Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem43.1 |
⊢ 𝐾 = ( 𝑠 ∈ ( - π [,] π ) ↦ if ( 𝑠 = 0 , 1 , ( 𝑠 / ( 2 · ( sin ‘ ( 𝑠 / 2 ) ) ) ) ) ) |
2 |
|
1red |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ 𝑠 = 0 ) → 1 ∈ ℝ ) |
3 |
|
pire |
⊢ π ∈ ℝ |
4 |
3
|
a1i |
⊢ ( 𝑠 ∈ ( - π [,] π ) → π ∈ ℝ ) |
5 |
4
|
renegcld |
⊢ ( 𝑠 ∈ ( - π [,] π ) → - π ∈ ℝ ) |
6 |
|
id |
⊢ ( 𝑠 ∈ ( - π [,] π ) → 𝑠 ∈ ( - π [,] π ) ) |
7 |
|
eliccre |
⊢ ( ( - π ∈ ℝ ∧ π ∈ ℝ ∧ 𝑠 ∈ ( - π [,] π ) ) → 𝑠 ∈ ℝ ) |
8 |
5 4 6 7
|
syl3anc |
⊢ ( 𝑠 ∈ ( - π [,] π ) → 𝑠 ∈ ℝ ) |
9 |
8
|
adantr |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) → 𝑠 ∈ ℝ ) |
10 |
|
2re |
⊢ 2 ∈ ℝ |
11 |
10
|
a1i |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) → 2 ∈ ℝ ) |
12 |
9
|
rehalfcld |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) → ( 𝑠 / 2 ) ∈ ℝ ) |
13 |
12
|
resincld |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) → ( sin ‘ ( 𝑠 / 2 ) ) ∈ ℝ ) |
14 |
11 13
|
remulcld |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) → ( 2 · ( sin ‘ ( 𝑠 / 2 ) ) ) ∈ ℝ ) |
15 |
|
2cnd |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) → 2 ∈ ℂ ) |
16 |
13
|
recnd |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) → ( sin ‘ ( 𝑠 / 2 ) ) ∈ ℂ ) |
17 |
|
2ne0 |
⊢ 2 ≠ 0 |
18 |
17
|
a1i |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) → 2 ≠ 0 ) |
19 |
|
0xr |
⊢ 0 ∈ ℝ* |
20 |
19
|
a1i |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ 0 < 𝑠 ) → 0 ∈ ℝ* ) |
21 |
10 3
|
remulcli |
⊢ ( 2 · π ) ∈ ℝ |
22 |
21
|
rexri |
⊢ ( 2 · π ) ∈ ℝ* |
23 |
22
|
a1i |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ 0 < 𝑠 ) → ( 2 · π ) ∈ ℝ* ) |
24 |
8
|
adantr |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ 0 < 𝑠 ) → 𝑠 ∈ ℝ ) |
25 |
|
simpr |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ 0 < 𝑠 ) → 0 < 𝑠 ) |
26 |
21
|
a1i |
⊢ ( 𝑠 ∈ ( - π [,] π ) → ( 2 · π ) ∈ ℝ ) |
27 |
5
|
rexrd |
⊢ ( 𝑠 ∈ ( - π [,] π ) → - π ∈ ℝ* ) |
28 |
4
|
rexrd |
⊢ ( 𝑠 ∈ ( - π [,] π ) → π ∈ ℝ* ) |
29 |
|
iccleub |
⊢ ( ( - π ∈ ℝ* ∧ π ∈ ℝ* ∧ 𝑠 ∈ ( - π [,] π ) ) → 𝑠 ≤ π ) |
30 |
27 28 6 29
|
syl3anc |
⊢ ( 𝑠 ∈ ( - π [,] π ) → 𝑠 ≤ π ) |
31 |
|
pirp |
⊢ π ∈ ℝ+ |
32 |
|
2timesgt |
⊢ ( π ∈ ℝ+ → π < ( 2 · π ) ) |
33 |
31 32
|
ax-mp |
⊢ π < ( 2 · π ) |
34 |
33
|
a1i |
⊢ ( 𝑠 ∈ ( - π [,] π ) → π < ( 2 · π ) ) |
35 |
8 4 26 30 34
|
lelttrd |
⊢ ( 𝑠 ∈ ( - π [,] π ) → 𝑠 < ( 2 · π ) ) |
36 |
35
|
adantr |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ 0 < 𝑠 ) → 𝑠 < ( 2 · π ) ) |
37 |
20 23 24 25 36
|
eliood |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ 0 < 𝑠 ) → 𝑠 ∈ ( 0 (,) ( 2 · π ) ) ) |
38 |
|
sinaover2ne0 |
⊢ ( 𝑠 ∈ ( 0 (,) ( 2 · π ) ) → ( sin ‘ ( 𝑠 / 2 ) ) ≠ 0 ) |
39 |
37 38
|
syl |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ 0 < 𝑠 ) → ( sin ‘ ( 𝑠 / 2 ) ) ≠ 0 ) |
40 |
39
|
adantlr |
⊢ ( ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) ∧ 0 < 𝑠 ) → ( sin ‘ ( 𝑠 / 2 ) ) ≠ 0 ) |
41 |
8
|
ad2antrr |
⊢ ( ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) ∧ ¬ 0 < 𝑠 ) → 𝑠 ∈ ℝ ) |
42 |
|
iccgelb |
⊢ ( ( - π ∈ ℝ* ∧ π ∈ ℝ* ∧ 𝑠 ∈ ( - π [,] π ) ) → - π ≤ 𝑠 ) |
43 |
27 28 6 42
|
syl3anc |
⊢ ( 𝑠 ∈ ( - π [,] π ) → - π ≤ 𝑠 ) |
44 |
43
|
ad2antrr |
⊢ ( ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) ∧ ¬ 0 < 𝑠 ) → - π ≤ 𝑠 ) |
45 |
|
0red |
⊢ ( ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) ∧ ¬ 0 < 𝑠 ) → 0 ∈ ℝ ) |
46 |
|
neqne |
⊢ ( ¬ 𝑠 = 0 → 𝑠 ≠ 0 ) |
47 |
46
|
ad2antlr |
⊢ ( ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) ∧ ¬ 0 < 𝑠 ) → 𝑠 ≠ 0 ) |
48 |
|
simpr |
⊢ ( ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) ∧ ¬ 0 < 𝑠 ) → ¬ 0 < 𝑠 ) |
49 |
41 45 47 48
|
lttri5d |
⊢ ( ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) ∧ ¬ 0 < 𝑠 ) → 𝑠 < 0 ) |
50 |
5
|
ad2antrr |
⊢ ( ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) ∧ ¬ 0 < 𝑠 ) → - π ∈ ℝ ) |
51 |
|
elico2 |
⊢ ( ( - π ∈ ℝ ∧ 0 ∈ ℝ* ) → ( 𝑠 ∈ ( - π [,) 0 ) ↔ ( 𝑠 ∈ ℝ ∧ - π ≤ 𝑠 ∧ 𝑠 < 0 ) ) ) |
52 |
50 19 51
|
sylancl |
⊢ ( ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) ∧ ¬ 0 < 𝑠 ) → ( 𝑠 ∈ ( - π [,) 0 ) ↔ ( 𝑠 ∈ ℝ ∧ - π ≤ 𝑠 ∧ 𝑠 < 0 ) ) ) |
53 |
41 44 49 52
|
mpbir3and |
⊢ ( ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) ∧ ¬ 0 < 𝑠 ) → 𝑠 ∈ ( - π [,) 0 ) ) |
54 |
3
|
renegcli |
⊢ - π ∈ ℝ |
55 |
|
elicore |
⊢ ( ( - π ∈ ℝ ∧ 𝑠 ∈ ( - π [,) 0 ) ) → 𝑠 ∈ ℝ ) |
56 |
54 55
|
mpan |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → 𝑠 ∈ ℝ ) |
57 |
56
|
recnd |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → 𝑠 ∈ ℂ ) |
58 |
|
2cnd |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → 2 ∈ ℂ ) |
59 |
17
|
a1i |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → 2 ≠ 0 ) |
60 |
57 58 59
|
divnegd |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - ( 𝑠 / 2 ) = ( - 𝑠 / 2 ) ) |
61 |
60
|
eqcomd |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → ( - 𝑠 / 2 ) = - ( 𝑠 / 2 ) ) |
62 |
61
|
fveq2d |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → ( sin ‘ ( - 𝑠 / 2 ) ) = ( sin ‘ - ( 𝑠 / 2 ) ) ) |
63 |
62
|
negeqd |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - ( sin ‘ ( - 𝑠 / 2 ) ) = - ( sin ‘ - ( 𝑠 / 2 ) ) ) |
64 |
57
|
halfcld |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → ( 𝑠 / 2 ) ∈ ℂ ) |
65 |
|
sinneg |
⊢ ( ( 𝑠 / 2 ) ∈ ℂ → ( sin ‘ - ( 𝑠 / 2 ) ) = - ( sin ‘ ( 𝑠 / 2 ) ) ) |
66 |
64 65
|
syl |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → ( sin ‘ - ( 𝑠 / 2 ) ) = - ( sin ‘ ( 𝑠 / 2 ) ) ) |
67 |
66
|
negeqd |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - ( sin ‘ - ( 𝑠 / 2 ) ) = - - ( sin ‘ ( 𝑠 / 2 ) ) ) |
68 |
64
|
sincld |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → ( sin ‘ ( 𝑠 / 2 ) ) ∈ ℂ ) |
69 |
68
|
negnegd |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - - ( sin ‘ ( 𝑠 / 2 ) ) = ( sin ‘ ( 𝑠 / 2 ) ) ) |
70 |
63 67 69
|
3eqtrd |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - ( sin ‘ ( - 𝑠 / 2 ) ) = ( sin ‘ ( 𝑠 / 2 ) ) ) |
71 |
57
|
negcld |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - 𝑠 ∈ ℂ ) |
72 |
71
|
halfcld |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → ( - 𝑠 / 2 ) ∈ ℂ ) |
73 |
72
|
sincld |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → ( sin ‘ ( - 𝑠 / 2 ) ) ∈ ℂ ) |
74 |
19
|
a1i |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → 0 ∈ ℝ* ) |
75 |
22
|
a1i |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → ( 2 · π ) ∈ ℝ* ) |
76 |
56
|
renegcld |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - 𝑠 ∈ ℝ ) |
77 |
54
|
a1i |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - π ∈ ℝ ) |
78 |
77
|
rexrd |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - π ∈ ℝ* ) |
79 |
|
id |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → 𝑠 ∈ ( - π [,) 0 ) ) |
80 |
|
icoltub |
⊢ ( ( - π ∈ ℝ* ∧ 0 ∈ ℝ* ∧ 𝑠 ∈ ( - π [,) 0 ) ) → 𝑠 < 0 ) |
81 |
78 74 79 80
|
syl3anc |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → 𝑠 < 0 ) |
82 |
56
|
lt0neg1d |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → ( 𝑠 < 0 ↔ 0 < - 𝑠 ) ) |
83 |
81 82
|
mpbid |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → 0 < - 𝑠 ) |
84 |
3
|
a1i |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → π ∈ ℝ ) |
85 |
21
|
a1i |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → ( 2 · π ) ∈ ℝ ) |
86 |
|
icogelb |
⊢ ( ( - π ∈ ℝ* ∧ 0 ∈ ℝ* ∧ 𝑠 ∈ ( - π [,) 0 ) ) → - π ≤ 𝑠 ) |
87 |
78 74 79 86
|
syl3anc |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - π ≤ 𝑠 ) |
88 |
84 56 87
|
lenegcon1d |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - 𝑠 ≤ π ) |
89 |
33
|
a1i |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → π < ( 2 · π ) ) |
90 |
76 84 85 88 89
|
lelttrd |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - 𝑠 < ( 2 · π ) ) |
91 |
74 75 76 83 90
|
eliood |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - 𝑠 ∈ ( 0 (,) ( 2 · π ) ) ) |
92 |
|
sinaover2ne0 |
⊢ ( - 𝑠 ∈ ( 0 (,) ( 2 · π ) ) → ( sin ‘ ( - 𝑠 / 2 ) ) ≠ 0 ) |
93 |
91 92
|
syl |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → ( sin ‘ ( - 𝑠 / 2 ) ) ≠ 0 ) |
94 |
73 93
|
negne0d |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → - ( sin ‘ ( - 𝑠 / 2 ) ) ≠ 0 ) |
95 |
70 94
|
eqnetrrd |
⊢ ( 𝑠 ∈ ( - π [,) 0 ) → ( sin ‘ ( 𝑠 / 2 ) ) ≠ 0 ) |
96 |
53 95
|
syl |
⊢ ( ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) ∧ ¬ 0 < 𝑠 ) → ( sin ‘ ( 𝑠 / 2 ) ) ≠ 0 ) |
97 |
40 96
|
pm2.61dan |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) → ( sin ‘ ( 𝑠 / 2 ) ) ≠ 0 ) |
98 |
15 16 18 97
|
mulne0d |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) → ( 2 · ( sin ‘ ( 𝑠 / 2 ) ) ) ≠ 0 ) |
99 |
9 14 98
|
redivcld |
⊢ ( ( 𝑠 ∈ ( - π [,] π ) ∧ ¬ 𝑠 = 0 ) → ( 𝑠 / ( 2 · ( sin ‘ ( 𝑠 / 2 ) ) ) ) ∈ ℝ ) |
100 |
2 99
|
ifclda |
⊢ ( 𝑠 ∈ ( - π [,] π ) → if ( 𝑠 = 0 , 1 , ( 𝑠 / ( 2 · ( sin ‘ ( 𝑠 / 2 ) ) ) ) ) ∈ ℝ ) |
101 |
1 100
|
fmpti |
⊢ 𝐾 : ( - π [,] π ) ⟶ ℝ |