Step |
Hyp |
Ref |
Expression |
1 |
|
0lt1 |
⊢ 0 < 1 |
2 |
|
0re |
⊢ 0 ∈ ℝ |
3 |
|
1re |
⊢ 1 ∈ ℝ |
4 |
2 3
|
ltnsymi |
⊢ ( 0 < 1 → ¬ 1 < 0 ) |
5 |
1 4
|
ax-mp |
⊢ ¬ 1 < 0 |
6 |
|
lt0neg1 |
⊢ ( 1 ∈ ℝ → ( 1 < 0 ↔ 0 < - 1 ) ) |
7 |
3 6
|
ax-mp |
⊢ ( 1 < 0 ↔ 0 < - 1 ) |
8 |
5 7
|
mtbi |
⊢ ¬ 0 < - 1 |
9 |
|
pire |
⊢ π ∈ ℝ |
10 |
9
|
rehalfcli |
⊢ ( π / 2 ) ∈ ℝ |
11 |
|
2re |
⊢ 2 ∈ ℝ |
12 |
|
pipos |
⊢ 0 < π |
13 |
|
2pos |
⊢ 0 < 2 |
14 |
9 11 12 13
|
divgt0ii |
⊢ 0 < ( π / 2 ) |
15 |
|
4re |
⊢ 4 ∈ ℝ |
16 |
|
pigt2lt4 |
⊢ ( 2 < π ∧ π < 4 ) |
17 |
16
|
simpri |
⊢ π < 4 |
18 |
9 15 17
|
ltleii |
⊢ π ≤ 4 |
19 |
11 13
|
pm3.2i |
⊢ ( 2 ∈ ℝ ∧ 0 < 2 ) |
20 |
|
ledivmul |
⊢ ( ( π ∈ ℝ ∧ 2 ∈ ℝ ∧ ( 2 ∈ ℝ ∧ 0 < 2 ) ) → ( ( π / 2 ) ≤ 2 ↔ π ≤ ( 2 · 2 ) ) ) |
21 |
9 11 19 20
|
mp3an |
⊢ ( ( π / 2 ) ≤ 2 ↔ π ≤ ( 2 · 2 ) ) |
22 |
|
2t2e4 |
⊢ ( 2 · 2 ) = 4 |
23 |
22
|
breq2i |
⊢ ( π ≤ ( 2 · 2 ) ↔ π ≤ 4 ) |
24 |
21 23
|
bitr2i |
⊢ ( π ≤ 4 ↔ ( π / 2 ) ≤ 2 ) |
25 |
18 24
|
mpbi |
⊢ ( π / 2 ) ≤ 2 |
26 |
|
0xr |
⊢ 0 ∈ ℝ* |
27 |
|
elioc2 |
⊢ ( ( 0 ∈ ℝ* ∧ 2 ∈ ℝ ) → ( ( π / 2 ) ∈ ( 0 (,] 2 ) ↔ ( ( π / 2 ) ∈ ℝ ∧ 0 < ( π / 2 ) ∧ ( π / 2 ) ≤ 2 ) ) ) |
28 |
26 11 27
|
mp2an |
⊢ ( ( π / 2 ) ∈ ( 0 (,] 2 ) ↔ ( ( π / 2 ) ∈ ℝ ∧ 0 < ( π / 2 ) ∧ ( π / 2 ) ≤ 2 ) ) |
29 |
10 14 25 28
|
mpbir3an |
⊢ ( π / 2 ) ∈ ( 0 (,] 2 ) |
30 |
|
sin02gt0 |
⊢ ( ( π / 2 ) ∈ ( 0 (,] 2 ) → 0 < ( sin ‘ ( π / 2 ) ) ) |
31 |
29 30
|
ax-mp |
⊢ 0 < ( sin ‘ ( π / 2 ) ) |
32 |
|
breq2 |
⊢ ( ( sin ‘ ( π / 2 ) ) = - 1 → ( 0 < ( sin ‘ ( π / 2 ) ) ↔ 0 < - 1 ) ) |
33 |
31 32
|
mpbii |
⊢ ( ( sin ‘ ( π / 2 ) ) = - 1 → 0 < - 1 ) |
34 |
8 33
|
mto |
⊢ ¬ ( sin ‘ ( π / 2 ) ) = - 1 |
35 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
36 |
|
resincl |
⊢ ( ( π / 2 ) ∈ ℝ → ( sin ‘ ( π / 2 ) ) ∈ ℝ ) |
37 |
10 36
|
ax-mp |
⊢ ( sin ‘ ( π / 2 ) ) ∈ ℝ |
38 |
37 31
|
gt0ne0ii |
⊢ ( sin ‘ ( π / 2 ) ) ≠ 0 |
39 |
38
|
neii |
⊢ ¬ ( sin ‘ ( π / 2 ) ) = 0 |
40 |
|
2ne0 |
⊢ 2 ≠ 0 |
41 |
40
|
neii |
⊢ ¬ 2 = 0 |
42 |
9
|
recni |
⊢ π ∈ ℂ |
43 |
|
2cn |
⊢ 2 ∈ ℂ |
44 |
42 43 40
|
divcan2i |
⊢ ( 2 · ( π / 2 ) ) = π |
45 |
44
|
fveq2i |
⊢ ( sin ‘ ( 2 · ( π / 2 ) ) ) = ( sin ‘ π ) |
46 |
10
|
recni |
⊢ ( π / 2 ) ∈ ℂ |
47 |
|
sin2t |
⊢ ( ( π / 2 ) ∈ ℂ → ( sin ‘ ( 2 · ( π / 2 ) ) ) = ( 2 · ( ( sin ‘ ( π / 2 ) ) · ( cos ‘ ( π / 2 ) ) ) ) ) |
48 |
46 47
|
ax-mp |
⊢ ( sin ‘ ( 2 · ( π / 2 ) ) ) = ( 2 · ( ( sin ‘ ( π / 2 ) ) · ( cos ‘ ( π / 2 ) ) ) ) |
49 |
45 48
|
eqtr3i |
⊢ ( sin ‘ π ) = ( 2 · ( ( sin ‘ ( π / 2 ) ) · ( cos ‘ ( π / 2 ) ) ) ) |
50 |
|
sinpi |
⊢ ( sin ‘ π ) = 0 |
51 |
49 50
|
eqtr3i |
⊢ ( 2 · ( ( sin ‘ ( π / 2 ) ) · ( cos ‘ ( π / 2 ) ) ) ) = 0 |
52 |
|
sincl |
⊢ ( ( π / 2 ) ∈ ℂ → ( sin ‘ ( π / 2 ) ) ∈ ℂ ) |
53 |
46 52
|
ax-mp |
⊢ ( sin ‘ ( π / 2 ) ) ∈ ℂ |
54 |
|
coscl |
⊢ ( ( π / 2 ) ∈ ℂ → ( cos ‘ ( π / 2 ) ) ∈ ℂ ) |
55 |
46 54
|
ax-mp |
⊢ ( cos ‘ ( π / 2 ) ) ∈ ℂ |
56 |
53 55
|
mulcli |
⊢ ( ( sin ‘ ( π / 2 ) ) · ( cos ‘ ( π / 2 ) ) ) ∈ ℂ |
57 |
43 56
|
mul0ori |
⊢ ( ( 2 · ( ( sin ‘ ( π / 2 ) ) · ( cos ‘ ( π / 2 ) ) ) ) = 0 ↔ ( 2 = 0 ∨ ( ( sin ‘ ( π / 2 ) ) · ( cos ‘ ( π / 2 ) ) ) = 0 ) ) |
58 |
51 57
|
mpbi |
⊢ ( 2 = 0 ∨ ( ( sin ‘ ( π / 2 ) ) · ( cos ‘ ( π / 2 ) ) ) = 0 ) |
59 |
41 58
|
mtpor |
⊢ ( ( sin ‘ ( π / 2 ) ) · ( cos ‘ ( π / 2 ) ) ) = 0 |
60 |
53 55
|
mul0ori |
⊢ ( ( ( sin ‘ ( π / 2 ) ) · ( cos ‘ ( π / 2 ) ) ) = 0 ↔ ( ( sin ‘ ( π / 2 ) ) = 0 ∨ ( cos ‘ ( π / 2 ) ) = 0 ) ) |
61 |
59 60
|
mpbi |
⊢ ( ( sin ‘ ( π / 2 ) ) = 0 ∨ ( cos ‘ ( π / 2 ) ) = 0 ) |
62 |
39 61
|
mtpor |
⊢ ( cos ‘ ( π / 2 ) ) = 0 |
63 |
62
|
oveq1i |
⊢ ( ( cos ‘ ( π / 2 ) ) ↑ 2 ) = ( 0 ↑ 2 ) |
64 |
|
sq0 |
⊢ ( 0 ↑ 2 ) = 0 |
65 |
63 64
|
eqtri |
⊢ ( ( cos ‘ ( π / 2 ) ) ↑ 2 ) = 0 |
66 |
65
|
oveq2i |
⊢ ( ( ( sin ‘ ( π / 2 ) ) ↑ 2 ) + ( ( cos ‘ ( π / 2 ) ) ↑ 2 ) ) = ( ( ( sin ‘ ( π / 2 ) ) ↑ 2 ) + 0 ) |
67 |
|
sincossq |
⊢ ( ( π / 2 ) ∈ ℂ → ( ( ( sin ‘ ( π / 2 ) ) ↑ 2 ) + ( ( cos ‘ ( π / 2 ) ) ↑ 2 ) ) = 1 ) |
68 |
46 67
|
ax-mp |
⊢ ( ( ( sin ‘ ( π / 2 ) ) ↑ 2 ) + ( ( cos ‘ ( π / 2 ) ) ↑ 2 ) ) = 1 |
69 |
66 68
|
eqtr3i |
⊢ ( ( ( sin ‘ ( π / 2 ) ) ↑ 2 ) + 0 ) = 1 |
70 |
53
|
sqcli |
⊢ ( ( sin ‘ ( π / 2 ) ) ↑ 2 ) ∈ ℂ |
71 |
70
|
addid1i |
⊢ ( ( ( sin ‘ ( π / 2 ) ) ↑ 2 ) + 0 ) = ( ( sin ‘ ( π / 2 ) ) ↑ 2 ) |
72 |
35 69 71
|
3eqtr2ri |
⊢ ( ( sin ‘ ( π / 2 ) ) ↑ 2 ) = ( 1 ↑ 2 ) |
73 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
74 |
53 73
|
sqeqori |
⊢ ( ( ( sin ‘ ( π / 2 ) ) ↑ 2 ) = ( 1 ↑ 2 ) ↔ ( ( sin ‘ ( π / 2 ) ) = 1 ∨ ( sin ‘ ( π / 2 ) ) = - 1 ) ) |
75 |
72 74
|
mpbi |
⊢ ( ( sin ‘ ( π / 2 ) ) = 1 ∨ ( sin ‘ ( π / 2 ) ) = - 1 ) |
76 |
75
|
ori |
⊢ ( ¬ ( sin ‘ ( π / 2 ) ) = 1 → ( sin ‘ ( π / 2 ) ) = - 1 ) |
77 |
34 76
|
mt3 |
⊢ ( sin ‘ ( π / 2 ) ) = 1 |
78 |
77 62
|
pm3.2i |
⊢ ( ( sin ‘ ( π / 2 ) ) = 1 ∧ ( cos ‘ ( π / 2 ) ) = 0 ) |