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Theorem ang180lem3 26316

Description: Lemma for ang180 26319. Since ang180lem1 26314 shows that 𝑁 is an integer and ang180lem2 26315 shows that 𝑁 is strictly between -2 and 1, it follows that 𝑁 ∈ {-1, 0}, and these two cases correspond to the two possible values for 𝑇. (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
ang180lem3	⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑇 ∈ {-(i · π), (i · π)})

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

**Proof of Theorem ang180lem3**
Step	Hyp	Ref	Expression
1		ang180lem1.3	. . . . . . . . . 10 ⊢ 𝑁 = (((𝑇 / i) / (2 · π)) − (1 / 2))
2		ang.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
3		ang180lem1.2	. . . . . . . . . . . . . . . 16 ⊢ 𝑇 = (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))
4	2, 3, 1	ang180lem2 26315	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-2 < 𝑁 ∧ 𝑁 < 1))
5	4	simprd 497	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 < 1)
6		1e0p1 12719	. . . . . . . . . . . . . 14 ⊢ 1 = (0 + 1)
7	5, 6	breqtrdi 5190	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 < (0 + 1))
8	2, 3, 1	ang180lem1 26314	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 ∈ ℤ ∧ (𝑇 / i) ∈ ℝ))
9	8	simpld 496	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 ∈ ℤ)
10		0z 12569	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℤ
11		zleltp1 12613	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 ≤ 0 ↔ 𝑁 < (0 + 1)))
12	9, 10, 11	sylancl 587	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 ≤ 0 ↔ 𝑁 < (0 + 1)))
13	7, 12	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 ≤ 0)
14	13	adantr 482	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → 𝑁 ≤ 0)
15		zlem1lt 12614	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ≤ 𝑁 ↔ (0 − 1) < 𝑁))
16	10, 9, 15	sylancr 588	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (0 ≤ 𝑁 ↔ (0 − 1) < 𝑁))
17		df-neg 11447	. . . . . . . . . . . . . 14 ⊢ -1 = (0 − 1)
18	17	breq1i 5156	. . . . . . . . . . . . 13 ⊢ (-1 < 𝑁 ↔ (0 − 1) < 𝑁)
19	16, 18	bitr4di 289	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (0 ≤ 𝑁 ↔ -1 < 𝑁))
20	19	biimpar 479	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → 0 ≤ 𝑁)
21	9	zred 12666	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 ∈ ℝ)
22	21	adantr 482	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → 𝑁 ∈ ℝ)
23		0re 11216	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
24		letri3 11299	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 = 0 ↔ (𝑁 ≤ 0 ∧ 0 ≤ 𝑁)))
25	22, 23, 24	sylancl 587	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → (𝑁 = 0 ↔ (𝑁 ≤ 0 ∧ 0 ≤ 𝑁)))
26	14, 20, 25	mpbir2and 712	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → 𝑁 = 0)
27	1, 26	eqtr3id 2787	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → (((𝑇 / i) / (2 · π)) − (1 / 2)) = 0)
28		ax-1cn 11168	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
29		simp1 1137	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ∈ ℂ)
30		subcl 11459	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
31	28, 29, 30	sylancr 588	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 − 𝐴) ∈ ℂ)
32		simp3 1139	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ≠ 1)
33	32	necomd 2997	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 1 ≠ 𝐴)
34		subeq0 11486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
35	28, 29, 34	sylancr 588	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
36	35	necon3bid 2986	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 − 𝐴) ≠ 0 ↔ 1 ≠ 𝐴))
37	33, 36	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 − 𝐴) ≠ 0)
38	31, 37	reccld 11983	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) ∈ ℂ)
39	31, 37	recne0d 11984	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) ≠ 0)
40	38, 39	logcld 26079	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘(1 / (1 − 𝐴))) ∈ ℂ)
41		subcl 11459	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
42	29, 28, 41	sylancl 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝐴 − 1) ∈ ℂ)
43		simp2 1138	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ≠ 0)
44	42, 29, 43	divcld 11990	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) / 𝐴) ∈ ℂ)
45		subeq0 11486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1))
46	29, 28, 45	sylancl 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1))
47	46	necon3bid 2986	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) ≠ 0 ↔ 𝐴 ≠ 1))
48	32, 47	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝐴 − 1) ≠ 0)
49	42, 29, 48, 43	divne0d 12006	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) / 𝐴) ≠ 0)
50	44, 49	logcld 26079	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘((𝐴 − 1) / 𝐴)) ∈ ℂ)
51	40, 50	addcld 11233	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) ∈ ℂ)
52		logcl 26077	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
53	52	3adant3 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘𝐴) ∈ ℂ)
54	51, 53	addcld 11233	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴)) ∈ ℂ)
55	3, 54	eqeltrid 2838	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑇 ∈ ℂ)
56		ax-icn 11169	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
57	56	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → i ∈ ℂ)
58		ine0 11649	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
59	58	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → i ≠ 0)
60	55, 57, 59	divcld 11990	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 / i) ∈ ℂ)
61		2cn 12287	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
62		picn 25969	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
63	61, 62	mulcli 11221	. . . . . . . . . . . . 13 ⊢ (2 · π) ∈ ℂ
64	63	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (2 · π) ∈ ℂ)
65		2ne0 12316	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
66		pire 25968	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℝ
67		pipos 25970	. . . . . . . . . . . . . . 15 ⊢ 0 < π
68	66, 67	gt0ne0ii 11750	. . . . . . . . . . . . . 14 ⊢ π ≠ 0
69	61, 62, 65, 68	mulne0i 11857	. . . . . . . . . . . . 13 ⊢ (2 · π) ≠ 0
70	69	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (2 · π) ≠ 0)
71	60, 64, 70	divcld 11990	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) ∈ ℂ)
72	71	adantr 482	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → ((𝑇 / i) / (2 · π)) ∈ ℂ)
73		halfcn 12427	. . . . . . . . . 10 ⊢ (1 / 2) ∈ ℂ
74		subeq0 11486	. . . . . . . . . 10 ⊢ ((((𝑇 / i) / (2 · π)) ∈ ℂ ∧ (1 / 2) ∈ ℂ) → ((((𝑇 / i) / (2 · π)) − (1 / 2)) = 0 ↔ ((𝑇 / i) / (2 · π)) = (1 / 2)))
75	72, 73, 74	sylancl 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → ((((𝑇 / i) / (2 · π)) − (1 / 2)) = 0 ↔ ((𝑇 / i) / (2 · π)) = (1 / 2)))
76	27, 75	mpbid 231	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → ((𝑇 / i) / (2 · π)) = (1 / 2))
77	60	adantr 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → (𝑇 / i) ∈ ℂ)
78	63	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → (2 · π) ∈ ℂ)
79	73	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → (1 / 2) ∈ ℂ)
80	69	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → (2 · π) ≠ 0)
81	77, 78, 79, 80	divmuld 12012	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → (((𝑇 / i) / (2 · π)) = (1 / 2) ↔ ((2 · π) · (1 / 2)) = (𝑇 / i)))
82	76, 81	mpbid 231	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → ((2 · π) · (1 / 2)) = (𝑇 / i))
83	63, 61, 65	divreci 11959	. . . . . . . 8 ⊢ ((2 · π) / 2) = ((2 · π) · (1 / 2))
84	62, 61, 65	divcan3i 11960	. . . . . . . 8 ⊢ ((2 · π) / 2) = π
85	83, 84	eqtr3i 2763	. . . . . . 7 ⊢ ((2 · π) · (1 / 2)) = π
86	82, 85	eqtr3di 2788	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → (𝑇 / i) = π)
87	55	adantr 482	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → 𝑇 ∈ ℂ)
88	56	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → i ∈ ℂ)
89	62	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → π ∈ ℂ)
90	58	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → i ≠ 0)
91	87, 88, 89, 90	divmuld 12012	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → ((𝑇 / i) = π ↔ (i · π) = 𝑇))
92	86, 91	mpbid 231	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → (i · π) = 𝑇)
93	92	eqcomd 2739	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → 𝑇 = (i · π))
94	93	olcd 873	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 < 𝑁) → (𝑇 = -(i · π) ∨ 𝑇 = (i · π)))
95	62, 56	mulneg1i 11660	. . . . . . 7 ⊢ (-π · i) = -(π · i)
96	62, 56	mulcomi 11222	. . . . . . . 8 ⊢ (π · i) = (i · π)
97	96	negeqi 11453	. . . . . . 7 ⊢ -(π · i) = -(i · π)
98	95, 97	eqtri 2761	. . . . . 6 ⊢ (-π · i) = -(i · π)
99	73, 63	mulneg1i 11660	. . . . . . . . . 10 ⊢ (-(1 / 2) · (2 · π)) = -((1 / 2) · (2 · π))
100	28, 61, 65	divcan1i 11958	. . . . . . . . . . . . 13 ⊢ ((1 / 2) · 2) = 1
101	100	oveq1i 7419	. . . . . . . . . . . 12 ⊢ (((1 / 2) · 2) · π) = (1 · π)
102	73, 61, 62	mulassi 11225	. . . . . . . . . . . 12 ⊢ (((1 / 2) · 2) · π) = ((1 / 2) · (2 · π))
103	62	mullidi 11219	. . . . . . . . . . . 12 ⊢ (1 · π) = π
104	101, 102, 103	3eqtr3i 2769	. . . . . . . . . . 11 ⊢ ((1 / 2) · (2 · π)) = π
105	104	negeqi 11453	. . . . . . . . . 10 ⊢ -((1 / 2) · (2 · π)) = -π
106	99, 105	eqtri 2761	. . . . . . . . 9 ⊢ (-(1 / 2) · (2 · π)) = -π
107	28, 73	negsubdii 11545	. . . . . . . . . . . . 13 ⊢ -(1 − (1 / 2)) = (-1 + (1 / 2))
108		1mhlfehlf 12431	. . . . . . . . . . . . . 14 ⊢ (1 − (1 / 2)) = (1 / 2)
109	108	negeqi 11453	. . . . . . . . . . . . 13 ⊢ -(1 − (1 / 2)) = -(1 / 2)
110	107, 109	eqtr3i 2763	. . . . . . . . . . . 12 ⊢ (-1 + (1 / 2)) = -(1 / 2)
111		simpr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → -1 = 𝑁)
112	111, 1	eqtrdi 2789	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → -1 = (((𝑇 / i) / (2 · π)) − (1 / 2)))
113	112	oveq1d 7424	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → (-1 + (1 / 2)) = ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2)))
114	110, 113	eqtr3id 2787	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → -(1 / 2) = ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2)))
115		npcan 11469	. . . . . . . . . . . . 13 ⊢ ((((𝑇 / i) / (2 · π)) ∈ ℂ ∧ (1 / 2) ∈ ℂ) → ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2)) = ((𝑇 / i) / (2 · π)))
116	71, 73, 115	sylancl 587	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2)) = ((𝑇 / i) / (2 · π)))
117	116	adantr 482	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2)) = ((𝑇 / i) / (2 · π)))
118	114, 117	eqtrd 2773	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → -(1 / 2) = ((𝑇 / i) / (2 · π)))
119	118	oveq1d 7424	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → (-(1 / 2) · (2 · π)) = (((𝑇 / i) / (2 · π)) · (2 · π)))
120	106, 119	eqtr3id 2787	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → -π = (((𝑇 / i) / (2 · π)) · (2 · π)))
121	60, 64, 70	divcan1d 11991	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) · (2 · π)) = (𝑇 / i))
122	121	adantr 482	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → (((𝑇 / i) / (2 · π)) · (2 · π)) = (𝑇 / i))
123	120, 122	eqtrd 2773	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → -π = (𝑇 / i))
124	123	oveq1d 7424	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → (-π · i) = ((𝑇 / i) · i))
125	98, 124	eqtr3id 2787	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → -(i · π) = ((𝑇 / i) · i))
126	55, 57, 59	divcan1d 11991	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) · i) = 𝑇)
127	126	adantr 482	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → ((𝑇 / i) · i) = 𝑇)
128	125, 127	eqtr2d 2774	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → 𝑇 = -(i · π))
129	128	orcd 872	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ -1 = 𝑁) → (𝑇 = -(i · π) ∨ 𝑇 = (i · π)))
130		df-2 12275	. . . . . . . 8 ⊢ 2 = (1 + 1)
131	130	negeqi 11453	. . . . . . 7 ⊢ -2 = -(1 + 1)
132		negdi2 11518	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ) → -(1 + 1) = (-1 − 1))
133	28, 28, 132	mp2an 691	. . . . . . 7 ⊢ -(1 + 1) = (-1 − 1)
134	131, 133	eqtri 2761	. . . . . 6 ⊢ -2 = (-1 − 1)
135	4	simpld 496	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -2 < 𝑁)
136	134, 135	eqbrtrrid 5185	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-1 − 1) < 𝑁)
137		neg1z 12598	. . . . . 6 ⊢ -1 ∈ ℤ
138		zlem1lt 12614	. . . . . 6 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-1 ≤ 𝑁 ↔ (-1 − 1) < 𝑁))
139	137, 9, 138	sylancr 588	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-1 ≤ 𝑁 ↔ (-1 − 1) < 𝑁))
140	136, 139	mpbird 257	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -1 ≤ 𝑁)
141		neg1rr 12327	. . . . 5 ⊢ -1 ∈ ℝ
142		leloe 11300	. . . . 5 ⊢ ((-1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (-1 ≤ 𝑁 ↔ (-1 < 𝑁 ∨ -1 = 𝑁)))
143	141, 21, 142	sylancr 588	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-1 ≤ 𝑁 ↔ (-1 < 𝑁 ∨ -1 = 𝑁)))
144	140, 143	mpbid 231	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-1 < 𝑁 ∨ -1 = 𝑁))
145	94, 129, 144	mpjaodan 958	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 = -(i · π) ∨ 𝑇 = (i · π)))
146	3	ovexi 7443	. . 3 ⊢ 𝑇 ∈ V
147	146	elpr 4652	. 2 ⊢ (𝑇 ∈ {-(i · π), (i · π)} ↔ (𝑇 = -(i · π) ∨ 𝑇 = (i · π)))
148	145, 147	sylibr 233	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑇 ∈ {-(i · π), (i · π)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3946 {csn 4629 {cpr 4631 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 ici 11112 + caddc 11113 · cmul 11115 < clt 11248 ≤ cle 11249 − cmin 11444 -cneg 11445 / cdiv 11871 2c2 12267 ℤcz 12558 ℑcim 15045 πcpi 16010 logclog 26063

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065

This theorem is referenced by: ang180lem4 26317

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