ang180lem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ang180lem1		Structured version Visualization version GIF version

Theorem ang180lem1 26757

Description: Lemma for ang180 26762. Show that the "revolution number" 𝑁 is an integer, using efeq1 26478 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 26410	. . . . . . 7 ⊢ π ∈ ℂ
2		2re 12314	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
3		pire 26409	. . . . . . . . . 10 ⊢ π ∈ ℝ
4	2, 3	remulcli 11258	. . . . . . . . 9 ⊢ (2 · π) ∈ ℝ
5	4	recni 11256	. . . . . . . 8 ⊢ (2 · π) ∈ ℂ
6		2pos 12343	. . . . . . . . . 10 ⊢ 0 < 2
7		pipos 26411	. . . . . . . . . 10 ⊢ 0 < π
8	2, 3, 6, 7	mulgt0ii 11375	. . . . . . . . 9 ⊢ 0 < (2 · π)
9	4, 8	gt0ne0ii 11778	. . . . . . . 8 ⊢ (2 · π) ≠ 0
10	5, 9	pm3.2i 469	. . . . . . 7 ⊢ ((2 · π) ∈ ℂ ∧ (2 · π) ≠ 0)
11		ax-icn 11195	. . . . . . . 8 ⊢ i ∈ ℂ
12		ine0 11677	. . . . . . . 8 ⊢ i ≠ 0
13	11, 12	pm3.2i 469	. . . . . . 7 ⊢ (i ∈ ℂ ∧ i ≠ 0)
14		divcan5 11944	. . . . . . 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 11778	. . . . . . 7 ⊢ π ≠ 0
17		recdiv 11948	. . . . . . 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 11256	. . . . . . . 8 ⊢ 2 ∈ ℂ
20	19, 1, 16	divcan4i 11989	. . . . . . 7 ⊢ ((2 · π) / π) = 2
21	20	oveq2i 7426	. . . . . 6 ⊢ (1 / ((2 · π) / π)) = (1 / 2)
22	15, 18, 21	3eqtr2i 2759	. . . . 5 ⊢ ((i · π) / (i · (2 · π))) = (1 / 2)
23	22	oveq2i 7426	. . . 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 11194	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
26		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ∈ ℂ)
27		subcl 11487	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
28	25, 26, 27	sylancr 585	. . . . . . . . . 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 2986	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 1 ≠ 𝐴)
31		subeq0 11514	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
32	25, 26, 31	sylancr 585	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
33	32	necon3bid 2975	. . . . . . . . . . 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 12011	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) ∈ ℂ)
36	28, 34	recne0d 12012	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) ≠ 0)
37	35, 36	logcld 26520	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘(1 / (1 − 𝐴))) ∈ ℂ)
38		subcl 11487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
39	26, 25, 38	sylancl 584	. . . . . . . . . 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 12018	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) / 𝐴) ∈ ℂ)
42		subeq0 11514	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1))
43	26, 25, 42	sylancl 584	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) = 0 ↔ 𝐴 = 1))
44	43	necon3bid 2975	. . . . . . . . . . 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 12034	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝐴 − 1) / 𝐴) ≠ 0)
47	41, 46	logcld 26520	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘((𝐴 − 1) / 𝐴)) ∈ ℂ)
48	37, 47	addcld 11261	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) ∈ ℂ)
49	26, 40	logcld 26520	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘𝐴) ∈ ℂ)
50	48, 49	addcld 11261	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴)) ∈ ℂ)
51	24, 50	eqeltrid 2829	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑇 ∈ ℂ)
52	11, 1	mulcli 11249	. . . . . 6 ⊢ (i · π) ∈ ℂ
53	52	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (i · π) ∈ ℂ)
54	11, 5	mulcli 11249	. . . . . 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 11885	. . . . . 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 12057	. . . 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 11953	. . . . . . 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 7430	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) − (1 / 2)) = ((𝑇 / (i · (2 · π))) − (1 / 2)))
65	59, 64	eqtrid 2777	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 = ((𝑇 / (i · (2 · π))) − (1 / 2)))
66	23, 58, 65	3eqtr4a 2791	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 − (i · π)) / (i · (2 · π))) = 𝑁)
67		efsub 16074	. . . . . 6 ⊢ ((𝑇 ∈ ℂ ∧ (i · π) ∈ ℂ) → (exp‘(𝑇 − (i · π))) = ((exp‘𝑇) / (exp‘(i · π))))
68	51, 52, 67	sylancl 584	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(𝑇 − (i · π))) = ((exp‘𝑇) / (exp‘(i · π))))
69		efipi 26424	. . . . . . 7 ⊢ (exp‘(i · π)) = -1
70	69	oveq2i 7426	. . . . . 6 ⊢ ((exp‘𝑇) / (exp‘(i · π))) = ((exp‘𝑇) / -1)
71	24	fveq2i 6894	. . . . . . . . 9 ⊢ (exp‘𝑇) = (exp‘(((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴)))
72		efadd 16068	. . . . . . . . . . 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 582	. . . . . . . . . 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 16068	. . . . . . . . . . . . 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 582	. . . . . . . . . . . 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 26526	. . . . . . . . . . . . . 14 ⊢ (((1 / (1 − 𝐴)) ∈ ℂ ∧ (1 / (1 − 𝐴)) ≠ 0) → (exp‘(log‘(1 / (1 − 𝐴)))) = (1 / (1 − 𝐴)))
77	35, 36, 76	syl2anc 582	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(log‘(1 / (1 − 𝐴)))) = (1 / (1 − 𝐴)))
78		eflog 26526	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 − 1) / 𝐴) ∈ ℂ ∧ ((𝐴 − 1) / 𝐴) ≠ 0) → (exp‘(log‘((𝐴 − 1) / 𝐴))) = ((𝐴 − 1) / 𝐴))
79	41, 46, 78	syl2anc 582	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(log‘((𝐴 − 1) / 𝐴))) = ((𝐴 − 1) / 𝐴))
80	77, 79	oveq12d 7433	. . . . . . . . . . . 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 11263	. . . . . . . . . . . . 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 12033	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-1 / -(1 − 𝐴)) = (1 / (1 − 𝐴)))
84		negsubdi2 11547	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → -(1 − 𝐴) = (𝐴 − 1))
85	25, 26, 84	sylancr 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -(1 − 𝐴) = (𝐴 − 1))
86	85	oveq2d 7431	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (-1 / -(1 − 𝐴)) = (-1 / (𝐴 − 1)))
87	83, 86	eqtr3d 2767	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (1 / (1 − 𝐴)) = (-1 / (𝐴 − 1)))
88	87	oveq2d 7431	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝐴 − 1) / 𝐴) · (1 / (1 − 𝐴))) = (((𝐴 − 1) / 𝐴) · (-1 / (𝐴 − 1))))
89		neg1cn 12354	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
90	89	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → -1 ∈ ℂ)
91	90, 39, 26, 45, 40	dmdcand 12047	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝐴 − 1) / 𝐴) · (-1 / (𝐴 − 1))) = (-1 / 𝐴))
92	81, 88, 91	3eqtrd 2769	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((1 / (1 − 𝐴)) · ((𝐴 − 1) / 𝐴)) = (-1 / 𝐴))
93	75, 80, 92	3eqtrd 2769	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴)))) = (-1 / 𝐴))
94		eflog 26526	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
95	26, 40, 94	syl2anc 582	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(log‘𝐴)) = 𝐴)
96	93, 95	oveq12d 7433	. . . . . . . . . 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 12019	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((-1 / 𝐴) · 𝐴) = -1)
98	73, 96, 97	3eqtrd 2769	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(((log‘(1 / (1 − 𝐴))) + (log‘((𝐴 − 1) / 𝐴))) + (log‘𝐴))) = -1)
99	71, 98	eqtrid 2777	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘𝑇) = -1)
100	99	oveq1d 7430	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘𝑇) / -1) = (-1 / -1))
101		neg1ne0 12356	. . . . . . . 8 ⊢ -1 ≠ 0
102	89, 101	dividi 11975	. . . . . . 7 ⊢ (-1 / -1) = 1
103	100, 102	eqtrdi 2781	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘𝑇) / -1) = 1)
104	70, 103	eqtrid 2777	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((exp‘𝑇) / (exp‘(i · π))) = 1)
105	68, 104	eqtrd 2765	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (exp‘(𝑇 − (i · π))) = 1)
106		subcl 11487	. . . . . 6 ⊢ ((𝑇 ∈ ℂ ∧ (i · π) ∈ ℂ) → (𝑇 − (i · π)) ∈ ℂ)
107	51, 52, 106	sylancl 584	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 − (i · π)) ∈ ℂ)
108		efeq1 26478	. . . . 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 2826	. 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 12018	. . . 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 12019	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) · (2 · π)) = (𝑇 / i))
118	59	oveq1i 7425	. . . . . 6 ⊢ (𝑁 + (1 / 2)) = ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2))
119	114, 115, 116	divcld 12018	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) ∈ ℂ)
120		halfre 12454	. . . . . . . 8 ⊢ (1 / 2) ∈ ℝ
121	120	recni 11256	. . . . . . 7 ⊢ (1 / 2) ∈ ℂ
122		npcan 11497	. . . . . . 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 584	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((((𝑇 / i) / (2 · π)) − (1 / 2)) + (1 / 2)) = ((𝑇 / i) / (2 · π)))
124	118, 123	eqtrid 2777	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 + (1 / 2)) = ((𝑇 / i) / (2 · π)))
125	111	zred 12694	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝑁 ∈ ℝ)
126		readdcl 11219	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝑁 + (1 / 2)) ∈ ℝ)
127	125, 120, 126	sylancl 584	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑁 + (1 / 2)) ∈ ℝ)
128	124, 127	eqeltrrd 2826	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((𝑇 / i) / (2 · π)) ∈ ℝ)
129		remulcl 11221	. . . 4 ⊢ ((((𝑇 / i) / (2 · π)) ∈ ℝ ∧ (2 · π) ∈ ℝ) → (((𝑇 / i) / (2 · π)) · (2 · π)) ∈ ℝ)
130	128, 4, 129	sylancl 584	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (((𝑇 / i) / (2 · π)) · (2 · π)) ∈ ℝ)
131	117, 130	eqeltrrd 2826	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (𝑇 / i) ∈ ℝ)
132	111, 131	jca 510	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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∖ cdif 3937 {csn 4624 ‘cfv 6542 (class class class)co 7415 ∈ cmpo 7417 ℂcc 11134 ℝcr 11135 0cc0 11136 1c1 11137 ici 11138 + caddc 11139 · cmul 11141 − cmin 11472 -cneg 11473 / cdiv 11899 2c2 12295 ℤcz 12586 ℑcim 15075 expce 16035 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: ang180lem2 26758 ang180lem3 26759

W3C validator