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Theorem ang180lem1 26157

Description: Lemma for ang180 26162. Show that the "revolution number" 𝑁 is an integer, using efeq1 25882 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 25814	. . . . . . 7 ⊢ π ∈ ℂ
2		2re 12226	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
3		pire 25813	. . . . . . . . . 10 ⊢ π ∈ ℝ
4	2, 3	remulcli 11170	. . . . . . . . 9 ⊢ (2 · π) ∈ ℝ
5	4	recni 11168	. . . . . . . 8 ⊢ (2 · π) ∈ ℂ
6		2pos 12255	. . . . . . . . . 10 ⊢ 0 < 2
7		pipos 25815	. . . . . . . . . 10 ⊢ 0 < π
8	2, 3, 6, 7	mulgt0ii 11287	. . . . . . . . 9 ⊢ 0 < (2 · π)
9	4, 8	gt0ne0ii 11690	. . . . . . . 8 ⊢ (2 · π) ≠ 0
10	5, 9	pm3.2i 471	. . . . . . 7 ⊢ ((2 · π) ∈ ℂ ∧ (2 · π) ≠ 0)
11		ax-icn 11109	. . . . . . . 8 ⊢ i ∈ ℂ
12		ine0 11589	. . . . . . . 8 ⊢ i ≠ 0
13	11, 12	pm3.2i 471	. . . . . . 7 ⊢ (i ∈ ℂ ∧ i ≠ 0)
14		divcan5 11856	. . . . . . 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 1461	. . . . . 6 ⊢ ((i · π) / (i · (2 · π))) = (π / (2 · π))
16	3, 7	gt0ne0ii 11690	. . . . . . 7 ⊢ π ≠ 0
17		recdiv 11860	. . . . . . 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 11168	. . . . . . . 8 ⊢ 2 ∈ ℂ
20	19, 1, 16	divcan4i 11901	. . . . . . 7 ⊢ ((2 · π) / π) = 2
21	20	oveq2i 7367	. . . . . 6 ⊢ (1 / ((2 · π) / π)) = (1 / 2)
22	15, 18, 21	3eqtr2i 2770	. . . . 5 ⊢ ((i · π) / (i · (2 · π))) = (1 / 2)
23	22	oveq2i 7367	. . . 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 11108	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
26		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ∈ ℂ)
27		subcl 11399	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
28	25, 26, 27	sylancr 587	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 − 𝐴) ∈ ℂ)
29		simp3 1138	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ≠ 1)
30	29	necomd 2999	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 1 ≠ 𝐴)
31		subeq0 11426	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
32	25, 26, 31	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
33	32	necon3bid 2988	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 − 𝐴) ≠ 0 ↔ 1 ≠ 𝐴))
34	30, 33	mpbird 256	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 − 𝐴) ≠ 0)
35	28, 34	reccld 11923	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) ∈ ℂ)
36	28, 34	recne0d 11924	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) ≠ 0)
37	35, 36	logcld 25924	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘(1 / (1 − 𝐴))) ∈ ℂ)
38		subcl 11399	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
39	26, 25, 38	sylancl 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝐴 − 1) ∈ ℂ)
40		simp2 1137	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ≠ 0)
41	39, 26, 40	divcld 11930	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) / 𝐴) ∈ ℂ)
42		subeq0 11426	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1))
43	26, 25, 42	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1))
44	43	necon3bid 2988	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) ≠ 0 ↔ 𝐴 ≠ 1))
45	29, 44	mpbird 256	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝐴 − 1) ≠ 0)
46	39, 26, 45, 40	divne0d 11946	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) / 𝐴) ≠ 0)
47	41, 46	logcld 25924	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘((𝐴 − 1) / 𝐴)) ∈ ℂ)
48	37, 47	addcld 11173	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) ∈ ℂ)
49	26, 40	logcld 25924	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘𝐴) ∈ ℂ)
50	48, 49	addcld 11173	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴)) ∈ ℂ)
51	24, 50	eqeltrid 2842	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑇 ∈ ℂ)
52	11, 1	mulcli 11161	. . . . . 6 ⊢ (i · π) ∈ ℂ
53	52	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (i · π) ∈ ℂ)
54	11, 5	mulcli 11161	. . . . . 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 11797	. . . . . 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 11969	. . . 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 11865	. . . . . . 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 1371	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) = (𝑇 / (i · (2 · π))))
64	63	oveq1d 7371	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) − (1 / 2)) = ((𝑇 / (i · (2 · π))) − (1 / 2)))
65	59, 64	eqtrid 2788	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 = ((𝑇 / (i · (2 · π))) − (1 / 2)))
66	23, 58, 65	3eqtr4a 2802	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 − (i · π)) / (i · (2 · π))) = 𝑁)
67		efsub 15981	. . . . . 6 ⊢ ((𝑇 ∈ ℂ ∧ (i · π) ∈ ℂ) → (exp‘(𝑇 − (i · π))) = ((exp‘𝑇) / (exp‘(i · π))))
68	51, 52, 67	sylancl 586	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(𝑇 − (i · π))) = ((exp‘𝑇) / (exp‘(i · π))))
69		efipi 25828	. . . . . . 7 ⊢ (exp‘(i · π)) = -1
70	69	oveq2i 7367	. . . . . 6 ⊢ ((exp‘𝑇) / (exp‘(i · π))) = ((exp‘𝑇) / -1)
71	24	fveq2i 6845	. . . . . . . . 9 ⊢ (exp‘𝑇) = (exp‘(((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴)))
72		efadd 15975	. . . . . . . . . . 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 584	. . . . . . . . . 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 15975	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 25930	. . . . . . . . . . . . . 14 ⊢ (((1 / (1 − 𝐴)) ∈ ℂ ∧ (1 / (1 − 𝐴)) ≠ 0) → (exp‘(log‘(1 / (1 − 𝐴)))) = (1 / (1 − 𝐴)))
77	35, 36, 76	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(log‘(1 / (1 − 𝐴)))) = (1 / (1 − 𝐴)))
78		eflog 25930	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 − 1) / 𝐴) ∈ ℂ ∧ ((𝐴 − 1) / 𝐴) ≠ 0) → (exp‘(log‘((𝐴 − 1) / 𝐴))) = ((𝐴 − 1) / 𝐴))
79	41, 46, 78	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(log‘((𝐴 − 1) / 𝐴))) = ((𝐴 − 1) / 𝐴))
80	77, 79	oveq12d 7374	. . . . . . . . . . . 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 11175	. . . . . . . . . . . . 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 11945	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-1 / -(1 − 𝐴)) = (1 / (1 − 𝐴)))
84		negsubdi2 11459	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → -(1 − 𝐴) = (𝐴 − 1))
85	25, 26, 84	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -(1 − 𝐴) = (𝐴 − 1))
86	85	oveq2d 7372	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-1 / -(1 − 𝐴)) = (-1 / (𝐴 − 1)))
87	83, 86	eqtr3d 2778	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) = (-1 / (𝐴 − 1)))
88	87	oveq2d 7372	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝐴 − 1) / 𝐴) · (1 / (1 − 𝐴))) = (((𝐴 − 1) / 𝐴) · (-1 / (𝐴 − 1))))
89		neg1cn 12266	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
90	89	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -1 ∈ ℂ)
91	90, 39, 26, 45, 40	dmdcand 11959	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝐴 − 1) / 𝐴) · (-1 / (𝐴 − 1))) = (-1 / 𝐴))
92	81, 88, 91	3eqtrd 2780	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 / (1 − 𝐴)) · ((𝐴 − 1) / 𝐴)) = (-1 / 𝐴))
93	75, 80, 92	3eqtrd 2780	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) = (-1 / 𝐴))
94		eflog 25930	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
95	26, 40, 94	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(log‘𝐴)) = 𝐴)
96	93, 95	oveq12d 7374	. . . . . . . . . 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 11931	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((-1 / 𝐴) · 𝐴) = -1)
98	73, 96, 97	3eqtrd 2780	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))) = -1)
99	71, 98	eqtrid 2788	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘𝑇) = -1)
100	99	oveq1d 7371	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘𝑇) / -1) = (-1 / -1))
101		neg1ne0 12268	. . . . . . . 8 ⊢ -1 ≠ 0
102	89, 101	dividi 11887	. . . . . . 7 ⊢ (-1 / -1) = 1
103	100, 102	eqtrdi 2792	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘𝑇) / -1) = 1)
104	70, 103	eqtrid 2788	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘𝑇) / (exp‘(i · π))) = 1)
105	68, 104	eqtrd 2776	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(𝑇 − (i · π))) = 1)
106		subcl 11399	. . . . . 6 ⊢ ((𝑇 ∈ ℂ ∧ (i · π) ∈ ℂ) → (𝑇 − (i · π)) ∈ ℂ)
107	51, 52, 106	sylancl 586	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 − (i · π)) ∈ ℂ)
108		efeq1 25882	. . . . 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 231	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 − (i · π)) / (i · (2 · π))) ∈ ℤ)
111	66, 110	eqeltrrd 2839	. 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 11930	. . . 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 11931	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) · (2 · π)) = (𝑇 / i))
118	59	oveq1i 7366	. . . . . 6 ⊢ (𝑁 + (1 / 2)) = ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2))
119	114, 115, 116	divcld 11930	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) ∈ ℂ)
120		halfre 12366	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ
121	120	recni 11168	. . . . . . 7 ⊢ (1 / 2) ∈ ℂ
122		npcan 11409	. . . . . . 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 586	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2)) = ((𝑇 / i) / (2 · π)))
124	118, 123	eqtrid 2788	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 + (1 / 2)) = ((𝑇 / i) / (2 · π)))
125	111	zred 12606	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 ∈ ℝ)
126		readdcl 11133	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝑁 + (1 / 2)) ∈ ℝ)
127	125, 120, 126	sylancl 586	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 + (1 / 2)) ∈ ℝ)
128	124, 127	eqeltrrd 2839	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) ∈ ℝ)
129		remulcl 11135	. . . 4 ⊢ ((((𝑇 / i) / (2 · π)) ∈ ℝ ∧ (2 · π) ∈ ℝ) → (((𝑇 / i) / (2 · π)) · (2 · π)) ∈ ℝ)
130	128, 4, 129	sylancl 586	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) · (2 · π)) ∈ ℝ)
131	117, 130	eqeltrrd 2839	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 / i) ∈ ℝ)
132	111, 131	jca 512	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 ∈ ℤ ∧ (𝑇 / i) ∈ ℝ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3907 {csn 4586 ‘cfv 6496 (class class class)co 7356 ∈ cmpo 7358 ℂcc 11048 ℝcr 11049 0cc0 11050 1c1 11051 ici 11052 + caddc 11053 · cmul 11055 − cmin 11384 -cneg 11385 / cdiv 11811 2c2 12207 ℤcz 12498 ℑcim 14982 expce 15943 πcpi 15948 logclog 25908

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 ax-addf 11129 ax-mulf 11130

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-supp 8092 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8647 df-map 8766 df-pm 8767 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9305 df-fi 9346 df-sup 9377 df-inf 9378 df-oi 9445 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-q 12873 df-rp 12915 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13267 df-ioc 13268 df-ico 13269 df-icc 13270 df-fz 13424 df-fzo 13567 df-fl 13696 df-mod 13774 df-seq 13906 df-exp 13967 df-fac 14173 df-bc 14202 df-hash 14230 df-shft 14951 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-limsup 15352 df-clim 15369 df-rlim 15370 df-sum 15570 df-ef 15949 df-sin 15951 df-cos 15952 df-pi 15954 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-starv 17147 df-sca 17148 df-vsca 17149 df-ip 17150 df-tset 17151 df-ple 17152 df-ds 17154 df-unif 17155 df-hom 17156 df-cco 17157 df-rest 17303 df-topn 17304 df-0g 17322 df-gsum 17323 df-topgen 17324 df-pt 17325 df-prds 17328 df-xrs 17383 df-qtop 17388 df-imas 17389 df-xps 17391 df-mre 17465 df-mrc 17466 df-acs 17468 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-submnd 18601 df-mulg 18871 df-cntz 19095 df-cmn 19562 df-psmet 20786 df-xmet 20787 df-met 20788 df-bl 20789 df-mopn 20790 df-fbas 20791 df-fg 20792 df-cnfld 20795 df-top 22241 df-topon 22258 df-topsp 22280 df-bases 22294 df-cld 22368 df-ntr 22369 df-cls 22370 df-nei 22447 df-lp 22485 df-perf 22486 df-cn 22576 df-cnp 22577 df-haus 22664 df-tx 22911 df-hmeo 23104 df-fil 23195 df-fm 23287 df-flim 23288 df-flf 23289 df-xms 23671 df-ms 23672 df-tms 23673 df-cncf 24239 df-limc 25228 df-dv 25229 df-log 25910

This theorem is referenced by: ang180lem2 26158 ang180lem3 26159

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