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Theorem sineq0ALT 39487
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 39487. The Virtual Deduction proof is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 24318. 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
StepHypRef Expression
1 pire 24255 . . . . 5 π ∈ ℝ
2 pipos 24257 . . . . 5 0 < π
31, 2elrpii 11873 . . . 4 π ∈ ℝ+
4 2ne0 11151 . . . . . 6 2 ≠ 0
54a1i 11 . . . . 5 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 2 ≠ 0)
6 2cn 11129 . . . . . . 7 2 ∈ ℂ
7 2re 11128 . . . . . . . 8 2 ∈ ℝ
87a1i 11 . . . . . . 7 (2 ∈ ℂ → 2 ∈ ℝ)
96, 8ax-mp 5 . . . . . 6 2 ∈ ℝ
109a1i 11 . . . . 5 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 2 ∈ ℝ)
11 id 22 . . . . . 6 (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
1211adantr 480 . . . . 5 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 𝐴 ∈ ℂ)
136a1i 11 . . . . . . 7 (𝐴 ∈ ℂ → 2 ∈ ℂ)
1413, 11mulcld 10098 . . . . . 6 (𝐴 ∈ ℂ → (2 · 𝐴) ∈ ℂ)
15 ax-icn 10033 . . . . . . . . . . . . . . 15 i ∈ ℂ
1615a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → i ∈ ℂ)
1713, 16, 11mul12d 10283 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (2 · (i · 𝐴)) = (i · (2 · 𝐴)))
1816, 11mulcld 10098 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ)
19182timesd 11313 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (2 · (i · 𝐴)) = ((i · 𝐴) + (i · 𝐴)))
2017, 19eqtr3d 2687 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (i · (2 · 𝐴)) = ((i · 𝐴) + (i · 𝐴)))
2120fveq2d 6233 . . . . . . . . . . 11 (𝐴 ∈ ℂ → (exp‘(i · (2 · 𝐴))) = (exp‘((i · 𝐴) + (i · 𝐴))))
22 efadd 14868 . . . . . . . . . . . 12 (((i · 𝐴) ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (exp‘((i · 𝐴) + (i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
2318, 18, 22syl2anc 694 . . . . . . . . . . 11 (𝐴 ∈ ℂ → (exp‘((i · 𝐴) + (i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
2421, 23eqtrd 2685 . . . . . . . . . 10 (𝐴 ∈ ℂ → (exp‘(i · (2 · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
2524adantr 480 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘(i · (2 · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
26 sinval 14896 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)))
27 id 22 . . . . . . . . . . . . . . 15 ((sin‘𝐴) = 0 → (sin‘𝐴) = 0)
2826, 27sylan9req 2706 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = 0)
29 efcl 14857 . . . . . . . . . . . . . . . . . 18 ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ)
3018, 29syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ)
31 negicn 10320 . . . . . . . . . . . . . . . . . . . 20 -i ∈ ℂ
3231a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ ℂ → -i ∈ ℂ)
3332, 11mulcld 10098 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ)
34 efcl 14857 . . . . . . . . . . . . . . . . . 18 ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ)
3533, 34syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ)
3630, 35subcld 10430 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℂ → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ)
37 2mulicn 11293 . . . . . . . . . . . . . . . . 17 (2 · i) ∈ ℂ
3837a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℂ → (2 · i) ∈ ℂ)
39 2muline0 11294 . . . . . . . . . . . . . . . . 17 (2 · i) ≠ 0
4039a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℂ → (2 · i) ≠ 0)
4136, 38, 40diveq0ad 10849 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → ((((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = 0 ↔ ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0))
4241adantr 480 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = 0 ↔ ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0))
4328, 42mpbid 222 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0)
4430, 35subeq0ad 10440 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0 ↔ (exp‘(i · 𝐴)) = (exp‘(-i · 𝐴))))
4544adantr 480 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0 ↔ (exp‘(i · 𝐴)) = (exp‘(-i · 𝐴))))
4643, 45mpbid 222 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘(i · 𝐴)) = (exp‘(-i · 𝐴)))
4746oveq2d 6706 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴))))
48 efadd 14868 . . . . . . . . . . . . 13 (((i · 𝐴) ∈ ℂ ∧ (-i · 𝐴) ∈ ℂ) → (exp‘((i · 𝐴) + (-i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴))))
4918, 33, 48syl2anc 694 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (exp‘((i · 𝐴) + (-i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴))))
5049adantr 480 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘((i · 𝐴) + (-i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴))))
5147, 50eqtr4d 2688 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) = (exp‘((i · 𝐴) + (-i · 𝐴))))
5215negidi 10388 . . . . . . . . . . . . . . 15 (i + -i) = 0
5352oveq1i 6700 . . . . . . . . . . . . . 14 ((i + -i) · 𝐴) = (0 · 𝐴)
5416, 32, 11adddird 10103 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → ((i + -i) · 𝐴) = ((i · 𝐴) + (-i · 𝐴)))
5553, 54syl5reqr 2700 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → ((i · 𝐴) + (-i · 𝐴)) = (0 · 𝐴))
5611mul02d 10272 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (0 · 𝐴) = 0)
5755, 56eqtrd 2685 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → ((i · 𝐴) + (-i · 𝐴)) = 0)
5857fveq2d 6233 . . . . . . . . . . 11 (𝐴 ∈ ℂ → (exp‘((i · 𝐴) + (-i · 𝐴))) = (exp‘0))
5958adantr 480 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘((i · 𝐴) + (-i · 𝐴))) = (exp‘0))
60 ef0 14865 . . . . . . . . . . 11 (exp‘0) = 1
6160a1i 11 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘0) = 1)
6251, 59, 613eqtrd 2689 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) = 1)
6325, 62eqtrd 2685 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘(i · (2 · 𝐴))) = 1)
6463fveq2d 6233 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(exp‘(i · (2 · 𝐴)))) = (abs‘1))
65 abs1 14081 . . . . . . 7 (abs‘1) = 1
6664, 65syl6eq 2701 . . . . . 6 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(exp‘(i · (2 · 𝐴)))) = 1)
67 absefib 14972 . . . . . . . 8 ((2 · 𝐴) ∈ ℂ → ((2 · 𝐴) ∈ ℝ ↔ (abs‘(exp‘(i · (2 · 𝐴)))) = 1))
6867biimparc 503 . . . . . . 7 (((abs‘(exp‘(i · (2 · 𝐴)))) = 1 ∧ (2 · 𝐴) ∈ ℂ) → (2 · 𝐴) ∈ ℝ)
6968ancoms 468 . . . . . 6 (((2 · 𝐴) ∈ ℂ ∧ (abs‘(exp‘(i · (2 · 𝐴)))) = 1) → (2 · 𝐴) ∈ ℝ)
7014, 66, 69syl2an2r 893 . . . . 5 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (2 · 𝐴) ∈ ℝ)
71 mulre 13905 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 2 ∈ ℝ ∧ 2 ≠ 0) → (𝐴 ∈ ℝ ↔ (2 · 𝐴) ∈ ℝ))
72714animp1 39020 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 2 ∈ ℝ) ∧ 2 ≠ 0) ∧ (2 · 𝐴) ∈ ℝ) → 𝐴 ∈ ℝ)
73724an31 39021 . . . . 5 ((((2 ≠ 0 ∧ 2 ∈ ℝ) ∧ 𝐴 ∈ ℂ) ∧ (2 · 𝐴) ∈ ℝ) → 𝐴 ∈ ℝ)
745, 10, 12, 70, 73eel1111 39264 . . . 4 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 𝐴 ∈ ℝ)
753a1i 11 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → π ∈ ℝ+)
7674, 75modcld 12714 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 mod π) ∈ ℝ)
7776recnd 10106 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 mod π) ∈ ℂ)
7877sincld 14904 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (sin‘(𝐴 mod π)) ∈ ℂ)
791a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → π ∈ ℝ)
80 0re 10078 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ
8180, 1, 2ltleii 10198 . . . . . . . . . . . . . . . . . . . . 21 0 ≤ π
82 gt0ne0 10531 . . . . . . . . . . . . . . . . . . . . . . 23 ((π ∈ ℝ ∧ 0 < π) → π ≠ 0)
83823adant3 1101 . . . . . . . . . . . . . . . . . . . . . 22 ((π ∈ ℝ ∧ 0 < π ∧ 0 ≤ π) → π ≠ 0)
84833com23 1291 . . . . . . . . . . . . . . . . . . . . 21 ((π ∈ ℝ ∧ 0 ≤ π ∧ 0 < π) → π ≠ 0)
851, 81, 2, 84mp3an 1464 . . . . . . . . . . . . . . . . . . . 20 π ≠ 0
8685a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → π ≠ 0)
8774, 79, 86redivcld 10891 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 / π) ∈ ℝ)
8887flcld 12639 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (⌊‘(𝐴 / π)) ∈ ℤ)
8988znegcld 11522 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → -(⌊‘(𝐴 / π)) ∈ ℤ)
90 abssinper 24315 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ -(⌊‘(𝐴 / π)) ∈ ℤ) → (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))) = (abs‘(sin‘𝐴)))
9190eqcomd 2657 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ -(⌊‘(𝐴 / π)) ∈ ℤ) → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))))
9291ex 449 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℂ → (-(⌊‘(𝐴 / π)) ∈ ℤ → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π))))))
9392adantr 480 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (-(⌊‘(𝐴 / π)) ∈ ℤ → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π))))))
9489, 93mpd 15 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))))
9588zcnd 11521 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (⌊‘(𝐴 / π)) ∈ ℂ)
9695negcld 10417 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → -(⌊‘(𝐴 / π)) ∈ ℂ)
971recni 10090 . . . . . . . . . . . . . . . . . . . . 21 π ∈ ℂ
9897a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → π ∈ ℂ)
9996, 98mulcld 10098 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (-(⌊‘(𝐴 / π)) · π) ∈ ℂ)
10098, 95mulcld 10098 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (π · (⌊‘(𝐴 / π))) ∈ ℂ)
101100negcld 10417 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → -(π · (⌊‘(𝐴 / π))) ∈ ℂ)
10295, 98mulneg1d 10521 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (-(⌊‘(𝐴 / π)) · π) = -((⌊‘(𝐴 / π)) · π))
10395, 98mulcomd 10099 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((⌊‘(𝐴 / π)) · π) = (π · (⌊‘(𝐴 / π))))
104103negeqd 10313 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → -((⌊‘(𝐴 / π)) · π) = -(π · (⌊‘(𝐴 / π))))
105102, 104eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (-(⌊‘(𝐴 / π)) · π) = -(π · (⌊‘(𝐴 / π))))
106 oveq2 6698 . . . . . . . . . . . . . . . . . . . . 21 ((-(⌊‘(𝐴 / π)) · π) = -(π · (⌊‘(𝐴 / π))) → (𝐴 + (-(⌊‘(𝐴 / π)) · π)) = (𝐴 + -(π · (⌊‘(𝐴 / π)))))
107106ad3antrrr 766 . . . . . . . . . . . . . . . . . . . 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)))))
1081074an4132 39022 . . . . . . . . . . . . . . . . . . 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)))))
10912, 99, 101, 105, 108eel1111 39264 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 + (-(⌊‘(𝐴 / π)) · π)) = (𝐴 + -(π · (⌊‘(𝐴 / π)))))
11012, 100negsubd 10436 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 + -(π · (⌊‘(𝐴 / π)))) = (𝐴 − (π · (⌊‘(𝐴 / π)))))
111109, 110eqtrd 2685 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 + (-(⌊‘(𝐴 / π)) · π)) = (𝐴 − (π · (⌊‘(𝐴 / π)))))
112111fveq2d 6233 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π))) = (sin‘(𝐴 − (π · (⌊‘(𝐴 / π))))))
113112fveq2d 6233 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
11494, 113eqtrd 2685 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
115 modval 12710 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → (𝐴 mod π) = (𝐴 − (π · (⌊‘(𝐴 / π)))))
116115fveq2d 6233 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → (sin‘(𝐴 mod π)) = (sin‘(𝐴 − (π · (⌊‘(𝐴 / π))))))
117116fveq2d 6233 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → (abs‘(sin‘(𝐴 mod π))) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
1183, 117mpan2 707 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℝ → (abs‘(sin‘(𝐴 mod π))) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
11974, 118syl 17 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘(𝐴 mod π))) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
120114, 119eqtr4d 2688 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 mod π))))
12127fveq2d 6233 . . . . . . . . . . . . . . 15 ((sin‘𝐴) = 0 → (abs‘(sin‘𝐴)) = (abs‘0))
122 abs0 14069 . . . . . . . . . . . . . . 15 (abs‘0) = 0
123121, 122syl6eq 2701 . . . . . . . . . . . . . 14 ((sin‘𝐴) = 0 → (abs‘(sin‘𝐴)) = 0)
124123adantl 481 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘𝐴)) = 0)
125120, 124eqtr3d 2687 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘(𝐴 mod π))) = 0)
12678, 125abs00d 14229 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (sin‘(𝐴 mod π)) = 0)
127 notnotb 304 . . . . . . . . . . . . 13 ((sin‘(𝐴 mod π)) = 0 ↔ ¬ ¬ (sin‘(𝐴 mod π)) = 0)
128127bicomi 214 . . . . . . . . . . . 12 (¬ ¬ (sin‘(𝐴 mod π)) = 0 ↔ (sin‘(𝐴 mod π)) = 0)
129 ltne 10172 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ 0 < (sin‘(𝐴 mod π))) → (sin‘(𝐴 mod π)) ≠ 0)
130129neneqd 2828 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ ∧ 0 < (sin‘(𝐴 mod π))) → ¬ (sin‘(𝐴 mod π)) = 0)
131130expcom 450 . . . . . . . . . . . . . 14 (0 < (sin‘(𝐴 mod π)) → (0 ∈ ℝ → ¬ (sin‘(𝐴 mod π)) = 0))
13280, 131mpi 20 . . . . . . . . . . . . 13 (0 < (sin‘(𝐴 mod π)) → ¬ (sin‘(𝐴 mod π)) = 0)
133132con3i 150 . . . . . . . . . . . 12 (¬ ¬ (sin‘(𝐴 mod π)) = 0 → ¬ 0 < (sin‘(𝐴 mod π)))
134128, 133sylbir 225 . . . . . . . . . . 11 ((sin‘(𝐴 mod π)) = 0 → ¬ 0 < (sin‘(𝐴 mod π)))
135126, 134syl 17 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ¬ 0 < (sin‘(𝐴 mod π)))
136 sinq12gt0 24304 . . . . . . . . . 10 ((𝐴 mod π) ∈ (0(,)π) → 0 < (sin‘(𝐴 mod π)))
137135, 136nsyl 135 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ¬ (𝐴 mod π) ∈ (0(,)π))
13880rexri 10135 . . . . . . . . . . 11 0 ∈ ℝ*
1391rexri 10135 . . . . . . . . . . 11 π ∈ ℝ*
140 elioo2 12254 . . . . . . . . . . 11 ((0 ∈ ℝ* ∧ π ∈ ℝ*) → ((𝐴 mod π) ∈ (0(,)π) ↔ ((𝐴 mod π) ∈ ℝ ∧ 0 < (𝐴 mod π) ∧ (𝐴 mod π) < π)))
141138, 139, 140mp2an 708 . . . . . . . . . 10 ((𝐴 mod π) ∈ (0(,)π) ↔ ((𝐴 mod π) ∈ ℝ ∧ 0 < (𝐴 mod π) ∧ (𝐴 mod π) < π))
142141notbii 309 . . . . . . . . 9 (¬ (𝐴 mod π) ∈ (0(,)π) ↔ ¬ ((𝐴 mod π) ∈ ℝ ∧ 0 < (𝐴 mod π) ∧ (𝐴 mod π) < π))
143137, 142sylib 208 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ¬ ((𝐴 mod π) ∈ ℝ ∧ 0 < (𝐴 mod π) ∧ (𝐴 mod π) < π))
144 3anan12 1069 . . . . . . . . 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)))
145144notbii 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)))
146143, 145sylib 208 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ¬ (0 < (𝐴 mod π) ∧ ((𝐴 mod π) ∈ ℝ ∧ (𝐴 mod π) < π)))
147 modlt 12719 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → (𝐴 mod π) < π)
148147ancoms 468 . . . . . . . . 9 ((π ∈ ℝ+𝐴 ∈ ℝ) → (𝐴 mod π) < π)
1493, 74, 148sylancr 696 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 mod π) < π)
15076, 149jca 553 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((𝐴 mod π) ∈ ℝ ∧ (𝐴 mod π) < π))
151 not12an2impnot1 39101 . . . . . . 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))
152146, 150, 151syl2anc 694 . . . . . 6 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ¬ 0 < (𝐴 mod π))
153 modge0 12718 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → 0 ≤ (𝐴 mod π))
154153ancoms 468 . . . . . . . 8 ((π ∈ ℝ+𝐴 ∈ ℝ) → 0 ≤ (𝐴 mod π))
1553, 74, 154sylancr 696 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 0 ≤ (𝐴 mod π))
156 leloe 10162 . . . . . . . . 9 ((0 ∈ ℝ ∧ (𝐴 mod π) ∈ ℝ) → (0 ≤ (𝐴 mod π) ↔ (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π))))
157156biimp3a 1472 . . . . . . . 8 ((0 ∈ ℝ ∧ (𝐴 mod π) ∈ ℝ ∧ 0 ≤ (𝐴 mod π)) → (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)))
158157idiALT 39000 . . . . . . 7 ((0 ∈ ℝ ∧ (𝐴 mod π) ∈ ℝ ∧ 0 ≤ (𝐴 mod π)) → (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)))
15980, 76, 155, 158mp3an2i 1469 . . . . . 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)))
161160imp 444 . . . . . . 7 (((0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)) ∧ ¬ 0 < (𝐴 mod π)) → 0 = (𝐴 mod π))
162161ancoms 468 . . . . . 6 ((¬ 0 < (𝐴 mod π) ∧ (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π))) → 0 = (𝐴 mod π))
163152, 159, 162syl2anc 694 . . . . 5 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 0 = (𝐴 mod π))
164163eqcomd 2657 . . . 4 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 mod π) = 0)
165 mod0 12715 . . . . . 6 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → ((𝐴 mod π) = 0 ↔ (𝐴 / π) ∈ ℤ))
166165biimp3a 1472 . . . . 5 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+ ∧ (𝐴 mod π) = 0) → (𝐴 / π) ∈ ℤ)
1671663com12 1288 . . . 4 ((π ∈ ℝ+𝐴 ∈ ℝ ∧ (𝐴 mod π) = 0) → (𝐴 / π) ∈ ℤ)
1683, 74, 164, 167mp3an2i 1469 . . 3 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 / π) ∈ ℤ)
169168ex 449 . 2 (𝐴 ∈ ℂ → ((sin‘𝐴) = 0 → (𝐴 / π) ∈ ℤ))
17097a1i 11 . . . . . 6 (𝐴 ∈ ℂ → π ∈ ℂ)
17185a1i 11 . . . . . 6 (𝐴 ∈ ℂ → π ≠ 0)
17211, 170, 171divcan1d 10840 . . . . 5 (𝐴 ∈ ℂ → ((𝐴 / π) · π) = 𝐴)
173172fveq2d 6233 . . . 4 (𝐴 ∈ ℂ → (sin‘((𝐴 / π) · π)) = (sin‘𝐴))
174 id 22 . . . . 5 ((𝐴 / π) ∈ ℤ → (𝐴 / π) ∈ ℤ)
175 sinkpi 24316 . . . . 5 ((𝐴 / π) ∈ ℤ → (sin‘((𝐴 / π) · π)) = 0)
176174, 175syl 17 . . . 4 ((𝐴 / π) ∈ ℤ → (sin‘((𝐴 / π) · π)) = 0)
177173, 176sylan9req 2706 . . 3 ((𝐴 ∈ ℂ ∧ (𝐴 / π) ∈ ℤ) → (sin‘𝐴) = 0)
178177ex 449 . 2 (𝐴 ∈ ℂ → ((𝐴 / π) ∈ ℤ → (sin‘𝐴) = 0))
179169, 178impbid 202 1 (𝐴 ∈ ℂ → ((sin‘𝐴) = 0 ↔ (𝐴 / π) ∈ ℤ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  cfv 5926  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974  1c1 9975  ici 9976   + caddc 9977   · cmul 9979  *cxr 10111   < clt 10112  cle 10113  cmin 10304  -cneg 10305   / cdiv 10722  2c2 11108  cz 11415  +crp 11870  (,)cioo 12213  cfl 12631   mod cmo 12708  abscabs 14018  expce 14836  sincsin 14838  πcpi 14841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-sin 14844  df-cos 14845  df-pi 14847  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676
This theorem is referenced by: (None)
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