Theorem isosctrlem2 26796

Description: Lemma for isosctr 26798. 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 11130	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℂ)
2		simpl1 1193	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 𝐴 ∈ ℂ)
3	1, 2	negsubd 11502	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 + -𝐴) = (1 − 𝐴))
4		1rp 12937	. . . . . . . 8 ⊢ 1 ∈ ℝ+
5	4	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℝ+)
6		simpl3 1195	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ¬ 1 = 𝐴)
7		simpl2 1194	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (abs‘𝐴) = 1)
8	1, 2, 1	sub32d 11528	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) = ((1 − 1) − 𝐴))
9		1m1e0 12244	. . . . . . . . . . . . . . . . 17 ⊢ (1 − 1) = 0
10	9	oveq1i 7370	. . . . . . . . . . . . . . . 16 ⊢ ((1 − 1) − 𝐴) = (0 − 𝐴)
11		df-neg 11371	. . . . . . . . . . . . . . . 16 ⊢ -𝐴 = (0 − 𝐴)
12	10, 11	eqtr4i 2763	. . . . . . . . . . . . . . 15 ⊢ ((1 − 1) − 𝐴) = -𝐴
13	8, 12	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) = -𝐴)
14		1cnd 11130	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 1 ∈ ℂ)
15		simp1 1137	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 𝐴 ∈ ℂ)
16	14, 15	subcld 11496	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ)
17	16	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℂ)
18		ax-1cn 11087	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℂ
19		subeq0 11411	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
20	18, 19	mpan 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
21	20	biimpd 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 → 1 = 𝐴))
22	21	con3dimp 408	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → ¬ (1 − 𝐴) = 0)
23	22	neqned 2940	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
24	23	3adant2 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
25	24	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ≠ 0)
26	17, 25	recrecd 11919	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 / (1 − 𝐴))) = (1 − 𝐴))
27	14, 16, 24	div2negd 11937	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / -(1 − 𝐴)) = (1 / (1 − 𝐴)))
28	27	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-1 / -(1 − 𝐴)) = (1 / (1 − 𝐴)))
29	15	negcld 11483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -𝐴 ∈ ℂ)
30	29, 16, 24	cjdivd 15176	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = ((∗‘-𝐴) / (∗‘(1 − 𝐴))))
31	15	cjnegd 15164	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘-𝐴) = -(∗‘𝐴))
32		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
33		abs0 15238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (abs‘0) = 0
34	32, 33	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0)
35		eqtr2 2758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((abs‘𝐴) = 1 ∧ (abs‘𝐴) = 0) → 1 = 0)
36	34, 35	sylan2 594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((abs‘𝐴) = 1 ∧ 𝐴 = 0) → 1 = 0)
37		ax-1ne0 11098	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 1 ≠ 0
38		neneq 2939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (1 ≠ 0 → ¬ 1 = 0)
39	37, 38	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((abs‘𝐴) = 1 ∧ 𝐴 = 0) → ¬ 1 = 0)
40	36, 39	pm2.65da 817	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((abs‘𝐴) = 1 → ¬ 𝐴 = 0)
41	40	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ¬ 𝐴 = 0)
42		df-ne 2934	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
43		oveq1 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴)↑2) = (1↑2))
44		sq1 14148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (1↑2) = 1
45	43, 44	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴)↑2) = 1)
46	45	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
47		absvalsq 15233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
48	47	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
49	46, 48	eqtr3d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 1 = (𝐴 · (∗‘𝐴)))
50	49	3adant3 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 1 = (𝐴 · (∗‘𝐴)))
51	50	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (1 / 𝐴) = ((𝐴 · (∗‘𝐴)) / 𝐴))
52		simp1 1137	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
53	52	cjcld 15149	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (∗‘𝐴) ∈ ℂ)
54		simp3 1139	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
55	53, 52, 54	divcan3d 11927	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
56	51, 55	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (1 / 𝐴) = (∗‘𝐴))
57	42, 56	syl3an3br 1411	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 𝐴 = 0) → (1 / 𝐴) = (∗‘𝐴))
58	41, 57	mpd3an3 1465	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 / 𝐴) = (∗‘𝐴))
59	58	eqcomd 2743	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘𝐴) = (1 / 𝐴))
60	59	3adant3 1133	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘𝐴) = (1 / 𝐴))
61	60	negeqd 11378	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -(∗‘𝐴) = -(1 / 𝐴))
62	31, 61	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘-𝐴) = -(1 / 𝐴))
63	62	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((∗‘-𝐴) / (∗‘(1 − 𝐴))) = (-(1 / 𝐴) / (∗‘(1 − 𝐴))))
64		cjsub 15102	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (∗‘(1 − 𝐴)) = ((∗‘1) − (∗‘𝐴)))
65	18, 64	mpan 691	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 ∈ ℂ → (∗‘(1 − 𝐴)) = ((∗‘1) − (∗‘𝐴)))
66		1red 11136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℝ)
67	66	cjred 15179	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐴 ∈ ℂ → (∗‘1) = 1)
68	67	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 ∈ ℂ → ((∗‘1) − (∗‘𝐴)) = (1 − (∗‘𝐴)))
69	65, 68	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 ∈ ℂ → (∗‘(1 − 𝐴)) = (1 − (∗‘𝐴)))
70	69	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘(1 − 𝐴)) = (1 − (∗‘𝐴)))
71	59	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 − (∗‘𝐴)) = (1 − (1 / 𝐴)))
72	70, 71	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘(1 − 𝐴)) = (1 − (1 / 𝐴)))
73	72	3adant3 1133	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(1 − 𝐴)) = (1 − (1 / 𝐴)))
74	73	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (∗‘(1 − 𝐴))) = (-(1 / 𝐴) / (1 − (1 / 𝐴))))
75	30, 63, 74	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = (-(1 / 𝐴) / (1 − (1 / 𝐴))))
76	40	3ad2ant2 1135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ¬ 𝐴 = 0)
77	76	neqned 2940	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 𝐴 ≠ 0)
78		1cnd 11130	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 1 ∈ ℂ)
79		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
80		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
81	78, 79, 80	divnegd 11935	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -(1 / 𝐴) = (-1 / 𝐴))
82	81	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
83	15, 77, 82	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
84	14	negcld 11483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -1 ∈ ℂ)
85	84, 15, 77	divcld 11922	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / 𝐴) ∈ ℂ)
86	15, 77	reccld 11915	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / 𝐴) ∈ ℂ)
87	14, 86	subcld 11496	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − (1 / 𝐴)) ∈ ℂ)
88	16, 24	cjne0d 15156	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(1 − 𝐴)) ≠ 0)
89	73, 88	eqnetrrd 3001	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − (1 / 𝐴)) ≠ 0)
90	85, 87, 15, 89, 77	divcan5d 11948	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · (-1 / 𝐴)) / (𝐴 · (1 − (1 / 𝐴)))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
91	84, 15, 77	divcan2d 11924	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (-1 / 𝐴)) = -1)
92	15, 14, 86	subdid 11597	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 − (1 / 𝐴))) = ((𝐴 · 1) − (𝐴 · (1 / 𝐴))))
93	15	mulridd 11153	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · 1) = 𝐴)
94	15, 77	recidd 11917	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 / 𝐴)) = 1)
95	93, 94	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · 1) − (𝐴 · (1 / 𝐴))) = (𝐴 − 1))
96	92, 95	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 − (1 / 𝐴))) = (𝐴 − 1))
97	91, 96	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · (-1 / 𝐴)) / (𝐴 · (1 − (1 / 𝐴)))) = (-1 / (𝐴 − 1)))
98	83, 90, 97	3eqtr2d 2778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = (-1 / (𝐴 − 1)))
99		subcl 11383	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
100	99	negnegd 11487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → --(𝐴 − 1) = (𝐴 − 1))
101		negsubdi2 11444	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (1 − 𝐴))
102	101	negeqd 11378	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → --(𝐴 − 1) = -(1 − 𝐴))
103	100, 102	eqtr3d 2774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) = -(1 − 𝐴))
104	15, 14, 103	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 − 1) = -(1 − 𝐴))
105	104	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / (𝐴 − 1)) = (-1 / -(1 − 𝐴)))
106	75, 98, 105	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = (-1 / -(1 − 𝐴)))
107	106	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) = (-1 / -(1 − 𝐴)))
108	29, 16, 24	divcld 11922	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-𝐴 / (1 − 𝐴)) ∈ ℂ)
109	108	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℂ)
110		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(-𝐴 / (1 − 𝐴))) = 0)
111	109, 110	reim0bd 15153	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℝ)
112	111	cjred 15179	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) = (-𝐴 / (1 − 𝐴)))
113	112, 111	eqeltrd 2837	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) ∈ ℝ)
114	107, 113	eqeltrrd 2838	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-1 / -(1 − 𝐴)) ∈ ℝ)
115	28, 114	eqeltrrd 2838	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 − 𝐴)) ∈ ℝ)
116	16, 24	recne0d 11916	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / (1 − 𝐴)) ≠ 0)
117	116	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 − 𝐴)) ≠ 0)
118	115, 117	rereccld 11973	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 / (1 − 𝐴))) ∈ ℝ)
119	26, 118	eqeltrrd 2838	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℝ)
120		1red 11136	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℝ)
121	119, 120	resubcld 11569	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) ∈ ℝ)
122	13, 121	eqeltrrd 2838	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → -𝐴 ∈ ℝ)
123	2, 122	negrebd 11495	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 𝐴 ∈ ℝ)
124	123	absord 15369	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))
125		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴) = 𝐴 ↔ 1 = 𝐴))
126	125	biimpd 229	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴) = 𝐴 → 1 = 𝐴))
127		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴) = -𝐴 ↔ 1 = -𝐴))
128	127	biimpd 229	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴) = -𝐴 → 1 = -𝐴))
129	126, 128	orim12d 967	. . . . . . . . . . 11 ⊢ ((abs‘𝐴) = 1 → (((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴) → (1 = 𝐴 ∨ 1 = -𝐴)))
130	7, 124, 129	sylc 65	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 = 𝐴 ∨ 1 = -𝐴))
131	130	ord 865	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (¬ 1 = 𝐴 → 1 = -𝐴))
132	6, 131	mpd 15	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 = -𝐴)
133	132, 5	eqeltrrd 2838	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → -𝐴 ∈ ℝ+)
134	5, 133	rpaddcld 12992	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 + -𝐴) ∈ ℝ+)
135	3, 134	eqeltrrd 2838	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℝ+)
136	135	relogcld 26600	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (log‘(1 − 𝐴)) ∈ ℝ)
137	136	reim0d 15178	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(1 − 𝐴))) = 0)
138	133, 135	rpdivcld 12994	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℝ+)
139	138	relogcld 26600	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (log‘(-𝐴 / (1 − 𝐴))) ∈ ℝ)
140	139	reim0d 15178	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) = 0)
141	137, 140	eqtr4d 2775	. 2 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
142	16, 24	logcld 26547	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) ∈ ℂ)
143	142	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(1 − 𝐴)) ∈ ℂ)
144	143	imcld 15148	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
145	144	recnd 11164	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℂ)
146	108	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (-𝐴 / (1 − 𝐴)) ∈ ℂ)
147	15, 77	negne0d 11494	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -𝐴 ≠ 0)
148	29, 16, 147, 24	divne0d 11938	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-𝐴 / (1 − 𝐴)) ≠ 0)
149	148	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (-𝐴 / (1 − 𝐴)) ≠ 0)
150	146, 149	logcld 26547	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(-𝐴 / (1 − 𝐴))) ∈ ℂ)
151	150	imcld 15148	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) ∈ ℝ)
152	151	recnd 11164	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) ∈ ℂ)
153	106	fveq2d 6838	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (log‘(-1 / -(1 − 𝐴))))
154	153	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (log‘(-1 / -(1 − 𝐴))))
155		logcj 26583	. . . . . . 7 ⊢ (((-𝐴 / (1 − 𝐴)) ∈ ℂ ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
156	108, 155	sylan 581	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
157	16, 24	reccld 11915	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / (1 − 𝐴)) ∈ ℂ)
158	157, 116	logcld 26547	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 / (1 − 𝐴))) ∈ ℂ)
159	158	negnegd 11487	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → --(log‘(1 / (1 − 𝐴))) = (log‘(1 / (1 − 𝐴))))
160		isosctrlem1 26795	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)
161		logrec 26740	. . . . . . . . . 10 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ (ℑ‘(log‘(1 − 𝐴))) ≠ π) → (log‘(1 − 𝐴)) = -(log‘(1 / (1 − 𝐴))))
162	16, 24, 160, 161	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) = -(log‘(1 / (1 − 𝐴))))
163	162	negeqd 11378	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -(log‘(1 − 𝐴)) = --(log‘(1 / (1 − 𝐴))))
164	27	fveq2d 6838	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(-1 / -(1 − 𝐴))) = (log‘(1 / (1 − 𝐴))))
165	159, 163, 164	3eqtr4rd 2783	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(-1 / -(1 − 𝐴))) = -(log‘(1 − 𝐴)))
166	165	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(-1 / -(1 − 𝐴))) = -(log‘(1 − 𝐴)))
167	154, 156, 166	3eqtr3rd 2781	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → -(log‘(1 − 𝐴)) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
168	167	fveq2d 6838	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘-(log‘(1 − 𝐴))) = (ℑ‘(∗‘(log‘(-𝐴 / (1 − 𝐴))))))
169	143	imnegd 15163	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘-(log‘(1 − 𝐴))) = -(ℑ‘(log‘(1 − 𝐴))))
170	150	imcjd 15158	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(∗‘(log‘(-𝐴 / (1 − 𝐴))))) = -(ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
171	168, 169, 170	3eqtr3d 2780	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → -(ℑ‘(log‘(1 − 𝐴))) = -(ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
172	145, 152, 171	neg11d 11508	. 2 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))
173	141, 172	pm2.61dane 3020	1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) = (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ‘cfv 6492  (class class class)co 7360  ℂcc 11027  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   · cmul 11034   − cmin 11368  -cneg 11369   / cdiv 11798  2c2 12227  ℝ⁺crp 12933  ↑cexp 14014  ∗ccj 15049  ℑcim 15051  abscabs 15187  πcpi 16022  logclog 26531

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107  ax-addf 11108

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-pm 8769  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-fi 9317  df-sup 9348  df-inf 9349  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-q 12890  df-rp 12934  df-xneg 13054  df-xadd 13055  df-xmul 13056  df-ioo 13293  df-ioc 13294  df-ico 13295  df-icc 13296  df-fz 13453  df-fzo 13600  df-fl 13742  df-mod 13820  df-seq 13955  df-exp 14015  df-fac 14227  df-bc 14256  df-hash 14284  df-shft 15020  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-limsup 15424  df-clim 15441  df-rlim 15442  df-sum 15640  df-ef 16023  df-sin 16025  df-cos 16026  df-pi 16028  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-starv 17226  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-unif 17234  df-hom 17235  df-cco 17236  df-rest 17376  df-topn 17377  df-0g 17395  df-gsum 17396  df-topgen 17397  df-pt 17398  df-prds 17401  df-xrs 17457  df-qtop 17462  df-imas 17463  df-xps 17465  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18743  df-mulg 19035  df-cntz 19283  df-cmn 19748  df-psmet 21336  df-xmet 21337  df-met 21338  df-bl 21339  df-mopn 21340  df-fbas 21341  df-fg 21342  df-cnfld 21345  df-top 22869  df-topon 22886  df-topsp 22908  df-bases 22921  df-cld 22994  df-ntr 22995  df-cls 22996  df-nei 23073  df-lp 23111  df-perf 23112  df-cn 23202  df-cnp 23203  df-haus 23290  df-tx 23537  df-hmeo 23730  df-fil 23821  df-fm 23913  df-flim 23914  df-flf 23915  df-xms 24295  df-ms 24296  df-tms 24297  df-cncf 24855  df-limc 25843  df-dv 25844  df-log 26533

This theorem is referenced by:  isosctrlem3  26797