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Theorem ang180lem1 25387

Description: Lemma for ang180 25392. Show that the "revolution number" 𝑁 is an integer, using efeq1 25113 to show that since the product of the three arguments 𝐴, 1 / (1 − 𝐴), (𝐴 − 1) / 𝐴 is -1, the sum of the logarithms must be an integer multiple of 2πi away from πi = log(-1). (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
ang180lem1	⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 ∈ ℤ ∧ (𝑇 / i) ∈ ℝ))

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

**Proof of Theorem ang180lem1**
Step	Hyp	Ref	Expression
1		picn 25045	. . . . . . 7 ⊢ π ∈ ℂ
2		2re 11712	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
3		pire 25044	. . . . . . . . . 10 ⊢ π ∈ ℝ
4	2, 3	remulcli 10657	. . . . . . . . 9 ⊢ (2 · π) ∈ ℝ
5	4	recni 10655	. . . . . . . 8 ⊢ (2 · π) ∈ ℂ
6		2pos 11741	. . . . . . . . . 10 ⊢ 0 < 2
7		pipos 25046	. . . . . . . . . 10 ⊢ 0 < π
8	2, 3, 6, 7	mulgt0ii 10773	. . . . . . . . 9 ⊢ 0 < (2 · π)
9	4, 8	gt0ne0ii 11176	. . . . . . . 8 ⊢ (2 · π) ≠ 0
10	5, 9	pm3.2i 473	. . . . . . 7 ⊢ ((2 · π) ∈ ℂ ∧ (2 · π) ≠ 0)
11		ax-icn 10596	. . . . . . . 8 ⊢ i ∈ ℂ
12		ine0 11075	. . . . . . . 8 ⊢ i ≠ 0
13	11, 12	pm3.2i 473	. . . . . . 7 ⊢ (i ∈ ℂ ∧ i ≠ 0)
14		divcan5 11342	. . . . . . 7 ⊢ ((π ∈ ℂ ∧ ((2 · π) ∈ ℂ ∧ (2 · π) ≠ 0) ∧ (i ∈ ℂ ∧ i ≠ 0)) → ((i · π) / (i · (2 · π))) = (π / (2 · π)))
15	1, 10, 13, 14	mp3an 1457	. . . . . 6 ⊢ ((i · π) / (i · (2 · π))) = (π / (2 · π))
16	3, 7	gt0ne0ii 11176	. . . . . . 7 ⊢ π ≠ 0
17		recdiv 11346	. . . . . . 7 ⊢ ((((2 · π) ∈ ℂ ∧ (2 · π) ≠ 0) ∧ (π ∈ ℂ ∧ π ≠ 0)) → (1 / ((2 · π) / π)) = (π / (2 · π)))
18	5, 9, 1, 16, 17	mp4an 691	. . . . . 6 ⊢ (1 / ((2 · π) / π)) = (π / (2 · π))
19	2	recni 10655	. . . . . . . 8 ⊢ 2 ∈ ℂ
20	19, 1, 16	divcan4i 11387	. . . . . . 7 ⊢ ((2 · π) / π) = 2
21	20	oveq2i 7167	. . . . . 6 ⊢ (1 / ((2 · π) / π)) = (1 / 2)
22	15, 18, 21	3eqtr2i 2850	. . . . 5 ⊢ ((i · π) / (i · (2 · π))) = (1 / 2)
23	22	oveq2i 7167	. . . 4 ⊢ ((𝑇 / (i · (2 · π))) − ((i · π) / (i · (2 · π)))) = ((𝑇 / (i · (2 · π))) − (1 / 2))
24		ang180lem1.2	. . . . . 6 ⊢ 𝑇 = (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))
25		ax-1cn 10595	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
26		simp1 1132	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ∈ ℂ)
27		subcl 10885	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
28	25, 26, 27	sylancr 589	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 − 𝐴) ∈ ℂ)
29		simp3 1134	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ≠ 1)
30	29	necomd 3071	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 1 ≠ 𝐴)
31		subeq0 10912	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
32	25, 26, 31	sylancr 589	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
33	32	necon3bid 3060	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 − 𝐴) ≠ 0 ↔ 1 ≠ 𝐴))
34	30, 33	mpbird 259	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 − 𝐴) ≠ 0)
35	28, 34	reccld 11409	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) ∈ ℂ)
36	28, 34	recne0d 11410	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) ≠ 0)
37	35, 36	logcld 25154	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘(1 / (1 − 𝐴))) ∈ ℂ)
38		subcl 10885	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
39	26, 25, 38	sylancl 588	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝐴 − 1) ∈ ℂ)
40		simp2 1133	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ≠ 0)
41	39, 26, 40	divcld 11416	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) / 𝐴) ∈ ℂ)
42		subeq0 10912	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1))
43	26, 25, 42	sylancl 588	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1))
44	43	necon3bid 3060	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) ≠ 0 ↔ 𝐴 ≠ 1))
45	29, 44	mpbird 259	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝐴 − 1) ≠ 0)
46	39, 26, 45, 40	divne0d 11432	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) / 𝐴) ≠ 0)
47	41, 46	logcld 25154	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘((𝐴 − 1) / 𝐴)) ∈ ℂ)
48	37, 47	addcld 10660	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) ∈ ℂ)
49	26, 40	logcld 25154	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘𝐴) ∈ ℂ)
50	48, 49	addcld 10660	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴)) ∈ ℂ)
51	24, 50	eqeltrid 2917	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑇 ∈ ℂ)
52	11, 1	mulcli 10648	. . . . . 6 ⊢ (i · π) ∈ ℂ
53	52	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (i · π) ∈ ℂ)
54	11, 5	mulcli 10648	. . . . . 6 ⊢ (i · (2 · π)) ∈ ℂ
55	54	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (i · (2 · π)) ∈ ℂ)
56	11, 5, 12, 9	mulne0i 11283	. . . . . 6 ⊢ (i · (2 · π)) ≠ 0
57	56	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (i · (2 · π)) ≠ 0)
58	51, 53, 55, 57	divsubdird 11455	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 − (i · π)) / (i · (2 · π))) = ((𝑇 / (i · (2 · π))) − ((i · π) / (i · (2 · π)))))
59		ang180lem1.3	. . . . 5 ⊢ 𝑁 = (((𝑇 / i) / (2 · π)) − (1 / 2))
60	13	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (i ∈ ℂ ∧ i ≠ 0))
61	10	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((2 · π) ∈ ℂ ∧ (2 · π) ≠ 0))
62		divdiv1 11351	. . . . . . 7 ⊢ ((𝑇 ∈ ℂ ∧ (i ∈ ℂ ∧ i ≠ 0) ∧ ((2 · π) ∈ ℂ ∧ (2 · π) ≠ 0)) → ((𝑇 / i) / (2 · π)) = (𝑇 / (i · (2 · π))))
63	51, 60, 61, 62	syl3anc 1367	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) = (𝑇 / (i · (2 · π))))
64	63	oveq1d 7171	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) − (1 / 2)) = ((𝑇 / (i · (2 · π))) − (1 / 2)))
65	59, 64	syl5eq 2868	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 = ((𝑇 / (i · (2 · π))) − (1 / 2)))
66	23, 58, 65	3eqtr4a 2882	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 − (i · π)) / (i · (2 · π))) = 𝑁)
67		efsub 15453	. . . . . 6 ⊢ ((𝑇 ∈ ℂ ∧ (i · π) ∈ ℂ) → (exp‘(𝑇 − (i · π))) = ((exp‘𝑇) / (exp‘(i · π))))
68	51, 52, 67	sylancl 588	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(𝑇 − (i · π))) = ((exp‘𝑇) / (exp‘(i · π))))
69		efipi 25059	. . . . . . 7 ⊢ (exp‘(i · π)) = -1
70	69	oveq2i 7167	. . . . . 6 ⊢ ((exp‘𝑇) / (exp‘(i · π))) = ((exp‘𝑇) / -1)
71	24	fveq2i 6673	. . . . . . . . 9 ⊢ (exp‘𝑇) = (exp‘(((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴)))
72		efadd 15447	. . . . . . . . . . 11 ⊢ ((((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → (exp‘(((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))) = ((exp‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) · (exp‘(log‘𝐴))))
73	48, 49, 72	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))) = ((exp‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) · (exp‘(log‘𝐴))))
74		efadd 15447	. . . . . . . . . . . . 13 ⊢ (((log‘(1 / (1 − 𝐴))) ∈ ℂ ∧ (log‘((𝐴 − 1) / 𝐴)) ∈ ℂ) → (exp‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) = ((exp‘(log‘(1 / (1 − 𝐴)))) · (exp‘(log‘((𝐴 − 1) / 𝐴)))))
75	37, 47, 74	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) = ((exp‘(log‘(1 / (1 − 𝐴)))) · (exp‘(log‘((𝐴 − 1) / 𝐴)))))
76		eflog 25160	. . . . . . . . . . . . . 14 ⊢ (((1 / (1 − 𝐴)) ∈ ℂ ∧ (1 / (1 − 𝐴)) ≠ 0) → (exp‘(log‘(1 / (1 − 𝐴)))) = (1 / (1 − 𝐴)))
77	35, 36, 76	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(log‘(1 / (1 − 𝐴)))) = (1 / (1 − 𝐴)))
78		eflog 25160	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 − 1) / 𝐴) ∈ ℂ ∧ ((𝐴 − 1) / 𝐴) ≠ 0) → (exp‘(log‘((𝐴 − 1) / 𝐴))) = ((𝐴 − 1) / 𝐴))
79	41, 46, 78	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(log‘((𝐴 − 1) / 𝐴))) = ((𝐴 − 1) / 𝐴))
80	77, 79	oveq12d 7174	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘(log‘(1 / (1 − 𝐴)))) · (exp‘(log‘((𝐴 − 1) / 𝐴)))) = ((1 / (1 − 𝐴)) · ((𝐴 − 1) / 𝐴)))
81	35, 41	mulcomd 10662	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 / (1 − 𝐴)) · ((𝐴 − 1) / 𝐴)) = (((𝐴 − 1) / 𝐴) · (1 / (1 − 𝐴))))
82	25	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 1 ∈ ℂ)
83	82, 28, 34	div2negd 11431	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-1 / -(1 − 𝐴)) = (1 / (1 − 𝐴)))
84		negsubdi2 10945	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → -(1 − 𝐴) = (𝐴 − 1))
85	25, 26, 84	sylancr 589	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -(1 − 𝐴) = (𝐴 − 1))
86	85	oveq2d 7172	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-1 / -(1 − 𝐴)) = (-1 / (𝐴 − 1)))
87	83, 86	eqtr3d 2858	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) = (-1 / (𝐴 − 1)))
88	87	oveq2d 7172	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝐴 − 1) / 𝐴) · (1 / (1 − 𝐴))) = (((𝐴 − 1) / 𝐴) · (-1 / (𝐴 − 1))))
89		neg1cn 11752	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
90	89	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -1 ∈ ℂ)
91	90, 39, 26, 45, 40	dmdcand 11445	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝐴 − 1) / 𝐴) · (-1 / (𝐴 − 1))) = (-1 / 𝐴))
92	81, 88, 91	3eqtrd 2860	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 / (1 − 𝐴)) · ((𝐴 − 1) / 𝐴)) = (-1 / 𝐴))
93	75, 80, 92	3eqtrd 2860	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) = (-1 / 𝐴))
94		eflog 25160	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
95	26, 40, 94	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(log‘𝐴)) = 𝐴)
96	93, 95	oveq12d 7174	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) · (exp‘(log‘𝐴))) = ((-1 / 𝐴) · 𝐴))
97	90, 26, 40	divcan1d 11417	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((-1 / 𝐴) · 𝐴) = -1)
98	73, 96, 97	3eqtrd 2860	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))) = -1)
99	71, 98	syl5eq 2868	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘𝑇) = -1)
100	99	oveq1d 7171	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘𝑇) / -1) = (-1 / -1))
101		neg1ne0 11754	. . . . . . . 8 ⊢ -1 ≠ 0
102	89, 101	dividi 11373	. . . . . . 7 ⊢ (-1 / -1) = 1
103	100, 102	syl6eq 2872	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘𝑇) / -1) = 1)
104	70, 103	syl5eq 2868	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘𝑇) / (exp‘(i · π))) = 1)
105	68, 104	eqtrd 2856	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(𝑇 − (i · π))) = 1)
106		subcl 10885	. . . . . 6 ⊢ ((𝑇 ∈ ℂ ∧ (i · π) ∈ ℂ) → (𝑇 − (i · π)) ∈ ℂ)
107	51, 52, 106	sylancl 588	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 − (i · π)) ∈ ℂ)
108		efeq1 25113	. . . . 5 ⊢ ((𝑇 − (i · π)) ∈ ℂ → ((exp‘(𝑇 − (i · π))) = 1 ↔ ((𝑇 − (i · π)) / (i · (2 · π))) ∈ ℤ))
109	107, 108	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘(𝑇 − (i · π))) = 1 ↔ ((𝑇 − (i · π)) / (i · (2 · π))) ∈ ℤ))
110	105, 109	mpbid 234	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 − (i · π)) / (i · (2 · π))) ∈ ℤ)
111	66, 110	eqeltrrd 2914	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 ∈ ℤ)
112	11	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → i ∈ ℂ)
113	12	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → i ≠ 0)
114	51, 112, 113	divcld 11416	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 / i) ∈ ℂ)
115	5	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (2 · π) ∈ ℂ)
116	9	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (2 · π) ≠ 0)
117	114, 115, 116	divcan1d 11417	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) · (2 · π)) = (𝑇 / i))
118	59	oveq1i 7166	. . . . . 6 ⊢ (𝑁 + (1 / 2)) = ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2))
119	114, 115, 116	divcld 11416	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) ∈ ℂ)
120		halfre 11852	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ
121	120	recni 10655	. . . . . . 7 ⊢ (1 / 2) ∈ ℂ
122		npcan 10895	. . . . . . 7 ⊢ ((((𝑇 / i) / (2 · π)) ∈ ℂ ∧ (1 / 2) ∈ ℂ) → ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2)) = ((𝑇 / i) / (2 · π)))
123	119, 121, 122	sylancl 588	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2)) = ((𝑇 / i) / (2 · π)))
124	118, 123	syl5eq 2868	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 + (1 / 2)) = ((𝑇 / i) / (2 · π)))
125	111	zred 12088	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 ∈ ℝ)
126		readdcl 10620	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝑁 + (1 / 2)) ∈ ℝ)
127	125, 120, 126	sylancl 588	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 + (1 / 2)) ∈ ℝ)
128	124, 127	eqeltrrd 2914	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) ∈ ℝ)
129		remulcl 10622	. . . 4 ⊢ ((((𝑇 / i) / (2 · π)) ∈ ℝ ∧ (2 · π) ∈ ℝ) → (((𝑇 / i) / (2 · π)) · (2 · π)) ∈ ℝ)
130	128, 4, 129	sylancl 588	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) · (2 · π)) ∈ ℝ)
131	117, 130	eqeltrrd 2914	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 / i) ∈ ℝ)
132	111, 131	jca 514	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 ∈ ℤ ∧ (𝑇 / i) ∈ ℝ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 {csn 4567 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 ici 10539 + caddc 10540 · cmul 10542 − cmin 10870 -cneg 10871 / cdiv 11297 2c2 11693 ℤcz 11982 ℑcim 14457 expce 15415 πcpi 15420 logclog 25138

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140

This theorem is referenced by: ang180lem2 25388 ang180lem3 25389

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