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Theorem ang180lem2 26758

Description: Lemma for ang180 26762. Show that the revolution number 𝑁 is strictly between -2 and 1. Both bounds are established by iterating using the bounds on the imaginary part of the logarithm, logimcl 26519, but the resulting bound gives only 𝑁 ≤ 1 for the upper bound. The case 𝑁 = 1 is not ruled out here, but it is in some sense an "edge case" that can only happen under very specific conditions; in particular we show that all the angle arguments 𝐴, 1 / (1 − 𝐴), (𝐴 − 1) / 𝐴 must lie on the negative real axis, which is a contradiction because clearly if 𝐴 is negative then the other two are positive real. (Contributed by Mario Carneiro, 23-Sep-2014.)

**Hypotheses**
Ref	Expression
ang.1	⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
ang180lem1.2	⊢ 𝑇 = (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))
ang180lem1.3	⊢ 𝑁 = (((𝑇 / i) / (2 · π)) − (1 / 2))

**Assertion**
Ref	Expression
ang180lem2	⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-2 < 𝑁 ∧ 𝑁 < 1))

Distinct variable group: 𝑥,𝑦,𝐴 Allowed substitution hints: 𝑇(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝑁(𝑥,𝑦)

**Proof of Theorem ang180lem2**
Step	Hyp	Ref	Expression
1		2cn 12315	. . . . . . 7 ⊢ 2 ∈ ℂ
2		1re 11242	. . . . . . . . 9 ⊢ 1 ∈ ℝ
3	2	rehalfcli 12489	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ
4	3	recni 11256	. . . . . . 7 ⊢ (1 / 2) ∈ ℂ
5	1, 4	negsubdii 11573	. . . . . 6 ⊢ -(2 − (1 / 2)) = (-2 + (1 / 2))
6		4d2e2 12410	. . . . . . . . 9 ⊢ (4 / 2) = 2
7	6	oveq1i 7425	. . . . . . . 8 ⊢ ((4 / 2) − (1 / 2)) = (2 − (1 / 2))
8		4cn 12325	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
9		ax-1cn 11194	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
10		2cnne0 12450	. . . . . . . . . 10 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
11		divsubdir 11936	. . . . . . . . . 10 ⊢ ((4 ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((4 − 1) / 2) = ((4 / 2) − (1 / 2)))
12	8, 9, 10, 11	mp3an 1457	. . . . . . . . 9 ⊢ ((4 − 1) / 2) = ((4 / 2) − (1 / 2))
13		4m1e3 12369	. . . . . . . . . 10 ⊢ (4 − 1) = 3
14	13	oveq1i 7425	. . . . . . . . 9 ⊢ ((4 − 1) / 2) = (3 / 2)
15	12, 14	eqtr3i 2755	. . . . . . . 8 ⊢ ((4 / 2) − (1 / 2)) = (3 / 2)
16	7, 15	eqtr3i 2755	. . . . . . 7 ⊢ (2 − (1 / 2)) = (3 / 2)
17	16	negeqi 11481	. . . . . 6 ⊢ -(2 − (1 / 2)) = -(3 / 2)
18	5, 17	eqtr3i 2755	. . . . 5 ⊢ (-2 + (1 / 2)) = -(3 / 2)
19		3re 12320	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
20	19	rehalfcli 12489	. . . . . . . . . . . 12 ⊢ (3 / 2) ∈ ℝ
21	20	recni 11256	. . . . . . . . . . 11 ⊢ (3 / 2) ∈ ℂ
22		picn 26410	. . . . . . . . . . 11 ⊢ π ∈ ℂ
23	21, 1, 22	mulassi 11253	. . . . . . . . . 10 ⊢ (((3 / 2) · 2) · π) = ((3 / 2) · (2 · π))
24		3cn 12321	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
25		2ne0 12344	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
26	24, 1, 25	divcan1i 11986	. . . . . . . . . . 11 ⊢ ((3 / 2) · 2) = 3
27	26	oveq1i 7425	. . . . . . . . . 10 ⊢ (((3 / 2) · 2) · π) = (3 · π)
28	23, 27	eqtr3i 2755	. . . . . . . . 9 ⊢ ((3 / 2) · (2 · π)) = (3 · π)
29	28	negeqi 11481	. . . . . . . 8 ⊢ -((3 / 2) · (2 · π)) = -(3 · π)
30		2re 12314	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
31		pire 26409	. . . . . . . . . . 11 ⊢ π ∈ ℝ
32	30, 31	remulcli 11258	. . . . . . . . . 10 ⊢ (2 · π) ∈ ℝ
33	32	recni 11256	. . . . . . . . 9 ⊢ (2 · π) ∈ ℂ
34	21, 33	mulneg1i 11688	. . . . . . . 8 ⊢ (-(3 / 2) · (2 · π)) = -((3 / 2) · (2 · π))
35	24, 22	mulneg2i 11689	. . . . . . . 8 ⊢ (3 · -π) = -(3 · π)
36	29, 34, 35	3eqtr4i 2763	. . . . . . 7 ⊢ (-(3 / 2) · (2 · π)) = (3 · -π)
37	31	renegcli 11549	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ
38	30, 37	remulcli 11258	. . . . . . . . . . 11 ⊢ (2 · -π) ∈ ℝ
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (2 · -π) ∈ ℝ)
40	37	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -π ∈ ℝ)
41		simp1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ∈ ℂ)
42		subcl 11487	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
43	9, 41, 42	sylancr 585	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 − 𝐴) ∈ ℂ)
44		simp3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ≠ 1)
45	44	necomd 2986	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 1 ≠ 𝐴)
46		subeq0 11514	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
47	9, 41, 46	sylancr 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
48	47	necon3bid 2975	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 − 𝐴) ≠ 0 ↔ 1 ≠ 𝐴))
49	45, 48	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 − 𝐴) ≠ 0)
50	43, 49	reccld 12011	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) ∈ ℂ)
51	43, 49	recne0d 12012	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) ≠ 0)
52	50, 51	logcld 26520	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘(1 / (1 − 𝐴))) ∈ ℂ)
53		subcl 11487	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
54	41, 9, 53	sylancl 584	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝐴 − 1) ∈ ℂ)
55		simp2 1134	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ≠ 0)
56	54, 41, 55	divcld 12018	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) / 𝐴) ∈ ℂ)
57		subeq0 11514	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1))
58	41, 9, 57	sylancl 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1))
59	58	necon3bid 2975	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) ≠ 0 ↔ 𝐴 ≠ 1))
60	44, 59	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝐴 − 1) ≠ 0)
61	54, 41, 60, 55	divne0d 12034	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) / 𝐴) ≠ 0)
62	56, 61	logcld 26520	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘((𝐴 − 1) / 𝐴)) ∈ ℂ)
63	52, 62	addcld 11261	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) ∈ ℂ)
64	63	imcld 15172	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) ∈ ℝ)
65		logcl 26518	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
66	65	3adant3 1129	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘𝐴) ∈ ℂ)
67	66	imcld 15172	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘(log‘𝐴)) ∈ ℝ)
68	52	imcld 15172	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘(log‘(1 / (1 − 𝐴)))) ∈ ℝ)
69	62	imcld 15172	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘(log‘((𝐴 − 1) / 𝐴))) ∈ ℝ)
70	50, 51	logimcld 26521	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-π < (ℑ‘(log‘(1 / (1 − 𝐴)))) ∧ (ℑ‘(log‘(1 / (1 − 𝐴)))) ≤ π))
71	70	simpld 493	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -π < (ℑ‘(log‘(1 / (1 − 𝐴)))))
72	56, 61	logimcld 26521	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-π < (ℑ‘(log‘((𝐴 − 1) / 𝐴))) ∧ (ℑ‘(log‘((𝐴 − 1) / 𝐴))) ≤ π))
73	72	simpld 493	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -π < (ℑ‘(log‘((𝐴 − 1) / 𝐴))))
74	40, 40, 68, 69, 71, 73	lt2addd 11865	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-π + -π) < ((ℑ‘(log‘(1 / (1 − 𝐴)))) + (ℑ‘(log‘((𝐴 − 1) / 𝐴)))))
75		negpicn 26413	. . . . . . . . . . . . 13 ⊢ -π ∈ ℂ
76	75	2timesi 12378	. . . . . . . . . . . 12 ⊢ (2 · -π) = (-π + -π)
77	76	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (2 · -π) = (-π + -π))
78	52, 62	imaddd 15192	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) = ((ℑ‘(log‘(1 / (1 − 𝐴)))) + (ℑ‘(log‘((𝐴 − 1) / 𝐴)))))
79	74, 77, 78	3brtr4d 5175	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (2 · -π) < (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))))
80		logimcl 26519	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
81	80	3adant3 1129	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
82	81	simpld 493	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -π < (ℑ‘(log‘𝐴)))
83	39, 40, 64, 67, 79, 82	lt2addd 11865	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((2 · -π) + -π) < ((ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) + (ℑ‘(log‘𝐴))))
84		df-3 12304	. . . . . . . . . . . 12 ⊢ 3 = (2 + 1)
85	84	oveq1i 7425	. . . . . . . . . . 11 ⊢ (3 · -π) = ((2 + 1) · -π)
86	1, 9, 75	adddiri 11255	. . . . . . . . . . 11 ⊢ ((2 + 1) · -π) = ((2 · -π) + (1 · -π))
87	75	mullidi 11247	. . . . . . . . . . . 12 ⊢ (1 · -π) = -π
88	87	oveq2i 7426	. . . . . . . . . . 11 ⊢ ((2 · -π) + (1 · -π)) = ((2 · -π) + -π)
89	85, 86, 88	3eqtri 2757	. . . . . . . . . 10 ⊢ (3 · -π) = ((2 · -π) + -π)
90	89	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (3 · -π) = ((2 · -π) + -π))
91		ang180lem1.2	. . . . . . . . . . 11 ⊢ 𝑇 = (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))
92	91	fveq2i 6894	. . . . . . . . . 10 ⊢ (ℑ‘𝑇) = (ℑ‘(((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴)))
93	63, 66	imaddd 15192	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘(((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))) = ((ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) + (ℑ‘(log‘𝐴))))
94	92, 93	eqtrid 2777	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘𝑇) = ((ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) + (ℑ‘(log‘𝐴))))
95	83, 90, 94	3brtr4d 5175	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (3 · -π) < (ℑ‘𝑇))
96	63, 66	addcld 11261	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴)) ∈ ℂ)
97	91, 96	eqeltrid 2829	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑇 ∈ ℂ)
98		imval 15084	. . . . . . . . . 10 ⊢ (𝑇 ∈ ℂ → (ℑ‘𝑇) = (ℜ‘(𝑇 / i)))
99	97, 98	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘𝑇) = (ℜ‘(𝑇 / i)))
100		ang.1	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
101		ang180lem1.3	. . . . . . . . . . . 12 ⊢ 𝑁 = (((𝑇 / i) / (2 · π)) − (1 / 2))
102	100, 91, 101	ang180lem1 26757	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 ∈ ℤ ∧ (𝑇 / i) ∈ ℝ))
103	102	simprd 494	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 / i) ∈ ℝ)
104	103	rered 15201	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℜ‘(𝑇 / i)) = (𝑇 / i))
105	99, 104	eqtrd 2765	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘𝑇) = (𝑇 / i))
106	95, 105	breqtrd 5169	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (3 · -π) < (𝑇 / i))
107	36, 106	eqbrtrid 5178	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-(3 / 2) · (2 · π)) < (𝑇 / i))
108	20	renegcli 11549	. . . . . . . 8 ⊢ -(3 / 2) ∈ ℝ
109	108	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -(3 / 2) ∈ ℝ)
110	32	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (2 · π) ∈ ℝ)
111		2pos 12343	. . . . . . . . 9 ⊢ 0 < 2
112		pipos 26411	. . . . . . . . 9 ⊢ 0 < π
113	30, 31, 111, 112	mulgt0ii 11375	. . . . . . . 8 ⊢ 0 < (2 · π)
114	113	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 0 < (2 · π))
115		ltmuldiv 12115	. . . . . . 7 ⊢ ((-(3 / 2) ∈ ℝ ∧ (𝑇 / i) ∈ ℝ ∧ ((2 · π) ∈ ℝ ∧ 0 < (2 · π))) → ((-(3 / 2) · (2 · π)) < (𝑇 / i) ↔ -(3 / 2) < ((𝑇 / i) / (2 · π))))
116	109, 103, 110, 114, 115	syl112anc 1371	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((-(3 / 2) · (2 · π)) < (𝑇 / i) ↔ -(3 / 2) < ((𝑇 / i) / (2 · π))))
117	107, 116	mpbid 231	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -(3 / 2) < ((𝑇 / i) / (2 · π)))
118	18, 117	eqbrtrid 5178	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-2 + (1 / 2)) < ((𝑇 / i) / (2 · π)))
119	30	renegcli 11549	. . . . . 6 ⊢ -2 ∈ ℝ
120	119	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -2 ∈ ℝ)
121	3	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / 2) ∈ ℝ)
122	32, 113	gt0ne0ii 11778	. . . . . . 7 ⊢ (2 · π) ≠ 0
123	122	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (2 · π) ≠ 0)
124	103, 110, 123	redivcld 12070	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) ∈ ℝ)
125	120, 121, 124	ltaddsubd 11842	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((-2 + (1 / 2)) < ((𝑇 / i) / (2 · π)) ↔ -2 < (((𝑇 / i) / (2 · π)) − (1 / 2))))
126	118, 125	mpbid 231	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -2 < (((𝑇 / i) / (2 · π)) − (1 / 2)))
127	126, 101	breqtrrdi 5185	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -2 < 𝑁)
128	31	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → π ∈ ℝ)
129	70	simprd 494	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘(log‘(1 / (1 − 𝐴)))) ≤ π)
130	72	simprd 494	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘(log‘((𝐴 − 1) / 𝐴))) ≤ π)
131	68, 69, 128, 128, 129, 130	le2addd 11861	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((ℑ‘(log‘(1 / (1 − 𝐴)))) + (ℑ‘(log‘((𝐴 − 1) / 𝐴)))) ≤ (π + π))
132	22	2timesi 12378	. . . . . . . . . . . 12 ⊢ (2 · π) = (π + π)
133	132	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (2 · π) = (π + π))
134	131, 78, 133	3brtr4d 5175	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) ≤ (2 · π))
135	81	simprd 494	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘(log‘𝐴)) ≤ π)
136	64, 67, 110, 128, 134, 135	le2addd 11861	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) + (ℑ‘(log‘𝐴))) ≤ ((2 · π) + π))
137	105, 94	eqtr3d 2767	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 / i) = ((ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) + (ℑ‘(log‘𝐴))))
138	84	oveq1i 7425	. . . . . . . . . . 11 ⊢ (3 · π) = ((2 + 1) · π)
139	1, 9, 22	adddiri 11255	. . . . . . . . . . 11 ⊢ ((2 + 1) · π) = ((2 · π) + (1 · π))
140	22	mullidi 11247	. . . . . . . . . . . 12 ⊢ (1 · π) = π
141	140	oveq2i 7426	. . . . . . . . . . 11 ⊢ ((2 · π) + (1 · π)) = ((2 · π) + π)
142	138, 139, 141	3eqtri 2757	. . . . . . . . . 10 ⊢ (3 · π) = ((2 · π) + π)
143	142	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (3 · π) = ((2 · π) + π))
144	136, 137, 143	3brtr4d 5175	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 / i) ≤ (3 · π))
145	33	subid1i 11560	. . . . . . . . . 10 ⊢ ((2 · π) − 0) = (2 · π)
146	145, 122	eqnetri 3001	. . . . . . . . 9 ⊢ ((2 · π) − 0) ≠ 0
147		negsub 11536	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + -𝐴) = (1 − 𝐴))
148	9, 41, 147	sylancr 585	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 + -𝐴) = (1 − 𝐴))
149	148	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (1 + -𝐴) = (1 − 𝐴))
150		1rp 13008	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ⁺
151	143, 137	oveq12d 7433	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((3 · π) − (𝑇 / i)) = (((2 · π) + π) − ((ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) + (ℑ‘(log‘𝐴)))))
152	33	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (2 · π) ∈ ℂ)
153	22	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → π ∈ ℂ)
154	64	recnd 11270	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) ∈ ℂ)
155	67	recnd 11270	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (ℑ‘(log‘𝐴)) ∈ ℂ)
156	152, 153, 154, 155	addsub4d 11646	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((2 · π) + π) − ((ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) + (ℑ‘(log‘𝐴)))) = (((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) + (π − (ℑ‘(log‘𝐴)))))
157	151, 156	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((3 · π) − (𝑇 / i)) = (((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) + (π − (ℑ‘(log‘𝐴)))))
158	157	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → ((3 · π) − (𝑇 / i)) = (((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) + (π − (ℑ‘(log‘𝐴)))))
159	19, 31	remulcli 11258	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (3 · π) ∈ ℝ
160	159	recni 11256	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (3 · π) ∈ ℂ
161		ax-icn 11195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ i ∈ ℂ
162	161	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → i ∈ ℂ)
163		ine0 11677	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ i ≠ 0
164	163	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → i ≠ 0)
165	97, 162, 164	divcld 12018	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 / i) ∈ ℂ)
166		subeq0 11514	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((3 · π) ∈ ℂ ∧ (𝑇 / i) ∈ ℂ) → (((3 · π) − (𝑇 / i)) = 0 ↔ (3 · π) = (𝑇 / i)))
167	160, 165, 166	sylancr 585	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((3 · π) − (𝑇 / i)) = 0 ↔ (3 · π) = (𝑇 / i)))
168	167	biimpar 476	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → ((3 · π) − (𝑇 / i)) = 0)
169	158, 168	eqtr3d 2767	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) + (π − (ℑ‘(log‘𝐴)))) = 0)
170		resubcl 11552	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2 · π) ∈ ℝ ∧ (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) ∈ ℝ) → ((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) ∈ ℝ)
171	32, 64, 170	sylancr 585	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) ∈ ℝ)
172		subge0 11755	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((2 · π) ∈ ℝ ∧ (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) ∈ ℝ) → (0 ≤ ((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) ↔ (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) ≤ (2 · π)))
173	32, 64, 172	sylancr 585	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (0 ≤ ((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) ↔ (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) ≤ (2 · π)))
174	134, 173	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 0 ≤ ((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))))
175		resubcl 11552	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (π − (ℑ‘(log‘𝐴))) ∈ ℝ)
176	31, 67, 175	sylancr 585	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (π − (ℑ‘(log‘𝐴))) ∈ ℝ)
177		subge0 11755	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (0 ≤ (π − (ℑ‘(log‘𝐴))) ↔ (ℑ‘(log‘𝐴)) ≤ π))
178	31, 67, 177	sylancr 585	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (0 ≤ (π − (ℑ‘(log‘𝐴))) ↔ (ℑ‘(log‘𝐴)) ≤ π))
179	135, 178	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 0 ≤ (π − (ℑ‘(log‘𝐴))))
180		add20 11754	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) ∈ ℝ ∧ 0 ≤ ((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))))) ∧ ((π − (ℑ‘(log‘𝐴))) ∈ ℝ ∧ 0 ≤ (π − (ℑ‘(log‘𝐴))))) → ((((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) + (π − (ℑ‘(log‘𝐴)))) = 0 ↔ (((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) = 0 ∧ (π − (ℑ‘(log‘𝐴))) = 0)))
181	171, 174, 176, 179, 180	syl22anc 837	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) + (π − (ℑ‘(log‘𝐴)))) = 0 ↔ (((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) = 0 ∧ (π − (ℑ‘(log‘𝐴))) = 0)))
182	181	biimpa 475	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) + (π − (ℑ‘(log‘𝐴)))) = 0) → (((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) = 0 ∧ (π − (ℑ‘(log‘𝐴))) = 0))
183	169, 182	syldan 589	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) = 0 ∧ (π − (ℑ‘(log‘𝐴))) = 0))
184	183	simprd 494	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (π − (ℑ‘(log‘𝐴))) = 0)
185	155	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (ℑ‘(log‘𝐴)) ∈ ℂ)
186		subeq0 11514	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((π ∈ ℂ ∧ (ℑ‘(log‘𝐴)) ∈ ℂ) → ((π − (ℑ‘(log‘𝐴))) = 0 ↔ π = (ℑ‘(log‘𝐴))))
187	22, 185, 186	sylancr 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → ((π − (ℑ‘(log‘𝐴))) = 0 ↔ π = (ℑ‘(log‘𝐴))))
188	184, 187	mpbid 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → π = (ℑ‘(log‘𝐴)))
189	188	eqcomd 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (ℑ‘(log‘𝐴)) = π)
190		lognegb 26540	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-𝐴 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝐴)) = π))
191	190	3adant3 1129	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-𝐴 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝐴)) = π))
192	191	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (-𝐴 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝐴)) = π))
193	189, 192	mpbird 256	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → -𝐴 ∈ ℝ⁺)
194		rpaddcl 13026	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℝ⁺ ∧ -𝐴 ∈ ℝ⁺) → (1 + -𝐴) ∈ ℝ⁺)
195	150, 193, 194	sylancr 585	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (1 + -𝐴) ∈ ℝ⁺)
196	149, 195	eqeltrrd 2826	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (1 − 𝐴) ∈ ℝ⁺)
197	196	rpreccld 13056	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (1 / (1 − 𝐴)) ∈ ℝ⁺)
198	197	relogcld 26573	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (log‘(1 / (1 − 𝐴))) ∈ ℝ)
199		negsubdi2 11547	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (1 − 𝐴))
200	41, 9, 199	sylancl 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -(𝐴 − 1) = (1 − 𝐴))
201	200	oveq1d 7430	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-(𝐴 − 1) / -𝐴) = ((1 − 𝐴) / -𝐴))
202	54, 41, 55	div2negd 12033	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-(𝐴 − 1) / -𝐴) = ((𝐴 − 1) / 𝐴))
203	201, 202	eqtr3d 2767	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 − 𝐴) / -𝐴) = ((𝐴 − 1) / 𝐴))
204	203	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → ((1 − 𝐴) / -𝐴) = ((𝐴 − 1) / 𝐴))
205	196, 193	rpdivcld 13063	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → ((1 − 𝐴) / -𝐴) ∈ ℝ⁺)
206	204, 205	eqeltrrd 2826	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → ((𝐴 − 1) / 𝐴) ∈ ℝ⁺)
207	206	relogcld 26573	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (log‘((𝐴 − 1) / 𝐴)) ∈ ℝ)
208	198, 207	readdcld 11271	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → ((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) ∈ ℝ)
209	208	reim0d 15202	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) = 0)
210	209	oveq2d 7431	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → ((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) = ((2 · π) − 0))
211	183	simpld 493	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → ((2 · π) − (ℑ‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))))) = 0)
212	210, 211	eqtr3d 2767	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (3 · π) = (𝑇 / i)) → ((2 · π) − 0) = 0)
213	212	ex 411	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((3 · π) = (𝑇 / i) → ((2 · π) − 0) = 0))
214	213	necon3d 2951	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((2 · π) − 0) ≠ 0 → (3 · π) ≠ (𝑇 / i)))
215	146, 214	mpi 20	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (3 · π) ≠ (𝑇 / i))
216		ltlen 11343	. . . . . . . . 9 ⊢ (((𝑇 / i) ∈ ℝ ∧ (3 · π) ∈ ℝ) → ((𝑇 / i) < (3 · π) ↔ ((𝑇 / i) ≤ (3 · π) ∧ (3 · π) ≠ (𝑇 / i))))
217	103, 159, 216	sylancl 584	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) < (3 · π) ↔ ((𝑇 / i) ≤ (3 · π) ∧ (3 · π) ≠ (𝑇 / i))))
218	144, 215, 217	mpbir2and 711	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 / i) < (3 · π))
219	218, 28	breqtrrdi 5185	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 / i) < ((3 / 2) · (2 · π)))
220	20	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (3 / 2) ∈ ℝ)
221		ltdivmul2 12119	. . . . . . 7 ⊢ (((𝑇 / i) ∈ ℝ ∧ (3 / 2) ∈ ℝ ∧ ((2 · π) ∈ ℝ ∧ 0 < (2 · π))) → (((𝑇 / i) / (2 · π)) < (3 / 2) ↔ (𝑇 / i) < ((3 / 2) · (2 · π))))
222	103, 220, 110, 114, 221	syl112anc 1371	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) < (3 / 2) ↔ (𝑇 / i) < ((3 / 2) · (2 · π))))
223	219, 222	mpbird 256	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) < (3 / 2))
224	84	oveq1i 7425	. . . . . 6 ⊢ (3 / 2) = ((2 + 1) / 2)
225	1, 9, 1, 25	divdiri 11999	. . . . . 6 ⊢ ((2 + 1) / 2) = ((2 / 2) + (1 / 2))
226		2div2e1 12381	. . . . . . 7 ⊢ (2 / 2) = 1
227	226	oveq1i 7425	. . . . . 6 ⊢ ((2 / 2) + (1 / 2)) = (1 + (1 / 2))
228	224, 225, 227	3eqtri 2757	. . . . 5 ⊢ (3 / 2) = (1 + (1 / 2))
229	223, 228	breqtrdi 5184	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) < (1 + (1 / 2)))
230	2	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 1 ∈ ℝ)
231	124, 121, 230	ltsubaddd 11838	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((((𝑇 / i) / (2 · π)) − (1 / 2)) < 1 ↔ ((𝑇 / i) / (2 · π)) < (1 + (1 / 2))))
232	229, 231	mpbird 256	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) − (1 / 2)) < 1)
233	101, 232	eqbrtrid 5178	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 < 1)
234	127, 233	jca 510	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-2 < 𝑁 ∧ 𝑁 < 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∖ cdif 3937 {csn 4624 class class class wbr 5143 ‘cfv 6542 (class class class)co 7415 ∈ cmpo 7417 ℂcc 11134 ℝcr 11135 0cc0 11136 1c1 11137 ici 11138 + caddc 11139 · cmul 11141 < clt 11276 ≤ cle 11277 − cmin 11472 -cneg 11473 / cdiv 11899 2c2 12295 3c3 12296 4c4 12297 ℤcz 12586 ℝ⁺crp 13004 ℜcre 15074 ℑcim 15075 πcpi 16040 logclog 26504

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-fi 9432 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ioo 13358 df-ioc 13359 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-fl 13787 df-mod 13865 df-seq 13997 df-exp 14057 df-fac 14263 df-bc 14292 df-hash 14320 df-shft 15044 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-limsup 15445 df-clim 15462 df-rlim 15463 df-sum 15663 df-ef 16041 df-sin 16043 df-cos 16044 df-pi 16046 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-rest 17401 df-topn 17402 df-0g 17420 df-gsum 17421 df-topgen 17422 df-pt 17423 df-prds 17426 df-xrs 17481 df-qtop 17486 df-imas 17487 df-xps 17489 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-mulg 19026 df-cntz 19270 df-cmn 19739 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-fbas 21278 df-fg 21279 df-cnfld 21282 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24242 df-ms 24243 df-tms 24244 df-cncf 24814 df-limc 25811 df-dv 25812 df-log 26506

This theorem is referenced by: ang180lem3 26759

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