Step |
Hyp |
Ref |
Expression |
1 |
|
pire |
⊢ π ∈ ℝ |
2 |
|
2re |
⊢ 2 ∈ ℝ |
3 |
2 1
|
remulcli |
⊢ ( 2 · π ) ∈ ℝ |
4 |
3
|
rexri |
⊢ ( 2 · π ) ∈ ℝ* |
5 |
|
elico2 |
⊢ ( ( π ∈ ℝ ∧ ( 2 · π ) ∈ ℝ* ) → ( 𝐴 ∈ ( π [,) ( 2 · π ) ) ↔ ( 𝐴 ∈ ℝ ∧ π ≤ 𝐴 ∧ 𝐴 < ( 2 · π ) ) ) ) |
6 |
1 4 5
|
mp2an |
⊢ ( 𝐴 ∈ ( π [,) ( 2 · π ) ) ↔ ( 𝐴 ∈ ℝ ∧ π ≤ 𝐴 ∧ 𝐴 < ( 2 · π ) ) ) |
7 |
6
|
simp1bi |
⊢ ( 𝐴 ∈ ( π [,) ( 2 · π ) ) → 𝐴 ∈ ℝ ) |
8 |
|
0red |
⊢ ( 𝐴 ∈ ( π [,) ( 2 · π ) ) → 0 ∈ ℝ ) |
9 |
1
|
a1i |
⊢ ( 𝐴 ∈ ( π [,) ( 2 · π ) ) → π ∈ ℝ ) |
10 |
|
pipos |
⊢ 0 < π |
11 |
10
|
a1i |
⊢ ( 𝐴 ∈ ( π [,) ( 2 · π ) ) → 0 < π ) |
12 |
6
|
simp2bi |
⊢ ( 𝐴 ∈ ( π [,) ( 2 · π ) ) → π ≤ 𝐴 ) |
13 |
8 9 7 11 12
|
ltletrd |
⊢ ( 𝐴 ∈ ( π [,) ( 2 · π ) ) → 0 < 𝐴 ) |
14 |
6
|
simp3bi |
⊢ ( 𝐴 ∈ ( π [,) ( 2 · π ) ) → 𝐴 < ( 2 · π ) ) |
15 |
|
0xr |
⊢ 0 ∈ ℝ* |
16 |
|
elioo2 |
⊢ ( ( 0 ∈ ℝ* ∧ ( 2 · π ) ∈ ℝ* ) → ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) ↔ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < ( 2 · π ) ) ) ) |
17 |
15 4 16
|
mp2an |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) ↔ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < ( 2 · π ) ) ) |
18 |
7 13 14 17
|
syl3anbrc |
⊢ ( 𝐴 ∈ ( π [,) ( 2 · π ) ) → 𝐴 ∈ ( 0 (,) ( 2 · π ) ) ) |
19 |
|
cos02pilt1 |
⊢ ( 𝐴 ∈ ( 0 (,) ( 2 · π ) ) → ( cos ‘ 𝐴 ) < 1 ) |
20 |
18 19
|
syl |
⊢ ( 𝐴 ∈ ( π [,) ( 2 · π ) ) → ( cos ‘ 𝐴 ) < 1 ) |