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Theorem ang180lem1 26696

Description: Lemma for ang180 26701. Show that the "revolution number" 𝑁 is an integer, using efeq1 26417 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 26349	. . . . . . 7 ⊢ π ∈ ℂ
2		2re 12290	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
3		pire 26348	. . . . . . . . . 10 ⊢ π ∈ ℝ
4	2, 3	remulcli 11234	. . . . . . . . 9 ⊢ (2 · π) ∈ ℝ
5	4	recni 11232	. . . . . . . 8 ⊢ (2 · π) ∈ ℂ
6		2pos 12319	. . . . . . . . . 10 ⊢ 0 < 2
7		pipos 26350	. . . . . . . . . 10 ⊢ 0 < π
8	2, 3, 6, 7	mulgt0ii 11351	. . . . . . . . 9 ⊢ 0 < (2 · π)
9	4, 8	gt0ne0ii 11754	. . . . . . . 8 ⊢ (2 · π) ≠ 0
10	5, 9	pm3.2i 470	. . . . . . 7 ⊢ ((2 · π) ∈ ℂ ∧ (2 · π) ≠ 0)
11		ax-icn 11171	. . . . . . . 8 ⊢ i ∈ ℂ
12		ine0 11653	. . . . . . . 8 ⊢ i ≠ 0
13	11, 12	pm3.2i 470	. . . . . . 7 ⊢ (i ∈ ℂ ∧ i ≠ 0)
14		divcan5 11920	. . . . . . 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 11754	. . . . . . 7 ⊢ π ≠ 0
17		recdiv 11924	. . . . . . 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 690	. . . . . 6 ⊢ (1 / ((2 · π) / π)) = (π / (2 · π))
19	2	recni 11232	. . . . . . . 8 ⊢ 2 ∈ ℂ
20	19, 1, 16	divcan4i 11965	. . . . . . 7 ⊢ ((2 · π) / π) = 2
21	20	oveq2i 7416	. . . . . 6 ⊢ (1 / ((2 · π) / π)) = (1 / 2)
22	15, 18, 21	3eqtr2i 2760	. . . . 5 ⊢ ((i · π) / (i · (2 · π))) = (1 / 2)
23	22	oveq2i 7416	. . . 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 11170	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
26		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ∈ ℂ)
27		subcl 11463	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
28	25, 26, 27	sylancr 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 − 𝐴) ∈ ℂ)
29		simp3 1135	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ≠ 1)
30	29	necomd 2990	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 1 ≠ 𝐴)
31		subeq0 11490	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
32	25, 26, 31	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
33	32	necon3bid 2979	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 − 𝐴) ≠ 0 ↔ 1 ≠ 𝐴))
34	30, 33	mpbird 257	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 − 𝐴) ≠ 0)
35	28, 34	reccld 11987	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) ∈ ℂ)
36	28, 34	recne0d 11988	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) ≠ 0)
37	35, 36	logcld 26459	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘(1 / (1 − 𝐴))) ∈ ℂ)
38		subcl 11463	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
39	26, 25, 38	sylancl 585	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝐴 − 1) ∈ ℂ)
40		simp2 1134	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ≠ 0)
41	39, 26, 40	divcld 11994	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) / 𝐴) ∈ ℂ)
42		subeq0 11490	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1))
43	26, 25, 42	sylancl 585	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1))
44	43	necon3bid 2979	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) ≠ 0 ↔ 𝐴 ≠ 1))
45	29, 44	mpbird 257	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝐴 − 1) ≠ 0)
46	39, 26, 45, 40	divne0d 12010	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) / 𝐴) ≠ 0)
47	41, 46	logcld 26459	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘((𝐴 − 1) / 𝐴)) ∈ ℂ)
48	37, 47	addcld 11237	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) ∈ ℂ)
49	26, 40	logcld 26459	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘𝐴) ∈ ℂ)
50	48, 49	addcld 11237	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴)) ∈ ℂ)
51	24, 50	eqeltrid 2831	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑇 ∈ ℂ)
52	11, 1	mulcli 11225	. . . . . 6 ⊢ (i · π) ∈ ℂ
53	52	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (i · π) ∈ ℂ)
54	11, 5	mulcli 11225	. . . . . 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 11861	. . . . . 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 12033	. . . 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 11929	. . . . . . 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 1368	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) = (𝑇 / (i · (2 · π))))
64	63	oveq1d 7420	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) − (1 / 2)) = ((𝑇 / (i · (2 · π))) − (1 / 2)))
65	59, 64	eqtrid 2778	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 = ((𝑇 / (i · (2 · π))) − (1 / 2)))
66	23, 58, 65	3eqtr4a 2792	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 − (i · π)) / (i · (2 · π))) = 𝑁)
67		efsub 16050	. . . . . 6 ⊢ ((𝑇 ∈ ℂ ∧ (i · π) ∈ ℂ) → (exp‘(𝑇 − (i · π))) = ((exp‘𝑇) / (exp‘(i · π))))
68	51, 52, 67	sylancl 585	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(𝑇 − (i · π))) = ((exp‘𝑇) / (exp‘(i · π))))
69		efipi 26363	. . . . . . 7 ⊢ (exp‘(i · π)) = -1
70	69	oveq2i 7416	. . . . . 6 ⊢ ((exp‘𝑇) / (exp‘(i · π))) = ((exp‘𝑇) / -1)
71	24	fveq2i 6888	. . . . . . . . 9 ⊢ (exp‘𝑇) = (exp‘(((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴)))
72		efadd 16044	. . . . . . . . . . 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 583	. . . . . . . . . 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 16044	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 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 26465	. . . . . . . . . . . . . 14 ⊢ (((1 / (1 − 𝐴)) ∈ ℂ ∧ (1 / (1 − 𝐴)) ≠ 0) → (exp‘(log‘(1 / (1 − 𝐴)))) = (1 / (1 − 𝐴)))
77	35, 36, 76	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(log‘(1 / (1 − 𝐴)))) = (1 / (1 − 𝐴)))
78		eflog 26465	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 − 1) / 𝐴) ∈ ℂ ∧ ((𝐴 − 1) / 𝐴) ≠ 0) → (exp‘(log‘((𝐴 − 1) / 𝐴))) = ((𝐴 − 1) / 𝐴))
79	41, 46, 78	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(log‘((𝐴 − 1) / 𝐴))) = ((𝐴 − 1) / 𝐴))
80	77, 79	oveq12d 7423	. . . . . . . . . . . 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 11239	. . . . . . . . . . . . 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 12009	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-1 / -(1 − 𝐴)) = (1 / (1 − 𝐴)))
84		negsubdi2 11523	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → -(1 − 𝐴) = (𝐴 − 1))
85	25, 26, 84	sylancr 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -(1 − 𝐴) = (𝐴 − 1))
86	85	oveq2d 7421	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-1 / -(1 − 𝐴)) = (-1 / (𝐴 − 1)))
87	83, 86	eqtr3d 2768	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) = (-1 / (𝐴 − 1)))
88	87	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝐴 − 1) / 𝐴) · (1 / (1 − 𝐴))) = (((𝐴 − 1) / 𝐴) · (-1 / (𝐴 − 1))))
89		neg1cn 12330	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
90	89	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -1 ∈ ℂ)
91	90, 39, 26, 45, 40	dmdcand 12023	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝐴 − 1) / 𝐴) · (-1 / (𝐴 − 1))) = (-1 / 𝐴))
92	81, 88, 91	3eqtrd 2770	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 / (1 − 𝐴)) · ((𝐴 − 1) / 𝐴)) = (-1 / 𝐴))
93	75, 80, 92	3eqtrd 2770	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) = (-1 / 𝐴))
94		eflog 26465	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
95	26, 40, 94	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(log‘𝐴)) = 𝐴)
96	93, 95	oveq12d 7423	. . . . . . . . . 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 11995	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((-1 / 𝐴) · 𝐴) = -1)
98	73, 96, 97	3eqtrd 2770	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))) = -1)
99	71, 98	eqtrid 2778	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘𝑇) = -1)
100	99	oveq1d 7420	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘𝑇) / -1) = (-1 / -1))
101		neg1ne0 12332	. . . . . . . 8 ⊢ -1 ≠ 0
102	89, 101	dividi 11951	. . . . . . 7 ⊢ (-1 / -1) = 1
103	100, 102	eqtrdi 2782	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘𝑇) / -1) = 1)
104	70, 103	eqtrid 2778	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘𝑇) / (exp‘(i · π))) = 1)
105	68, 104	eqtrd 2766	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(𝑇 − (i · π))) = 1)
106		subcl 11463	. . . . . 6 ⊢ ((𝑇 ∈ ℂ ∧ (i · π) ∈ ℂ) → (𝑇 − (i · π)) ∈ ℂ)
107	51, 52, 106	sylancl 585	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 − (i · π)) ∈ ℂ)
108		efeq1 26417	. . . . 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 2828	. 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 11994	. . . 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 11995	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) · (2 · π)) = (𝑇 / i))
118	59	oveq1i 7415	. . . . . 6 ⊢ (𝑁 + (1 / 2)) = ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2))
119	114, 115, 116	divcld 11994	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) ∈ ℂ)
120		halfre 12430	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ
121	120	recni 11232	. . . . . . 7 ⊢ (1 / 2) ∈ ℂ
122		npcan 11473	. . . . . . 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 585	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2)) = ((𝑇 / i) / (2 · π)))
124	118, 123	eqtrid 2778	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 + (1 / 2)) = ((𝑇 / i) / (2 · π)))
125	111	zred 12670	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 ∈ ℝ)
126		readdcl 11195	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝑁 + (1 / 2)) ∈ ℝ)
127	125, 120, 126	sylancl 585	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 + (1 / 2)) ∈ ℝ)
128	124, 127	eqeltrrd 2828	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) ∈ ℝ)
129		remulcl 11197	. . . 4 ⊢ ((((𝑇 / i) / (2 · π)) ∈ ℝ ∧ (2 · π) ∈ ℝ) → (((𝑇 / i) / (2 · π)) · (2 · π)) ∈ ℝ)
130	128, 4, 129	sylancl 585	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) · (2 · π)) ∈ ℝ)
131	117, 130	eqeltrrd 2828	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 / i) ∈ ℝ)
132	111, 131	jca 511	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 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∖ cdif 3940 {csn 4623 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 ici 11114 + caddc 11115 · cmul 11117 − cmin 11448 -cneg 11449 / cdiv 11875 2c2 12271 ℤcz 12562 ℑcim 15051 expce 16011 πcpi 16016 logclog 26443

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-ef 16017 df-sin 16019 df-cos 16020 df-pi 16022 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-limc 25750 df-dv 25751 df-log 26445

This theorem is referenced by: ang180lem2 26697 ang180lem3 26698

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