Proof of Theorem sineq0
| Step | Hyp | Ref
| Expression |
| 1 | | sinval 16158 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ∈ ℂ →
(sin‘𝐴) =
(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 ·
i))) |
| 2 | 1 | eqeq1d 2739 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ℂ →
((sin‘𝐴) = 0 ↔
(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) =
0)) |
| 3 | | ax-icn 11214 |
. . . . . . . . . . . . . . . . . . . 20
⊢ i ∈
ℂ |
| 4 | | mulcl 11239 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((i
∈ ℂ ∧ 𝐴
∈ ℂ) → (i · 𝐴) ∈ ℂ) |
| 5 | 3, 4 | mpan 690 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ → (i
· 𝐴) ∈
ℂ) |
| 6 | | efcl 16118 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((i
· 𝐴) ∈ ℂ
→ (exp‘(i · 𝐴)) ∈ ℂ) |
| 7 | 5, 6 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ ℂ →
(exp‘(i · 𝐴))
∈ ℂ) |
| 8 | | negicn 11509 |
. . . . . . . . . . . . . . . . . . . 20
⊢ -i ∈
ℂ |
| 9 | | mulcl 11239 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((-i
∈ ℂ ∧ 𝐴
∈ ℂ) → (-i · 𝐴) ∈ ℂ) |
| 10 | 8, 9 | mpan 690 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ → (-i
· 𝐴) ∈
ℂ) |
| 11 | | efcl 16118 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((-i
· 𝐴) ∈ ℂ
→ (exp‘(-i · 𝐴)) ∈ ℂ) |
| 12 | 10, 11 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ ℂ →
(exp‘(-i · 𝐴))
∈ ℂ) |
| 13 | 7, 12 | subcld 11620 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ∈ ℂ →
((exp‘(i · 𝐴))
− (exp‘(-i · 𝐴))) ∈ ℂ) |
| 14 | | 2mulicn 12489 |
. . . . . . . . . . . . . . . . . 18
⊢ (2
· i) ∈ ℂ |
| 15 | | 2muline0 12490 |
. . . . . . . . . . . . . . . . . 18
⊢ (2
· i) ≠ 0 |
| 16 | | diveq0 11932 |
. . . . . . . . . . . . . . . . . 18
⊢
((((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ ∧ (2
· i) ∈ ℂ ∧ (2 · i) ≠ 0) → ((((exp‘(i
· 𝐴)) −
(exp‘(-i · 𝐴))) / (2 · i)) = 0 ↔
((exp‘(i · 𝐴))
− (exp‘(-i · 𝐴))) = 0)) |
| 17 | 14, 15, 16 | mp3an23 1455 |
. . . . . . . . . . . . . . . . 17
⊢
(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) ∈ ℂ →
((((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = 0 ↔
((exp‘(i · 𝐴))
− (exp‘(-i · 𝐴))) = 0)) |
| 18 | 13, 17 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ℂ →
((((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) = 0 ↔
((exp‘(i · 𝐴))
− (exp‘(-i · 𝐴))) = 0)) |
| 19 | 7, 12 | subeq0ad 11630 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ℂ →
(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0 ↔ (exp‘(i
· 𝐴)) =
(exp‘(-i · 𝐴)))) |
| 20 | 2, 18, 19 | 3bitrd 305 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ ℂ →
((sin‘𝐴) = 0 ↔
(exp‘(i · 𝐴))
= (exp‘(-i · 𝐴)))) |
| 21 | | oveq2 7439 |
. . . . . . . . . . . . . . . 16
⊢
((exp‘(i · 𝐴)) = (exp‘(-i · 𝐴)) → ((exp‘(i
· 𝐴)) ·
(exp‘(i · 𝐴)))
= ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴)))) |
| 22 | | 2cn 12341 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
ℂ |
| 23 | | mul12 11426 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((i
∈ ℂ ∧ 2 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · (2
· 𝐴)) = (2 ·
(i · 𝐴))) |
| 24 | 3, 22, 23 | mp3an12 1453 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ∈ ℂ → (i
· (2 · 𝐴)) =
(2 · (i · 𝐴))) |
| 25 | 5 | 2timesd 12509 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ∈ ℂ → (2
· (i · 𝐴)) =
((i · 𝐴) + (i
· 𝐴))) |
| 26 | 24, 25 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ∈ ℂ → (i
· (2 · 𝐴)) =
((i · 𝐴) + (i
· 𝐴))) |
| 27 | 26 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ →
(exp‘(i · (2 · 𝐴))) = (exp‘((i · 𝐴) + (i · 𝐴)))) |
| 28 | | efadd 16130 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((i
· 𝐴) ∈ ℂ
∧ (i · 𝐴) ∈
ℂ) → (exp‘((i · 𝐴) + (i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i
· 𝐴)))) |
| 29 | 5, 5, 28 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ →
(exp‘((i · 𝐴)
+ (i · 𝐴))) =
((exp‘(i · 𝐴))
· (exp‘(i · 𝐴)))) |
| 30 | 27, 29 | eqtr2d 2778 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ ℂ →
((exp‘(i · 𝐴))
· (exp‘(i · 𝐴))) = (exp‘(i · (2 ·
𝐴)))) |
| 31 | | efadd 16130 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((i
· 𝐴) ∈ ℂ
∧ (-i · 𝐴)
∈ ℂ) → (exp‘((i · 𝐴) + (-i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i
· 𝐴)))) |
| 32 | 5, 10, 31 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ →
(exp‘((i · 𝐴)
+ (-i · 𝐴))) =
((exp‘(i · 𝐴))
· (exp‘(-i · 𝐴)))) |
| 33 | 3 | negidi 11578 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (i + -i)
= 0 |
| 34 | 33 | oveq1i 7441 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((i + -i)
· 𝐴) = (0 ·
𝐴) |
| 35 | | adddir 11252 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((i
∈ ℂ ∧ -i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((i + -i) ·
𝐴) = ((i · 𝐴) + (-i · 𝐴))) |
| 36 | 3, 8, 35 | mp3an12 1453 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐴 ∈ ℂ → ((i + -i)
· 𝐴) = ((i ·
𝐴) + (-i · 𝐴))) |
| 37 | | mul02 11439 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐴 ∈ ℂ → (0
· 𝐴) =
0) |
| 38 | 34, 36, 37 | 3eqtr3a 2801 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ∈ ℂ → ((i
· 𝐴) + (-i ·
𝐴)) = 0) |
| 39 | 38 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ∈ ℂ →
(exp‘((i · 𝐴)
+ (-i · 𝐴))) =
(exp‘0)) |
| 40 | | ef0 16127 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(exp‘0) = 1 |
| 41 | 39, 40 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ →
(exp‘((i · 𝐴)
+ (-i · 𝐴))) =
1) |
| 42 | 32, 41 | eqtr3d 2779 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ ℂ →
((exp‘(i · 𝐴))
· (exp‘(-i · 𝐴))) = 1) |
| 43 | 30, 42 | eqeq12d 2753 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ∈ ℂ →
(((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) = ((exp‘(i ·
𝐴)) · (exp‘(-i
· 𝐴))) ↔
(exp‘(i · (2 · 𝐴))) = 1)) |
| 44 | | fveq2 6906 |
. . . . . . . . . . . . . . . . 17
⊢
((exp‘(i · (2 · 𝐴))) = 1 → (abs‘(exp‘(i
· (2 · 𝐴))))
= (abs‘1)) |
| 45 | 43, 44 | biimtrdi 253 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ℂ →
(((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) = ((exp‘(i ·
𝐴)) · (exp‘(-i
· 𝐴))) →
(abs‘(exp‘(i · (2 · 𝐴)))) = (abs‘1))) |
| 46 | 21, 45 | syl5 34 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ ℂ →
((exp‘(i · 𝐴))
= (exp‘(-i · 𝐴)) → (abs‘(exp‘(i ·
(2 · 𝐴)))) =
(abs‘1))) |
| 47 | 20, 46 | sylbid 240 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℂ →
((sin‘𝐴) = 0 →
(abs‘(exp‘(i · (2 · 𝐴)))) = (abs‘1))) |
| 48 | | abs1 15336 |
. . . . . . . . . . . . . . . 16
⊢
(abs‘1) = 1 |
| 49 | 48 | eqeq2i 2750 |
. . . . . . . . . . . . . . 15
⊢
((abs‘(exp‘(i · (2 · 𝐴)))) = (abs‘1) ↔
(abs‘(exp‘(i · (2 · 𝐴)))) = 1) |
| 50 | | 2re 12340 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℝ |
| 51 | | 2ne0 12370 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ≠
0 |
| 52 | | mulre 15160 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℂ ∧ 2 ∈
ℝ ∧ 2 ≠ 0) → (𝐴 ∈ ℝ ↔ (2 · 𝐴) ∈
ℝ)) |
| 53 | 50, 51, 52 | mp3an23 1455 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (2
· 𝐴) ∈
ℝ)) |
| 54 | | mulcl 11239 |
. . . . . . . . . . . . . . . . . 18
⊢ ((2
∈ ℂ ∧ 𝐴
∈ ℂ) → (2 · 𝐴) ∈ ℂ) |
| 55 | 22, 54 | mpan 690 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ∈ ℂ → (2
· 𝐴) ∈
ℂ) |
| 56 | | absefib 16234 |
. . . . . . . . . . . . . . . . 17
⊢ ((2
· 𝐴) ∈ ℂ
→ ((2 · 𝐴)
∈ ℝ ↔ (abs‘(exp‘(i · (2 · 𝐴)))) = 1)) |
| 57 | 55, 56 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ℂ → ((2
· 𝐴) ∈ ℝ
↔ (abs‘(exp‘(i · (2 · 𝐴)))) = 1)) |
| 58 | 53, 57 | bitr2d 280 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ ℂ →
((abs‘(exp‘(i · (2 · 𝐴)))) = 1 ↔ 𝐴 ∈ ℝ)) |
| 59 | 49, 58 | bitrid 283 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℂ →
((abs‘(exp‘(i · (2 · 𝐴)))) = (abs‘1) ↔ 𝐴 ∈
ℝ)) |
| 60 | 47, 59 | sylibd 239 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ ℂ →
((sin‘𝐴) = 0 →
𝐴 ∈
ℝ)) |
| 61 | 60 | imp 406 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
𝐴 ∈
ℝ) |
| 62 | | pirp 26503 |
. . . . . . . . . . . 12
⊢ π
∈ ℝ+ |
| 63 | | modval 13911 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ π
∈ ℝ+) → (𝐴 mod π) = (𝐴 − (π ·
(⌊‘(𝐴 /
π))))) |
| 64 | 61, 62, 63 | sylancl 586 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 mod π) = (𝐴 − (π ·
(⌊‘(𝐴 /
π))))) |
| 65 | | picn 26501 |
. . . . . . . . . . . . 13
⊢ π
∈ ℂ |
| 66 | | pire 26500 |
. . . . . . . . . . . . . . . . 17
⊢ π
∈ ℝ |
| 67 | | pipos 26502 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 <
π |
| 68 | 66, 67 | gt0ne0ii 11799 |
. . . . . . . . . . . . . . . . 17
⊢ π ≠
0 |
| 69 | | redivcl 11986 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℝ ∧ π
∈ ℝ ∧ π ≠ 0) → (𝐴 / π) ∈ ℝ) |
| 70 | 66, 68, 69 | mp3an23 1455 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ℝ → (𝐴 / π) ∈
ℝ) |
| 71 | 61, 70 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 / π) ∈
ℝ) |
| 72 | 71 | flcld 13838 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(⌊‘(𝐴 / π))
∈ ℤ) |
| 73 | 72 | zcnd 12723 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(⌊‘(𝐴 / π))
∈ ℂ) |
| 74 | | mulcl 11239 |
. . . . . . . . . . . . 13
⊢ ((π
∈ ℂ ∧ (⌊‘(𝐴 / π)) ∈ ℂ) → (π
· (⌊‘(𝐴
/ π))) ∈ ℂ) |
| 75 | 65, 73, 74 | sylancr 587 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(π · (⌊‘(𝐴 / π))) ∈ ℂ) |
| 76 | | negsub 11557 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧ (π
· (⌊‘(𝐴
/ π))) ∈ ℂ) → (𝐴 + -(π · (⌊‘(𝐴 / π)))) = (𝐴 − (π ·
(⌊‘(𝐴 /
π))))) |
| 77 | 75, 76 | syldan 591 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 + -(π ·
(⌊‘(𝐴 /
π)))) = (𝐴 − (π
· (⌊‘(𝐴
/ π))))) |
| 78 | | mulcom 11241 |
. . . . . . . . . . . . . . 15
⊢ ((π
∈ ℂ ∧ (⌊‘(𝐴 / π)) ∈ ℂ) → (π
· (⌊‘(𝐴
/ π))) = ((⌊‘(𝐴 / π)) · π)) |
| 79 | 65, 73, 78 | sylancr 587 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(π · (⌊‘(𝐴 / π))) = ((⌊‘(𝐴 / π)) ·
π)) |
| 80 | 79 | negeqd 11502 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
-(π · (⌊‘(𝐴 / π))) = -((⌊‘(𝐴 / π)) ·
π)) |
| 81 | | mulneg1 11699 |
. . . . . . . . . . . . . 14
⊢
(((⌊‘(𝐴
/ π)) ∈ ℂ ∧ π ∈ ℂ) →
(-(⌊‘(𝐴 /
π)) · π) = -((⌊‘(𝐴 / π)) · π)) |
| 82 | 73, 65, 81 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(-(⌊‘(𝐴 /
π)) · π) = -((⌊‘(𝐴 / π)) · π)) |
| 83 | 80, 82 | eqtr4d 2780 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
-(π · (⌊‘(𝐴 / π))) = (-(⌊‘(𝐴 / π)) ·
π)) |
| 84 | 83 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 + -(π ·
(⌊‘(𝐴 /
π)))) = (𝐴 +
(-(⌊‘(𝐴 /
π)) · π))) |
| 85 | 64, 77, 84 | 3eqtr2d 2783 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 mod π) = (𝐴 + (-(⌊‘(𝐴 / π)) ·
π))) |
| 86 | 85 | fveq2d 6910 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(sin‘(𝐴 mod π)) =
(sin‘(𝐴 +
(-(⌊‘(𝐴 /
π)) · π)))) |
| 87 | 86 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(abs‘(sin‘(𝐴
mod π))) = (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) ·
π))))) |
| 88 | 72 | znegcld 12724 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
-(⌊‘(𝐴 / π))
∈ ℤ) |
| 89 | | abssinper 26563 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
-(⌊‘(𝐴 / π))
∈ ℤ) → (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))) =
(abs‘(sin‘𝐴))) |
| 90 | 88, 89 | syldan 591 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(abs‘(sin‘(𝐴 +
(-(⌊‘(𝐴 /
π)) · π)))) = (abs‘(sin‘𝐴))) |
| 91 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(sin‘𝐴) =
0) |
| 92 | 91 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(abs‘(sin‘𝐴)) =
(abs‘0)) |
| 93 | 87, 90, 92 | 3eqtrd 2781 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(abs‘(sin‘(𝐴
mod π))) = (abs‘0)) |
| 94 | | abs0 15324 |
. . . . . . 7
⊢
(abs‘0) = 0 |
| 95 | 93, 94 | eqtrdi 2793 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(abs‘(sin‘(𝐴
mod π))) = 0) |
| 96 | | modcl 13913 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ π
∈ ℝ+) → (𝐴 mod π) ∈ ℝ) |
| 97 | 61, 62, 96 | sylancl 586 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 mod π) ∈
ℝ) |
| 98 | | modlt 13920 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ π
∈ ℝ+) → (𝐴 mod π) < π) |
| 99 | 61, 62, 98 | sylancl 586 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 mod π) <
π) |
| 100 | 97, 99 | jca 511 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
((𝐴 mod π) ∈
ℝ ∧ (𝐴 mod π)
< π)) |
| 101 | 100 | biantrurd 532 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(0 < (𝐴 mod π) ↔
(((𝐴 mod π) ∈
ℝ ∧ (𝐴 mod π)
< π) ∧ 0 < (𝐴
mod π)))) |
| 102 | | 0re 11263 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℝ |
| 103 | | rexr 11307 |
. . . . . . . . . . . . 13
⊢ (0 ∈
ℝ → 0 ∈ ℝ*) |
| 104 | | rexr 11307 |
. . . . . . . . . . . . 13
⊢ (π
∈ ℝ → π ∈ ℝ*) |
| 105 | | elioo2 13428 |
. . . . . . . . . . . . 13
⊢ ((0
∈ ℝ* ∧ π ∈ ℝ*) →
((𝐴 mod π) ∈
(0(,)π) ↔ ((𝐴 mod
π) ∈ ℝ ∧ 0 < (𝐴 mod π) ∧ (𝐴 mod π) < π))) |
| 106 | 103, 104,
105 | syl2an 596 |
. . . . . . . . . . . 12
⊢ ((0
∈ ℝ ∧ π ∈ ℝ) → ((𝐴 mod π) ∈ (0(,)π) ↔ ((𝐴 mod π) ∈ ℝ ∧
0 < (𝐴 mod π) ∧
(𝐴 mod π) <
π))) |
| 107 | 102, 66, 106 | mp2an 692 |
. . . . . . . . . . 11
⊢ ((𝐴 mod π) ∈ (0(,)π)
↔ ((𝐴 mod π) ∈
ℝ ∧ 0 < (𝐴 mod
π) ∧ (𝐴 mod π)
< π)) |
| 108 | | 3anan32 1097 |
. . . . . . . . . . 11
⊢ (((𝐴 mod π) ∈ ℝ ∧
0 < (𝐴 mod π) ∧
(𝐴 mod π) < π)
↔ (((𝐴 mod π)
∈ ℝ ∧ (𝐴 mod
π) < π) ∧ 0 < (𝐴 mod π))) |
| 109 | 107, 108 | bitri 275 |
. . . . . . . . . 10
⊢ ((𝐴 mod π) ∈ (0(,)π)
↔ (((𝐴 mod π)
∈ ℝ ∧ (𝐴 mod
π) < π) ∧ 0 < (𝐴 mod π))) |
| 110 | 101, 109 | bitr4di 289 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(0 < (𝐴 mod π) ↔
(𝐴 mod π) ∈
(0(,)π))) |
| 111 | | sinq12gt0 26549 |
. . . . . . . . . 10
⊢ ((𝐴 mod π) ∈ (0(,)π)
→ 0 < (sin‘(𝐴
mod π))) |
| 112 | | elioore 13417 |
. . . . . . . . . . . 12
⊢ ((𝐴 mod π) ∈ (0(,)π)
→ (𝐴 mod π) ∈
ℝ) |
| 113 | 112 | resincld 16179 |
. . . . . . . . . . 11
⊢ ((𝐴 mod π) ∈ (0(,)π)
→ (sin‘(𝐴 mod
π)) ∈ ℝ) |
| 114 | | ltle 11349 |
. . . . . . . . . . . . 13
⊢ ((0
∈ ℝ ∧ (sin‘(𝐴 mod π)) ∈ ℝ) → (0 <
(sin‘(𝐴 mod π))
→ 0 ≤ (sin‘(𝐴
mod π)))) |
| 115 | 102, 113,
114 | sylancr 587 |
. . . . . . . . . . . 12
⊢ ((𝐴 mod π) ∈ (0(,)π)
→ (0 < (sin‘(𝐴 mod π)) → 0 ≤ (sin‘(𝐴 mod π)))) |
| 116 | 111, 115 | mpd 15 |
. . . . . . . . . . 11
⊢ ((𝐴 mod π) ∈ (0(,)π)
→ 0 ≤ (sin‘(𝐴
mod π))) |
| 117 | 113, 116 | absidd 15461 |
. . . . . . . . . 10
⊢ ((𝐴 mod π) ∈ (0(,)π)
→ (abs‘(sin‘(𝐴 mod π))) = (sin‘(𝐴 mod π))) |
| 118 | 111, 117 | breqtrrd 5171 |
. . . . . . . . 9
⊢ ((𝐴 mod π) ∈ (0(,)π)
→ 0 < (abs‘(sin‘(𝐴 mod π)))) |
| 119 | 110, 118 | biimtrdi 253 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(0 < (𝐴 mod π) →
0 < (abs‘(sin‘(𝐴 mod π))))) |
| 120 | | ltne 11358 |
. . . . . . . . 9
⊢ ((0
∈ ℝ ∧ 0 < (abs‘(sin‘(𝐴 mod π)))) →
(abs‘(sin‘(𝐴
mod π))) ≠ 0) |
| 121 | 102, 120 | mpan 690 |
. . . . . . . 8
⊢ (0 <
(abs‘(sin‘(𝐴
mod π))) → (abs‘(sin‘(𝐴 mod π))) ≠ 0) |
| 122 | 119, 121 | syl6 35 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(0 < (𝐴 mod π) →
(abs‘(sin‘(𝐴
mod π))) ≠ 0)) |
| 123 | 122 | necon2bd 2956 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
((abs‘(sin‘(𝐴
mod π))) = 0 → ¬ 0 < (𝐴 mod π))) |
| 124 | 95, 123 | mpd 15 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
¬ 0 < (𝐴 mod
π)) |
| 125 | | modge0 13919 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ π
∈ ℝ+) → 0 ≤ (𝐴 mod π)) |
| 126 | 61, 62, 125 | sylancl 586 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) → 0
≤ (𝐴 mod
π)) |
| 127 | | leloe 11347 |
. . . . . . . 8
⊢ ((0
∈ ℝ ∧ (𝐴 mod
π) ∈ ℝ) → (0 ≤ (𝐴 mod π) ↔ (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)))) |
| 128 | 102, 97, 127 | sylancr 587 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(0 ≤ (𝐴 mod π) ↔
(0 < (𝐴 mod π) ∨ 0
= (𝐴 mod
π)))) |
| 129 | 126, 128 | mpbid 232 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(0 < (𝐴 mod π) ∨ 0
= (𝐴 mod
π))) |
| 130 | 129 | ord 865 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(¬ 0 < (𝐴 mod π)
→ 0 = (𝐴 mod
π))) |
| 131 | 124, 130 | mpd 15 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) → 0
= (𝐴 mod
π)) |
| 132 | 131 | eqcomd 2743 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 mod π) =
0) |
| 133 | | mod0 13916 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ π
∈ ℝ+) → ((𝐴 mod π) = 0 ↔ (𝐴 / π) ∈ ℤ)) |
| 134 | 61, 62, 133 | sylancl 586 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
((𝐴 mod π) = 0 ↔
(𝐴 / π) ∈
ℤ)) |
| 135 | 132, 134 | mpbid 232 |
. 2
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 / π) ∈
ℤ) |
| 136 | | divcan1 11931 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ π
∈ ℂ ∧ π ≠ 0) → ((𝐴 / π) · π) = 𝐴) |
| 137 | 65, 68, 136 | mp3an23 1455 |
. . . 4
⊢ (𝐴 ∈ ℂ → ((𝐴 / π) · π) = 𝐴) |
| 138 | 137 | fveq2d 6910 |
. . 3
⊢ (𝐴 ∈ ℂ →
(sin‘((𝐴 / π)
· π)) = (sin‘𝐴)) |
| 139 | | sinkpi 26564 |
. . 3
⊢ ((𝐴 / π) ∈ ℤ →
(sin‘((𝐴 / π)
· π)) = 0) |
| 140 | 138, 139 | sylan9req 2798 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ (𝐴 / π) ∈ ℤ) →
(sin‘𝐴) =
0) |
| 141 | 135, 140 | impbida 801 |
1
⊢ (𝐴 ∈ ℂ →
((sin‘𝐴) = 0 ↔
(𝐴 / π) ∈
ℤ)) |