Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sineq0ALT Structured version   Visualization version   GIF version

Theorem sineq0ALT 38656
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 38656. The Virtual Deduction proof is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 24177. 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 24114 . . . . 5 π ∈ ℝ
2 pipos 24116 . . . . 5 0 < π
31, 2elrpii 11779 . . . 4 π ∈ ℝ+
4 2ne0 11057 . . . . . 6 2 ≠ 0
54a1i 11 . . . . 5 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 2 ≠ 0)
6 2cn 11035 . . . . . . 7 2 ∈ ℂ
7 2re 11034 . . . . . . . 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 481 . . . . 5 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 𝐴 ∈ ℂ)
136a1i 11 . . . . . . 7 (𝐴 ∈ ℂ → 2 ∈ ℂ)
1413, 11mulcld 10004 . . . . . 6 (𝐴 ∈ ℂ → (2 · 𝐴) ∈ ℂ)
15 ax-icn 9939 . . . . . . . . . . . . . . 15 i ∈ ℂ
1615a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → i ∈ ℂ)
1713, 16, 11mul12d 10189 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (2 · (i · 𝐴)) = (i · (2 · 𝐴)))
1816, 11mulcld 10004 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ)
19182timesd 11219 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (2 · (i · 𝐴)) = ((i · 𝐴) + (i · 𝐴)))
2017, 19eqtr3d 2657 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (i · (2 · 𝐴)) = ((i · 𝐴) + (i · 𝐴)))
2120fveq2d 6152 . . . . . . . . . . 11 (𝐴 ∈ ℂ → (exp‘(i · (2 · 𝐴))) = (exp‘((i · 𝐴) + (i · 𝐴))))
22 efadd 14749 . . . . . . . . . . . 12 (((i · 𝐴) ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (exp‘((i · 𝐴) + (i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
2318, 18, 22syl2anc 692 . . . . . . . . . . 11 (𝐴 ∈ ℂ → (exp‘((i · 𝐴) + (i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
2421, 23eqtrd 2655 . . . . . . . . . 10 (𝐴 ∈ ℂ → (exp‘(i · (2 · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
2524adantr 481 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘(i · (2 · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
26 sinval 14777 . . . . . . . . . . . . . . 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 2676 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = 0)
29 efcl 14738 . . . . . . . . . . . . . . . . . 18 ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ)
3018, 29syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ)
31 negicn 10226 . . . . . . . . . . . . . . . . . . . 20 -i ∈ ℂ
3231a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ ℂ → -i ∈ ℂ)
3332, 11mulcld 10004 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ)
34 efcl 14738 . . . . . . . . . . . . . . . . . 18 ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ)
3533, 34syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ)
3630, 35subcld 10336 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℂ → ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ)
37 2mulicn 11199 . . . . . . . . . . . . . . . . 17 (2 · i) ∈ ℂ
3837a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℂ → (2 · i) ∈ ℂ)
39 2muline0 11200 . . . . . . . . . . . . . . . . 17 (2 · i) ≠ 0
4039a1i 11 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℂ → (2 · i) ≠ 0)
4136, 38, 40diveq0ad 10755 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℂ → ((((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = 0 ↔ ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0))
4241adantr 481 . . . . . . . . . . . . . 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 10346 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0 ↔ (exp‘(i · 𝐴)) = (exp‘(-i · 𝐴))))
4544adantr 481 . . . . . . . . . . . . 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 6620 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴))))
48 efadd 14749 . . . . . . . . . . . . 13 (((i · 𝐴) ∈ ℂ ∧ (-i · 𝐴) ∈ ℂ) → (exp‘((i · 𝐴) + (-i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴))))
4918, 33, 48syl2anc 692 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (exp‘((i · 𝐴) + (-i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴))))
5049adantr 481 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘((i · 𝐴) + (-i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴))))
5147, 50eqtr4d 2658 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) = (exp‘((i · 𝐴) + (-i · 𝐴))))
5215negidi 10294 . . . . . . . . . . . . . . 15 (i + -i) = 0
5352oveq1i 6614 . . . . . . . . . . . . . 14 ((i + -i) · 𝐴) = (0 · 𝐴)
5416, 32, 11adddird 10009 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → ((i + -i) · 𝐴) = ((i · 𝐴) + (-i · 𝐴)))
5553, 54syl5reqr 2670 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → ((i · 𝐴) + (-i · 𝐴)) = (0 · 𝐴))
5611mul02d 10178 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (0 · 𝐴) = 0)
5755, 56eqtrd 2655 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → ((i · 𝐴) + (-i · 𝐴)) = 0)
5857fveq2d 6152 . . . . . . . . . . 11 (𝐴 ∈ ℂ → (exp‘((i · 𝐴) + (-i · 𝐴))) = (exp‘0))
5958adantr 481 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘((i · 𝐴) + (-i · 𝐴))) = (exp‘0))
60 ef0 14746 . . . . . . . . . . 11 (exp‘0) = 1
6160a1i 11 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘0) = 1)
6251, 59, 613eqtrd 2659 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) = 1)
6325, 62eqtrd 2655 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (exp‘(i · (2 · 𝐴))) = 1)
6463fveq2d 6152 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(exp‘(i · (2 · 𝐴)))) = (abs‘1))
65 abs1 13971 . . . . . . 7 (abs‘1) = 1
6664, 65syl6eq 2671 . . . . . 6 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(exp‘(i · (2 · 𝐴)))) = 1)
67 absefib 14853 . . . . . . . 8 ((2 · 𝐴) ∈ ℂ → ((2 · 𝐴) ∈ ℝ ↔ (abs‘(exp‘(i · (2 · 𝐴)))) = 1))
6867biimparc 504 . . . . . . 7 (((abs‘(exp‘(i · (2 · 𝐴)))) = 1 ∧ (2 · 𝐴) ∈ ℂ) → (2 · 𝐴) ∈ ℝ)
6968ancoms 469 . . . . . 6 (((2 · 𝐴) ∈ ℂ ∧ (abs‘(exp‘(i · (2 · 𝐴)))) = 1) → (2 · 𝐴) ∈ ℝ)
7014, 66, 69syl2an2r 875 . . . . 5 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (2 · 𝐴) ∈ ℝ)
71 mulre 13795 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 2 ∈ ℝ ∧ 2 ≠ 0) → (𝐴 ∈ ℝ ↔ (2 · 𝐴) ∈ ℝ))
72714animp1 38185 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 2 ∈ ℝ) ∧ 2 ≠ 0) ∧ (2 · 𝐴) ∈ ℝ) → 𝐴 ∈ ℝ)
73724an31 38186 . . . . 5 ((((2 ≠ 0 ∧ 2 ∈ ℝ) ∧ 𝐴 ∈ ℂ) ∧ (2 · 𝐴) ∈ ℝ) → 𝐴 ∈ ℝ)
745, 10, 12, 70, 73eel1111 38429 . . . 4 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 𝐴 ∈ ℝ)
753a1i 11 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → π ∈ ℝ+)
7674, 75modcld 12614 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 mod π) ∈ ℝ)
7776recnd 10012 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 mod π) ∈ ℂ)
7877sincld 14785 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (sin‘(𝐴 mod π)) ∈ ℂ)
791a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → π ∈ ℝ)
80 0re 9984 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ
8180, 1, 2ltleii 10104 . . . . . . . . . . . . . . . . . . . . 21 0 ≤ π
82 gt0ne0 10437 . . . . . . . . . . . . . . . . . . . . . . 23 ((π ∈ ℝ ∧ 0 < π) → π ≠ 0)
83823adant3 1079 . . . . . . . . . . . . . . . . . . . . . 22 ((π ∈ ℝ ∧ 0 < π ∧ 0 ≤ π) → π ≠ 0)
84833com23 1268 . . . . . . . . . . . . . . . . . . . . 21 ((π ∈ ℝ ∧ 0 ≤ π ∧ 0 < π) → π ≠ 0)
851, 81, 2, 84mp3an 1421 . . . . . . . . . . . . . . . . . . . 20 π ≠ 0
8685a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → π ≠ 0)
8774, 79, 86redivcld 10797 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 / π) ∈ ℝ)
8887flcld 12539 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (⌊‘(𝐴 / π)) ∈ ℤ)
8988znegcld 11428 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → -(⌊‘(𝐴 / π)) ∈ ℤ)
90 abssinper 24174 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ -(⌊‘(𝐴 / π)) ∈ ℤ) → (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))) = (abs‘(sin‘𝐴)))
9190eqcomd 2627 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ -(⌊‘(𝐴 / π)) ∈ ℤ) → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))))
9291ex 450 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℂ → (-(⌊‘(𝐴 / π)) ∈ ℤ → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π))))))
9392adantr 481 . . . . . . . . . . . . . . . 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 11427 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (⌊‘(𝐴 / π)) ∈ ℂ)
9695negcld 10323 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → -(⌊‘(𝐴 / π)) ∈ ℂ)
971recni 9996 . . . . . . . . . . . . . . . . . . . . 21 π ∈ ℂ
9897a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → π ∈ ℂ)
9996, 98mulcld 10004 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (-(⌊‘(𝐴 / π)) · π) ∈ ℂ)
10098, 95mulcld 10004 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (π · (⌊‘(𝐴 / π))) ∈ ℂ)
101100negcld 10323 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → -(π · (⌊‘(𝐴 / π))) ∈ ℂ)
10295, 98mulneg1d 10427 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (-(⌊‘(𝐴 / π)) · π) = -((⌊‘(𝐴 / π)) · π))
10395, 98mulcomd 10005 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((⌊‘(𝐴 / π)) · π) = (π · (⌊‘(𝐴 / π))))
104103negeqd 10219 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → -((⌊‘(𝐴 / π)) · π) = -(π · (⌊‘(𝐴 / π))))
105102, 104eqtrd 2655 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (-(⌊‘(𝐴 / π)) · π) = -(π · (⌊‘(𝐴 / π))))
106 oveq2 6612 . . . . . . . . . . . . . . . . . . . . 21 ((-(⌊‘(𝐴 / π)) · π) = -(π · (⌊‘(𝐴 / π))) → (𝐴 + (-(⌊‘(𝐴 / π)) · π)) = (𝐴 + -(π · (⌊‘(𝐴 / π)))))
107106ad3antrrr 765 . . . . . . . . . . . . . . . . . . . 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 38187 . . . . . . . . . . . . . . . . . . 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 38429 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 + (-(⌊‘(𝐴 / π)) · π)) = (𝐴 + -(π · (⌊‘(𝐴 / π)))))
11012, 100negsubd 10342 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 + -(π · (⌊‘(𝐴 / π)))) = (𝐴 − (π · (⌊‘(𝐴 / π)))))
111109, 110eqtrd 2655 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 + (-(⌊‘(𝐴 / π)) · π)) = (𝐴 − (π · (⌊‘(𝐴 / π)))))
112111fveq2d 6152 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π))) = (sin‘(𝐴 − (π · (⌊‘(𝐴 / π))))))
113112fveq2d 6152 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
11494, 113eqtrd 2655 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
115 modval 12610 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → (𝐴 mod π) = (𝐴 − (π · (⌊‘(𝐴 / π)))))
116115fveq2d 6152 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → (sin‘(𝐴 mod π)) = (sin‘(𝐴 − (π · (⌊‘(𝐴 / π))))))
117116fveq2d 6152 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → (abs‘(sin‘(𝐴 mod π))) = (abs‘(sin‘(𝐴 − (π · (⌊‘(𝐴 / π)))))))
1183, 117mpan2 706 . . . . . . . . . . . . . . 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 2658 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘𝐴)) = (abs‘(sin‘(𝐴 mod π))))
12127fveq2d 6152 . . . . . . . . . . . . . . 15 ((sin‘𝐴) = 0 → (abs‘(sin‘𝐴)) = (abs‘0))
122 abs0 13959 . . . . . . . . . . . . . . 15 (abs‘0) = 0
123121, 122syl6eq 2671 . . . . . . . . . . . . . 14 ((sin‘𝐴) = 0 → (abs‘(sin‘𝐴)) = 0)
124123adantl 482 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘𝐴)) = 0)
125120, 124eqtr3d 2657 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (abs‘(sin‘(𝐴 mod π))) = 0)
12678, 125abs00d 14119 . . . . . . . . . . 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 10078 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ 0 < (sin‘(𝐴 mod π))) → (sin‘(𝐴 mod π)) ≠ 0)
130129neneqd 2795 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ ∧ 0 < (sin‘(𝐴 mod π))) → ¬ (sin‘(𝐴 mod π)) = 0)
131130expcom 451 . . . . . . . . . . . . . 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 24163 . . . . . . . . . 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 10041 . . . . . . . . . . 11 0 ∈ ℝ*
1391rexri 10041 . . . . . . . . . . 11 π ∈ ℝ*
140 elioo2 12158 . . . . . . . . . . 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 707 . . . . . . . . . 10 ((𝐴 mod π) ∈ (0(,)π) ↔ ((𝐴 mod π) ∈ ℝ ∧ 0 < (𝐴 mod π) ∧ (𝐴 mod π) < π))
142141notbii 310 . . . . . . . . 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 1049 . . . . . . . . 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 310 . . . . . . . 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 12619 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → (𝐴 mod π) < π)
148147ancoms 469 . . . . . . . . 9 ((π ∈ ℝ+𝐴 ∈ ℝ) → (𝐴 mod π) < π)
1493, 74, 148sylancr 694 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 mod π) < π)
15076, 149jca 554 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ((𝐴 mod π) ∈ ℝ ∧ (𝐴 mod π) < π))
151 not12an2impnot1 38266 . . . . . . 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 692 . . . . . 6 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → ¬ 0 < (𝐴 mod π))
153 modge0 12618 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → 0 ≤ (𝐴 mod π))
154153ancoms 469 . . . . . . . 8 ((π ∈ ℝ+𝐴 ∈ ℝ) → 0 ≤ (𝐴 mod π))
1553, 74, 154sylancr 694 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 0 ≤ (𝐴 mod π))
156 leloe 10068 . . . . . . . . 9 ((0 ∈ ℝ ∧ (𝐴 mod π) ∈ ℝ) → (0 ≤ (𝐴 mod π) ↔ (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π))))
157156biimp3a 1429 . . . . . . . 8 ((0 ∈ ℝ ∧ (𝐴 mod π) ∈ ℝ ∧ 0 ≤ (𝐴 mod π)) → (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)))
158157idiALT 38165 . . . . . . 7 ((0 ∈ ℝ ∧ (𝐴 mod π) ∈ ℝ ∧ 0 ≤ (𝐴 mod π)) → (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)))
15980, 76, 155, 158mp3an2i 1426 . . . . . 6 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)))
160 pm2.53 388 . . . . . . . 8 ((0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)) → (¬ 0 < (𝐴 mod π) → 0 = (𝐴 mod π)))
161160imp 445 . . . . . . 7 (((0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)) ∧ ¬ 0 < (𝐴 mod π)) → 0 = (𝐴 mod π))
162161ancoms 469 . . . . . 6 ((¬ 0 < (𝐴 mod π) ∧ (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π))) → 0 = (𝐴 mod π))
163152, 159, 162syl2anc 692 . . . . 5 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → 0 = (𝐴 mod π))
164163eqcomd 2627 . . . 4 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 mod π) = 0)
165 mod0 12615 . . . . . 6 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+) → ((𝐴 mod π) = 0 ↔ (𝐴 / π) ∈ ℤ))
166165biimp3a 1429 . . . . 5 ((𝐴 ∈ ℝ ∧ π ∈ ℝ+ ∧ (𝐴 mod π) = 0) → (𝐴 / π) ∈ ℤ)
1671663com12 1266 . . . 4 ((π ∈ ℝ+𝐴 ∈ ℝ ∧ (𝐴 mod π) = 0) → (𝐴 / π) ∈ ℤ)
1683, 74, 164, 167mp3an2i 1426 . . 3 ((𝐴 ∈ ℂ ∧ (sin‘𝐴) = 0) → (𝐴 / π) ∈ ℤ)
169168ex 450 . 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 10746 . . . . 5 (𝐴 ∈ ℂ → ((𝐴 / π) · π) = 𝐴)
173172fveq2d 6152 . . . 4 (𝐴 ∈ ℂ → (sin‘((𝐴 / π) · π)) = (sin‘𝐴))
174 id 22 . . . . 5 ((𝐴 / π) ∈ ℤ → (𝐴 / π) ∈ ℤ)
175 sinkpi 24175 . . . . 5 ((𝐴 / π) ∈ ℤ → (sin‘((𝐴 / π) · π)) = 0)
176174, 175syl 17 . . . 4 ((𝐴 / π) ∈ ℤ → (sin‘((𝐴 / π) · π)) = 0)
177173, 176sylan9req 2676 . . 3 ((𝐴 ∈ ℂ ∧ (𝐴 / π) ∈ ℤ) → (sin‘𝐴) = 0)
178177ex 450 . 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 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4613  cfv 5847  (class class class)co 6604  cc 9878  cr 9879  0cc0 9880  1c1 9881  ici 9882   + caddc 9883   · cmul 9885  *cxr 10017   < clt 10018  cle 10019  cmin 10210  -cneg 10211   / cdiv 10628  2c2 11014  cz 11321  +crp 11776  (,)cioo 12117  cfl 12531   mod cmo 12608  abscabs 13908  expce 14717  sincsin 14719  πcpi 14722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ioc 12122  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-seq 12742  df-exp 12801  df-fac 13001  df-bc 13030  df-hash 13058  df-shft 13741  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-limsup 14136  df-clim 14153  df-rlim 14154  df-sum 14351  df-ef 14723  df-sin 14725  df-cos 14726  df-pi 14728  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-fbas 19662  df-fg 19663  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-lp 20850  df-perf 20851  df-cn 20941  df-cnp 20942  df-haus 21029  df-tx 21275  df-hmeo 21468  df-fil 21560  df-fm 21652  df-flim 21653  df-flf 21654  df-xms 22035  df-ms 22036  df-tms 22037  df-cncf 22589  df-limc 23536  df-dv 23537
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator