Proof of Theorem sineq0
Step | Hyp | Ref
| Expression |
1 | | sinval 15580 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ∈ ℂ →
(sin‘𝐴) =
(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 ·
i))) |
2 | 1 | eqeq1d 2741 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ℂ →
((sin‘𝐴) = 0 ↔
(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) =
0)) |
3 | | ax-icn 10687 |
. . . . . . . . . . . . . . . . . . . 20
⊢ i ∈
ℂ |
4 | | mulcl 10712 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((i
∈ ℂ ∧ 𝐴
∈ ℂ) → (i · 𝐴) ∈ ℂ) |
5 | 3, 4 | mpan 690 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ → (i
· 𝐴) ∈
ℂ) |
6 | | efcl 15541 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((i
· 𝐴) ∈ ℂ
→ (exp‘(i · 𝐴)) ∈ ℂ) |
7 | 5, 6 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ ℂ →
(exp‘(i · 𝐴))
∈ ℂ) |
8 | | negicn 10978 |
. . . . . . . . . . . . . . . . . . . 20
⊢ -i ∈
ℂ |
9 | | mulcl 10712 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((-i
∈ ℂ ∧ 𝐴
∈ ℂ) → (-i · 𝐴) ∈ ℂ) |
10 | 8, 9 | mpan 690 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ → (-i
· 𝐴) ∈
ℂ) |
11 | | efcl 15541 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((-i
· 𝐴) ∈ ℂ
→ (exp‘(-i · 𝐴)) ∈ ℂ) |
12 | 10, 11 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ ℂ →
(exp‘(-i · 𝐴))
∈ ℂ) |
13 | 7, 12 | subcld 11088 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ∈ ℂ →
((exp‘(i · 𝐴))
− (exp‘(-i · 𝐴))) ∈ ℂ) |
14 | | 2mulicn 11952 |
. . . . . . . . . . . . . . . . . 18
⊢ (2
· i) ∈ ℂ |
15 | | 2muline0 11953 |
. . . . . . . . . . . . . . . . . 18
⊢ (2
· i) ≠ 0 |
16 | | diveq0 11399 |
. . . . . . . . . . . . . . . . . 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 1454 |
. . . . . . . . . . . . . . . . 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 11098 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ℂ →
(((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) = 0 ↔ (exp‘(i
· 𝐴)) =
(exp‘(-i · 𝐴)))) |
20 | 2, 18, 19 | 3bitrd 308 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ ℂ →
((sin‘𝐴) = 0 ↔
(exp‘(i · 𝐴))
= (exp‘(-i · 𝐴)))) |
21 | | oveq2 7191 |
. . . . . . . . . . . . . . . 16
⊢
((exp‘(i · 𝐴)) = (exp‘(-i · 𝐴)) → ((exp‘(i
· 𝐴)) ·
(exp‘(i · 𝐴)))
= ((exp‘(i · 𝐴)) · (exp‘(-i · 𝐴)))) |
22 | | 2cn 11804 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
ℂ |
23 | | mul12 10896 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((i
∈ ℂ ∧ 2 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · (2
· 𝐴)) = (2 ·
(i · 𝐴))) |
24 | 3, 22, 23 | mp3an12 1452 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ∈ ℂ → (i
· (2 · 𝐴)) =
(2 · (i · 𝐴))) |
25 | 5 | 2timesd 11972 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ∈ ℂ → (2
· (i · 𝐴)) =
((i · 𝐴) + (i
· 𝐴))) |
26 | 24, 25 | eqtrd 2774 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ∈ ℂ → (i
· (2 · 𝐴)) =
((i · 𝐴) + (i
· 𝐴))) |
27 | 26 | fveq2d 6691 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ →
(exp‘(i · (2 · 𝐴))) = (exp‘((i · 𝐴) + (i · 𝐴)))) |
28 | | efadd 15552 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((i
· 𝐴) ∈ ℂ
∧ (i · 𝐴) ∈
ℂ) → (exp‘((i · 𝐴) + (i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(i
· 𝐴)))) |
29 | 5, 5, 28 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ →
(exp‘((i · 𝐴)
+ (i · 𝐴))) =
((exp‘(i · 𝐴))
· (exp‘(i · 𝐴)))) |
30 | 27, 29 | eqtr2d 2775 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ ℂ →
((exp‘(i · 𝐴))
· (exp‘(i · 𝐴))) = (exp‘(i · (2 ·
𝐴)))) |
31 | | efadd 15552 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((i
· 𝐴) ∈ ℂ
∧ (-i · 𝐴)
∈ ℂ) → (exp‘((i · 𝐴) + (-i · 𝐴))) = ((exp‘(i · 𝐴)) · (exp‘(-i
· 𝐴)))) |
32 | 5, 10, 31 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ →
(exp‘((i · 𝐴)
+ (-i · 𝐴))) =
((exp‘(i · 𝐴))
· (exp‘(-i · 𝐴)))) |
33 | 3 | negidi 11046 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (i + -i)
= 0 |
34 | 33 | oveq1i 7193 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((i + -i)
· 𝐴) = (0 ·
𝐴) |
35 | | adddir 10723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((i
∈ ℂ ∧ -i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((i + -i) ·
𝐴) = ((i · 𝐴) + (-i · 𝐴))) |
36 | 3, 8, 35 | mp3an12 1452 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐴 ∈ ℂ → ((i + -i)
· 𝐴) = ((i ·
𝐴) + (-i · 𝐴))) |
37 | | mul02 10909 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐴 ∈ ℂ → (0
· 𝐴) =
0) |
38 | 34, 36, 37 | 3eqtr3a 2798 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ∈ ℂ → ((i
· 𝐴) + (-i ·
𝐴)) = 0) |
39 | 38 | fveq2d 6691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ∈ ℂ →
(exp‘((i · 𝐴)
+ (-i · 𝐴))) =
(exp‘0)) |
40 | | ef0 15549 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(exp‘0) = 1 |
41 | 39, 40 | eqtrdi 2790 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ →
(exp‘((i · 𝐴)
+ (-i · 𝐴))) =
1) |
42 | 32, 41 | eqtr3d 2776 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ ℂ →
((exp‘(i · 𝐴))
· (exp‘(-i · 𝐴))) = 1) |
43 | 30, 42 | eqeq12d 2755 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ∈ ℂ →
(((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) = ((exp‘(i ·
𝐴)) · (exp‘(-i
· 𝐴))) ↔
(exp‘(i · (2 · 𝐴))) = 1)) |
44 | | fveq2 6687 |
. . . . . . . . . . . . . . . . 17
⊢
((exp‘(i · (2 · 𝐴))) = 1 → (abs‘(exp‘(i
· (2 · 𝐴))))
= (abs‘1)) |
45 | 43, 44 | syl6bi 256 |
. . . . . . . . . . . . . . . 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 243 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℂ →
((sin‘𝐴) = 0 →
(abs‘(exp‘(i · (2 · 𝐴)))) = (abs‘1))) |
48 | | abs1 14760 |
. . . . . . . . . . . . . . . 16
⊢
(abs‘1) = 1 |
49 | 48 | eqeq2i 2752 |
. . . . . . . . . . . . . . 15
⊢
((abs‘(exp‘(i · (2 · 𝐴)))) = (abs‘1) ↔
(abs‘(exp‘(i · (2 · 𝐴)))) = 1) |
50 | | 2re 11803 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℝ |
51 | | 2ne0 11833 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ≠
0 |
52 | | mulre 14583 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℂ ∧ 2 ∈
ℝ ∧ 2 ≠ 0) → (𝐴 ∈ ℝ ↔ (2 · 𝐴) ∈
ℝ)) |
53 | 50, 51, 52 | mp3an23 1454 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (2
· 𝐴) ∈
ℝ)) |
54 | | mulcl 10712 |
. . . . . . . . . . . . . . . . . 18
⊢ ((2
∈ ℂ ∧ 𝐴
∈ ℂ) → (2 · 𝐴) ∈ ℂ) |
55 | 22, 54 | mpan 690 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ∈ ℂ → (2
· 𝐴) ∈
ℂ) |
56 | | absefib 15656 |
. . . . . . . . . . . . . . . . 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 283 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ ℂ →
((abs‘(exp‘(i · (2 · 𝐴)))) = 1 ↔ 𝐴 ∈ ℝ)) |
59 | 49, 58 | syl5bb 286 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℂ →
((abs‘(exp‘(i · (2 · 𝐴)))) = (abs‘1) ↔ 𝐴 ∈
ℝ)) |
60 | 47, 59 | sylibd 242 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ ℂ →
((sin‘𝐴) = 0 →
𝐴 ∈
ℝ)) |
61 | 60 | imp 410 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
𝐴 ∈
ℝ) |
62 | | pirp 25219 |
. . . . . . . . . . . 12
⊢ π
∈ ℝ+ |
63 | | modval 13343 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ π
∈ ℝ+) → (𝐴 mod π) = (𝐴 − (π ·
(⌊‘(𝐴 /
π))))) |
64 | 61, 62, 63 | sylancl 589 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 mod π) = (𝐴 − (π ·
(⌊‘(𝐴 /
π))))) |
65 | | picn 25217 |
. . . . . . . . . . . . 13
⊢ π
∈ ℂ |
66 | | pire 25216 |
. . . . . . . . . . . . . . . . 17
⊢ π
∈ ℝ |
67 | | pipos 25218 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 <
π |
68 | 66, 67 | gt0ne0ii 11267 |
. . . . . . . . . . . . . . . . 17
⊢ π ≠
0 |
69 | | redivcl 11450 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℝ ∧ π
∈ ℝ ∧ π ≠ 0) → (𝐴 / π) ∈ ℝ) |
70 | 66, 68, 69 | mp3an23 1454 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ℝ → (𝐴 / π) ∈
ℝ) |
71 | 61, 70 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 / π) ∈
ℝ) |
72 | 71 | flcld 13272 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(⌊‘(𝐴 / π))
∈ ℤ) |
73 | 72 | zcnd 12182 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(⌊‘(𝐴 / π))
∈ ℂ) |
74 | | mulcl 10712 |
. . . . . . . . . . . . 13
⊢ ((π
∈ ℂ ∧ (⌊‘(𝐴 / π)) ∈ ℂ) → (π
· (⌊‘(𝐴
/ π))) ∈ ℂ) |
75 | 65, 73, 74 | sylancr 590 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(π · (⌊‘(𝐴 / π))) ∈ ℂ) |
76 | | negsub 11025 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧ (π
· (⌊‘(𝐴
/ π))) ∈ ℂ) → (𝐴 + -(π · (⌊‘(𝐴 / π)))) = (𝐴 − (π ·
(⌊‘(𝐴 /
π))))) |
77 | 75, 76 | syldan 594 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 + -(π ·
(⌊‘(𝐴 /
π)))) = (𝐴 − (π
· (⌊‘(𝐴
/ π))))) |
78 | | mulcom 10714 |
. . . . . . . . . . . . . . 15
⊢ ((π
∈ ℂ ∧ (⌊‘(𝐴 / π)) ∈ ℂ) → (π
· (⌊‘(𝐴
/ π))) = ((⌊‘(𝐴 / π)) · π)) |
79 | 65, 73, 78 | sylancr 590 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(π · (⌊‘(𝐴 / π))) = ((⌊‘(𝐴 / π)) ·
π)) |
80 | 79 | negeqd 10971 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
-(π · (⌊‘(𝐴 / π))) = -((⌊‘(𝐴 / π)) ·
π)) |
81 | | mulneg1 11167 |
. . . . . . . . . . . . . 14
⊢
(((⌊‘(𝐴
/ π)) ∈ ℂ ∧ π ∈ ℂ) →
(-(⌊‘(𝐴 /
π)) · π) = -((⌊‘(𝐴 / π)) · π)) |
82 | 73, 65, 81 | sylancl 589 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(-(⌊‘(𝐴 /
π)) · π) = -((⌊‘(𝐴 / π)) · π)) |
83 | 80, 82 | eqtr4d 2777 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
-(π · (⌊‘(𝐴 / π))) = (-(⌊‘(𝐴 / π)) ·
π)) |
84 | 83 | oveq2d 7199 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 + -(π ·
(⌊‘(𝐴 /
π)))) = (𝐴 +
(-(⌊‘(𝐴 /
π)) · π))) |
85 | 64, 77, 84 | 3eqtr2d 2780 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 mod π) = (𝐴 + (-(⌊‘(𝐴 / π)) ·
π))) |
86 | 85 | fveq2d 6691 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(sin‘(𝐴 mod π)) =
(sin‘(𝐴 +
(-(⌊‘(𝐴 /
π)) · π)))) |
87 | 86 | fveq2d 6691 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(abs‘(sin‘(𝐴
mod π))) = (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) ·
π))))) |
88 | 72 | znegcld 12183 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
-(⌊‘(𝐴 / π))
∈ ℤ) |
89 | | abssinper 25278 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
-(⌊‘(𝐴 / π))
∈ ℤ) → (abs‘(sin‘(𝐴 + (-(⌊‘(𝐴 / π)) · π)))) =
(abs‘(sin‘𝐴))) |
90 | 88, 89 | syldan 594 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(abs‘(sin‘(𝐴 +
(-(⌊‘(𝐴 /
π)) · π)))) = (abs‘(sin‘𝐴))) |
91 | | simpr 488 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(sin‘𝐴) =
0) |
92 | 91 | fveq2d 6691 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(abs‘(sin‘𝐴)) =
(abs‘0)) |
93 | 87, 90, 92 | 3eqtrd 2778 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(abs‘(sin‘(𝐴
mod π))) = (abs‘0)) |
94 | | abs0 14748 |
. . . . . . 7
⊢
(abs‘0) = 0 |
95 | 93, 94 | eqtrdi 2790 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(abs‘(sin‘(𝐴
mod π))) = 0) |
96 | | modcl 13345 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ π
∈ ℝ+) → (𝐴 mod π) ∈ ℝ) |
97 | 61, 62, 96 | sylancl 589 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 mod π) ∈
ℝ) |
98 | | modlt 13352 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ π
∈ ℝ+) → (𝐴 mod π) < π) |
99 | 61, 62, 98 | sylancl 589 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 mod π) <
π) |
100 | 97, 99 | jca 515 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
((𝐴 mod π) ∈
ℝ ∧ (𝐴 mod π)
< π)) |
101 | 100 | biantrurd 536 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(0 < (𝐴 mod π) ↔
(((𝐴 mod π) ∈
ℝ ∧ (𝐴 mod π)
< π) ∧ 0 < (𝐴
mod π)))) |
102 | | 0re 10734 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℝ |
103 | | rexr 10778 |
. . . . . . . . . . . . 13
⊢ (0 ∈
ℝ → 0 ∈ ℝ*) |
104 | | rexr 10778 |
. . . . . . . . . . . . 13
⊢ (π
∈ ℝ → π ∈ ℝ*) |
105 | | elioo2 12875 |
. . . . . . . . . . . . 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 599 |
. . . . . . . . . . . 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 1098 |
. . . . . . . . . . 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 278 |
. . . . . . . . . 10
⊢ ((𝐴 mod π) ∈ (0(,)π)
↔ (((𝐴 mod π)
∈ ℝ ∧ (𝐴 mod
π) < π) ∧ 0 < (𝐴 mod π))) |
110 | 101, 109 | bitr4di 292 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(0 < (𝐴 mod π) ↔
(𝐴 mod π) ∈
(0(,)π))) |
111 | | sinq12gt0 25265 |
. . . . . . . . . 10
⊢ ((𝐴 mod π) ∈ (0(,)π)
→ 0 < (sin‘(𝐴
mod π))) |
112 | | elioore 12864 |
. . . . . . . . . . . 12
⊢ ((𝐴 mod π) ∈ (0(,)π)
→ (𝐴 mod π) ∈
ℝ) |
113 | 112 | resincld 15601 |
. . . . . . . . . . 11
⊢ ((𝐴 mod π) ∈ (0(,)π)
→ (sin‘(𝐴 mod
π)) ∈ ℝ) |
114 | | ltle 10820 |
. . . . . . . . . . . . 13
⊢ ((0
∈ ℝ ∧ (sin‘(𝐴 mod π)) ∈ ℝ) → (0 <
(sin‘(𝐴 mod π))
→ 0 ≤ (sin‘(𝐴
mod π)))) |
115 | 102, 113,
114 | sylancr 590 |
. . . . . . . . . . . 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 14885 |
. . . . . . . . . 10
⊢ ((𝐴 mod π) ∈ (0(,)π)
→ (abs‘(sin‘(𝐴 mod π))) = (sin‘(𝐴 mod π))) |
118 | 111, 117 | breqtrrd 5068 |
. . . . . . . . 9
⊢ ((𝐴 mod π) ∈ (0(,)π)
→ 0 < (abs‘(sin‘(𝐴 mod π)))) |
119 | 110, 118 | syl6bi 256 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(0 < (𝐴 mod π) →
0 < (abs‘(sin‘(𝐴 mod π))))) |
120 | | ltne 10828 |
. . . . . . . . 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 2951 |
. . . . . 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 13351 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ π
∈ ℝ+) → 0 ≤ (𝐴 mod π)) |
126 | 61, 62, 125 | sylancl 589 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) → 0
≤ (𝐴 mod
π)) |
127 | | leloe 10818 |
. . . . . . . 8
⊢ ((0
∈ ℝ ∧ (𝐴 mod
π) ∈ ℝ) → (0 ≤ (𝐴 mod π) ↔ (0 < (𝐴 mod π) ∨ 0 = (𝐴 mod π)))) |
128 | 102, 97, 127 | sylancr 590 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(0 ≤ (𝐴 mod π) ↔
(0 < (𝐴 mod π) ∨ 0
= (𝐴 mod
π)))) |
129 | 126, 128 | mpbid 235 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(0 < (𝐴 mod π) ∨ 0
= (𝐴 mod
π))) |
130 | 129 | ord 863 |
. . . . 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 2745 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 mod π) =
0) |
133 | | mod0 13348 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ π
∈ ℝ+) → ((𝐴 mod π) = 0 ↔ (𝐴 / π) ∈ ℤ)) |
134 | 61, 62, 133 | sylancl 589 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
((𝐴 mod π) = 0 ↔
(𝐴 / π) ∈
ℤ)) |
135 | 132, 134 | mpbid 235 |
. 2
⊢ ((𝐴 ∈ ℂ ∧
(sin‘𝐴) = 0) →
(𝐴 / π) ∈
ℤ) |
136 | | divcan1 11398 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ π
∈ ℂ ∧ π ≠ 0) → ((𝐴 / π) · π) = 𝐴) |
137 | 65, 68, 136 | mp3an23 1454 |
. . . 4
⊢ (𝐴 ∈ ℂ → ((𝐴 / π) · π) = 𝐴) |
138 | 137 | fveq2d 6691 |
. . 3
⊢ (𝐴 ∈ ℂ →
(sin‘((𝐴 / π)
· π)) = (sin‘𝐴)) |
139 | | sinkpi 25279 |
. . 3
⊢ ((𝐴 / π) ∈ ℤ →
(sin‘((𝐴 / π)
· π)) = 0) |
140 | 138, 139 | sylan9req 2795 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ (𝐴 / π) ∈ ℤ) →
(sin‘𝐴) =
0) |
141 | 135, 140 | impbida 801 |
1
⊢ (𝐴 ∈ ℂ →
((sin‘𝐴) = 0 ↔
(𝐴 / π) ∈
ℤ)) |