Theorem isosctrlem2 26786

Description: Lemma for isosctr 26788. 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 11235	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℂ)
2		simpl1 1192	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 𝐴 ∈ ℂ)
3	1, 2	negsubd 11605	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 + -𝐴) = (1 − 𝐴))
4		1rp 13017	. . . . . . . 8 ⊢ 1 ∈ ℝ+
5	4	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℝ+)
6		simpl3 1194	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ¬ 1 = 𝐴)
7		simpl2 1193	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (abs‘𝐴) = 1)
8	1, 2, 1	sub32d 11631	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) = ((1 − 1) − 𝐴))
9		1m1e0 12317	. . . . . . . . . . . . . . . . 17 ⊢ (1 − 1) = 0
10	9	oveq1i 7420	. . . . . . . . . . . . . . . 16 ⊢ ((1 − 1) − 𝐴) = (0 − 𝐴)
11		df-neg 11474	. . . . . . . . . . . . . . . 16 ⊢ -𝐴 = (0 − 𝐴)
12	10, 11	eqtr4i 2762	. . . . . . . . . . . . . . 15 ⊢ ((1 − 1) − 𝐴) = -𝐴
13	8, 12	eqtrdi 2787	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) = -𝐴)
14		1cnd 11235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 1 ∈ ℂ)
15		simp1 1136	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 𝐴 ∈ ℂ)
16	14, 15	subcld 11599	. . . . . . . . . . . . . . . . . 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 11192	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℂ
19		subeq0 11514	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
20	18, 19	mpan 690	. . . . . . . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . . . 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 12019	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 / (1 − 𝐴))) = (1 − 𝐴))
27	14, 16, 24	div2negd 12037	. . . . . . . . . . . . . . . . . . 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 11586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -𝐴 ∈ ℂ)
30	29, 16, 24	cjdivd 15247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = ((∗‘-𝐴) / (∗‘(1 − 𝐴))))
31	15	cjnegd 15235	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘-𝐴) = -(∗‘𝐴))
32		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
33		abs0 15309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (abs‘0) = 0
34	32, 33	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0)
35		eqtr2 2757	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((abs‘𝐴) = 1 ∧ (abs‘𝐴) = 0) → 1 = 0)
36	34, 35	sylan2 593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((abs‘𝐴) = 1 ∧ 𝐴 = 0) → 1 = 0)
37		ax-1ne0 11203	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 816	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴)↑2) = (1↑2))
44		sq1 14218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (1↑2) = 1
45	43, 44	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴)↑2) = 1)
46	45	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
47		absvalsq 15304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 1 = (𝐴 · (∗‘𝐴)))
50	49	3adant3 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 1 = (𝐴 · (∗‘𝐴)))
51	50	oveq1d 7425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (1 / 𝐴) = ((𝐴 · (∗‘𝐴)) / 𝐴))
52		simp1 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
53	52	cjcld 15220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (∗‘𝐴) ∈ ℂ)
54		simp3 1138	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
55	53, 52, 54	divcan3d 12027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
56	51, 55	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ 𝐴 ≠ 0) → (1 / 𝐴) = (∗‘𝐴))
57	42, 56	syl3an3br 1410	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 𝐴 = 0) → (1 / 𝐴) = (∗‘𝐴))
58	41, 57	mpd3an3 1464	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 / 𝐴) = (∗‘𝐴))
59	58	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘𝐴) = (1 / 𝐴))
60	59	3adant3 1132	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘𝐴) = (1 / 𝐴))
61	60	negeqd 11481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -(∗‘𝐴) = -(1 / 𝐴))
62	31, 61	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘-𝐴) = -(1 / 𝐴))
63	62	oveq1d 7425	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((∗‘-𝐴) / (∗‘(1 − 𝐴))) = (-(1 / 𝐴) / (∗‘(1 − 𝐴))))
64		cjsub 15173	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (∗‘(1 − 𝐴)) = ((∗‘1) − (∗‘𝐴)))
65	18, 64	mpan 690	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 ∈ ℂ → (∗‘(1 − 𝐴)) = ((∗‘1) − (∗‘𝐴)))
66		1red 11241	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℝ)
67	66	cjred 15250	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐴 ∈ ℂ → (∗‘1) = 1)
68	67	oveq1d 7425	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 ∈ ℂ → ((∗‘1) − (∗‘𝐴)) = (1 − (∗‘𝐴)))
69	65, 68	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 ∈ ℂ → (∗‘(1 − 𝐴)) = (1 − (∗‘𝐴)))
70	69	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘(1 − 𝐴)) = (1 − (∗‘𝐴)))
71	59	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 − (∗‘𝐴)) = (1 − (1 / 𝐴)))
72	70, 71	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (∗‘(1 − 𝐴)) = (1 − (1 / 𝐴)))
73	72	3adant3 1132	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(1 − 𝐴)) = (1 − (1 / 𝐴)))
74	73	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (∗‘(1 − 𝐴))) = (-(1 / 𝐴) / (1 − (1 / 𝐴))))
75	30, 63, 74	3eqtrd 2775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (∗‘(-𝐴 / (1 − 𝐴))) = (-(1 / 𝐴) / (1 − (1 / 𝐴))))
76	40	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . . . . . . 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 11235	. . . . . . . . . . . . . . . . . . . . . . . . 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 12035	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -(1 / 𝐴) = (-1 / 𝐴))
82	81	oveq1d 7425	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
83	15, 77, 82	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = ((-1 / 𝐴) / (1 − (1 / 𝐴))))
84	14	negcld 11586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -1 ∈ ℂ)
85	84, 15, 77	divcld 12022	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / 𝐴) ∈ ℂ)
86	15, 77	reccld 12015	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / 𝐴) ∈ ℂ)
87	14, 86	subcld 11599	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − (1 / 𝐴)) ∈ ℂ)
88	16, 24	cjne0d 15227	. . . . . . . . . . . . . . . . . . . . . . . 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 12048	. . . . . . . . . . . . . . . . . . . . . 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 12024	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (-1 / 𝐴)) = -1)
92	15, 14, 86	subdid 11698	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 − (1 / 𝐴))) = ((𝐴 · 1) − (𝐴 · (1 / 𝐴))))
93	15	mulridd 11257	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · 1) = 𝐴)
94	15, 77	recidd 12017	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 / 𝐴)) = 1)
95	93, 94	oveq12d 7428	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · 1) − (𝐴 · (1 / 𝐴))) = (𝐴 − 1))
96	92, 95	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 · (1 − (1 / 𝐴))) = (𝐴 − 1))
97	91, 96	oveq12d 7428	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((𝐴 · (-1 / 𝐴)) / (𝐴 · (1 − (1 / 𝐴)))) = (-1 / (𝐴 − 1)))
98	83, 90, 97	3eqtr2d 2777	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-(1 / 𝐴) / (1 − (1 / 𝐴))) = (-1 / (𝐴 − 1)))
99		subcl 11486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
100	99	negnegd 11590	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → --(𝐴 − 1) = (𝐴 − 1))
101		negsubdi2 11547	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (1 − 𝐴))
102	101	negeqd 11481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → --(𝐴 − 1) = -(1 − 𝐴))
103	100, 102	eqtr3d 2773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) = -(1 − 𝐴))
104	15, 14, 103	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (𝐴 − 1) = -(1 − 𝐴))
105	104	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (-1 / (𝐴 − 1)) = (-1 / -(1 − 𝐴)))
106	75, 98, 105	3eqtrd 2775	. . . . . . . . . . . . . . . . . . . 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 12022	. . . . . . . . . . . . . . . . . . . . . . 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 15224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℝ)
112	111	cjred 15250	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) = (-𝐴 / (1 − 𝐴)))
113	112, 111	eqeltrd 2835	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (∗‘(-𝐴 / (1 − 𝐴))) ∈ ℝ)
114	107, 113	eqeltrrd 2836	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-1 / -(1 − 𝐴)) ∈ ℝ)
115	28, 114	eqeltrrd 2836	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 − 𝐴)) ∈ ℝ)
116	16, 24	recne0d 12016	. . . . . . . . . . . . . . . . . 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 12073	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 / (1 / (1 − 𝐴))) ∈ ℝ)
119	26, 118	eqeltrrd 2836	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℝ)
120		1red 11241	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 1 ∈ ℝ)
121	119, 120	resubcld 11670	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((1 − 𝐴) − 1) ∈ ℝ)
122	13, 121	eqeltrrd 2836	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → -𝐴 ∈ ℝ)
123	2, 122	negrebd 11598	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → 𝐴 ∈ ℝ)
124	123	absord 15439	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))
125		eqeq1 2740	. . . . . . . . . . . . 13 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴) = 𝐴 ↔ 1 = 𝐴))
126	125	biimpd 229	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴) = 𝐴 → 1 = 𝐴))
127		eqeq1 2740	. . . . . . . . . . . . 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 966	. . . . . . . . . . 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 864	. . . . . . . . 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 2836	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → -𝐴 ∈ ℝ+)
134	5, 133	rpaddcld 13071	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 + -𝐴) ∈ ℝ+)
135	3, 134	eqeltrrd 2836	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (1 − 𝐴) ∈ ℝ+)
136	135	relogcld 26589	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (log‘(1 − 𝐴)) ∈ ℝ)
137	136	reim0d 15249	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(1 − 𝐴))) = 0)
138	133, 135	rpdivcld 13073	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (-𝐴 / (1 − 𝐴)) ∈ ℝ+)
139	138	relogcld 26589	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (log‘(-𝐴 / (1 − 𝐴))) ∈ ℝ)
140	139	reim0d 15249	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) = 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) = 0)
141	137, 140	eqtr4d 2774	. 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 26536	. . . . . 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 15219	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
145	144	recnd 11268	. . 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 11597	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -𝐴 ≠ 0)
148	29, 16, 147, 24	divne0d 12038	. . . . . . 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 26536	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(-𝐴 / (1 − 𝐴))) ∈ ℂ)
151	150	imcld 15219	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) ∈ ℝ)
152	151	recnd 11268	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘(log‘(-𝐴 / (1 − 𝐴)))) ∈ ℂ)
153	106	fveq2d 6885	. . . . . . 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 26572	. . . . . . 7 ⊢ (((-𝐴 / (1 − 𝐴)) ∈ ℂ ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (log‘(∗‘(-𝐴 / (1 − 𝐴)))) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
156	108, 155	sylan 580	. . . . . 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 12015	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 / (1 − 𝐴)) ∈ ℂ)
158	157, 116	logcld 26536	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 / (1 − 𝐴))) ∈ ℂ)
159	158	negnegd 11590	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → --(log‘(1 / (1 − 𝐴))) = (log‘(1 / (1 − 𝐴))))
160		isosctrlem1 26785	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)
161		logrec 26730	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) = -(log‘(1 / (1 − 𝐴))))
163	162	negeqd 11481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → -(log‘(1 − 𝐴)) = --(log‘(1 / (1 − 𝐴))))
164	27	fveq2d 6885	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (log‘(-1 / -(1 − 𝐴))) = (log‘(1 / (1 − 𝐴))))
165	159, 163, 164	3eqtr4rd 2782	. . . . . . 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 2780	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → -(log‘(1 − 𝐴)) = (∗‘(log‘(-𝐴 / (1 − 𝐴)))))
168	167	fveq2d 6885	. . . 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 15234	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) ∧ (ℑ‘(-𝐴 / (1 − 𝐴))) ≠ 0) → (ℑ‘-(log‘(1 − 𝐴))) = -(ℑ‘(log‘(1 − 𝐴))))
170	150	imcjd 15229	. . . 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 2779	. . 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 11611	. 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 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ‘cfv 6536  (class class class)co 7410  ℂcc 11132  ℝcr 11133  0cc0 11134  1c1 11135   + caddc 11137   · cmul 11139   − cmin 11471  -cneg 11472   / cdiv 11899  2c2 12300  ℝ⁺crp 13013  ↑cexp 14084  ∗ccj 15120  ℑcim 15122  abscabs 15258  πcpi 16087  logclog 26520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212  ax-addf 11213

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7676  df-om 7867  df-1st 7993  df-2nd 7994  df-supp 8165  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9379  df-fi 9428  df-sup 9459  df-inf 9460  df-oi 9529  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-q 12970  df-rp 13014  df-xneg 13133  df-xadd 13134  df-xmul 13135  df-ioo 13371  df-ioc 13372  df-ico 13373  df-icc 13374  df-fz 13530  df-fzo 13677  df-fl 13814  df-mod 13892  df-seq 14025  df-exp 14085  df-fac 14297  df-bc 14326  df-hash 14354  df-shft 15091  df-cj 15123  df-re 15124  df-im 15125  df-sqrt 15259  df-abs 15260  df-limsup 15492  df-clim 15509  df-rlim 15510  df-sum 15708  df-ef 16088  df-sin 16090  df-cos 16091  df-pi 16093  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-mulr 17290  df-starv 17291  df-sca 17292  df-vsca 17293  df-ip 17294  df-tset 17295  df-ple 17296  df-ds 17298  df-unif 17299  df-hom 17300  df-cco 17301  df-rest 17441  df-topn 17442  df-0g 17460  df-gsum 17461  df-topgen 17462  df-pt 17463  df-prds 17466  df-xrs 17521  df-qtop 17526  df-imas 17527  df-xps 17529  df-mre 17603  df-mrc 17604  df-acs 17606  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-submnd 18767  df-mulg 19056  df-cntz 19305  df-cmn 19768  df-psmet 21312  df-xmet 21313  df-met 21314  df-bl 21315  df-mopn 21316  df-fbas 21317  df-fg 21318  df-cnfld 21321  df-top 22837  df-topon 22854  df-topsp 22876  df-bases 22889  df-cld 22962  df-ntr 22963  df-cls 22964  df-nei 23041  df-lp 23079  df-perf 23080  df-cn 23170  df-cnp 23171  df-haus 23258  df-tx 23505  df-hmeo 23698  df-fil 23789  df-fm 23881  df-flim 23882  df-flf 23883  df-xms 24264  df-ms 24265  df-tms 24266  df-cncf 24827  df-limc 25824  df-dv 25825  df-log 26522

This theorem is referenced by:  isosctrlem3  26787