Step |
Hyp |
Ref |
Expression |
1 |
|
fourierswlem.t |
⊢ 𝑇 = ( 2 · π ) |
2 |
|
fourierswlem.f |
⊢ 𝐹 = ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 mod 𝑇 ) < π , 1 , - 1 ) ) |
3 |
|
fourierswlem.x |
⊢ 𝑋 ∈ ℝ |
4 |
|
fourierswlem.y |
⊢ 𝑌 = if ( ( 𝑋 mod π ) = 0 , 0 , ( 𝐹 ‘ 𝑋 ) ) |
5 |
|
simpr |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ 2 ∥ ( 𝑋 / π ) ) → 2 ∥ ( 𝑋 / π ) ) |
6 |
|
2z |
⊢ 2 ∈ ℤ |
7 |
6
|
a1i |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ 2 ∥ ( 𝑋 / π ) ) → 2 ∈ ℤ ) |
8 |
|
pirp |
⊢ π ∈ ℝ+ |
9 |
|
mod0 |
⊢ ( ( 𝑋 ∈ ℝ ∧ π ∈ ℝ+ ) → ( ( 𝑋 mod π ) = 0 ↔ ( 𝑋 / π ) ∈ ℤ ) ) |
10 |
3 8 9
|
mp2an |
⊢ ( ( 𝑋 mod π ) = 0 ↔ ( 𝑋 / π ) ∈ ℤ ) |
11 |
10
|
biimpi |
⊢ ( ( 𝑋 mod π ) = 0 → ( 𝑋 / π ) ∈ ℤ ) |
12 |
11
|
adantr |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ 2 ∥ ( 𝑋 / π ) ) → ( 𝑋 / π ) ∈ ℤ ) |
13 |
|
divides |
⊢ ( ( 2 ∈ ℤ ∧ ( 𝑋 / π ) ∈ ℤ ) → ( 2 ∥ ( 𝑋 / π ) ↔ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) ) |
14 |
7 12 13
|
syl2anc |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ 2 ∥ ( 𝑋 / π ) ) → ( 2 ∥ ( 𝑋 / π ) ↔ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) ) |
15 |
5 14
|
mpbid |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ 2 ∥ ( 𝑋 / π ) ) → ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) |
16 |
|
2cnd |
⊢ ( 𝑘 ∈ ℤ → 2 ∈ ℂ ) |
17 |
|
picn |
⊢ π ∈ ℂ |
18 |
17
|
a1i |
⊢ ( 𝑘 ∈ ℤ → π ∈ ℂ ) |
19 |
|
zcn |
⊢ ( 𝑘 ∈ ℤ → 𝑘 ∈ ℂ ) |
20 |
16 18 19
|
mulassd |
⊢ ( 𝑘 ∈ ℤ → ( ( 2 · π ) · 𝑘 ) = ( 2 · ( π · 𝑘 ) ) ) |
21 |
18 19
|
mulcld |
⊢ ( 𝑘 ∈ ℤ → ( π · 𝑘 ) ∈ ℂ ) |
22 |
16 21
|
mulcomd |
⊢ ( 𝑘 ∈ ℤ → ( 2 · ( π · 𝑘 ) ) = ( ( π · 𝑘 ) · 2 ) ) |
23 |
20 22
|
eqtrd |
⊢ ( 𝑘 ∈ ℤ → ( ( 2 · π ) · 𝑘 ) = ( ( π · 𝑘 ) · 2 ) ) |
24 |
23
|
adantr |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → ( ( 2 · π ) · 𝑘 ) = ( ( π · 𝑘 ) · 2 ) ) |
25 |
18 19 16
|
mulassd |
⊢ ( 𝑘 ∈ ℤ → ( ( π · 𝑘 ) · 2 ) = ( π · ( 𝑘 · 2 ) ) ) |
26 |
25
|
adantr |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → ( ( π · 𝑘 ) · 2 ) = ( π · ( 𝑘 · 2 ) ) ) |
27 |
|
id |
⊢ ( ( 𝑘 · 2 ) = ( 𝑋 / π ) → ( 𝑘 · 2 ) = ( 𝑋 / π ) ) |
28 |
27
|
eqcomd |
⊢ ( ( 𝑘 · 2 ) = ( 𝑋 / π ) → ( 𝑋 / π ) = ( 𝑘 · 2 ) ) |
29 |
28
|
adantl |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → ( 𝑋 / π ) = ( 𝑘 · 2 ) ) |
30 |
3
|
recni |
⊢ 𝑋 ∈ ℂ |
31 |
30
|
a1i |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → 𝑋 ∈ ℂ ) |
32 |
17
|
a1i |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → π ∈ ℂ ) |
33 |
19
|
adantr |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → 𝑘 ∈ ℂ ) |
34 |
|
2cnd |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → 2 ∈ ℂ ) |
35 |
33 34
|
mulcld |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → ( 𝑘 · 2 ) ∈ ℂ ) |
36 |
|
pire |
⊢ π ∈ ℝ |
37 |
|
pipos |
⊢ 0 < π |
38 |
36 37
|
gt0ne0ii |
⊢ π ≠ 0 |
39 |
38
|
a1i |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → π ≠ 0 ) |
40 |
31 32 35 39
|
divmuld |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → ( ( 𝑋 / π ) = ( 𝑘 · 2 ) ↔ ( π · ( 𝑘 · 2 ) ) = 𝑋 ) ) |
41 |
29 40
|
mpbid |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → ( π · ( 𝑘 · 2 ) ) = 𝑋 ) |
42 |
24 26 41
|
3eqtrrd |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → 𝑋 = ( ( 2 · π ) · 𝑘 ) ) |
43 |
1
|
a1i |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → 𝑇 = ( 2 · π ) ) |
44 |
42 43
|
oveq12d |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → ( 𝑋 / 𝑇 ) = ( ( ( 2 · π ) · 𝑘 ) / ( 2 · π ) ) ) |
45 |
16 18
|
mulcld |
⊢ ( 𝑘 ∈ ℤ → ( 2 · π ) ∈ ℂ ) |
46 |
|
2ne0 |
⊢ 2 ≠ 0 |
47 |
46
|
a1i |
⊢ ( 𝑘 ∈ ℤ → 2 ≠ 0 ) |
48 |
38
|
a1i |
⊢ ( 𝑘 ∈ ℤ → π ≠ 0 ) |
49 |
16 18 47 48
|
mulne0d |
⊢ ( 𝑘 ∈ ℤ → ( 2 · π ) ≠ 0 ) |
50 |
19 45 49
|
divcan3d |
⊢ ( 𝑘 ∈ ℤ → ( ( ( 2 · π ) · 𝑘 ) / ( 2 · π ) ) = 𝑘 ) |
51 |
50
|
adantr |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → ( ( ( 2 · π ) · 𝑘 ) / ( 2 · π ) ) = 𝑘 ) |
52 |
44 51
|
eqtrd |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → ( 𝑋 / 𝑇 ) = 𝑘 ) |
53 |
|
simpl |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → 𝑘 ∈ ℤ ) |
54 |
52 53
|
eqeltrd |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( 𝑘 · 2 ) = ( 𝑋 / π ) ) → ( 𝑋 / 𝑇 ) ∈ ℤ ) |
55 |
54
|
ex |
⊢ ( 𝑘 ∈ ℤ → ( ( 𝑘 · 2 ) = ( 𝑋 / π ) → ( 𝑋 / 𝑇 ) ∈ ℤ ) ) |
56 |
55
|
a1i |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ 2 ∥ ( 𝑋 / π ) ) → ( 𝑘 ∈ ℤ → ( ( 𝑘 · 2 ) = ( 𝑋 / π ) → ( 𝑋 / 𝑇 ) ∈ ℤ ) ) ) |
57 |
56
|
rexlimdv |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ 2 ∥ ( 𝑋 / π ) ) → ( ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( 𝑋 / π ) → ( 𝑋 / 𝑇 ) ∈ ℤ ) ) |
58 |
15 57
|
mpd |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ 2 ∥ ( 𝑋 / π ) ) → ( 𝑋 / 𝑇 ) ∈ ℤ ) |
59 |
|
2re |
⊢ 2 ∈ ℝ |
60 |
59 36
|
remulcli |
⊢ ( 2 · π ) ∈ ℝ |
61 |
1 60
|
eqeltri |
⊢ 𝑇 ∈ ℝ |
62 |
|
2pos |
⊢ 0 < 2 |
63 |
59 36 62 37
|
mulgt0ii |
⊢ 0 < ( 2 · π ) |
64 |
63 1
|
breqtrri |
⊢ 0 < 𝑇 |
65 |
61 64
|
elrpii |
⊢ 𝑇 ∈ ℝ+ |
66 |
|
mod0 |
⊢ ( ( 𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ+ ) → ( ( 𝑋 mod 𝑇 ) = 0 ↔ ( 𝑋 / 𝑇 ) ∈ ℤ ) ) |
67 |
3 65 66
|
mp2an |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 ↔ ( 𝑋 / 𝑇 ) ∈ ℤ ) |
68 |
58 67
|
sylibr |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ 2 ∥ ( 𝑋 / π ) ) → ( 𝑋 mod 𝑇 ) = 0 ) |
69 |
68
|
orcd |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ 2 ∥ ( 𝑋 / π ) ) → ( ( 𝑋 mod 𝑇 ) = 0 ∨ ( 𝑋 mod 𝑇 ) = π ) ) |
70 |
|
odd2np1 |
⊢ ( ( 𝑋 / π ) ∈ ℤ → ( ¬ 2 ∥ ( 𝑋 / π ) ↔ ∃ 𝑘 ∈ ℤ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) ) |
71 |
10 70
|
sylbi |
⊢ ( ( 𝑋 mod π ) = 0 → ( ¬ 2 ∥ ( 𝑋 / π ) ↔ ∃ 𝑘 ∈ ℤ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) ) |
72 |
71
|
biimpa |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ¬ 2 ∥ ( 𝑋 / π ) ) → ∃ 𝑘 ∈ ℤ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) |
73 |
16 19
|
mulcld |
⊢ ( 𝑘 ∈ ℤ → ( 2 · 𝑘 ) ∈ ℂ ) |
74 |
73
|
adantr |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( 2 · 𝑘 ) ∈ ℂ ) |
75 |
|
1cnd |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → 1 ∈ ℂ ) |
76 |
17
|
a1i |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → π ∈ ℂ ) |
77 |
74 75 76
|
adddird |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( ( ( 2 · 𝑘 ) + 1 ) · π ) = ( ( ( 2 · 𝑘 ) · π ) + ( 1 · π ) ) ) |
78 |
16 19
|
mulcomd |
⊢ ( 𝑘 ∈ ℤ → ( 2 · 𝑘 ) = ( 𝑘 · 2 ) ) |
79 |
78
|
oveq1d |
⊢ ( 𝑘 ∈ ℤ → ( ( 2 · 𝑘 ) · π ) = ( ( 𝑘 · 2 ) · π ) ) |
80 |
19 16 18
|
mulassd |
⊢ ( 𝑘 ∈ ℤ → ( ( 𝑘 · 2 ) · π ) = ( 𝑘 · ( 2 · π ) ) ) |
81 |
1
|
eqcomi |
⊢ ( 2 · π ) = 𝑇 |
82 |
81
|
a1i |
⊢ ( 𝑘 ∈ ℤ → ( 2 · π ) = 𝑇 ) |
83 |
82
|
oveq2d |
⊢ ( 𝑘 ∈ ℤ → ( 𝑘 · ( 2 · π ) ) = ( 𝑘 · 𝑇 ) ) |
84 |
79 80 83
|
3eqtrd |
⊢ ( 𝑘 ∈ ℤ → ( ( 2 · 𝑘 ) · π ) = ( 𝑘 · 𝑇 ) ) |
85 |
17
|
mulid2i |
⊢ ( 1 · π ) = π |
86 |
85
|
a1i |
⊢ ( 𝑘 ∈ ℤ → ( 1 · π ) = π ) |
87 |
84 86
|
oveq12d |
⊢ ( 𝑘 ∈ ℤ → ( ( ( 2 · 𝑘 ) · π ) + ( 1 · π ) ) = ( ( 𝑘 · 𝑇 ) + π ) ) |
88 |
87
|
adantr |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( ( ( 2 · 𝑘 ) · π ) + ( 1 · π ) ) = ( ( 𝑘 · 𝑇 ) + π ) ) |
89 |
1 45
|
eqeltrid |
⊢ ( 𝑘 ∈ ℤ → 𝑇 ∈ ℂ ) |
90 |
19 89
|
mulcld |
⊢ ( 𝑘 ∈ ℤ → ( 𝑘 · 𝑇 ) ∈ ℂ ) |
91 |
90 18
|
addcomd |
⊢ ( 𝑘 ∈ ℤ → ( ( 𝑘 · 𝑇 ) + π ) = ( π + ( 𝑘 · 𝑇 ) ) ) |
92 |
91
|
adantr |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( ( 𝑘 · 𝑇 ) + π ) = ( π + ( 𝑘 · 𝑇 ) ) ) |
93 |
77 88 92
|
3eqtrrd |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( π + ( 𝑘 · 𝑇 ) ) = ( ( ( 2 · 𝑘 ) + 1 ) · π ) ) |
94 |
|
peano2cn |
⊢ ( ( 2 · 𝑘 ) ∈ ℂ → ( ( 2 · 𝑘 ) + 1 ) ∈ ℂ ) |
95 |
73 94
|
syl |
⊢ ( 𝑘 ∈ ℤ → ( ( 2 · 𝑘 ) + 1 ) ∈ ℂ ) |
96 |
95 18
|
mulcomd |
⊢ ( 𝑘 ∈ ℤ → ( ( ( 2 · 𝑘 ) + 1 ) · π ) = ( π · ( ( 2 · 𝑘 ) + 1 ) ) ) |
97 |
96
|
adantr |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( ( ( 2 · 𝑘 ) + 1 ) · π ) = ( π · ( ( 2 · 𝑘 ) + 1 ) ) ) |
98 |
|
id |
⊢ ( ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) → ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) |
99 |
98
|
eqcomd |
⊢ ( ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) → ( 𝑋 / π ) = ( ( 2 · 𝑘 ) + 1 ) ) |
100 |
99
|
adantl |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( 𝑋 / π ) = ( ( 2 · 𝑘 ) + 1 ) ) |
101 |
30
|
a1i |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → 𝑋 ∈ ℂ ) |
102 |
95
|
adantr |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( ( 2 · 𝑘 ) + 1 ) ∈ ℂ ) |
103 |
38
|
a1i |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → π ≠ 0 ) |
104 |
101 76 102 103
|
divmuld |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( ( 𝑋 / π ) = ( ( 2 · 𝑘 ) + 1 ) ↔ ( π · ( ( 2 · 𝑘 ) + 1 ) ) = 𝑋 ) ) |
105 |
100 104
|
mpbid |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( π · ( ( 2 · 𝑘 ) + 1 ) ) = 𝑋 ) |
106 |
93 97 105
|
3eqtrrd |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → 𝑋 = ( π + ( 𝑘 · 𝑇 ) ) ) |
107 |
106
|
oveq1d |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( 𝑋 mod 𝑇 ) = ( ( π + ( 𝑘 · 𝑇 ) ) mod 𝑇 ) ) |
108 |
|
modcyc |
⊢ ( ( π ∈ ℝ ∧ 𝑇 ∈ ℝ+ ∧ 𝑘 ∈ ℤ ) → ( ( π + ( 𝑘 · 𝑇 ) ) mod 𝑇 ) = ( π mod 𝑇 ) ) |
109 |
36 65 108
|
mp3an12 |
⊢ ( 𝑘 ∈ ℤ → ( ( π + ( 𝑘 · 𝑇 ) ) mod 𝑇 ) = ( π mod 𝑇 ) ) |
110 |
109
|
adantr |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( ( π + ( 𝑘 · 𝑇 ) ) mod 𝑇 ) = ( π mod 𝑇 ) ) |
111 |
36
|
a1i |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → π ∈ ℝ ) |
112 |
65
|
a1i |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → 𝑇 ∈ ℝ+ ) |
113 |
|
0re |
⊢ 0 ∈ ℝ |
114 |
113 36 37
|
ltleii |
⊢ 0 ≤ π |
115 |
114
|
a1i |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → 0 ≤ π ) |
116 |
|
2timesgt |
⊢ ( π ∈ ℝ+ → π < ( 2 · π ) ) |
117 |
8 116
|
ax-mp |
⊢ π < ( 2 · π ) |
118 |
117 1
|
breqtrri |
⊢ π < 𝑇 |
119 |
118
|
a1i |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → π < 𝑇 ) |
120 |
|
modid |
⊢ ( ( ( π ∈ ℝ ∧ 𝑇 ∈ ℝ+ ) ∧ ( 0 ≤ π ∧ π < 𝑇 ) ) → ( π mod 𝑇 ) = π ) |
121 |
111 112 115 119 120
|
syl22anc |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( π mod 𝑇 ) = π ) |
122 |
107 110 121
|
3eqtrd |
⊢ ( ( 𝑘 ∈ ℤ ∧ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) ) → ( 𝑋 mod 𝑇 ) = π ) |
123 |
122
|
ex |
⊢ ( 𝑘 ∈ ℤ → ( ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) → ( 𝑋 mod 𝑇 ) = π ) ) |
124 |
123
|
a1i |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ¬ 2 ∥ ( 𝑋 / π ) ) → ( 𝑘 ∈ ℤ → ( ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) → ( 𝑋 mod 𝑇 ) = π ) ) ) |
125 |
124
|
rexlimdv |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ¬ 2 ∥ ( 𝑋 / π ) ) → ( ∃ 𝑘 ∈ ℤ ( ( 2 · 𝑘 ) + 1 ) = ( 𝑋 / π ) → ( 𝑋 mod 𝑇 ) = π ) ) |
126 |
72 125
|
mpd |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ¬ 2 ∥ ( 𝑋 / π ) ) → ( 𝑋 mod 𝑇 ) = π ) |
127 |
126
|
olcd |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ¬ 2 ∥ ( 𝑋 / π ) ) → ( ( 𝑋 mod 𝑇 ) = 0 ∨ ( 𝑋 mod 𝑇 ) = π ) ) |
128 |
69 127
|
pm2.61dan |
⊢ ( ( 𝑋 mod π ) = 0 → ( ( 𝑋 mod 𝑇 ) = 0 ∨ ( 𝑋 mod 𝑇 ) = π ) ) |
129 |
|
0xr |
⊢ 0 ∈ ℝ* |
130 |
36
|
rexri |
⊢ π ∈ ℝ* |
131 |
|
iocgtlb |
⊢ ( ( 0 ∈ ℝ* ∧ π ∈ ℝ* ∧ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ) → 0 < ( 𝑋 mod 𝑇 ) ) |
132 |
129 130 131
|
mp3an12 |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) → 0 < ( 𝑋 mod 𝑇 ) ) |
133 |
132
|
gt0ne0d |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) → ( 𝑋 mod 𝑇 ) ≠ 0 ) |
134 |
133
|
neneqd |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) → ¬ ( 𝑋 mod 𝑇 ) = 0 ) |
135 |
|
pm2.53 |
⊢ ( ( ( 𝑋 mod 𝑇 ) = 0 ∨ ( 𝑋 mod 𝑇 ) = π ) → ( ¬ ( 𝑋 mod 𝑇 ) = 0 → ( 𝑋 mod 𝑇 ) = π ) ) |
136 |
135
|
imp |
⊢ ( ( ( ( 𝑋 mod 𝑇 ) = 0 ∨ ( 𝑋 mod 𝑇 ) = π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ( 𝑋 mod 𝑇 ) = π ) |
137 |
128 134 136
|
syl2anr |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ( 𝑋 mod π ) = 0 ) → ( 𝑋 mod 𝑇 ) = π ) |
138 |
129
|
a1i |
⊢ ( ( 𝑋 mod 𝑇 ) = π → 0 ∈ ℝ* ) |
139 |
130
|
a1i |
⊢ ( ( 𝑋 mod 𝑇 ) = π → π ∈ ℝ* ) |
140 |
|
modcl |
⊢ ( ( 𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ+ ) → ( 𝑋 mod 𝑇 ) ∈ ℝ ) |
141 |
3 65 140
|
mp2an |
⊢ ( 𝑋 mod 𝑇 ) ∈ ℝ |
142 |
141
|
rexri |
⊢ ( 𝑋 mod 𝑇 ) ∈ ℝ* |
143 |
142
|
a1i |
⊢ ( ( 𝑋 mod 𝑇 ) = π → ( 𝑋 mod 𝑇 ) ∈ ℝ* ) |
144 |
|
id |
⊢ ( ( 𝑋 mod 𝑇 ) = π → ( 𝑋 mod 𝑇 ) = π ) |
145 |
37 144
|
breqtrrid |
⊢ ( ( 𝑋 mod 𝑇 ) = π → 0 < ( 𝑋 mod 𝑇 ) ) |
146 |
36
|
eqlei2 |
⊢ ( ( 𝑋 mod 𝑇 ) = π → ( 𝑋 mod 𝑇 ) ≤ π ) |
147 |
138 139 143 145 146
|
eliocd |
⊢ ( ( 𝑋 mod 𝑇 ) = π → ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ) |
148 |
147
|
iftrued |
⊢ ( ( 𝑋 mod 𝑇 ) = π → if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) = 1 ) |
149 |
148
|
adantl |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ( 𝑋 mod 𝑇 ) = π ) → if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) = 1 ) |
150 |
|
oveq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 mod 𝑇 ) = ( 𝑋 mod 𝑇 ) ) |
151 |
150
|
breq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 mod 𝑇 ) < π ↔ ( 𝑋 mod 𝑇 ) < π ) ) |
152 |
151
|
ifbid |
⊢ ( 𝑥 = 𝑋 → if ( ( 𝑥 mod 𝑇 ) < π , 1 , - 1 ) = if ( ( 𝑋 mod 𝑇 ) < π , 1 , - 1 ) ) |
153 |
|
1ex |
⊢ 1 ∈ V |
154 |
|
negex |
⊢ - 1 ∈ V |
155 |
153 154
|
ifex |
⊢ if ( ( 𝑋 mod 𝑇 ) < π , 1 , - 1 ) ∈ V |
156 |
152 2 155
|
fvmpt |
⊢ ( 𝑋 ∈ ℝ → ( 𝐹 ‘ 𝑋 ) = if ( ( 𝑋 mod 𝑇 ) < π , 1 , - 1 ) ) |
157 |
3 156
|
ax-mp |
⊢ ( 𝐹 ‘ 𝑋 ) = if ( ( 𝑋 mod 𝑇 ) < π , 1 , - 1 ) |
158 |
141
|
a1i |
⊢ ( ( 𝑋 mod 𝑇 ) < π → ( 𝑋 mod 𝑇 ) ∈ ℝ ) |
159 |
|
id |
⊢ ( ( 𝑋 mod 𝑇 ) < π → ( 𝑋 mod 𝑇 ) < π ) |
160 |
158 159
|
ltned |
⊢ ( ( 𝑋 mod 𝑇 ) < π → ( 𝑋 mod 𝑇 ) ≠ π ) |
161 |
160
|
necon2bi |
⊢ ( ( 𝑋 mod 𝑇 ) = π → ¬ ( 𝑋 mod 𝑇 ) < π ) |
162 |
161
|
iffalsed |
⊢ ( ( 𝑋 mod 𝑇 ) = π → if ( ( 𝑋 mod 𝑇 ) < π , 1 , - 1 ) = - 1 ) |
163 |
157 162
|
syl5eq |
⊢ ( ( 𝑋 mod 𝑇 ) = π → ( 𝐹 ‘ 𝑋 ) = - 1 ) |
164 |
163
|
adantl |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ( 𝑋 mod 𝑇 ) = π ) → ( 𝐹 ‘ 𝑋 ) = - 1 ) |
165 |
149 164
|
oveq12d |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ( 𝑋 mod 𝑇 ) = π ) → ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) = ( 1 + - 1 ) ) |
166 |
|
1pneg1e0 |
⊢ ( 1 + - 1 ) = 0 |
167 |
165 166
|
eqtrdi |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ( 𝑋 mod 𝑇 ) = π ) → ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) = 0 ) |
168 |
167
|
oveq1d |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ( 𝑋 mod 𝑇 ) = π ) → ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) = ( 0 / 2 ) ) |
169 |
168
|
adantll |
⊢ ( ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ( 𝑋 mod π ) = 0 ) ∧ ( 𝑋 mod 𝑇 ) = π ) → ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) = ( 0 / 2 ) ) |
170 |
|
2cn |
⊢ 2 ∈ ℂ |
171 |
170 46
|
div0i |
⊢ ( 0 / 2 ) = 0 |
172 |
171
|
a1i |
⊢ ( ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ( 𝑋 mod π ) = 0 ) ∧ ( 𝑋 mod 𝑇 ) = π ) → ( 0 / 2 ) = 0 ) |
173 |
|
iftrue |
⊢ ( ( 𝑋 mod π ) = 0 → if ( ( 𝑋 mod π ) = 0 , 0 , ( 𝐹 ‘ 𝑋 ) ) = 0 ) |
174 |
4 173
|
eqtr2id |
⊢ ( ( 𝑋 mod π ) = 0 → 0 = 𝑌 ) |
175 |
174
|
ad2antlr |
⊢ ( ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ( 𝑋 mod π ) = 0 ) ∧ ( 𝑋 mod 𝑇 ) = π ) → 0 = 𝑌 ) |
176 |
169 172 175
|
3eqtrrd |
⊢ ( ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ( 𝑋 mod π ) = 0 ) ∧ ( 𝑋 mod 𝑇 ) = π ) → 𝑌 = ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) ) |
177 |
137 176
|
mpdan |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ( 𝑋 mod π ) = 0 ) → 𝑌 = ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) ) |
178 |
|
iftrue |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) → if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) = 1 ) |
179 |
178
|
adantr |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod π ) = 0 ) → if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) = 1 ) |
180 |
141
|
a1i |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod π ) = 0 ) → ( 𝑋 mod 𝑇 ) ∈ ℝ ) |
181 |
36
|
a1i |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod π ) = 0 ) → π ∈ ℝ ) |
182 |
|
iocleub |
⊢ ( ( 0 ∈ ℝ* ∧ π ∈ ℝ* ∧ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ) → ( 𝑋 mod 𝑇 ) ≤ π ) |
183 |
129 130 182
|
mp3an12 |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) → ( 𝑋 mod 𝑇 ) ≤ π ) |
184 |
183
|
adantr |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod π ) = 0 ) → ( 𝑋 mod 𝑇 ) ≤ π ) |
185 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
186 |
185 17
|
mulcomi |
⊢ ( 1 · π ) = ( π · 1 ) |
187 |
85 186
|
eqtr3i |
⊢ π = ( π · 1 ) |
188 |
187
|
oveq1i |
⊢ ( π + ( π · ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) = ( ( π · 1 ) + ( π · ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) |
189 |
170 17
|
mulcomi |
⊢ ( 2 · π ) = ( π · 2 ) |
190 |
1 189
|
eqtri |
⊢ 𝑇 = ( π · 2 ) |
191 |
190
|
oveq1i |
⊢ ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) = ( ( π · 2 ) · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) |
192 |
113 64
|
gtneii |
⊢ 𝑇 ≠ 0 |
193 |
3 61 192
|
redivcli |
⊢ ( 𝑋 / 𝑇 ) ∈ ℝ |
194 |
|
flcl |
⊢ ( ( 𝑋 / 𝑇 ) ∈ ℝ → ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ∈ ℤ ) |
195 |
193 194
|
ax-mp |
⊢ ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ∈ ℤ |
196 |
|
zcn |
⊢ ( ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ∈ ℤ → ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ∈ ℂ ) |
197 |
195 196
|
ax-mp |
⊢ ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ∈ ℂ |
198 |
17 170 197
|
mulassi |
⊢ ( ( π · 2 ) · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) = ( π · ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) |
199 |
191 198
|
eqtri |
⊢ ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) = ( π · ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) |
200 |
199
|
oveq2i |
⊢ ( π + ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) = ( π + ( π · ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) |
201 |
170 197
|
mulcli |
⊢ ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ∈ ℂ |
202 |
17 185 201
|
adddii |
⊢ ( π · ( 1 + ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) = ( ( π · 1 ) + ( π · ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) |
203 |
188 200 202
|
3eqtr4ri |
⊢ ( π · ( 1 + ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) = ( π + ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) |
204 |
203
|
a1i |
⊢ ( π = ( 𝑋 mod 𝑇 ) → ( π · ( 1 + ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) = ( π + ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) |
205 |
|
id |
⊢ ( π = ( 𝑋 mod 𝑇 ) → π = ( 𝑋 mod 𝑇 ) ) |
206 |
|
modval |
⊢ ( ( 𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ+ ) → ( 𝑋 mod 𝑇 ) = ( 𝑋 − ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) |
207 |
3 65 206
|
mp2an |
⊢ ( 𝑋 mod 𝑇 ) = ( 𝑋 − ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) |
208 |
205 207
|
eqtrdi |
⊢ ( π = ( 𝑋 mod 𝑇 ) → π = ( 𝑋 − ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) |
209 |
208
|
oveq1d |
⊢ ( π = ( 𝑋 mod 𝑇 ) → ( π + ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) = ( ( 𝑋 − ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) + ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) |
210 |
30
|
a1i |
⊢ ( π = ( 𝑋 mod 𝑇 ) → 𝑋 ∈ ℂ ) |
211 |
61
|
recni |
⊢ 𝑇 ∈ ℂ |
212 |
211 197
|
mulcli |
⊢ ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ∈ ℂ |
213 |
212
|
a1i |
⊢ ( π = ( 𝑋 mod 𝑇 ) → ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ∈ ℂ ) |
214 |
210 213
|
npcand |
⊢ ( π = ( 𝑋 mod 𝑇 ) → ( ( 𝑋 − ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) + ( 𝑇 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) = 𝑋 ) |
215 |
204 209 214
|
3eqtrrd |
⊢ ( π = ( 𝑋 mod 𝑇 ) → 𝑋 = ( π · ( 1 + ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) ) |
216 |
215
|
oveq1d |
⊢ ( π = ( 𝑋 mod 𝑇 ) → ( 𝑋 / π ) = ( ( π · ( 1 + ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) / π ) ) |
217 |
185 201
|
addcli |
⊢ ( 1 + ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ∈ ℂ |
218 |
217 17 38
|
divcan3i |
⊢ ( ( π · ( 1 + ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) / π ) = ( 1 + ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) |
219 |
216 218
|
eqtrdi |
⊢ ( π = ( 𝑋 mod 𝑇 ) → ( 𝑋 / π ) = ( 1 + ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ) |
220 |
|
1z |
⊢ 1 ∈ ℤ |
221 |
|
zmulcl |
⊢ ( ( 2 ∈ ℤ ∧ ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ∈ ℤ ) → ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ∈ ℤ ) |
222 |
6 195 221
|
mp2an |
⊢ ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ∈ ℤ |
223 |
|
zaddcl |
⊢ ( ( 1 ∈ ℤ ∧ ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ∈ ℤ ) → ( 1 + ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ∈ ℤ ) |
224 |
220 222 223
|
mp2an |
⊢ ( 1 + ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ∈ ℤ |
225 |
224
|
a1i |
⊢ ( π = ( 𝑋 mod 𝑇 ) → ( 1 + ( 2 · ( ⌊ ‘ ( 𝑋 / 𝑇 ) ) ) ) ∈ ℤ ) |
226 |
219 225
|
eqeltrd |
⊢ ( π = ( 𝑋 mod 𝑇 ) → ( 𝑋 / π ) ∈ ℤ ) |
227 |
226 10
|
sylibr |
⊢ ( π = ( 𝑋 mod 𝑇 ) → ( 𝑋 mod π ) = 0 ) |
228 |
227
|
necon3bi |
⊢ ( ¬ ( 𝑋 mod π ) = 0 → π ≠ ( 𝑋 mod 𝑇 ) ) |
229 |
228
|
adantl |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod π ) = 0 ) → π ≠ ( 𝑋 mod 𝑇 ) ) |
230 |
180 181 184 229
|
leneltd |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod π ) = 0 ) → ( 𝑋 mod 𝑇 ) < π ) |
231 |
|
iftrue |
⊢ ( ( 𝑋 mod 𝑇 ) < π → if ( ( 𝑋 mod 𝑇 ) < π , 1 , - 1 ) = 1 ) |
232 |
157 231
|
syl5eq |
⊢ ( ( 𝑋 mod 𝑇 ) < π → ( 𝐹 ‘ 𝑋 ) = 1 ) |
233 |
230 232
|
syl |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod π ) = 0 ) → ( 𝐹 ‘ 𝑋 ) = 1 ) |
234 |
179 233
|
oveq12d |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod π ) = 0 ) → ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) = ( 1 + 1 ) ) |
235 |
234
|
oveq1d |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod π ) = 0 ) → ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) = ( ( 1 + 1 ) / 2 ) ) |
236 |
|
1p1e2 |
⊢ ( 1 + 1 ) = 2 |
237 |
236
|
oveq1i |
⊢ ( ( 1 + 1 ) / 2 ) = ( 2 / 2 ) |
238 |
|
2div2e1 |
⊢ ( 2 / 2 ) = 1 |
239 |
237 238
|
eqtr2i |
⊢ 1 = ( ( 1 + 1 ) / 2 ) |
240 |
233 239
|
eqtr2di |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod π ) = 0 ) → ( ( 1 + 1 ) / 2 ) = ( 𝐹 ‘ 𝑋 ) ) |
241 |
|
iffalse |
⊢ ( ¬ ( 𝑋 mod π ) = 0 → if ( ( 𝑋 mod π ) = 0 , 0 , ( 𝐹 ‘ 𝑋 ) ) = ( 𝐹 ‘ 𝑋 ) ) |
242 |
4 241
|
eqtr2id |
⊢ ( ¬ ( 𝑋 mod π ) = 0 → ( 𝐹 ‘ 𝑋 ) = 𝑌 ) |
243 |
242
|
adantl |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod π ) = 0 ) → ( 𝐹 ‘ 𝑋 ) = 𝑌 ) |
244 |
235 240 243
|
3eqtrrd |
⊢ ( ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod π ) = 0 ) → 𝑌 = ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) ) |
245 |
177 244
|
pm2.61dan |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) → 𝑌 = ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) ) |
246 |
133
|
necon2bi |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ) |
247 |
246
|
iffalsed |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) = - 1 ) |
248 |
|
id |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → ( 𝑋 mod 𝑇 ) = 0 ) |
249 |
248 37
|
eqbrtrdi |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → ( 𝑋 mod 𝑇 ) < π ) |
250 |
249
|
iftrued |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → if ( ( 𝑋 mod 𝑇 ) < π , 1 , - 1 ) = 1 ) |
251 |
157 250
|
syl5eq |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → ( 𝐹 ‘ 𝑋 ) = 1 ) |
252 |
247 251
|
oveq12d |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) = ( - 1 + 1 ) ) |
253 |
252
|
oveq1d |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) = ( ( - 1 + 1 ) / 2 ) ) |
254 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
255 |
185 254 166
|
addcomli |
⊢ ( - 1 + 1 ) = 0 |
256 |
255
|
oveq1i |
⊢ ( ( - 1 + 1 ) / 2 ) = ( 0 / 2 ) |
257 |
256 171
|
eqtri |
⊢ ( ( - 1 + 1 ) / 2 ) = 0 |
258 |
257
|
a1i |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → ( ( - 1 + 1 ) / 2 ) = 0 ) |
259 |
1
|
oveq2i |
⊢ ( 𝑋 / 𝑇 ) = ( 𝑋 / ( 2 · π ) ) |
260 |
|
2cnne0 |
⊢ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) |
261 |
17 38
|
pm3.2i |
⊢ ( π ∈ ℂ ∧ π ≠ 0 ) |
262 |
|
divdiv1 |
⊢ ( ( 𝑋 ∈ ℂ ∧ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ∧ ( π ∈ ℂ ∧ π ≠ 0 ) ) → ( ( 𝑋 / 2 ) / π ) = ( 𝑋 / ( 2 · π ) ) ) |
263 |
30 260 261 262
|
mp3an |
⊢ ( ( 𝑋 / 2 ) / π ) = ( 𝑋 / ( 2 · π ) ) |
264 |
30 170 17 46 38
|
divdiv32i |
⊢ ( ( 𝑋 / 2 ) / π ) = ( ( 𝑋 / π ) / 2 ) |
265 |
259 263 264
|
3eqtr2i |
⊢ ( 𝑋 / 𝑇 ) = ( ( 𝑋 / π ) / 2 ) |
266 |
265
|
oveq2i |
⊢ ( 2 · ( 𝑋 / 𝑇 ) ) = ( 2 · ( ( 𝑋 / π ) / 2 ) ) |
267 |
30 17 38
|
divcli |
⊢ ( 𝑋 / π ) ∈ ℂ |
268 |
267 170 46
|
divcan2i |
⊢ ( 2 · ( ( 𝑋 / π ) / 2 ) ) = ( 𝑋 / π ) |
269 |
266 268
|
eqtr2i |
⊢ ( 𝑋 / π ) = ( 2 · ( 𝑋 / 𝑇 ) ) |
270 |
6
|
a1i |
⊢ ( ( 𝑋 / 𝑇 ) ∈ ℤ → 2 ∈ ℤ ) |
271 |
|
id |
⊢ ( ( 𝑋 / 𝑇 ) ∈ ℤ → ( 𝑋 / 𝑇 ) ∈ ℤ ) |
272 |
270 271
|
zmulcld |
⊢ ( ( 𝑋 / 𝑇 ) ∈ ℤ → ( 2 · ( 𝑋 / 𝑇 ) ) ∈ ℤ ) |
273 |
269 272
|
eqeltrid |
⊢ ( ( 𝑋 / 𝑇 ) ∈ ℤ → ( 𝑋 / π ) ∈ ℤ ) |
274 |
67 273
|
sylbi |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → ( 𝑋 / π ) ∈ ℤ ) |
275 |
274 10
|
sylibr |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → ( 𝑋 mod π ) = 0 ) |
276 |
275
|
iftrued |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → if ( ( 𝑋 mod π ) = 0 , 0 , ( 𝐹 ‘ 𝑋 ) ) = 0 ) |
277 |
4 276
|
eqtr2id |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → 0 = 𝑌 ) |
278 |
253 258 277
|
3eqtrrd |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → 𝑌 = ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) ) |
279 |
278
|
adantl |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ( 𝑋 mod 𝑇 ) = 0 ) → 𝑌 = ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) ) |
280 |
130
|
a1i |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → π ∈ ℝ* ) |
281 |
61
|
rexri |
⊢ 𝑇 ∈ ℝ* |
282 |
281
|
a1i |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → 𝑇 ∈ ℝ* ) |
283 |
141
|
a1i |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ( 𝑋 mod 𝑇 ) ∈ ℝ ) |
284 |
|
pm4.56 |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) ↔ ¬ ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∨ ( 𝑋 mod 𝑇 ) = 0 ) ) |
285 |
284
|
biimpi |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ¬ ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∨ ( 𝑋 mod 𝑇 ) = 0 ) ) |
286 |
|
olc |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∨ ( 𝑋 mod 𝑇 ) = 0 ) ) |
287 |
286
|
adantl |
⊢ ( ( ( 𝑋 mod 𝑇 ) ≤ π ∧ ( 𝑋 mod 𝑇 ) = 0 ) → ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∨ ( 𝑋 mod 𝑇 ) = 0 ) ) |
288 |
129
|
a1i |
⊢ ( ( ( 𝑋 mod 𝑇 ) ≤ π ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → 0 ∈ ℝ* ) |
289 |
130
|
a1i |
⊢ ( ( ( 𝑋 mod 𝑇 ) ≤ π ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → π ∈ ℝ* ) |
290 |
142
|
a1i |
⊢ ( ( ( 𝑋 mod 𝑇 ) ≤ π ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ( 𝑋 mod 𝑇 ) ∈ ℝ* ) |
291 |
|
0red |
⊢ ( ¬ ( 𝑋 mod 𝑇 ) = 0 → 0 ∈ ℝ ) |
292 |
141
|
a1i |
⊢ ( ¬ ( 𝑋 mod 𝑇 ) = 0 → ( 𝑋 mod 𝑇 ) ∈ ℝ ) |
293 |
|
modge0 |
⊢ ( ( 𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ+ ) → 0 ≤ ( 𝑋 mod 𝑇 ) ) |
294 |
3 65 293
|
mp2an |
⊢ 0 ≤ ( 𝑋 mod 𝑇 ) |
295 |
294
|
a1i |
⊢ ( ¬ ( 𝑋 mod 𝑇 ) = 0 → 0 ≤ ( 𝑋 mod 𝑇 ) ) |
296 |
|
neqne |
⊢ ( ¬ ( 𝑋 mod 𝑇 ) = 0 → ( 𝑋 mod 𝑇 ) ≠ 0 ) |
297 |
291 292 295 296
|
leneltd |
⊢ ( ¬ ( 𝑋 mod 𝑇 ) = 0 → 0 < ( 𝑋 mod 𝑇 ) ) |
298 |
297
|
adantl |
⊢ ( ( ( 𝑋 mod 𝑇 ) ≤ π ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → 0 < ( 𝑋 mod 𝑇 ) ) |
299 |
|
simpl |
⊢ ( ( ( 𝑋 mod 𝑇 ) ≤ π ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ( 𝑋 mod 𝑇 ) ≤ π ) |
300 |
288 289 290 298 299
|
eliocd |
⊢ ( ( ( 𝑋 mod 𝑇 ) ≤ π ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ) |
301 |
300
|
orcd |
⊢ ( ( ( 𝑋 mod 𝑇 ) ≤ π ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∨ ( 𝑋 mod 𝑇 ) = 0 ) ) |
302 |
287 301
|
pm2.61dan |
⊢ ( ( 𝑋 mod 𝑇 ) ≤ π → ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∨ ( 𝑋 mod 𝑇 ) = 0 ) ) |
303 |
285 302
|
nsyl |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ¬ ( 𝑋 mod 𝑇 ) ≤ π ) |
304 |
36
|
a1i |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → π ∈ ℝ ) |
305 |
304 283
|
ltnled |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ( π < ( 𝑋 mod 𝑇 ) ↔ ¬ ( 𝑋 mod 𝑇 ) ≤ π ) ) |
306 |
303 305
|
mpbird |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → π < ( 𝑋 mod 𝑇 ) ) |
307 |
|
modlt |
⊢ ( ( 𝑋 ∈ ℝ ∧ 𝑇 ∈ ℝ+ ) → ( 𝑋 mod 𝑇 ) < 𝑇 ) |
308 |
3 65 307
|
mp2an |
⊢ ( 𝑋 mod 𝑇 ) < 𝑇 |
309 |
308
|
a1i |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ( 𝑋 mod 𝑇 ) < 𝑇 ) |
310 |
280 282 283 306 309
|
eliood |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) ) |
311 |
129
|
a1i |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → 0 ∈ ℝ* ) |
312 |
36
|
a1i |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → π ∈ ℝ ) |
313 |
142
|
a1i |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → ( 𝑋 mod 𝑇 ) ∈ ℝ* ) |
314 |
|
ioogtlb |
⊢ ( ( π ∈ ℝ* ∧ 𝑇 ∈ ℝ* ∧ ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) ) → π < ( 𝑋 mod 𝑇 ) ) |
315 |
130 281 314
|
mp3an12 |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → π < ( 𝑋 mod 𝑇 ) ) |
316 |
311 312 313 315
|
gtnelioc |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ) |
317 |
316
|
iffalsed |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) = - 1 ) |
318 |
141
|
a1i |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → ( 𝑋 mod 𝑇 ) ∈ ℝ ) |
319 |
312 318 315
|
ltnsymd |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → ¬ ( 𝑋 mod 𝑇 ) < π ) |
320 |
319
|
iffalsed |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → if ( ( 𝑋 mod 𝑇 ) < π , 1 , - 1 ) = - 1 ) |
321 |
157 320
|
syl5eq |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → ( 𝐹 ‘ 𝑋 ) = - 1 ) |
322 |
317 321
|
oveq12d |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) = ( - 1 + - 1 ) ) |
323 |
322
|
oveq1d |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) = ( ( - 1 + - 1 ) / 2 ) ) |
324 |
|
df-2 |
⊢ 2 = ( 1 + 1 ) |
325 |
324
|
negeqi |
⊢ - 2 = - ( 1 + 1 ) |
326 |
185 185
|
negdii |
⊢ - ( 1 + 1 ) = ( - 1 + - 1 ) |
327 |
325 326
|
eqtr2i |
⊢ ( - 1 + - 1 ) = - 2 |
328 |
327
|
oveq1i |
⊢ ( ( - 1 + - 1 ) / 2 ) = ( - 2 / 2 ) |
329 |
|
divneg |
⊢ ( ( 2 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) → - ( 2 / 2 ) = ( - 2 / 2 ) ) |
330 |
170 170 46 329
|
mp3an |
⊢ - ( 2 / 2 ) = ( - 2 / 2 ) |
331 |
238
|
negeqi |
⊢ - ( 2 / 2 ) = - 1 |
332 |
328 330 331
|
3eqtr2i |
⊢ ( ( - 1 + - 1 ) / 2 ) = - 1 |
333 |
332
|
a1i |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → ( ( - 1 + - 1 ) / 2 ) = - 1 ) |
334 |
4
|
a1i |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → 𝑌 = if ( ( 𝑋 mod π ) = 0 , 0 , ( 𝐹 ‘ 𝑋 ) ) ) |
335 |
312 318
|
ltnled |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → ( π < ( 𝑋 mod 𝑇 ) ↔ ¬ ( 𝑋 mod 𝑇 ) ≤ π ) ) |
336 |
315 335
|
mpbid |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → ¬ ( 𝑋 mod 𝑇 ) ≤ π ) |
337 |
248 114
|
eqbrtrdi |
⊢ ( ( 𝑋 mod 𝑇 ) = 0 → ( 𝑋 mod 𝑇 ) ≤ π ) |
338 |
337
|
adantl |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ( 𝑋 mod 𝑇 ) = 0 ) → ( 𝑋 mod 𝑇 ) ≤ π ) |
339 |
128
|
orcanai |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ( 𝑋 mod 𝑇 ) = π ) |
340 |
339 146
|
syl |
⊢ ( ( ( 𝑋 mod π ) = 0 ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → ( 𝑋 mod 𝑇 ) ≤ π ) |
341 |
338 340
|
pm2.61dan |
⊢ ( ( 𝑋 mod π ) = 0 → ( 𝑋 mod 𝑇 ) ≤ π ) |
342 |
336 341
|
nsyl |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → ¬ ( 𝑋 mod π ) = 0 ) |
343 |
342
|
iffalsed |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → if ( ( 𝑋 mod π ) = 0 , 0 , ( 𝐹 ‘ 𝑋 ) ) = ( 𝐹 ‘ 𝑋 ) ) |
344 |
334 343 321
|
3eqtrrd |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → - 1 = 𝑌 ) |
345 |
323 333 344
|
3eqtrrd |
⊢ ( ( 𝑋 mod 𝑇 ) ∈ ( π (,) 𝑇 ) → 𝑌 = ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) ) |
346 |
310 345
|
syl |
⊢ ( ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) ∧ ¬ ( 𝑋 mod 𝑇 ) = 0 ) → 𝑌 = ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) ) |
347 |
279 346
|
pm2.61dan |
⊢ ( ¬ ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) → 𝑌 = ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) ) |
348 |
245 347
|
pm2.61i |
⊢ 𝑌 = ( ( if ( ( 𝑋 mod 𝑇 ) ∈ ( 0 (,] π ) , 1 , - 1 ) + ( 𝐹 ‘ 𝑋 ) ) / 2 ) |