Theorem sineq0ALT 38195

Description: A complex number whose sine is zero is an integer multiple of π. The Virtual Deduction form of the proof is http://us.metamath.org/other/completeusersproof/sineq0altvd.html. The Metamath form of the proof is sineq0ALT 38195. The Virtual Deduction proof is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 24077. The Virtual Deduction proof is verified by automatically transforming it into the Metamath form of the proof using completeusersproof, which is verified by the Metamath program. The proof of http://us.metamath.org/other/completeusersproof/sineq0altro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. (Contributed by Alan Sare, 13-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sineq0ALT	⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) = 0 ↔ (𝐴 / π) ∈ ℤ))

Proof of Theorem sineq0ALT

Step	Hyp	Ref	Expression
1		pire 24014	. . . . 5 ⊢ π ∈ ℝ
2		pipos 24016	. . . . 5 ⊢ 0 < π
3	1, 2	elrpii 11711	. . . 4 ⊢ π ∈ ℝ+
4		2ne0 10990	. . . . . 6 ⊢ 2 ≠ 0
5	4	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 2 ≠ 0)
6		2cn 10968	. . . . . . 7 ⊢ 2 ∈ ℂ
7		2re 10967	. . . . . . . 8 ⊢ 2 ∈ ℝ
8	7	a1i 11	. . . . . . 7 ⊢ (2 ∈ ℂ → 2 ∈ ℝ)
9	6, 8	ax-mp 5	. . . . . 6 ⊢ 2 ∈ ℝ
10	9	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 2 ∈ ℝ)
11		id 22	. . . . . 6 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
12	11	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 𝐴 ∈ ℂ)
13	6	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → 2 ∈ ℂ)
14	13, 11	mulcld 9939	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) ∈ ℂ)
15		ax-icn 9874	. . . . . . . . . . . . . . 15 ⊢ i ∈ ℂ
16	15	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → i ∈ ℂ)
17	13, 16, 11	mul12d 10124	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (2 · (i · 𝐴)) = (i · (2 · 𝐴)))
18	16, 11	mulcld 9939	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ)
19	18	2timesd 11152	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (2 · (i · 𝐴)) = ((i · 𝐴) + (i · 𝐴)))
20	17, 19	eqtr3d 2646	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (i · (2 · 𝐴)) = ((i · 𝐴) + (i · 𝐴)))
21	20	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (2 · 𝐴))) = (exp‘((i · 𝐴) + (i · 𝐴))))
22		efadd 14663	. . . . . . . . . . . 12 ⊢ (((i · 𝐴) ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (exp‘((i · 𝐴) + (i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
23	18, 18, 22	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (exp‘((i · 𝐴) + (i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
24	21, 23	eqtrd 2644	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (2 · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
25	24	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘(i · (2 · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
26		sinval 14691	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)))
27		id 22	. . . . . . . . . . . . . . 15 ⊢ ((sin‘𝐴) = 0 → (sin‘𝐴) = 0)
28	26, 27	sylan9req 2665	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = 0)
29		efcl 14652	. . . . . . . . . . . . . . . . . 18 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ)
30	18, 29	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ)
31		negicn 10161	. . . . . . . . . . . . . . . . . . . 20 ⊢ -i ∈ ℂ
32	31	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℂ → -i ∈ ℂ)
33	32, 11	mulcld 9939	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ)
34		efcl 14652	. . . . . . . . . . . . . . . . . 18 ⊢ ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ)
35	33, 34	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ)
36	30, 35	subcld 10271	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ)
37		2mulicn 11132	. . . . . . . . . . . . . . . . 17 ⊢ (2 · i) ∈ ℂ
38	37	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℂ → (2 · i) ∈ ℂ)
39		2muline0 11133	. . . . . . . . . . . . . . . . 17 ⊢ (2 · i) ≠ 0
40	39	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℂ → (2 · i) ≠ 0)
41	36, 38, 40	diveq0ad 10690	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → ((((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = 0 ↔ ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0))
42	41	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = 0 ↔ ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0))
43	28, 42	mpbid 221	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0)
44	30, 35	subeq0ad 10281	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0 ↔ (exp‘(i · 𝐴)) = (exp‘(-i · 𝐴))))
45	44	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0 ↔ (exp‘(i · 𝐴)) = (exp‘(-i · 𝐴))))
46	43, 45	mpbid 221	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘(i · 𝐴)) = (exp‘(-i · 𝐴)))
47	46	oveq2d 6565	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴))))
48		efadd 14663	. . . . . . . . . . . . 13 ⊢ (((i · 𝐴) ∈ ℂ ∧ (-i · 𝐴) ∈ ℂ) → (exp‘((i · 𝐴) + (-i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴))))
49	18, 33, 48	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (exp‘((i · 𝐴) + (-i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴))))
50	49	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘((i · 𝐴) + (-i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴))))
51	47, 50	eqtr4d 2647	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) = (exp‘((i · 𝐴) + (-i · 𝐴))))
52	15	negidi 10229	. . . . . . . . . . . . . . 15 ⊢ (i + -i) = 0
53	52	oveq1i 6559	. . . . . . . . . . . . . 14 ⊢ ((i + -i) · 𝐴) = (0 · 𝐴)
54	16, 32, 11	adddird 9944	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → ((i + -i) · 𝐴) = ((i · 𝐴) + (-i · 𝐴)))
55	53, 54	syl5reqr 2659	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (-i · 𝐴)) = (0 · 𝐴))
56	11	mul02d 10113	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0)
57	55, 56	eqtrd 2644	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (-i · 𝐴)) = 0)
58	57	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (exp‘((i · 𝐴) + (-i · 𝐴))) = (exp‘0))
59	58	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘((i · 𝐴) + (-i · 𝐴))) = (exp‘0))
60		ef0 14660	. . . . . . . . . . 11 ⊢ (exp‘0) = 1
61	60	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘0) = 1)
62	51, 59, 61	3eqtrd 2648	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) = 1)
63	25, 62	eqtrd 2644	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘(i · (2 · 𝐴))) = 1)
64	63	fveq2d 6107	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(exp‘(i · (2 · 𝐴)))) = (abs‘1))
65		abs1 13885	. . . . . . 7 ⊢ (abs‘1) = 1
66	64, 65	syl6eq 2660	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(exp‘(i · (2 · 𝐴)))) = 1)
67		absefib 14767	. . . . . . . 8 ⊢ ((2 · 𝐴) ∈ ℂ → ((2 · 𝐴) ∈ ℝ ↔ (abs‘(exp‘(i · (2 · 𝐴)))) = 1))
68	67	biimparc 503	. . . . . . 7 ⊢ (((abs‘(exp‘(i · (2 · 𝐴)))) = 1 ∧ (2 · 𝐴) ∈ ℂ) → (2 · 𝐴) ∈ ℝ)
69	68	ancoms 468	. . . . . 6 ⊢ (((2 · 𝐴) ∈ ℂ ∧ (abs‘(exp‘(i · (2 · 𝐴)))) = 1) → (2 · 𝐴) ∈ ℝ)
70	14, 66, 69	syl2an2r 872	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (2 · 𝐴) ∈ ℝ)
71		mulre 13709	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℝ ∧ 2 ≠ 0) → (𝐴 ∈ ℝ ↔ (2 · 𝐴) ∈ ℝ))
72	71	4animp1 37724	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 2 ∈ ℝ) ∧ 2 ≠ 0) ∧ (2 · 𝐴) ∈ ℝ) → 𝐴 ∈ ℝ)
73	72	4an31 37725	. . . . 5 ⊢ ((((2 ≠ 0 ∧ 2 ∈ ℝ) ∧ 𝐴 ∈ ℂ) ∧ (2 · 𝐴) ∈ ℝ) → 𝐴 ∈ ℝ)
74	5, 10, 12, 70, 73	eel1111 37968	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 𝐴 ∈ ℝ)
75	3	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → π ∈ ℝ+)
76	74, 75	modcld 12536	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 mod π) ∈ ℝ)
77	76	recnd 9947	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 mod π) ∈ ℂ)
78	77	sincld 14699	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (sin‘(𝐴 mod π)) ∈ ℂ)
79	1	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → π ∈ ℝ)
80		0re 9919	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℝ
81	80, 1, 2	ltleii 10039	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ≤ π
82		gt0ne0 10372	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((π ∈ ℝ ∧ 0 < π) → π ≠ 0)
83	82	3adant3 1074	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((π ∈ ℝ ∧ 0 < π ∧ 0 ≤ π) → π ≠ 0)
84	83	3com23 1263	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((π ∈ ℝ ∧ 0 ≤ π ∧ 0 < π) → π ≠ 0)
85	1, 81, 2, 84	mp3an 1416	. . . . . . . . . . . . . . . . . . . 20 ⊢ π ≠ 0
86	85	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → π ≠ 0)
87	74, 79, 86	redivcld 10732	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 / π) ∈ ℝ)
88	87	flcld 12461	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (⌊‘(𝐴 / π)) ∈ ℤ)
89	88	znegcld 11360	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → -(⌊‘(𝐴 / π)) ∈ ℤ)
90		abssinper 24074	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ -(⌊‘(𝐴 / π)) ∈ ℤ) → (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))) = (abs‘(sin‘𝐴)))
91	90	eqcomd 2616	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ -(⌊‘(𝐴 / π)) ∈ ℤ) → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))))
92	91	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (-(⌊‘(𝐴 / π)) ∈ ℤ → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π))))))
93	92	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (-(⌊‘(𝐴 / π)) ∈ ℤ → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π))))))
94	89, 93	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))))
95	88	zcnd 11359	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (⌊‘(𝐴 / π)) ∈ ℂ)
96	95	negcld 10258	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → -(⌊‘(𝐴 / π)) ∈ ℂ)
97	1	recni 9931	. . . . . . . . . . . . . . . . . . . . 21 ⊢ π ∈ ℂ
98	97	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → π ∈ ℂ)
99	96, 98	mulcld 9939	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (-(⌊‘(𝐴 / π)) · π) ∈ ℂ)
100	98, 95	mulcld 9939	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (π · (⌊‘(𝐴 / π))) ∈ ℂ)
101	100	negcld 10258	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → -(π · (⌊‘(𝐴 / π))) ∈ ℂ)
102	95, 98	mulneg1d 10362	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (-(⌊‘(𝐴 / π)) · π) = -((⌊‘(𝐴 / π)) · π))
103	95, 98	mulcomd 9940	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((⌊‘(𝐴 / π)) · π) = (π · (⌊‘(𝐴 / π))))
104	103	negeqd 10154	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → -((⌊‘(𝐴 / π)) · π) = -(π · (⌊‘(𝐴 / π))))
105	102, 104	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (-(⌊‘(𝐴 / π)) · π) = -(π · (⌊‘(𝐴 / π))))
106		oveq2 6557	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((-(⌊‘(𝐴 / π)) · π) = -(π · (⌊‘(𝐴 / π))) → (𝐴 + (-(⌊‘(𝐴 / π)) · π)) = (𝐴 + -(π · (⌊‘(𝐴 / π)))))
107	106	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((-(⌊‘(𝐴 / π)) · π) = -(π · (⌊‘(𝐴 / π))) ∧ -(π · (⌊‘(𝐴 / π))) ∈ ℂ) ∧ (-(⌊‘(𝐴 / π)) · π) ∈ ℂ) ∧ 𝐴 ∈ ℂ) → (𝐴 + (-(⌊‘(𝐴 / π)) · π)) = (𝐴 + -(π · (⌊‘(𝐴 / π)))))
108	107	4an4132 37726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ℂ ∧ (-(⌊‘(𝐴 / π)) · π) ∈ ℂ) ∧ -(π · (⌊‘(𝐴 / π))) ∈ ℂ) ∧ (-(⌊‘(𝐴 / π)) · π) = -(π · (⌊‘(𝐴 / π)))) → (𝐴 + (-(⌊‘(𝐴 / π)) · π)) = (𝐴 + -(π · (⌊‘(𝐴 / π)))))
109	12, 99, 101, 105, 108	eel1111 37968	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 + (-(⌊‘(𝐴 / π)) · π)) = (𝐴 + -(π · (⌊‘(𝐴 / π)))))
110	12, 100	negsubd 10277	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 + -(π · (⌊‘(𝐴 / π)))) = (𝐴 − (π · (⌊‘(𝐴 / π)))))
111	109, 110	eqtrd 2644	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 + (-(⌊‘(𝐴 / π)) · π)) = (𝐴 − (π · (⌊‘(𝐴 / π)))))
112	111	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π))) = (sin‘(𝐴 − (π · (⌊‘(𝐴 / π))))))
113	112	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
114	94, 113	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
115		modval 12532	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → (𝐴 mod π) = (𝐴 − (π · (⌊‘(𝐴 / π)))))
116	115	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → (sin‘(𝐴 mod π)) = (sin‘(𝐴 − (π · (⌊‘(𝐴 / π))))))
117	116	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → (abs‘(sin‘(𝐴 mod π))) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
118	3, 117	mpan2 703	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ → (abs‘(sin‘(𝐴 mod π))) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
119	74, 118	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘(𝐴 mod π))) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
120	114, 119	eqtr4d 2647	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 mod π))))
121	27	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ ((sin‘𝐴) = 0 → (abs‘(sin‘𝐴)) = (abs‘0))
122		abs0 13873	. . . . . . . . . . . . . . 15 ⊢ (abs‘0) = 0
123	121, 122	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ ((sin‘𝐴) = 0 → (abs‘(sin‘𝐴)) = 0)
124	123	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘𝐴)) = 0)
125	120, 124	eqtr3d 2646	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘(𝐴 mod π))) = 0)
126	78, 125	abs00d 14033	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (sin‘(𝐴 mod π)) = 0)
127		notnotb 303	. . . . . . . . . . . . 13 ⊢ ((sin‘(𝐴 mod π)) = 0 ↔ ¬ ¬ (sin‘(𝐴 mod π)) = 0)
128	127	bicomi 213	. . . . . . . . . . . 12 ⊢ (¬ ¬ (sin‘(𝐴 mod π)) = 0 ↔ (sin‘(𝐴 mod π)) = 0)
129		ltne 10013	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 0 < (sin‘(𝐴 mod π))) → (sin‘(𝐴 mod π)) ≠ 0)
130	129	neneqd 2787	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ 0 < (sin‘(𝐴 mod π))) → ¬ (sin‘(𝐴 mod π)) = 0)
131	130	expcom 450	. . . . . . . . . . . . . 14 ⊢ (0 < (sin‘(𝐴 mod π)) → (0 ∈ ℝ → ¬ (sin‘(𝐴 mod π)) = 0))
132	80, 131	mpi 20	. . . . . . . . . . . . 13 ⊢ (0 < (sin‘(𝐴 mod π)) → ¬ (sin‘(𝐴 mod π)) = 0)
133	132	con3i 149	. . . . . . . . . . . 12 ⊢ (¬ ¬ (sin‘(𝐴 mod π)) = 0 → ¬ 0 < (sin‘(𝐴 mod π)))
134	128, 133	sylbir 224	. . . . . . . . . . 11 ⊢ ((sin‘(𝐴 mod π)) = 0 → ¬ 0 < (sin‘(𝐴 mod π)))
135	126, 134	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ¬ 0 < (sin‘(𝐴 mod π)))
136		sinq12gt0 24063	. . . . . . . . . 10 ⊢ ((𝐴 mod π) ∈ (0(,)π) → 0 < (sin‘(𝐴 mod π)))
137	135, 136	nsyl 134	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ¬ (𝐴 mod π) ∈ (0(,)π))
138	80	rexri 9976	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ*
139	1	rexri 9976	. . . . . . . . . . 11 ⊢ π ∈ ℝ*
140		elioo2 12087	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ* ∧ π ∈ ℝ*) → ((𝐴 mod π) ∈ (0(,)π) ↔ ((𝐴 mod π) ∈ ℝ ∧ 0 < (𝐴 mod π) ∧ (𝐴 mod π) < π)))
141	138, 139, 140	mp2an 704	. . . . . . . . . 10 ⊢ ((𝐴 mod π) ∈ (0(,)π) ↔ ((𝐴 mod π) ∈ ℝ ∧ 0 < (𝐴 mod π) ∧ (𝐴 mod π) < π))
142	141	notbii 309	. . . . . . . . 9 ⊢ (¬ (𝐴 mod π) ∈ (0(,)π) ↔ ¬ ((𝐴 mod π) ∈ ℝ ∧ 0 < (𝐴 mod π) ∧ (𝐴 mod π) < π))
143	137, 142	sylib 207	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ¬ ((𝐴 mod π) ∈ ℝ ∧ 0 < (𝐴 mod π) ∧ (𝐴 mod π) < π))
144		3anan12 1044	. . . . . . . . 9 ⊢ (((𝐴 mod π) ∈ ℝ ∧ 0 < (𝐴 mod π) ∧ (𝐴 mod π) < π) ↔ (0 < (𝐴 mod π) ∧ ((𝐴 mod π) ∈ ℝ ∧ (𝐴 mod π) < π)))
145	144	notbii 309	. . . . . . . 8 ⊢ (¬ ((𝐴 mod π) ∈ ℝ ∧ 0 < (𝐴 mod π) ∧ (𝐴 mod π) < π) ↔ ¬ (0 < (𝐴 mod π) ∧ ((𝐴 mod π) ∈ ℝ ∧ (𝐴 mod π) < π)))
146	143, 145	sylib 207	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ¬ (0 < (𝐴 mod π) ∧ ((𝐴 mod π) ∈ ℝ ∧ (𝐴 mod π) < π)))
147		modlt 12541	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → (𝐴 mod π) < π)
148	147	ancoms 468	. . . . . . . . 9 ⊢ ((π ∈ ℝ+ ∧ 𝐴 ∈ ℝ) → (𝐴 mod π) < π)
149	3, 74, 148	sylancr 694	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 mod π) < π)
150	76, 149	jca 553	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((𝐴 mod π) ∈ ℝ ∧ (𝐴 mod π) < π))
151		not12an2impnot1 37805	. . . . . . 7 ⊢ ((¬ (0 < (𝐴 mod π) ∧ ((𝐴 mod π) ∈ ℝ ∧ (𝐴 mod π) < π)) ∧ ((𝐴 mod π) ∈ ℝ ∧ (𝐴 mod π) < π)) → ¬ 0 < (𝐴 mod π))
152	146, 150, 151	syl2anc 691	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ¬ 0 < (𝐴 mod π))
153		modge0 12540	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → 0 ≤ (𝐴 mod π))
154	153	ancoms 468	. . . . . . . 8 ⊢ ((π ∈ ℝ+ ∧ 𝐴 ∈ ℝ) → 0 ≤ (𝐴 mod π))
155	3, 74, 154	sylancr 694	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 0 ≤ (𝐴 mod π))
156		leloe 10003	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (𝐴 mod π) ∈ ℝ) → (0 ≤ (𝐴 mod π) ↔ (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π))))
157	156	biimp3a 1424	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ (𝐴 mod π) ∈ ℝ ∧ 0 ≤ (𝐴 mod π)) → (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)))
158	157	idiALT 37704	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝐴 mod π) ∈ ℝ ∧ 0 ≤ (𝐴 mod π)) → (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)))
159	80, 76, 155, 158	mp3an2i 1421	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)))
160		pm2.53 387	. . . . . . . 8 ⊢ ((0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)) → (¬ 0 < (𝐴 mod π) → 0 = (𝐴 mod π)))
161	160	imp 444	. . . . . . 7 ⊢ (((0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)) ∧ ¬ 0 < (𝐴 mod π)) → 0 = (𝐴 mod π))
162	161	ancoms 468	. . . . . 6 ⊢ ((¬ 0 < (𝐴 mod π) ∧ (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π))) → 0 = (𝐴 mod π))
163	152, 159, 162	syl2anc 691	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 0 = (𝐴 mod π))
164	163	eqcomd 2616	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 mod π) = 0)
165		mod0 12537	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → ((𝐴 mod π) = 0 ↔ (𝐴 / π) ∈ ℤ))
166	165	biimp3a 1424	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ π ∈ ℝ+ ∧ (𝐴 mod π) = 0) → (𝐴 / π) ∈ ℤ)
167	166	3com12 1261	. . . 4 ⊢ ((π ∈ ℝ+ ∧ 𝐴 ∈ ℝ ∧ (𝐴 mod π) = 0) → (𝐴 / π) ∈ ℤ)
168	3, 74, 164, 167	mp3an2i 1421	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 / π) ∈ ℤ)
169	168	ex 449	. 2 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) = 0 → (𝐴 / π) ∈ ℤ))
170	97	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → π ∈ ℂ)
171	85	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → π ≠ 0)
172	11, 170, 171	divcan1d 10681	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴 / π) · π) = 𝐴)
173	172	fveq2d 6107	. . . 4 ⊢ (𝐴 ∈ ℂ → (sin‘((𝐴 / π) · π)) = (sin‘𝐴))
174		id 22	. . . . 5 ⊢ ((𝐴 / π) ∈ ℤ → (𝐴 / π) ∈ ℤ)
175		sinkpi 24075	. . . . 5 ⊢ ((𝐴 / π) ∈ ℤ → (sin‘((𝐴 / π) · π)) = 0)
176	174, 175	syl 17	. . . 4 ⊢ ((𝐴 / π) ∈ ℤ → (sin‘((𝐴 / π) · π)) = 0)
177	173, 176	sylan9req 2665	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 / π) ∈ ℤ) → (sin‘𝐴) = 0)
178	177	ex 449	. 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 / π) ∈ ℤ → (sin‘𝐴) = 0))
179	169, 178	impbid 201	1 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) = 0 ↔ (𝐴 / π) ∈ ℤ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816  ici 9817   + caddc 9818   · cmul 9820  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  -cneg 10146   / cdiv 10563  2c2 10947  ℤcz 11254  ℝ⁺crp 11708  (,)cioo 12046  ⌊cfl 12453   mod cmo 12530  abscabs 13822  expce 14631  sincsin 14633  πcpi 14636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-pi 14642  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437

This theorem is referenced by: (None)