Step |
Hyp |
Ref |
Expression |
1 |
|
halfcn |
⊢ ( 1 / 2 ) ∈ ℂ |
2 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
3 |
|
2halves |
⊢ ( 1 ∈ ℂ → ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 ) |
4 |
2 3
|
ax-mp |
⊢ ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 |
5 |
|
sincosq1eq |
⊢ ( ( ( 1 / 2 ) ∈ ℂ ∧ ( 1 / 2 ) ∈ ℂ ∧ ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 ) → ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) = ( cos ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) ) |
6 |
1 1 4 5
|
mp3an |
⊢ ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) = ( cos ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) |
7 |
6
|
oveq2i |
⊢ ( ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) · ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) ) = ( ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) · ( cos ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) ) |
8 |
7
|
oveq2i |
⊢ ( 2 · ( ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) · ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) ) ) = ( 2 · ( ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) · ( cos ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) ) ) |
9 |
|
2cn |
⊢ 2 ∈ ℂ |
10 |
|
pire |
⊢ π ∈ ℝ |
11 |
10
|
recni |
⊢ π ∈ ℂ |
12 |
|
2ne0 |
⊢ 2 ≠ 0 |
13 |
2 9 11 9 12 12
|
divmuldivi |
⊢ ( ( 1 / 2 ) · ( π / 2 ) ) = ( ( 1 · π ) / ( 2 · 2 ) ) |
14 |
11
|
mulid2i |
⊢ ( 1 · π ) = π |
15 |
|
2t2e4 |
⊢ ( 2 · 2 ) = 4 |
16 |
14 15
|
oveq12i |
⊢ ( ( 1 · π ) / ( 2 · 2 ) ) = ( π / 4 ) |
17 |
13 16
|
eqtri |
⊢ ( ( 1 / 2 ) · ( π / 2 ) ) = ( π / 4 ) |
18 |
17
|
fveq2i |
⊢ ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) = ( sin ‘ ( π / 4 ) ) |
19 |
18 18
|
oveq12i |
⊢ ( ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) · ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) ) = ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) |
20 |
19
|
oveq2i |
⊢ ( 2 · ( ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) · ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) ) ) = ( 2 · ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) |
21 |
9 12
|
recidi |
⊢ ( 2 · ( 1 / 2 ) ) = 1 |
22 |
21
|
oveq1i |
⊢ ( ( 2 · ( 1 / 2 ) ) · ( π / 2 ) ) = ( 1 · ( π / 2 ) ) |
23 |
|
2re |
⊢ 2 ∈ ℝ |
24 |
10 23 12
|
redivcli |
⊢ ( π / 2 ) ∈ ℝ |
25 |
24
|
recni |
⊢ ( π / 2 ) ∈ ℂ |
26 |
9 1 25
|
mulassi |
⊢ ( ( 2 · ( 1 / 2 ) ) · ( π / 2 ) ) = ( 2 · ( ( 1 / 2 ) · ( π / 2 ) ) ) |
27 |
25
|
mulid2i |
⊢ ( 1 · ( π / 2 ) ) = ( π / 2 ) |
28 |
22 26 27
|
3eqtr3i |
⊢ ( 2 · ( ( 1 / 2 ) · ( π / 2 ) ) ) = ( π / 2 ) |
29 |
28
|
fveq2i |
⊢ ( sin ‘ ( 2 · ( ( 1 / 2 ) · ( π / 2 ) ) ) ) = ( sin ‘ ( π / 2 ) ) |
30 |
1 25
|
mulcli |
⊢ ( ( 1 / 2 ) · ( π / 2 ) ) ∈ ℂ |
31 |
|
sin2t |
⊢ ( ( ( 1 / 2 ) · ( π / 2 ) ) ∈ ℂ → ( sin ‘ ( 2 · ( ( 1 / 2 ) · ( π / 2 ) ) ) ) = ( 2 · ( ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) · ( cos ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) ) ) ) |
32 |
30 31
|
ax-mp |
⊢ ( sin ‘ ( 2 · ( ( 1 / 2 ) · ( π / 2 ) ) ) ) = ( 2 · ( ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) · ( cos ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) ) ) |
33 |
|
sinhalfpi |
⊢ ( sin ‘ ( π / 2 ) ) = 1 |
34 |
29 32 33
|
3eqtr3i |
⊢ ( 2 · ( ( sin ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) · ( cos ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) ) ) = 1 |
35 |
8 20 34
|
3eqtr3i |
⊢ ( 2 · ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) = 1 |
36 |
35
|
fveq2i |
⊢ ( √ ‘ ( 2 · ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) ) = ( √ ‘ 1 ) |
37 |
|
4re |
⊢ 4 ∈ ℝ |
38 |
|
4ne0 |
⊢ 4 ≠ 0 |
39 |
10 37 38
|
redivcli |
⊢ ( π / 4 ) ∈ ℝ |
40 |
|
resincl |
⊢ ( ( π / 4 ) ∈ ℝ → ( sin ‘ ( π / 4 ) ) ∈ ℝ ) |
41 |
39 40
|
ax-mp |
⊢ ( sin ‘ ( π / 4 ) ) ∈ ℝ |
42 |
41 41
|
remulcli |
⊢ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ∈ ℝ |
43 |
|
0le2 |
⊢ 0 ≤ 2 |
44 |
41
|
msqge0i |
⊢ 0 ≤ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) |
45 |
23 42 43 44
|
sqrtmulii |
⊢ ( √ ‘ ( 2 · ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) ) = ( ( √ ‘ 2 ) · ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) ) |
46 |
|
sqrt1 |
⊢ ( √ ‘ 1 ) = 1 |
47 |
36 45 46
|
3eqtr3ri |
⊢ 1 = ( ( √ ‘ 2 ) · ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) ) |
48 |
42
|
sqrtcli |
⊢ ( 0 ≤ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) → ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) ∈ ℝ ) |
49 |
44 48
|
ax-mp |
⊢ ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) ∈ ℝ |
50 |
49
|
recni |
⊢ ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) ∈ ℂ |
51 |
|
sqrt2re |
⊢ ( √ ‘ 2 ) ∈ ℝ |
52 |
51
|
recni |
⊢ ( √ ‘ 2 ) ∈ ℂ |
53 |
|
sqrt00 |
⊢ ( ( 2 ∈ ℝ ∧ 0 ≤ 2 ) → ( ( √ ‘ 2 ) = 0 ↔ 2 = 0 ) ) |
54 |
23 43 53
|
mp2an |
⊢ ( ( √ ‘ 2 ) = 0 ↔ 2 = 0 ) |
55 |
54
|
necon3bii |
⊢ ( ( √ ‘ 2 ) ≠ 0 ↔ 2 ≠ 0 ) |
56 |
12 55
|
mpbir |
⊢ ( √ ‘ 2 ) ≠ 0 |
57 |
52 56
|
pm3.2i |
⊢ ( ( √ ‘ 2 ) ∈ ℂ ∧ ( √ ‘ 2 ) ≠ 0 ) |
58 |
|
divmul2 |
⊢ ( ( 1 ∈ ℂ ∧ ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) ∈ ℂ ∧ ( ( √ ‘ 2 ) ∈ ℂ ∧ ( √ ‘ 2 ) ≠ 0 ) ) → ( ( 1 / ( √ ‘ 2 ) ) = ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) ↔ 1 = ( ( √ ‘ 2 ) · ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) ) ) ) |
59 |
2 50 57 58
|
mp3an |
⊢ ( ( 1 / ( √ ‘ 2 ) ) = ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) ↔ 1 = ( ( √ ‘ 2 ) · ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) ) ) |
60 |
47 59
|
mpbir |
⊢ ( 1 / ( √ ‘ 2 ) ) = ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) |
61 |
|
0re |
⊢ 0 ∈ ℝ |
62 |
|
pipos |
⊢ 0 < π |
63 |
|
4pos |
⊢ 0 < 4 |
64 |
10 37 62 63
|
divgt0ii |
⊢ 0 < ( π / 4 ) |
65 |
|
1re |
⊢ 1 ∈ ℝ |
66 |
|
pigt2lt4 |
⊢ ( 2 < π ∧ π < 4 ) |
67 |
66
|
simpri |
⊢ π < 4 |
68 |
10 37 37 63
|
ltdiv1ii |
⊢ ( π < 4 ↔ ( π / 4 ) < ( 4 / 4 ) ) |
69 |
67 68
|
mpbi |
⊢ ( π / 4 ) < ( 4 / 4 ) |
70 |
37
|
recni |
⊢ 4 ∈ ℂ |
71 |
70 38
|
dividi |
⊢ ( 4 / 4 ) = 1 |
72 |
69 71
|
breqtri |
⊢ ( π / 4 ) < 1 |
73 |
39 65 72
|
ltleii |
⊢ ( π / 4 ) ≤ 1 |
74 |
|
0xr |
⊢ 0 ∈ ℝ* |
75 |
|
elioc2 |
⊢ ( ( 0 ∈ ℝ* ∧ 1 ∈ ℝ ) → ( ( π / 4 ) ∈ ( 0 (,] 1 ) ↔ ( ( π / 4 ) ∈ ℝ ∧ 0 < ( π / 4 ) ∧ ( π / 4 ) ≤ 1 ) ) ) |
76 |
74 65 75
|
mp2an |
⊢ ( ( π / 4 ) ∈ ( 0 (,] 1 ) ↔ ( ( π / 4 ) ∈ ℝ ∧ 0 < ( π / 4 ) ∧ ( π / 4 ) ≤ 1 ) ) |
77 |
39 64 73 76
|
mpbir3an |
⊢ ( π / 4 ) ∈ ( 0 (,] 1 ) |
78 |
|
sin01gt0 |
⊢ ( ( π / 4 ) ∈ ( 0 (,] 1 ) → 0 < ( sin ‘ ( π / 4 ) ) ) |
79 |
77 78
|
ax-mp |
⊢ 0 < ( sin ‘ ( π / 4 ) ) |
80 |
61 41 79
|
ltleii |
⊢ 0 ≤ ( sin ‘ ( π / 4 ) ) |
81 |
41
|
sqrtmsqi |
⊢ ( 0 ≤ ( sin ‘ ( π / 4 ) ) → ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) = ( sin ‘ ( π / 4 ) ) ) |
82 |
80 81
|
ax-mp |
⊢ ( √ ‘ ( ( sin ‘ ( π / 4 ) ) · ( sin ‘ ( π / 4 ) ) ) ) = ( sin ‘ ( π / 4 ) ) |
83 |
60 82
|
eqtr2i |
⊢ ( sin ‘ ( π / 4 ) ) = ( 1 / ( √ ‘ 2 ) ) |
84 |
60 82
|
eqtri |
⊢ ( 1 / ( √ ‘ 2 ) ) = ( sin ‘ ( π / 4 ) ) |
85 |
17
|
fveq2i |
⊢ ( cos ‘ ( ( 1 / 2 ) · ( π / 2 ) ) ) = ( cos ‘ ( π / 4 ) ) |
86 |
6 18 85
|
3eqtr3i |
⊢ ( sin ‘ ( π / 4 ) ) = ( cos ‘ ( π / 4 ) ) |
87 |
84 86
|
eqtr2i |
⊢ ( cos ‘ ( π / 4 ) ) = ( 1 / ( √ ‘ 2 ) ) |
88 |
83 87
|
pm3.2i |
⊢ ( ( sin ‘ ( π / 4 ) ) = ( 1 / ( √ ‘ 2 ) ) ∧ ( cos ‘ ( π / 4 ) ) = ( 1 / ( √ ‘ 2 ) ) ) |