) ≠ 0)
118	117	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 − 𝐴)) ≠ 0)
119	116, 118	rereccld 11202	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 / (1 − 𝐴))) ∈ ℝ)
120	26, 119	eqeltrrd 2860	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℝ)
121		1red 10377	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℝ)
122	120, 121	resubcld 10803	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) ∈ ℝ)
123	13, 122	eqeltrrd 2860	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → -𝐴 ∈ ℝ)
124	2, 123	negrebd 10733	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 𝐴 ∈ ℝ)
125	124	absord 14562	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))
126		eqeq1 2782	. . . . . . . . . . . . 13 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴) = 𝐴 ↔ 1 = 𝐴))
127	126	biimpd 221	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴) = 𝐴 → 1 = 𝐴))
128		eqeq1 2782	. . . . . . . . . . . . 13 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴) = -𝐴 ↔ 1 = -𝐴))
129	128	biimpd 221	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴) = -𝐴 → 1 = -𝐴))
130	127, 129	orim12d 950	. . . . . . . . . . 11 ⊢ ((abs‘𝐴) = 1 → (((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴) → (1 = 𝐴 ∨ 1 = -𝐴)))
131	7, 125, 130	sylc 65	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 = 𝐴 ∨ 1 = -𝐴))
132	131	ord 853	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (¬ 1 = 𝐴 → 1 = -𝐴))
133	6, 132	mpd 15	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 = -𝐴)
134	133, 5	eqeltrrd 2860	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → -𝐴 ∈ ℝ+)
135	5, 134	rpaddcld 12196	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 + -𝐴) ∈ ℝ+)
136	3, 135	eqeltrrd 2860	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℝ+)
137	136	relogcld 24806	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (log‘(1 − 𝐴)) ∈ ℝ)
138	137	reim0d 14372	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(1 − 𝐴))) = 0)
139	134, 136	rpdivcld 12198	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℝ+)
140	139	relogcld 24806	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (log‘(-𝐴 / (1 − 𝐴))) ∈ ℝ)
141	140	reim0d 14372	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) = 0)
142	138, 141	eqtr4d 2817	. 2 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
143	16, 24	logcld 24754	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) ∈ ℂ)
144	143	adantr 474	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(1 − 𝐴)) ∈ ℂ)
145	144	imcld 14342	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
146	145	recnd 10405	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℂ)
147	109	adantr 474	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (-𝐴 / (1 − 𝐴)) ∈ ℂ)
148	15, 78	negne0d 10732	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -𝐴 ≠ 0)
149	29, 16, 148, 24	divne0d 11167	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-𝐴 / (1 − 𝐴)) ≠ 0)
150	149	adantr 474	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (-𝐴 / (1 − 𝐴)) ≠ 0)
151	147, 150	logcld 24754	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(-𝐴 / (1 − 𝐴))) ∈ ℂ)
152	151	imcld 14342	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) ∈ ℝ)
153	152	recnd 10405	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) ∈ ℂ)
154	107	fveq2d 6450	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (log‘(-1 / -(1 − 𝐴))))
155	154	adantr 474	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (log‘(-1 / -(1 − 𝐴))))
156		logcj 24789	. . . . . . 7 ⊢ (((-𝐴 / (1 − 𝐴)) ∈ ℂ ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
157	109, 156	sylan 575	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
158	16, 24	reccld 11144	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / (1 − 𝐴)) ∈ ℂ)
159	158, 117	logcld 24754	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 / (1 − 𝐴))) ∈ ℂ)
160	159	negnegd 10725	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → --(log‘(1 / (1 − 𝐴))) = (log‘(1 / (1 − 𝐴))))
161		isosctrlem1 24996	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)
162		logrec 24941	. . . . . . . . . 10 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ (ℑ‘(log‘(1 − 𝐴))) ≠ π) → (log‘(1 − 𝐴)) = -(log‘(1 / (1 − 𝐴))))
163	16, 24, 161, 162	syl3anc 1439	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) = -(log‘(1 / (1 − 𝐴))))
164	163	negeqd 10616	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -(log‘(1 − 𝐴)) = --(log‘(1 / (1 − 𝐴))))
165	27	fveq2d 6450	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(-1 / -(1 − 𝐴))) = (log‘(1 / (1 − 𝐴))))
166	160, 164, 165	3eqtr4rd 2825	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(-1 / -(1 − 𝐴))) = -(log‘(1 − 𝐴)))
167	166	adantr 474	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(-1 / -(1 − 𝐴))) = -(log‘(1 − 𝐴)))
168	155, 157, 167	3eqtr3rd 2823	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → -(log‘(1 − 𝐴)) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
169	168	fveq2d 6450	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘-(log‘(1 − 𝐴))) = (ℑ‘(∗‘(log‘(-𝐴 / (1 − 𝐴))))))
170	144	imnegd 14357	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘-(log‘(1 − 𝐴))) = -(ℑ‘(log‘(1 − 𝐴))))
171	151	imcjd 14352	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(∗‘(log‘(-𝐴 / (1 − 𝐴))))) = -(ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
172	169, 170, 171	3eqtr3d 2822	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → -(ℑ‘(log‘(1 − 𝐴))) = -(ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
173	146, 153, 172	neg11d 10746	. 2 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
174	142, 173	pm2.61dane 3057	1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ‘cfv 6135  (class class class)co 6922  ℂcc 10270  ℝcr 10271  0cc0 10272  1c1 10273   + caddc 10275   · cmul 10277   − cmin 10606  -cneg 10607   / cdiv 11032  2c2 11430  ℝ⁺crp 12137  ↑cexp 13178  ∗ccj 14243  ℑcim 14245  abscabs 14381  πcpi 15199  logclog 24738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350  ax-addf 10351  ax-mulf 10352

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-fi 8605  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-q 12096  df-rp 12138  df-xneg 12257  df-xadd 12258  df-xmul 12259  df-ioo 12491  df-ioc 12492  df-ico 12493  df-icc 12494  df-fz 12644  df-fzo 12785  df-fl 12912  df-mod 12988  df-seq 13120  df-exp 13179  df-fac 13379  df-bc 13408  df-hash 13436  df-shft 14214  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-limsup 14610  df-clim 14627  df-rlim 14628  df-sum 14825  df-ef 15200  df-sin 15202  df-cos 15203  df-pi 15205  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-mulr 16352  df-starv 16353  df-sca 16354  df-vsca 16355  df-ip 16356  df-tset 16357  df-ple 16358  df-ds 16360  df-unif 16361  df-hom 16362  df-cco 16363  df-rest 16469  df-topn 16470  df-0g 16488  df-gsum 16489  df-topgen 16490  df-pt 16491  df-prds 16494  df-xrs 16548  df-qtop 16553  df-imas 16554  df-xps 16556  df-mre 16632  df-mrc 16633  df-acs 16635  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-submnd 17722  df-mulg 17928  df-cntz 18133  df-cmn 18581  df-psmet 20134  df-xmet 20135  df-met 20136  df-bl 20137  df-mopn 20138  df-fbas 20139  df-fg 20140  df-cnfld 20143  df-top 21106  df-topon 21123  df-topsp 21145  df-bases 21158  df-cld 21231  df-ntr 21232  df-cls 21233  df-nei 21310  df-lp 21348  df-perf 21349  df-cn 21439  df-cnp 21440  df-haus 21527  df-tx 21774  df-hmeo 21967  df-fil 22058  df-fm 22150  df-flim 22151  df-flf 22152  df-xms 22533  df-ms 22534  df-tms 22535  df-cncf 23089  df-limc 24067  df-dv 24068  df-log 24740

This theorem is referenced by:  isosctrlem3  24998