Step |
Hyp |
Ref |
Expression |
1 |
|
elioore |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → 𝐴 ∈ ℝ ) |
2 |
1
|
recnd |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → 𝐴 ∈ ℂ ) |
3 |
|
2cnd |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → 2 ∈ ℂ ) |
4 |
|
picn |
⊢ π ∈ ℂ |
5 |
4
|
a1i |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → π ∈ ℂ ) |
6 |
|
2ne0 |
⊢ 2 ≠ 0 |
7 |
6
|
a1i |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → 2 ≠ 0 ) |
8 |
|
pire |
⊢ π ∈ ℝ |
9 |
|
pipos |
⊢ 0 < π |
10 |
8 9
|
gt0ne0ii |
⊢ π ≠ 0 |
11 |
10
|
a1i |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → π ≠ 0 ) |
12 |
2 3 5 7 11
|
divdiv1d |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ( ( 𝐴 / 2 ) / π ) = ( 𝐴 / ( 2 · π ) ) ) |
13 |
|
0zd |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → 0 ∈ ℤ ) |
14 |
|
2re |
⊢ 2 ∈ ℝ |
15 |
14 8
|
remulcli |
⊢ ( 2 · π ) ∈ ℝ |
16 |
15
|
a1i |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ( 2 · π ) ∈ ℝ ) |
17 |
|
0xr |
⊢ 0 ∈ ℝ* |
18 |
17
|
a1i |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → 0 ∈ ℝ* ) |
19 |
16
|
rexrd |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ( 2 · π ) ∈ ℝ* ) |
20 |
|
id |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → 𝐴 ∈ ( 0 (,) ( 2 · π ) ) ) |
21 |
|
ioogtlb |
⊢ ( ( 0 ∈ ℝ* ∧ ( 2 · π ) ∈ ℝ* ∧ 𝐴 ∈ ( 0 (,) ( 2 · π ) ) ) → 0 < 𝐴 ) |
22 |
18 19 20 21
|
syl3anc |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → 0 < 𝐴 ) |
23 |
|
2pos |
⊢ 0 < 2 |
24 |
14 8 23 9
|
mulgt0ii |
⊢ 0 < ( 2 · π ) |
25 |
24
|
a1i |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → 0 < ( 2 · π ) ) |
26 |
1 16 22 25
|
divgt0d |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → 0 < ( 𝐴 / ( 2 · π ) ) ) |
27 |
|
1rp |
⊢ 1 ∈ ℝ+ |
28 |
27
|
a1i |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → 1 ∈ ℝ+ ) |
29 |
16 25
|
elrpd |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ( 2 · π ) ∈ ℝ+ ) |
30 |
2
|
div1d |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ( 𝐴 / 1 ) = 𝐴 ) |
31 |
|
iooltub |
⊢ ( ( 0 ∈ ℝ* ∧ ( 2 · π ) ∈ ℝ* ∧ 𝐴 ∈ ( 0 (,) ( 2 · π ) ) ) → 𝐴 < ( 2 · π ) ) |
32 |
18 19 20 31
|
syl3anc |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → 𝐴 < ( 2 · π ) ) |
33 |
30 32
|
eqbrtrd |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ( 𝐴 / 1 ) < ( 2 · π ) ) |
34 |
1 28 29 33
|
ltdiv23d |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ( 𝐴 / ( 2 · π ) ) < 1 ) |
35 |
|
1e0p1 |
⊢ 1 = ( 0 + 1 ) |
36 |
34 35
|
breqtrdi |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ( 𝐴 / ( 2 · π ) ) < ( 0 + 1 ) ) |
37 |
|
btwnnz |
⊢ ( ( 0 ∈ ℤ ∧ 0 < ( 𝐴 / ( 2 · π ) ) ∧ ( 𝐴 / ( 2 · π ) ) < ( 0 + 1 ) ) → ¬ ( 𝐴 / ( 2 · π ) ) ∈ ℤ ) |
38 |
13 26 36 37
|
syl3anc |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ¬ ( 𝐴 / ( 2 · π ) ) ∈ ℤ ) |
39 |
12 38
|
eqneltrd |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ¬ ( ( 𝐴 / 2 ) / π ) ∈ ℤ ) |
40 |
2
|
halfcld |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ( 𝐴 / 2 ) ∈ ℂ ) |
41 |
|
sineq0 |
⊢ ( ( 𝐴 / 2 ) ∈ ℂ → ( ( sin ‘ ( 𝐴 / 2 ) ) = 0 ↔ ( ( 𝐴 / 2 ) / π ) ∈ ℤ ) ) |
42 |
40 41
|
syl |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ( ( sin ‘ ( 𝐴 / 2 ) ) = 0 ↔ ( ( 𝐴 / 2 ) / π ) ∈ ℤ ) ) |
43 |
39 42
|
mtbird |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ¬ ( sin ‘ ( 𝐴 / 2 ) ) = 0 ) |
44 |
43
|
neqned |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ( sin ‘ ( 𝐴 / 2 ) ) ≠ 0 ) |