Metamath Proof Explorer

Theorem fourierdlem104

Description: The half upper part of the integral equal to the fourier partial sum, converges to half the right limit of the original function. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses fourierdlem104.f ⊢ φ → F : ℝ ⟶ ℝ
fourierdlem104.xre ⊢ φ → X ∈ ℝ
fourierdlem104.p ⊢ P = m ∈ ℕ ⟼ p ∈ ℝ 0 … m | p ⁡ 0 = - π + X ∧ p ⁡ m = π + X ∧ ∀ i ∈ 0 ..^ m p ⁡ i < p ⁡ i + 1
fourierdlem104.m ⊢ φ → M ∈ ℕ
fourierdlem104.v ⊢ φ → V ∈ P ⁡ M
fourierdlem104.x ⊢ φ → X ∈ ran ⁡ V
fourierdlem104.fcn ⊢ φ ∧ i ∈ 0 ..^ M → F ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶cn ℂ
fourierdlem104.fbdioo ⊢ φ ∧ i ∈ 0 ..^ M → ∃ w ∈ ℝ ∀ t ∈ V ⁡ i V ⁡ i + 1 F ⁡ t ≤ w
fourierdlem104.fdvcn ⊢ φ ∧ i ∈ 0 ..^ M → F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶cn ℝ
fourierdlem104.fdvbd ⊢ φ ∧ i ∈ 0 ..^ M → ∃ z ∈ ℝ ∀ t ∈ V ⁡ i V ⁡ i + 1 F ℝ ′ ⁡ t ≤ z
fourierdlem104.r ⊢ φ ∧ i ∈ 0 ..^ M → R ∈ F ↾ V ⁡ i V ⁡ i + 1 lim ℂ V ⁡ i
fourierdlem104.l ⊢ φ ∧ i ∈ 0 ..^ M → L ∈ F ↾ V ⁡ i V ⁡ i + 1 lim ℂ V ⁡ i + 1
fourierdlem104.h ⊢ H = s ∈ − π π ⟼ if s = 0 0 F ⁡ X + s − if 0 < s Y W s
fourierdlem104.k ⊢ K = s ∈ − π π ⟼ if s = 0 1 s 2 ⁢ sin ⁡ s 2
fourierdlem104.u ⊢ U = s ∈ − π π ⟼ H ⁡ s ⁢ K ⁡ s
fourierdlem104.s ⊢ S = s ∈ − π π ⟼ sin ⁡ n + 1 2 ⁢ s
fourierdlem104.g ⊢ G = s ∈ − π π ⟼ U ⁡ s ⁢ S ⁡ s
fourierdlem104.z ⊢ Z = m ∈ ℕ ⟼ ∫ 0 π F ⁡ X + s ⁢ D ⁡ m ⁡ s ds
fourierdlem104.e ⊢ E = n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds π
fourierdlem104.y ⊢ φ → Y ∈ F ↾ X +∞ lim ℂ X
fourierdlem104.w ⊢ φ → W ∈ F ↾ −∞ X lim ℂ X
fourierdlem104.a ⊢ φ → A ∈ F ℝ ′ ↾ −∞ X lim ℂ X
fourierdlem104.b ⊢ φ → B ∈ F ℝ ′ ↾ X +∞ lim ℂ X
fourierdlem104.d ⊢ D = n ∈ ℕ ⟼ s ∈ ℝ ⟼ if s mod 2 ⁢ π = 0 2 ⁢ n + 1 2 ⁢ π sin ⁡ n + 1 2 ⁢ s 2 ⁢ π ⁢ sin ⁡ s 2
fourierdlem104.o ⊢ O = U ↾ d π
fourierdlem104.t ⊢ T = d π ∪ ran ⁡ Q ∩ d π
fourierdlem104.n ⊢ N = T − 1
fourierdlem104.j ⊢ J = ι f | f Isom < , < 0 … N T
fourierdlem104.q ⊢ Q = i ∈ 0 … M ⟼ V ⁡ i − X
fourierdlem104.1 ⊢ C = ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1
fourierdlem104.ch ⊢ χ ↔ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ k ∈ ℕ ∧ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
Assertion fourierdlem104 ⊢ φ → Z ⇝ Y 2

Proof

Step Hyp Ref Expression
1 fourierdlem104.f ⊢ φ → F : ℝ ⟶ ℝ
2 fourierdlem104.xre ⊢ φ → X ∈ ℝ
3 fourierdlem104.p ⊢ P = m ∈ ℕ ⟼ p ∈ ℝ 0 … m | p ⁡ 0 = - π + X ∧ p ⁡ m = π + X ∧ ∀ i ∈ 0 ..^ m p ⁡ i < p ⁡ i + 1
4 fourierdlem104.m ⊢ φ → M ∈ ℕ
5 fourierdlem104.v ⊢ φ → V ∈ P ⁡ M
6 fourierdlem104.x ⊢ φ → X ∈ ran ⁡ V
7 fourierdlem104.fcn ⊢ φ ∧ i ∈ 0 ..^ M → F ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶cn ℂ
8 fourierdlem104.fbdioo ⊢ φ ∧ i ∈ 0 ..^ M → ∃ w ∈ ℝ ∀ t ∈ V ⁡ i V ⁡ i + 1 F ⁡ t ≤ w
9 fourierdlem104.fdvcn ⊢ φ ∧ i ∈ 0 ..^ M → F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶cn ℝ
10 fourierdlem104.fdvbd ⊢ φ ∧ i ∈ 0 ..^ M → ∃ z ∈ ℝ ∀ t ∈ V ⁡ i V ⁡ i + 1 F ℝ ′ ⁡ t ≤ z
11 fourierdlem104.r ⊢ φ ∧ i ∈ 0 ..^ M → R ∈ F ↾ V ⁡ i V ⁡ i + 1 lim ℂ V ⁡ i
12 fourierdlem104.l ⊢ φ ∧ i ∈ 0 ..^ M → L ∈ F ↾ V ⁡ i V ⁡ i + 1 lim ℂ V ⁡ i + 1
13 fourierdlem104.h ⊢ H = s ∈ − π π ⟼ if s = 0 0 F ⁡ X + s − if 0 < s Y W s
14 fourierdlem104.k ⊢ K = s ∈ − π π ⟼ if s = 0 1 s 2 ⁢ sin ⁡ s 2
15 fourierdlem104.u ⊢ U = s ∈ − π π ⟼ H ⁡ s ⁢ K ⁡ s
16 fourierdlem104.s ⊢ S = s ∈ − π π ⟼ sin ⁡ n + 1 2 ⁢ s
17 fourierdlem104.g ⊢ G = s ∈ − π π ⟼ U ⁡ s ⁢ S ⁡ s
18 fourierdlem104.z ⊢ Z = m ∈ ℕ ⟼ ∫ 0 π F ⁡ X + s ⁢ D ⁡ m ⁡ s ds
19 fourierdlem104.e ⊢ E = n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds π
20 fourierdlem104.y ⊢ φ → Y ∈ F ↾ X +∞ lim ℂ X
21 fourierdlem104.w ⊢ φ → W ∈ F ↾ −∞ X lim ℂ X
22 fourierdlem104.a ⊢ φ → A ∈ F ℝ ′ ↾ −∞ X lim ℂ X
23 fourierdlem104.b ⊢ φ → B ∈ F ℝ ′ ↾ X +∞ lim ℂ X
24 fourierdlem104.d ⊢ D = n ∈ ℕ ⟼ s ∈ ℝ ⟼ if s mod 2 ⁢ π = 0 2 ⁢ n + 1 2 ⁢ π sin ⁡ n + 1 2 ⁢ s 2 ⁢ π ⁢ sin ⁡ s 2
25 fourierdlem104.o ⊢ O = U ↾ d π
26 fourierdlem104.t ⊢ T = d π ∪ ran ⁡ Q ∩ d π
27 fourierdlem104.n ⊢ N = T − 1
28 fourierdlem104.j ⊢ J = ι f | f Isom < , < 0 … N T
29 fourierdlem104.q ⊢ Q = i ∈ 0 … M ⟼ V ⁡ i − X
30 fourierdlem104.1 ⊢ C = ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1
31 fourierdlem104.ch ⊢ χ ↔ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ k ∈ ℕ ∧ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
32 eqid ⊢ ℤ ≥ 1 = ℤ ≥ 1
33 1zzd ⊢ φ → 1 ∈ ℤ
34 nfv ⊢ Ⅎ n φ
35 nfmpt1 ⊢ Ⅎ _ n n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds
36 nfmpt1 ⊢ Ⅎ _ n n ∈ ℕ ⟼ π
37 nfmpt1 ⊢ Ⅎ _ n n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds π
38 19 37 nfcxfr ⊢ Ⅎ _ n E
39 nnuz ⊢ ℕ = ℤ ≥ 1
40 elioore ⊢ d ∈ 0 π → d ∈ ℝ
41 40 adantl ⊢ φ ∧ d ∈ 0 π → d ∈ ℝ
42 pire ⊢ π ∈ ℝ
43 42 a1i ⊢ φ ∧ d ∈ 0 π → π ∈ ℝ
44 ioossre ⊢ X +∞ ⊆ ℝ
45 44 a1i ⊢ φ → X +∞ ⊆ ℝ
46 1 45 fssresd ⊢ φ → F ↾ X +∞ : X +∞ ⟶ ℝ
47 ioosscn ⊢ X +∞ ⊆ ℂ
48 47 a1i ⊢ φ → X +∞ ⊆ ℂ
49 eqid ⊢ TopOpen ⁡ ℂ fld = TopOpen ⁡ ℂ fld
50 pnfxr ⊢ +∞ ∈ ℝ *
51 50 a1i ⊢ φ → +∞ ∈ ℝ *
52 2 ltpnfd ⊢ φ → X < +∞
53 49 51 2 52 lptioo1cn ⊢ φ → X ∈ limPt ⁡ TopOpen ⁡ ℂ fld ⁡ X +∞
54 46 48 53 20 limcrecl ⊢ φ → Y ∈ ℝ
55 ioossre ⊢ −∞ X ⊆ ℝ
56 55 a1i ⊢ φ → −∞ X ⊆ ℝ
57 1 56 fssresd ⊢ φ → F ↾ −∞ X : −∞ X ⟶ ℝ
58 ioosscn ⊢ −∞ X ⊆ ℂ
59 58 a1i ⊢ φ → −∞ X ⊆ ℂ
60 mnfxr ⊢ −∞ ∈ ℝ *
61 60 a1i ⊢ φ → −∞ ∈ ℝ *
62 2 mnfltd ⊢ φ → −∞ < X
63 49 61 2 62 lptioo2cn ⊢ φ → X ∈ limPt ⁡ TopOpen ⁡ ℂ fld ⁡ −∞ X
64 57 59 63 21 limcrecl ⊢ φ → W ∈ ℝ
65 1 2 54 64 13 14 15 fourierdlem55 ⊢ φ → U : − π π ⟶ ℝ
66 ax-resscn ⊢ ℝ ⊆ ℂ
67 66 a1i ⊢ φ → ℝ ⊆ ℂ
68 65 67 fssd ⊢ φ → U : − π π ⟶ ℂ
69 68 adantr ⊢ φ ∧ d ∈ 0 π → U : − π π ⟶ ℂ
70 42 renegcli ⊢ − π ∈ ℝ
71 70 a1i ⊢ φ ∧ d ∈ 0 π → − π ∈ ℝ
72 70 a1i ⊢ d ∈ 0 π → − π ∈ ℝ
73 0red ⊢ d ∈ 0 π → 0 ∈ ℝ
74 negpilt0 ⊢ − π < 0
75 74 a1i ⊢ d ∈ 0 π → − π < 0
76 0xr ⊢ 0 ∈ ℝ *
77 42 rexri ⊢ π ∈ ℝ *
78 ioogtlb ⊢ 0 ∈ ℝ * ∧ π ∈ ℝ * ∧ d ∈ 0 π → 0 < d
79 76 77 78 mp3an12 ⊢ d ∈ 0 π → 0 < d
80 72 73 40 75 79 lttrd ⊢ d ∈ 0 π → − π < d
81 72 40 80 ltled ⊢ d ∈ 0 π → − π ≤ d
82 81 adantl ⊢ φ ∧ d ∈ 0 π → − π ≤ d
83 43 leidd ⊢ φ ∧ d ∈ 0 π → π ≤ π
84 iccss ⊢ − π ∈ ℝ ∧ π ∈ ℝ ∧ − π ≤ d ∧ π ≤ π → d π ⊆ − π π
85 71 43 82 83 84 syl22anc ⊢ φ ∧ d ∈ 0 π → d π ⊆ − π π
86 69 85 fssresd ⊢ φ ∧ d ∈ 0 π → U ↾ d π : d π ⟶ ℂ
87 25 a1i ⊢ φ ∧ d ∈ 0 π → O = U ↾ d π
88 87 feq1d ⊢ φ ∧ d ∈ 0 π → O : d π ⟶ ℂ ↔ U ↾ d π : d π ⟶ ℂ
89 86 88 mpbird ⊢ φ ∧ d ∈ 0 π → O : d π ⟶ ℂ
90 42 elexi ⊢ π ∈ V
91 90 prid2 ⊢ π ∈ d π
92 elun1 ⊢ π ∈ d π → π ∈ d π ∪ ran ⁡ Q ∩ d π
93 91 92 ax-mp ⊢ π ∈ d π ∪ ran ⁡ Q ∩ d π
94 93 26 eleqtrri ⊢ π ∈ T
95 94 ne0ii ⊢ T ≠ ∅
96 95 a1i ⊢ φ → T ≠ ∅
97 prfi ⊢ d π ∈ Fin
98 97 a1i ⊢ φ → d π ∈ Fin
99 fzfi ⊢ 0 … M ∈ Fin
100 29 rnmptfi ⊢ 0 … M ∈ Fin → ran ⁡ Q ∈ Fin
101 99 100 ax-mp ⊢ ran ⁡ Q ∈ Fin
102 infi ⊢ ran ⁡ Q ∈ Fin → ran ⁡ Q ∩ d π ∈ Fin
103 101 102 mp1i ⊢ φ → ran ⁡ Q ∩ d π ∈ Fin
104 unfi ⊢ d π ∈ Fin ∧ ran ⁡ Q ∩ d π ∈ Fin → d π ∪ ran ⁡ Q ∩ d π ∈ Fin
105 98 103 104 syl2anc ⊢ φ → d π ∪ ran ⁡ Q ∩ d π ∈ Fin
106 26 105 eqeltrid ⊢ φ → T ∈ Fin
107 hashnncl ⊢ T ∈ Fin → T ∈ ℕ ↔ T ≠ ∅
108 106 107 syl ⊢ φ → T ∈ ℕ ↔ T ≠ ∅
109 96 108 mpbird ⊢ φ → T ∈ ℕ
110 nnm1nn0 ⊢ T ∈ ℕ → T − 1 ∈ ℕ 0
111 109 110 syl ⊢ φ → T − 1 ∈ ℕ 0
112 27 111 eqeltrid ⊢ φ → N ∈ ℕ 0
113 112 adantr ⊢ φ ∧ d ∈ 0 π → N ∈ ℕ 0
114 0red ⊢ φ ∧ d ∈ 0 π → 0 ∈ ℝ
115 1red ⊢ φ ∧ d ∈ 0 π → 1 ∈ ℝ
116 113 nn0red ⊢ φ ∧ d ∈ 0 π → N ∈ ℝ
117 0lt1 ⊢ 0 < 1
118 117 a1i ⊢ φ ∧ d ∈ 0 π → 0 < 1
119 2re ⊢ 2 ∈ ℝ
120 119 a1i ⊢ φ ∧ d ∈ 0 π → 2 ∈ ℝ
121 109 nnred ⊢ φ → T ∈ ℝ
122 121 adantr ⊢ φ ∧ d ∈ 0 π → T ∈ ℝ
123 iooltub ⊢ 0 ∈ ℝ * ∧ π ∈ ℝ * ∧ d ∈ 0 π → d < π
124 76 77 123 mp3an12 ⊢ d ∈ 0 π → d < π
125 40 124 ltned ⊢ d ∈ 0 π → d ≠ π
126 125 adantl ⊢ φ ∧ d ∈ 0 π → d ≠ π
127 hashprg ⊢ d ∈ ℝ ∧ π ∈ ℝ → d ≠ π ↔ d π = 2
128 41 42 127 sylancl ⊢ φ ∧ d ∈ 0 π → d ≠ π ↔ d π = 2
129 126 128 mpbid ⊢ φ ∧ d ∈ 0 π → d π = 2
130 129 eqcomd ⊢ φ ∧ d ∈ 0 π → 2 = d π
131 106 adantr ⊢ φ ∧ d ∈ 0 π → T ∈ Fin
132 ssun1 ⊢ d π ⊆ d π ∪ ran ⁡ Q ∩ d π
133 132 26 sseqtrri ⊢ d π ⊆ T
134 hashssle ⊢ T ∈ Fin ∧ d π ⊆ T → d π ≤ T
135 131 133 134 sylancl ⊢ φ ∧ d ∈ 0 π → d π ≤ T
136 130 135 eqbrtrd ⊢ φ ∧ d ∈ 0 π → 2 ≤ T
137 120 122 115 136 lesub1dd ⊢ φ ∧ d ∈ 0 π → 2 − 1 ≤ T − 1
138 1e2m1 ⊢ 1 = 2 − 1
139 137 138 27 3brtr4g ⊢ φ ∧ d ∈ 0 π → 1 ≤ N
140 114 115 116 118 139 ltletrd ⊢ φ ∧ d ∈ 0 π → 0 < N
141 140 gt0ne0d ⊢ φ ∧ d ∈ 0 π → N ≠ 0
142 elnnne0 ⊢ N ∈ ℕ ↔ N ∈ ℕ 0 ∧ N ≠ 0
143 113 141 142 sylanbrc ⊢ φ ∧ d ∈ 0 π → N ∈ ℕ
144 41 leidd ⊢ φ ∧ d ∈ 0 π → d ≤ d
145 42 a1i ⊢ d ∈ 0 π → π ∈ ℝ
146 40 145 124 ltled ⊢ d ∈ 0 π → d ≤ π
147 146 adantl ⊢ φ ∧ d ∈ 0 π → d ≤ π
148 41 43 41 144 147 eliccd ⊢ φ ∧ d ∈ 0 π → d ∈ d π
149 41 43 43 147 83 eliccd ⊢ φ ∧ d ∈ 0 π → π ∈ d π
150 148 149 jca ⊢ φ ∧ d ∈ 0 π → d ∈ d π ∧ π ∈ d π
151 vex ⊢ d ∈ V
152 151 90 prss ⊢ d ∈ d π ∧ π ∈ d π ↔ d π ⊆ d π
153 150 152 sylib ⊢ φ ∧ d ∈ 0 π → d π ⊆ d π
154 inss2 ⊢ ran ⁡ Q ∩ d π ⊆ d π
155 154 a1i ⊢ φ ∧ d ∈ 0 π → ran ⁡ Q ∩ d π ⊆ d π
156 ioossicc ⊢ d π ⊆ d π
157 155 156 sstrdi ⊢ φ ∧ d ∈ 0 π → ran ⁡ Q ∩ d π ⊆ d π
158 153 157 unssd ⊢ φ ∧ d ∈ 0 π → d π ∪ ran ⁡ Q ∩ d π ⊆ d π
159 26 158 eqsstrid ⊢ φ ∧ d ∈ 0 π → T ⊆ d π
160 151 prid1 ⊢ d ∈ d π
161 elun1 ⊢ d ∈ d π → d ∈ d π ∪ ran ⁡ Q ∩ d π
162 160 161 ax-mp ⊢ d ∈ d π ∪ ran ⁡ Q ∩ d π
163 162 26 eleqtrri ⊢ d ∈ T
164 163 a1i ⊢ φ ∧ d ∈ 0 π → d ∈ T
165 94 a1i ⊢ φ ∧ d ∈ 0 π → π ∈ T
166 131 27 28 41 43 159 164 165 fourierdlem52 ⊢ φ ∧ d ∈ 0 π → J : 0 … N ⟶ d π ∧ J ⁡ 0 = d ∧ J ⁡ N = π
167 166 simplld ⊢ φ ∧ d ∈ 0 π → J : 0 … N ⟶ d π
168 166 simplrd ⊢ φ ∧ d ∈ 0 π → J ⁡ 0 = d
169 166 simprd ⊢ φ ∧ d ∈ 0 π → J ⁡ N = π
170 elfzoelz ⊢ k ∈ 0 ..^ N → k ∈ ℤ
171 170 zred ⊢ k ∈ 0 ..^ N → k ∈ ℝ
172 171 adantl ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → k ∈ ℝ
173 172 ltp1d ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → k < k + 1
174 40 145 jca ⊢ d ∈ 0 π → d ∈ ℝ ∧ π ∈ ℝ
175 151 90 prss ⊢ d ∈ ℝ ∧ π ∈ ℝ ↔ d π ⊆ ℝ
176 174 175 sylib ⊢ d ∈ 0 π → d π ⊆ ℝ
177 176 adantl ⊢ φ ∧ d ∈ 0 π → d π ⊆ ℝ
178 ioossre ⊢ d π ⊆ ℝ
179 154 178 sstri ⊢ ran ⁡ Q ∩ d π ⊆ ℝ
180 179 a1i ⊢ φ ∧ d ∈ 0 π → ran ⁡ Q ∩ d π ⊆ ℝ
181 177 180 unssd ⊢ φ ∧ d ∈ 0 π → d π ∪ ran ⁡ Q ∩ d π ⊆ ℝ
182 26 181 eqsstrid ⊢ φ ∧ d ∈ 0 π → T ⊆ ℝ
183 131 182 28 27 fourierdlem36 ⊢ φ ∧ d ∈ 0 π → J Isom < , < 0 … N T
184 183 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J Isom < , < 0 … N T
185 elfzofz ⊢ k ∈ 0 ..^ N → k ∈ 0 … N
186 185 adantl ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → k ∈ 0 … N
187 fzofzp1 ⊢ k ∈ 0 ..^ N → k + 1 ∈ 0 … N
188 187 adantl ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → k + 1 ∈ 0 … N
189 isorel ⊢ J Isom < , < 0 … N T ∧ k ∈ 0 … N ∧ k + 1 ∈ 0 … N → k < k + 1 ↔ J ⁡ k < J ⁡ k + 1
190 184 186 188 189 syl12anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → k < k + 1 ↔ J ⁡ k < J ⁡ k + 1
191 173 190 mpbid ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k < J ⁡ k + 1
192 65 adantr ⊢ φ ∧ d ∈ 0 π → U : − π π ⟶ ℝ
193 192 85 feqresmpt ⊢ φ ∧ d ∈ 0 π → U ↾ d π = s ∈ d π ⟼ U ⁡ s
194 85 sselda ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → s ∈ − π π
195 1 2 54 64 13 fourierdlem9 ⊢ φ → H : − π π ⟶ ℝ
196 195 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → H : − π π ⟶ ℝ
197 196 194 ffvelcdmd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → H ⁡ s ∈ ℝ
198 14 fourierdlem43 ⊢ K : − π π ⟶ ℝ
199 198 a1i ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → K : − π π ⟶ ℝ
200 199 194 ffvelcdmd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → K ⁡ s ∈ ℝ
201 197 200 remulcld ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → H ⁡ s ⁢ K ⁡ s ∈ ℝ
202 15 fvmpt2 ⊢ s ∈ − π π ∧ H ⁡ s ⁢ K ⁡ s ∈ ℝ → U ⁡ s = H ⁡ s ⁢ K ⁡ s
203 194 201 202 syl2anc ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → U ⁡ s = H ⁡ s ⁢ K ⁡ s
204 0red ⊢ d ∈ 0 π ∧ s ∈ d π → 0 ∈ ℝ
205 40 adantr ⊢ d ∈ 0 π ∧ s ∈ d π → d ∈ ℝ
206 42 a1i ⊢ d ∈ 0 π ∧ s ∈ d π → π ∈ ℝ
207 simpr ⊢ d ∈ 0 π ∧ s ∈ d π → s ∈ d π
208 eliccre ⊢ d ∈ ℝ ∧ π ∈ ℝ ∧ s ∈ d π → s ∈ ℝ
209 205 206 207 208 syl3anc ⊢ d ∈ 0 π ∧ s ∈ d π → s ∈ ℝ
210 79 adantr ⊢ d ∈ 0 π ∧ s ∈ d π → 0 < d
211 205 rexrd ⊢ d ∈ 0 π ∧ s ∈ d π → d ∈ ℝ *
212 77 a1i ⊢ d ∈ 0 π ∧ s ∈ d π → π ∈ ℝ *
213 iccgelb ⊢ d ∈ ℝ * ∧ π ∈ ℝ * ∧ s ∈ d π → d ≤ s
214 211 212 207 213 syl3anc ⊢ d ∈ 0 π ∧ s ∈ d π → d ≤ s
215 204 205 209 210 214 ltletrd ⊢ d ∈ 0 π ∧ s ∈ d π → 0 < s
216 215 gt0ne0d ⊢ d ∈ 0 π ∧ s ∈ d π → s ≠ 0
217 216 adantll ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → s ≠ 0
218 217 neneqd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → ¬ s = 0
219 218 iffalsed ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → if s = 0 0 F ⁡ X + s − if 0 < s Y W s = F ⁡ X + s − if 0 < s Y W s
220 215 adantll ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → 0 < s
221 220 iftrued ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → if 0 < s Y W = Y
222 221 oveq2d ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → F ⁡ X + s − if 0 < s Y W = F ⁡ X + s − Y
223 222 oveq1d ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → F ⁡ X + s − if 0 < s Y W s = F ⁡ X + s − Y s
224 219 223 eqtrd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → if s = 0 0 F ⁡ X + s − if 0 < s Y W s = F ⁡ X + s − Y s
225 1 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → F : ℝ ⟶ ℝ
226 2 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → X ∈ ℝ
227 iccssre ⊢ − π ∈ ℝ ∧ π ∈ ℝ → − π π ⊆ ℝ
228 70 42 227 mp2an ⊢ − π π ⊆ ℝ
229 228 194 sselid ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → s ∈ ℝ
230 226 229 readdcld ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → X + s ∈ ℝ
231 225 230 ffvelcdmd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → F ⁡ X + s ∈ ℝ
232 54 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → Y ∈ ℝ
233 231 232 resubcld ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → F ⁡ X + s − Y ∈ ℝ
234 233 229 217 redivcld ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → F ⁡ X + s − Y s ∈ ℝ
235 224 234 eqeltrd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → if s = 0 0 F ⁡ X + s − if 0 < s Y W s ∈ ℝ
236 13 fvmpt2 ⊢ s ∈ − π π ∧ if s = 0 0 F ⁡ X + s − if 0 < s Y W s ∈ ℝ → H ⁡ s = if s = 0 0 F ⁡ X + s − if 0 < s Y W s
237 194 235 236 syl2anc ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → H ⁡ s = if s = 0 0 F ⁡ X + s − if 0 < s Y W s
238 237 219 223 3eqtrd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → H ⁡ s = F ⁡ X + s − Y s
239 206 renegcld ⊢ d ∈ 0 π ∧ s ∈ d π → − π ∈ ℝ
240 74 a1i ⊢ d ∈ 0 π ∧ s ∈ d π → − π < 0
241 239 204 209 240 215 lttrd ⊢ d ∈ 0 π ∧ s ∈ d π → − π < s
242 239 209 241 ltled ⊢ d ∈ 0 π ∧ s ∈ d π → − π ≤ s
243 iccleub ⊢ d ∈ ℝ * ∧ π ∈ ℝ * ∧ s ∈ d π → s ≤ π
244 211 212 207 243 syl3anc ⊢ d ∈ 0 π ∧ s ∈ d π → s ≤ π
245 239 206 209 242 244 eliccd ⊢ d ∈ 0 π ∧ s ∈ d π → s ∈ − π π
246 216 neneqd ⊢ d ∈ 0 π ∧ s ∈ d π → ¬ s = 0
247 246 iffalsed ⊢ d ∈ 0 π ∧ s ∈ d π → if s = 0 1 s 2 ⁢ sin ⁡ s 2 = s 2 ⁢ sin ⁡ s 2
248 119 a1i ⊢ d ∈ 0 π ∧ s ∈ d π → 2 ∈ ℝ
249 209 rehalfcld ⊢ d ∈ 0 π ∧ s ∈ d π → s 2 ∈ ℝ
250 249 resincld ⊢ d ∈ 0 π ∧ s ∈ d π → sin ⁡ s 2 ∈ ℝ
251 248 250 remulcld ⊢ d ∈ 0 π ∧ s ∈ d π → 2 ⁢ sin ⁡ s 2 ∈ ℝ
252 2cnd ⊢ d ∈ 0 π ∧ s ∈ d π → 2 ∈ ℂ
253 209 recnd ⊢ d ∈ 0 π ∧ s ∈ d π → s ∈ ℂ
254 253 halfcld ⊢ d ∈ 0 π ∧ s ∈ d π → s 2 ∈ ℂ
255 254 sincld ⊢ d ∈ 0 π ∧ s ∈ d π → sin ⁡ s 2 ∈ ℂ
256 2ne0 ⊢ 2 ≠ 0
257 256 a1i ⊢ d ∈ 0 π ∧ s ∈ d π → 2 ≠ 0
258 fourierdlem44 ⊢ s ∈ − π π ∧ s ≠ 0 → sin ⁡ s 2 ≠ 0
259 245 216 258 syl2anc ⊢ d ∈ 0 π ∧ s ∈ d π → sin ⁡ s 2 ≠ 0
260 252 255 257 259 mulne0d ⊢ d ∈ 0 π ∧ s ∈ d π → 2 ⁢ sin ⁡ s 2 ≠ 0
261 209 251 260 redivcld ⊢ d ∈ 0 π ∧ s ∈ d π → s 2 ⁢ sin ⁡ s 2 ∈ ℝ
262 247 261 eqeltrd ⊢ d ∈ 0 π ∧ s ∈ d π → if s = 0 1 s 2 ⁢ sin ⁡ s 2 ∈ ℝ
263 14 fvmpt2 ⊢ s ∈ − π π ∧ if s = 0 1 s 2 ⁢ sin ⁡ s 2 ∈ ℝ → K ⁡ s = if s = 0 1 s 2 ⁢ sin ⁡ s 2
264 245 262 263 syl2anc ⊢ d ∈ 0 π ∧ s ∈ d π → K ⁡ s = if s = 0 1 s 2 ⁢ sin ⁡ s 2
265 264 adantll ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → K ⁡ s = if s = 0 1 s 2 ⁢ sin ⁡ s 2
266 238 265 oveq12d ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → H ⁡ s ⁢ K ⁡ s = F ⁡ X + s − Y s ⁢ if s = 0 1 s 2 ⁢ sin ⁡ s 2
267 218 iffalsed ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → if s = 0 1 s 2 ⁢ sin ⁡ s 2 = s 2 ⁢ sin ⁡ s 2
268 267 oveq2d ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → F ⁡ X + s − Y s ⁢ if s = 0 1 s 2 ⁢ sin ⁡ s 2 = F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2
269 203 266 268 3eqtrd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → U ⁡ s = F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2
270 269 mpteq2dva ⊢ φ ∧ d ∈ 0 π → s ∈ d π ⟼ U ⁡ s = s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2
271 87 193 270 3eqtrd ⊢ φ ∧ d ∈ 0 π → O = s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2
272 271 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → O = s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2
273 272 reseq1d ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → O ↾ J ⁡ k J ⁡ k + 1 = s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 ↾ J ⁡ k J ⁡ k + 1
274 1 adantr ⊢ φ ∧ d ∈ 0 π → F : ℝ ⟶ ℝ
275 2 adantr ⊢ φ ∧ d ∈ 0 π → X ∈ ℝ
276 4 adantr ⊢ φ ∧ d ∈ 0 π → M ∈ ℕ
277 5 adantr ⊢ φ ∧ d ∈ 0 π → V ∈ P ⁡ M
278 7 adantlr ⊢ φ ∧ d ∈ 0 π ∧ i ∈ 0 ..^ M → F ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶cn ℂ
279 11 adantlr ⊢ φ ∧ d ∈ 0 π ∧ i ∈ 0 ..^ M → R ∈ F ↾ V ⁡ i V ⁡ i + 1 lim ℂ V ⁡ i
280 12 adantlr ⊢ φ ∧ d ∈ 0 π ∧ i ∈ 0 ..^ M → L ∈ F ↾ V ⁡ i V ⁡ i + 1 lim ℂ V ⁡ i + 1
281 124 adantl ⊢ φ ∧ d ∈ 0 π → d < π
282 73 40 ltnled ⊢ d ∈ 0 π → 0 < d ↔ ¬ d ≤ 0
283 79 282 mpbid ⊢ d ∈ 0 π → ¬ d ≤ 0
284 283 intn3an2d ⊢ d ∈ 0 π → ¬ 0 ∈ ℝ ∧ d ≤ 0 ∧ 0 ≤ π
285 elicc2 ⊢ d ∈ ℝ ∧ π ∈ ℝ → 0 ∈ d π ↔ 0 ∈ ℝ ∧ d ≤ 0 ∧ 0 ≤ π
286 40 42 285 sylancl ⊢ d ∈ 0 π → 0 ∈ d π ↔ 0 ∈ ℝ ∧ d ≤ 0 ∧ 0 ≤ π
287 284 286 mtbird ⊢ d ∈ 0 π → ¬ 0 ∈ d π
288 287 adantl ⊢ φ ∧ d ∈ 0 π → ¬ 0 ∈ d π
289 54 adantr ⊢ φ ∧ d ∈ 0 π → Y ∈ ℝ
290 eqid ⊢ s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 = s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2
291 eqid ⊢ if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2 = if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2
292 eqid ⊢ if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2 = if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2
293 fveq2 ⊢ l = i → Q ⁡ l = Q ⁡ i
294 oveq1 ⊢ l = i → l + 1 = i + 1
295 294 fveq2d ⊢ l = i → Q ⁡ l + 1 = Q ⁡ i + 1
296 293 295 oveq12d ⊢ l = i → Q ⁡ l Q ⁡ l + 1 = Q ⁡ i Q ⁡ i + 1
297 296 sseq2d ⊢ l = i → J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ↔ J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ i Q ⁡ i + 1
298 297 cbvriotavw ⊢ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 = ι i ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ i Q ⁡ i + 1
299 274 275 3 276 277 278 279 280 41 43 281 85 288 289 290 29 26 27 28 291 292 298 fourierdlem86 ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2 ∈ s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 ↾ J ⁡ k J ⁡ k + 1 lim ℂ J ⁡ k + 1 ∧ if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2 ∈ s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 ↾ J ⁡ k J ⁡ k + 1 lim ℂ J ⁡ k ∧ s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 ↾ J ⁡ k J ⁡ k + 1 : J ⁡ k J ⁡ k + 1 ⟶cn ℂ
300 299 simprd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 ↾ J ⁡ k J ⁡ k + 1 : J ⁡ k J ⁡ k + 1 ⟶cn ℂ
301 273 300 eqeltrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → O ↾ J ⁡ k J ⁡ k + 1 : J ⁡ k J ⁡ k + 1 ⟶cn ℂ
302 299 simplld ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2 ∈ s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 ↾ J ⁡ k J ⁡ k + 1 lim ℂ J ⁡ k + 1
303 272 eqcomd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 = O
304 303 reseq1d ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 ↾ J ⁡ k J ⁡ k + 1 = O ↾ J ⁡ k J ⁡ k + 1
305 304 oveq1d ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 ↾ J ⁡ k J ⁡ k + 1 lim ℂ J ⁡ k + 1 = O ↾ J ⁡ k J ⁡ k + 1 lim ℂ J ⁡ k + 1
306 302 305 eleqtrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2 ∈ O ↾ J ⁡ k J ⁡ k + 1 lim ℂ J ⁡ k + 1
307 299 simplrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2 ∈ s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 ↾ J ⁡ k J ⁡ k + 1 lim ℂ J ⁡ k
308 304 oveq1d ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 ↾ J ⁡ k J ⁡ k + 1 lim ℂ J ⁡ k = O ↾ J ⁡ k J ⁡ k + 1 lim ℂ J ⁡ k
309 307 308 eleqtrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2 ∈ O ↾ J ⁡ k J ⁡ k + 1 lim ℂ J ⁡ k
310 eqid ⊢ ℝ D O = ℝ D O
311 89 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → O : d π ⟶ ℂ
312 41 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → d ∈ ℝ
313 42 a1i ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → π ∈ ℝ
314 elioore ⊢ s ∈ J ⁡ k J ⁡ k + 1 → s ∈ ℝ
315 314 adantl ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → s ∈ ℝ
316 85 228 sstrdi ⊢ φ ∧ d ∈ 0 π → d π ⊆ ℝ
317 316 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → d π ⊆ ℝ
318 167 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J : 0 … N ⟶ d π
319 318 186 ffvelcdmd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k ∈ d π
320 317 319 sseldd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k ∈ ℝ
321 320 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → J ⁡ k ∈ ℝ
322 41 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → d ∈ ℝ
323 322 rexrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → d ∈ ℝ *
324 77 a1i ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → π ∈ ℝ *
325 iccgelb ⊢ d ∈ ℝ * ∧ π ∈ ℝ * ∧ J ⁡ k ∈ d π → d ≤ J ⁡ k
326 323 324 319 325 syl3anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → d ≤ J ⁡ k
327 326 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → d ≤ J ⁡ k
328 321 rexrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → J ⁡ k ∈ ℝ *
329 318 188 ffvelcdmd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k + 1 ∈ d π
330 317 329 sseldd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k + 1 ∈ ℝ
331 330 rexrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k + 1 ∈ ℝ *
332 331 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → J ⁡ k + 1 ∈ ℝ *
333 simpr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → s ∈ J ⁡ k J ⁡ k + 1
334 ioogtlb ⊢ J ⁡ k ∈ ℝ * ∧ J ⁡ k + 1 ∈ ℝ * ∧ s ∈ J ⁡ k J ⁡ k + 1 → J ⁡ k < s
335 328 332 333 334 syl3anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → J ⁡ k < s
336 312 321 315 327 335 lelttrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → d < s
337 312 315 336 ltled ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → d ≤ s
338 330 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → J ⁡ k + 1 ∈ ℝ
339 iooltub ⊢ J ⁡ k ∈ ℝ * ∧ J ⁡ k + 1 ∈ ℝ * ∧ s ∈ J ⁡ k J ⁡ k + 1 → s < J ⁡ k + 1
340 328 332 333 339 syl3anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → s < J ⁡ k + 1
341 iccleub ⊢ d ∈ ℝ * ∧ π ∈ ℝ * ∧ J ⁡ k + 1 ∈ d π → J ⁡ k + 1 ≤ π
342 323 324 329 341 syl3anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k + 1 ≤ π
343 342 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → J ⁡ k + 1 ≤ π
344 315 338 313 340 343 ltletrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → s < π
345 315 313 344 ltled ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → s ≤ π
346 312 313 315 337 345 eliccd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → s ∈ d π
347 346 ralrimiva ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ∀ s ∈ J ⁡ k J ⁡ k + 1 s ∈ d π
348 dfss3 ⊢ J ⁡ k J ⁡ k + 1 ⊆ d π ↔ ∀ s ∈ J ⁡ k J ⁡ k + 1 s ∈ d π
349 347 348 sylibr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k J ⁡ k + 1 ⊆ d π
350 311 349 feqresmpt ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → O ↾ J ⁡ k J ⁡ k + 1 = s ∈ J ⁡ k J ⁡ k + 1 ⟼ O ⁡ s
351 simplll ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → φ
352 simpllr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → d ∈ 0 π
353 25 fveq1i ⊢ O ⁡ s = U ↾ d π ⁡ s
354 353 a1i ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → O ⁡ s = U ↾ d π ⁡ s
355 fvres ⊢ s ∈ d π → U ↾ d π ⁡ s = U ⁡ s
356 355 adantl ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → U ↾ d π ⁡ s = U ⁡ s
357 265 267 eqtrd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → K ⁡ s = s 2 ⁢ sin ⁡ s 2
358 238 357 oveq12d ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → H ⁡ s ⁢ K ⁡ s = F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2
359 233 recnd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → F ⁡ X + s − Y ∈ ℂ
360 253 adantll ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → s ∈ ℂ
361 2cnd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → 2 ∈ ℂ
362 360 halfcld ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → s 2 ∈ ℂ
363 362 sincld ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → sin ⁡ s 2 ∈ ℂ
364 361 363 mulcld ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → 2 ⁢ sin ⁡ s 2 ∈ ℂ
365 260 adantll ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → 2 ⁢ sin ⁡ s 2 ≠ 0
366 359 360 364 217 365 dmdcan2d ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 = F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2
367 203 358 366 3eqtrd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → U ⁡ s = F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2
368 354 356 367 3eqtrd ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → O ⁡ s = F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2
369 351 352 346 368 syl21anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → O ⁡ s = F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2
370 351 352 346 366 syl21anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 = F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2
371 370 eqcomd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 = F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2
372 eqidd ⊢ φ ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t = t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t
373 oveq2 ⊢ t = s → X + t = X + s
374 373 fveq2d ⊢ t = s → F ⁡ X + t = F ⁡ X + s
375 374 oveq1d ⊢ t = s → F ⁡ X + t − Y = F ⁡ X + s − Y
376 id ⊢ t = s → t = s
377 375 376 oveq12d ⊢ t = s → F ⁡ X + t − Y t = F ⁡ X + s − Y s
378 377 adantl ⊢ φ ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 ∧ t = s → F ⁡ X + t − Y t = F ⁡ X + s − Y s
379 simpr ⊢ φ ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → s ∈ J ⁡ k J ⁡ k + 1
380 ovex ⊢ F ⁡ X + s − Y s ∈ V
381 380 a1i ⊢ φ ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → F ⁡ X + s − Y s ∈ V
382 372 378 379 381 fvmptd ⊢ φ ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t ⁡ s = F ⁡ X + s − Y s
383 eqidd ⊢ φ ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 = t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2
384 oveq1 ⊢ t = s → t 2 = s 2
385 384 fveq2d ⊢ t = s → sin ⁡ t 2 = sin ⁡ s 2
386 385 oveq2d ⊢ t = s → 2 ⁢ sin ⁡ t 2 = 2 ⁢ sin ⁡ s 2
387 376 386 oveq12d ⊢ t = s → t 2 ⁢ sin ⁡ t 2 = s 2 ⁢ sin ⁡ s 2
388 387 adantl ⊢ φ ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 ∧ t = s → t 2 ⁢ sin ⁡ t 2 = s 2 ⁢ sin ⁡ s 2
389 ovex ⊢ s 2 ⁢ sin ⁡ s 2 ∈ V
390 389 a1i ⊢ φ ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → s 2 ⁢ sin ⁡ s 2 ∈ V
391 383 388 379 390 fvmptd ⊢ φ ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 ⁡ s = s 2 ⁢ sin ⁡ s 2
392 382 391 oveq12d ⊢ φ ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t ⁡ s ⁢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 ⁡ s = F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2
393 392 eqcomd ⊢ φ ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 = t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t ⁡ s ⁢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 ⁡ s
394 393 adantllr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 = t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t ⁡ s ⁢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 ⁡ s
395 369 371 394 3eqtrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ s ∈ J ⁡ k J ⁡ k + 1 → O ⁡ s = t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t ⁡ s ⁢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 ⁡ s
396 395 mpteq2dva ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → s ∈ J ⁡ k J ⁡ k + 1 ⟼ O ⁡ s = s ∈ J ⁡ k J ⁡ k + 1 ⟼ t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t ⁡ s ⁢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 ⁡ s
397 350 396 eqtr2d ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → s ∈ J ⁡ k J ⁡ k + 1 ⟼ t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t ⁡ s ⁢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 ⁡ s = O ↾ J ⁡ k J ⁡ k + 1
398 397 oveq2d ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ds ∈ J ⁡ k J ⁡ k + 1 t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t ⁡ s ⁢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 ⁡ s d ℝ s = ℝ D O ↾ J ⁡ k J ⁡ k + 1
399 66 a1i ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ℝ ⊆ ℂ
400 349 317 sstrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k J ⁡ k + 1 ⊆ ℝ
401 tgioo4 ⊢ topGen ⁡ ran ⁡ . = TopOpen ⁡ ℂ fld ↾ 𝑡 ℝ
402 49 401 dvres ⊢ ℝ ⊆ ℂ ∧ O : d π ⟶ ℂ ∧ d π ⊆ ℝ ∧ J ⁡ k J ⁡ k + 1 ⊆ ℝ → ℝ D O ↾ J ⁡ k J ⁡ k + 1 = O ℝ ′ ↾ int ⁡ topGen ⁡ ran ⁡ . ⁡ J ⁡ k J ⁡ k + 1
403 399 311 317 400 402 syl22anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ℝ D O ↾ J ⁡ k J ⁡ k + 1 = O ℝ ′ ↾ int ⁡ topGen ⁡ ran ⁡ . ⁡ J ⁡ k J ⁡ k + 1
404 ioontr ⊢ int ⁡ topGen ⁡ ran ⁡ . ⁡ J ⁡ k J ⁡ k + 1 = J ⁡ k J ⁡ k + 1
405 404 a1i ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → int ⁡ topGen ⁡ ran ⁡ . ⁡ J ⁡ k J ⁡ k + 1 = J ⁡ k J ⁡ k + 1
406 405 reseq2d ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → O ℝ ′ ↾ int ⁡ topGen ⁡ ran ⁡ . ⁡ J ⁡ k J ⁡ k + 1 = O ℝ ′ ↾ J ⁡ k J ⁡ k + 1
407 398 403 406 3eqtrrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → O ℝ ′ ↾ J ⁡ k J ⁡ k + 1 = ds ∈ J ⁡ k J ⁡ k + 1 t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t ⁡ s ⁢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 ⁡ s d ℝ s
408 1 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → F : ℝ ⟶ ℝ
409 2 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → X ∈ ℝ
410 4 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → M ∈ ℕ
411 5 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → V ∈ P ⁡ M
412 9 ad4ant14 ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ i ∈ 0 ..^ M → F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶cn ℝ
413 85 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → d π ⊆ − π π
414 349 413 sstrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k J ⁡ k + 1 ⊆ − π π
415 76 a1i ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → 0 ∈ ℝ *
416 0red ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → 0 ∈ ℝ
417 79 ad2antlr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → 0 < d
418 416 322 320 417 326 ltletrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → 0 < J ⁡ k
419 320 331 415 418 ltnelicc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ¬ 0 ∈ J ⁡ k J ⁡ k + 1
420 54 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → Y ∈ ℝ
421 42 a1i ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → π ∈ ℝ
422 281 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → d < π
423 simpr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → k ∈ 0 ..^ N
424 biid ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ i ∈ 0 ..^ M ∧ J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ i Q ⁡ i + 1 ∧ v ∈ 0 ..^ M ∧ J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ v Q ⁡ v + 1 ↔ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ i ∈ 0 ..^ M ∧ J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ i Q ⁡ i + 1 ∧ v ∈ 0 ..^ M ∧ J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ v Q ⁡ v + 1
425 409 3 410 411 322 421 422 413 29 26 27 28 423 298 424 fourierdlem50 ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ∈ 0 ..^ M ∧ J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1
426 425 simpld ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ∈ 0 ..^ M
427 425 simprd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1
428 377 cbvmptv ⊢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t = s ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + s − Y s
429 387 cbvmptv ⊢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 = s ∈ J ⁡ k J ⁡ k + 1 ⟼ s 2 ⁢ sin ⁡ s 2
430 eqid ⊢ s ∈ J ⁡ k J ⁡ k + 1 ⟼ t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t ⁡ s ⁢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 ⁡ s = s ∈ J ⁡ k J ⁡ k + 1 ⟼ t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t ⁡ s ⁢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 ⁡ s
431 408 409 3 410 411 412 320 330 191 414 419 420 29 426 427 428 429 430 fourierdlem72 ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ds ∈ J ⁡ k J ⁡ k + 1 t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y t ⁡ s ⁢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ t 2 ⁢ sin ⁡ t 2 ⁡ s d ℝ s : J ⁡ k J ⁡ k + 1 ⟶cn ℂ
432 407 431 eqeltrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → O ℝ ′ ↾ J ⁡ k J ⁡ k + 1 : J ⁡ k J ⁡ k + 1 ⟶cn ℂ
433 eqid ⊢ s ∈ d π ⟼ F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 = s ∈ d π ⟼ F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2
434 eqid ⊢ X + J ⁡ k X + J ⁡ k + 1 = X + J ⁡ k X + J ⁡ k + 1
435 30 426 eqeltrid ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → C ∈ 0 ..^ M
436 simpll ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → φ
437 436 435 jca ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → φ ∧ C ∈ 0 ..^ M
438 eleq1 ⊢ i = C → i ∈ 0 ..^ M ↔ C ∈ 0 ..^ M
439 438 anbi2d ⊢ i = C → φ ∧ i ∈ 0 ..^ M ↔ φ ∧ C ∈ 0 ..^ M
440 fveq2 ⊢ i = C → V ⁡ i = V ⁡ C
441 oveq1 ⊢ i = C → i + 1 = C + 1
442 441 fveq2d ⊢ i = C → V ⁡ i + 1 = V ⁡ C + 1
443 440 442 oveq12d ⊢ i = C → V ⁡ i V ⁡ i + 1 = V ⁡ C V ⁡ C + 1
444 raleq ⊢ V ⁡ i V ⁡ i + 1 = V ⁡ C V ⁡ C + 1 → ∀ t ∈ V ⁡ i V ⁡ i + 1 F ⁡ t ≤ w ↔ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w
445 443 444 syl ⊢ i = C → ∀ t ∈ V ⁡ i V ⁡ i + 1 F ⁡ t ≤ w ↔ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w
446 445 rexbidv ⊢ i = C → ∃ w ∈ ℝ ∀ t ∈ V ⁡ i V ⁡ i + 1 F ⁡ t ≤ w ↔ ∃ w ∈ ℝ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w
447 439 446 imbi12d ⊢ i = C → φ ∧ i ∈ 0 ..^ M → ∃ w ∈ ℝ ∀ t ∈ V ⁡ i V ⁡ i + 1 F ⁡ t ≤ w ↔ φ ∧ C ∈ 0 ..^ M → ∃ w ∈ ℝ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w
448 447 8 vtoclg ⊢ C ∈ 0 ..^ M → φ ∧ C ∈ 0 ..^ M → ∃ w ∈ ℝ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w
449 435 437 448 sylc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ∃ w ∈ ℝ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w
450 nfv ⊢ Ⅎ t φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N
451 nfra1 ⊢ Ⅎ t ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w
452 450 451 nfan ⊢ Ⅎ t φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w
453 simplr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w
454 70 a1i ⊢ φ → − π ∈ ℝ
455 454 2 readdcld ⊢ φ → - π + X ∈ ℝ
456 42 a1i ⊢ φ → π ∈ ℝ
457 456 2 readdcld ⊢ φ → π + X ∈ ℝ
458 455 457 iccssred ⊢ φ → - π + X π + X ⊆ ℝ
459 ressxr ⊢ ℝ ⊆ ℝ *
460 458 459 sstrdi ⊢ φ → - π + X π + X ⊆ ℝ *
461 460 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → - π + X π + X ⊆ ℝ *
462 3 410 411 fourierdlem15 ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → V : 0 … M ⟶ - π + X π + X
463 elfzofz ⊢ C ∈ 0 ..^ M → C ∈ 0 … M
464 435 463 syl ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → C ∈ 0 … M
465 462 464 ffvelcdmd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → V ⁡ C ∈ - π + X π + X
466 461 465 sseldd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → V ⁡ C ∈ ℝ *
467 466 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → V ⁡ C ∈ ℝ *
468 fzofzp1 ⊢ C ∈ 0 ..^ M → C + 1 ∈ 0 … M
469 435 468 syl ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → C + 1 ∈ 0 … M
470 462 469 ffvelcdmd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → V ⁡ C + 1 ∈ - π + X π + X
471 461 470 sseldd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → V ⁡ C + 1 ∈ ℝ *
472 471 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → V ⁡ C + 1 ∈ ℝ *
473 elioore ⊢ t ∈ X + J ⁡ k X + J ⁡ k + 1 → t ∈ ℝ
474 473 adantl ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → t ∈ ℝ
475 70 a1i ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → − π ∈ ℝ
476 475 421 409 3 410 411 464 29 fourierdlem13 ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → Q ⁡ C = V ⁡ C − X ∧ V ⁡ C = X + Q ⁡ C
477 476 simprd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → V ⁡ C = X + Q ⁡ C
478 477 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → V ⁡ C = X + Q ⁡ C
479 458 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → - π + X π + X ⊆ ℝ
480 479 465 sseldd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → V ⁡ C ∈ ℝ
481 480 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → V ⁡ C ∈ ℝ
482 478 481 eqeltrrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → X + Q ⁡ C ∈ ℝ
483 409 320 readdcld ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → X + J ⁡ k ∈ ℝ
484 483 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → X + J ⁡ k ∈ ℝ
485 476 simpld ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → Q ⁡ C = V ⁡ C − X
486 480 409 resubcld ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → V ⁡ C − X ∈ ℝ
487 485 486 eqeltrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → Q ⁡ C ∈ ℝ
488 475 421 409 3 410 411 469 29 fourierdlem13 ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → Q ⁡ C + 1 = V ⁡ C + 1 − X ∧ V ⁡ C + 1 = X + Q ⁡ C + 1
489 488 simpld ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → Q ⁡ C + 1 = V ⁡ C + 1 − X
490 479 470 sseldd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → V ⁡ C + 1 ∈ ℝ
491 490 409 resubcld ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → V ⁡ C + 1 − X ∈ ℝ
492 489 491 eqeltrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → Q ⁡ C + 1 ∈ ℝ
493 30 eqcomi ⊢ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 = C
494 493 fveq2i ⊢ Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 = Q ⁡ C
495 493 oveq1i ⊢ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 = C + 1
496 495 fveq2i ⊢ Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 = Q ⁡ C + 1
497 494 496 oveq12i ⊢ Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 = Q ⁡ C Q ⁡ C + 1
498 427 497 sseqtrdi ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ C Q ⁡ C + 1
499 487 492 320 330 191 498 fourierdlem10 ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → Q ⁡ C ≤ J ⁡ k ∧ J ⁡ k + 1 ≤ Q ⁡ C + 1
500 499 simpld ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → Q ⁡ C ≤ J ⁡ k
501 487 320 409 500 leadd2dd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → X + Q ⁡ C ≤ X + J ⁡ k
502 501 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → X + Q ⁡ C ≤ X + J ⁡ k
503 484 rexrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → X + J ⁡ k ∈ ℝ *
504 409 330 readdcld ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → X + J ⁡ k + 1 ∈ ℝ
505 504 rexrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → X + J ⁡ k + 1 ∈ ℝ *
506 505 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → X + J ⁡ k + 1 ∈ ℝ *
507 simpr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → t ∈ X + J ⁡ k X + J ⁡ k + 1
508 ioogtlb ⊢ X + J ⁡ k ∈ ℝ * ∧ X + J ⁡ k + 1 ∈ ℝ * ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → X + J ⁡ k < t
509 503 506 507 508 syl3anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → X + J ⁡ k < t
510 482 484 474 502 509 lelttrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → X + Q ⁡ C < t
511 478 510 eqbrtrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → V ⁡ C < t
512 504 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → X + J ⁡ k + 1 ∈ ℝ
513 488 simprd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → V ⁡ C + 1 = X + Q ⁡ C + 1
514 513 490 eqeltrrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → X + Q ⁡ C + 1 ∈ ℝ
515 514 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → X + Q ⁡ C + 1 ∈ ℝ
516 iooltub ⊢ X + J ⁡ k ∈ ℝ * ∧ X + J ⁡ k + 1 ∈ ℝ * ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → t < X + J ⁡ k + 1
517 503 506 507 516 syl3anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → t < X + J ⁡ k + 1
518 499 simprd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k + 1 ≤ Q ⁡ C + 1
519 330 492 409 518 leadd2dd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → X + J ⁡ k + 1 ≤ X + Q ⁡ C + 1
520 519 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → X + J ⁡ k + 1 ≤ X + Q ⁡ C + 1
521 474 512 515 517 520 ltletrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → t < X + Q ⁡ C + 1
522 513 eqcomd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → X + Q ⁡ C + 1 = V ⁡ C + 1
523 522 adantr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → X + Q ⁡ C + 1 = V ⁡ C + 1
524 521 523 breqtrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → t < V ⁡ C + 1
525 467 472 474 511 524 eliood ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → t ∈ V ⁡ C V ⁡ C + 1
526 525 adantlr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → t ∈ V ⁡ C V ⁡ C + 1
527 rspa ⊢ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w ∧ t ∈ V ⁡ C V ⁡ C + 1 → F ⁡ t ≤ w
528 453 526 527 syl2anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → F ⁡ t ≤ w
529 528 ex ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w → t ∈ X + J ⁡ k X + J ⁡ k + 1 → F ⁡ t ≤ w
530 452 529 ralrimi ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w → ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ⁡ t ≤ w
531 530 ex ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w → ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ⁡ t ≤ w
532 531 reximdv ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ∃ w ∈ ℝ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ⁡ t ≤ w → ∃ w ∈ ℝ ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ⁡ t ≤ w
533 449 532 mpd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ∃ w ∈ ℝ ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ⁡ t ≤ w
534 443 raleqdv ⊢ i = C → ∀ t ∈ V ⁡ i V ⁡ i + 1 F ℝ ′ ⁡ t ≤ z ↔ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z
535 534 rexbidv ⊢ i = C → ∃ z ∈ ℝ ∀ t ∈ V ⁡ i V ⁡ i + 1 F ℝ ′ ⁡ t ≤ z ↔ ∃ z ∈ ℝ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z
536 439 535 imbi12d ⊢ i = C → φ ∧ i ∈ 0 ..^ M → ∃ z ∈ ℝ ∀ t ∈ V ⁡ i V ⁡ i + 1 F ℝ ′ ⁡ t ≤ z ↔ φ ∧ C ∈ 0 ..^ M → ∃ z ∈ ℝ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z
537 536 10 vtoclg ⊢ C ∈ 0 ..^ M → φ ∧ C ∈ 0 ..^ M → ∃ z ∈ ℝ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z
538 435 437 537 sylc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ∃ z ∈ ℝ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z
539 nfra1 ⊢ Ⅎ t ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z
540 450 539 nfan ⊢ Ⅎ t φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z
541 1 67 fssd ⊢ φ → F : ℝ ⟶ ℂ
542 ssid ⊢ ℝ ⊆ ℝ
543 542 a1i ⊢ φ → ℝ ⊆ ℝ
544 ioossre ⊢ X + J ⁡ k X + J ⁡ k + 1 ⊆ ℝ
545 544 a1i ⊢ φ → X + J ⁡ k X + J ⁡ k + 1 ⊆ ℝ
546 49 401 dvres ⊢ ℝ ⊆ ℂ ∧ F : ℝ ⟶ ℂ ∧ ℝ ⊆ ℝ ∧ X + J ⁡ k X + J ⁡ k + 1 ⊆ ℝ → ℝ D F ↾ X + J ⁡ k X + J ⁡ k + 1 = F ℝ ′ ↾ int ⁡ topGen ⁡ ran ⁡ . ⁡ X + J ⁡ k X + J ⁡ k + 1
547 67 541 543 545 546 syl22anc ⊢ φ → ℝ D F ↾ X + J ⁡ k X + J ⁡ k + 1 = F ℝ ′ ↾ int ⁡ topGen ⁡ ran ⁡ . ⁡ X + J ⁡ k X + J ⁡ k + 1
548 ioontr ⊢ int ⁡ topGen ⁡ ran ⁡ . ⁡ X + J ⁡ k X + J ⁡ k + 1 = X + J ⁡ k X + J ⁡ k + 1
549 548 reseq2i ⊢ F ℝ ′ ↾ int ⁡ topGen ⁡ ran ⁡ . ⁡ X + J ⁡ k X + J ⁡ k + 1 = F ℝ ′ ↾ X + J ⁡ k X + J ⁡ k + 1
550 547 549 eqtrdi ⊢ φ → ℝ D F ↾ X + J ⁡ k X + J ⁡ k + 1 = F ℝ ′ ↾ X + J ⁡ k X + J ⁡ k + 1
551 550 fveq1d ⊢ φ → F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t = F ℝ ′ ↾ X + J ⁡ k X + J ⁡ k + 1 ⁡ t
552 fvres ⊢ t ∈ X + J ⁡ k X + J ⁡ k + 1 → F ℝ ′ ↾ X + J ⁡ k X + J ⁡ k + 1 ⁡ t = F ℝ ′ ⁡ t
553 551 552 sylan9eq ⊢ φ ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t = F ℝ ′ ⁡ t
554 553 ad4ant14 ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t = F ℝ ′ ⁡ t
555 554 fveq2d ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t = F ℝ ′ ⁡ t
556 555 adantlr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t = F ℝ ′ ⁡ t
557 simplr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z
558 525 adantlr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → t ∈ V ⁡ C V ⁡ C + 1
559 rspa ⊢ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z ∧ t ∈ V ⁡ C V ⁡ C + 1 → F ℝ ′ ⁡ t ≤ z
560 557 558 559 syl2anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → F ℝ ′ ⁡ t ≤ z
561 556 560 eqbrtrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z ∧ t ∈ X + J ⁡ k X + J ⁡ k + 1 → F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t ≤ z
562 561 ex ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z → t ∈ X + J ⁡ k X + J ⁡ k + 1 → F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t ≤ z
563 540 562 ralrimi ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z → ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t ≤ z
564 563 ex ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z → ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t ≤ z
565 564 reximdv ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ∃ z ∈ ℝ ∀ t ∈ V ⁡ C V ⁡ C + 1 F ℝ ′ ⁡ t ≤ z → ∃ z ∈ ℝ ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t ≤ z
566 538 565 mpd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ∃ z ∈ ℝ ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t ≤ z
567 323 324 318 423 fourierdlem8 ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → J ⁡ k J ⁡ k + 1 ⊆ d π
568 143 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ r ∈ d π ∧ ¬ r ∈ ran ⁡ J → N ∈ ℕ
569 167 316 fssd ⊢ φ ∧ d ∈ 0 π → J : 0 … N ⟶ ℝ
570 569 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ r ∈ d π ∧ ¬ r ∈ ran ⁡ J → J : 0 … N ⟶ ℝ
571 simpr ⊢ φ ∧ d ∈ 0 π ∧ r ∈ d π → r ∈ d π
572 168 eqcomd ⊢ φ ∧ d ∈ 0 π → d = J ⁡ 0
573 169 eqcomd ⊢ φ ∧ d ∈ 0 π → π = J ⁡ N
574 572 573 oveq12d ⊢ φ ∧ d ∈ 0 π → d π = J ⁡ 0 J ⁡ N
575 574 adantr ⊢ φ ∧ d ∈ 0 π ∧ r ∈ d π → d π = J ⁡ 0 J ⁡ N
576 571 575 eleqtrd ⊢ φ ∧ d ∈ 0 π ∧ r ∈ d π → r ∈ J ⁡ 0 J ⁡ N
577 576 adantr ⊢ φ ∧ d ∈ 0 π ∧ r ∈ d π ∧ ¬ r ∈ ran ⁡ J → r ∈ J ⁡ 0 J ⁡ N
578 simpr ⊢ φ ∧ d ∈ 0 π ∧ r ∈ d π ∧ ¬ r ∈ ran ⁡ J → ¬ r ∈ ran ⁡ J
579 fveq2 ⊢ j = k → J ⁡ j = J ⁡ k
580 579 breq1d ⊢ j = k → J ⁡ j < r ↔ J ⁡ k < r
581 580 cbvrabv ⊢ j ∈ 0 ..^ N | J ⁡ j < r = k ∈ 0 ..^ N | J ⁡ k < r
582 581 supeq1i ⊢ sup j ∈ 0 ..^ N | J ⁡ j < r ℝ < = sup k ∈ 0 ..^ N | J ⁡ k < r ℝ <
583 568 570 577 578 582 fourierdlem25 ⊢ φ ∧ d ∈ 0 π ∧ r ∈ d π ∧ ¬ r ∈ ran ⁡ J → ∃ m ∈ 0 ..^ N r ∈ J ⁡ m J ⁡ m + 1
584 541 ad2antrr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → F : ℝ ⟶ ℂ
585 542 a1i ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ℝ ⊆ ℝ
586 544 a1i ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → X + J ⁡ k X + J ⁡ k + 1 ⊆ ℝ
587 399 584 585 586 546 syl22anc ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ℝ D F ↾ X + J ⁡ k X + J ⁡ k + 1 = F ℝ ′ ↾ int ⁡ topGen ⁡ ran ⁡ . ⁡ X + J ⁡ k X + J ⁡ k + 1
588 525 ralrimiva ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 t ∈ V ⁡ C V ⁡ C + 1
589 dfss3 ⊢ X + J ⁡ k X + J ⁡ k + 1 ⊆ V ⁡ C V ⁡ C + 1 ↔ ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 t ∈ V ⁡ C V ⁡ C + 1
590 588 589 sylibr ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → X + J ⁡ k X + J ⁡ k + 1 ⊆ V ⁡ C V ⁡ C + 1
591 resabs2 ⊢ X + J ⁡ k X + J ⁡ k + 1 ⊆ V ⁡ C V ⁡ C + 1 → F ℝ ′ ↾ X + J ⁡ k X + J ⁡ k + 1 ↾ V ⁡ C V ⁡ C + 1 = F ℝ ′ ↾ X + J ⁡ k X + J ⁡ k + 1
592 590 591 syl ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → F ℝ ′ ↾ X + J ⁡ k X + J ⁡ k + 1 ↾ V ⁡ C V ⁡ C + 1 = F ℝ ′ ↾ X + J ⁡ k X + J ⁡ k + 1
593 549 587 592 3eqtr4a ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → ℝ D F ↾ X + J ⁡ k X + J ⁡ k + 1 = F ℝ ′ ↾ X + J ⁡ k X + J ⁡ k + 1 ↾ V ⁡ C V ⁡ C + 1
594 590 resabs1d ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → F ℝ ′ ↾ V ⁡ C V ⁡ C + 1 ↾ X + J ⁡ k X + J ⁡ k + 1 = F ℝ ′ ↾ X + J ⁡ k X + J ⁡ k + 1
595 594 eqcomd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → F ℝ ′ ↾ X + J ⁡ k X + J ⁡ k + 1 = F ℝ ′ ↾ V ⁡ C V ⁡ C + 1 ↾ X + J ⁡ k X + J ⁡ k + 1
596 593 592 595 3eqtrrd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → F ℝ ′ ↾ V ⁡ C V ⁡ C + 1 ↾ X + J ⁡ k X + J ⁡ k + 1 = ℝ D F ↾ X + J ⁡ k X + J ⁡ k + 1
597 443 reseq2d ⊢ i = C → F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 = F ℝ ′ ↾ V ⁡ C V ⁡ C + 1
598 597 443 feq12d ⊢ i = C → F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶ ℝ ↔ F ℝ ′ ↾ V ⁡ C V ⁡ C + 1 : V ⁡ C V ⁡ C + 1 ⟶ ℝ
599 439 598 imbi12d ⊢ i = C → φ ∧ i ∈ 0 ..^ M → F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶ ℝ ↔ φ ∧ C ∈ 0 ..^ M → F ℝ ′ ↾ V ⁡ C V ⁡ C + 1 : V ⁡ C V ⁡ C + 1 ⟶ ℝ
600 cncff ⊢ F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶cn ℝ → F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶ ℝ
601 9 600 syl ⊢ φ ∧ i ∈ 0 ..^ M → F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶ ℝ
602 599 601 vtoclg ⊢ C ∈ 0 ..^ M → φ ∧ C ∈ 0 ..^ M → F ℝ ′ ↾ V ⁡ C V ⁡ C + 1 : V ⁡ C V ⁡ C + 1 ⟶ ℝ
603 602 anabsi7 ⊢ φ ∧ C ∈ 0 ..^ M → F ℝ ′ ↾ V ⁡ C V ⁡ C + 1 : V ⁡ C V ⁡ C + 1 ⟶ ℝ
604 437 603 syl ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → F ℝ ′ ↾ V ⁡ C V ⁡ C + 1 : V ⁡ C V ⁡ C + 1 ⟶ ℝ
605 604 590 fssresd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → F ℝ ′ ↾ V ⁡ C V ⁡ C + 1 ↾ X + J ⁡ k X + J ⁡ k + 1 : X + J ⁡ k X + J ⁡ k + 1 ⟶ ℝ
606 596 605 feq1dd ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N → F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ : X + J ⁡ k X + J ⁡ k + 1 ⟶ ℝ
607 375 386 oveq12d ⊢ t = s → F ⁡ X + t − Y 2 ⁢ sin ⁡ t 2 = F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2
608 607 cbvmptv ⊢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + t − Y 2 ⁢ sin ⁡ t 2 = s ∈ J ⁡ k J ⁡ k + 1 ⟼ F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2
609 fveq2 ⊢ r = t → F ⁡ r = F ⁡ t
610 609 fveq2d ⊢ r = t → F ⁡ r = F ⁡ t
611 610 breq1d ⊢ r = t → F ⁡ r ≤ w ↔ F ⁡ t ≤ w
612 611 cbvralvw ⊢ ∀ r ∈ X + J ⁡ k X + J ⁡ k + 1 F ⁡ r ≤ w ↔ ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ⁡ t ≤ w
613 612 anbi2i ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ w ∈ ℝ ∧ z ∈ ℝ ∧ ∀ r ∈ X + J ⁡ k X + J ⁡ k + 1 F ⁡ r ≤ w ↔ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ w ∈ ℝ ∧ z ∈ ℝ ∧ ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ⁡ t ≤ w
614 fveq2 ⊢ r = t → F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ r = F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t
615 614 fveq2d ⊢ r = t → F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ r = F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t
616 615 breq1d ⊢ r = t → F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ r ≤ z ↔ F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t ≤ z
617 616 cbvralvw ⊢ ∀ r ∈ X + J ⁡ k X + J ⁡ k + 1 F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ r ≤ z ↔ ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t ≤ z
618 613 617 anbi12i ⊢ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ w ∈ ℝ ∧ z ∈ ℝ ∧ ∀ r ∈ X + J ⁡ k X + J ⁡ k + 1 F ⁡ r ≤ w ∧ ∀ r ∈ X + J ⁡ k X + J ⁡ k + 1 F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ r ≤ z ↔ φ ∧ d ∈ 0 π ∧ k ∈ 0 ..^ N ∧ w ∈ ℝ ∧ z ∈ ℝ ∧ ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ⁡ t ≤ w ∧ ∀ t ∈ X + J ⁡ k X + J ⁡ k + 1 F ↾ X + J ⁡ k X + J ⁡ k + 1 ℝ ′ ⁡ t ≤ z
619 274 275 41 43 85 288 289 433 434 533 566 167 191 567 583 606 608 618 fourierdlem80 ⊢ φ ∧ d ∈ 0 π → ∃ b ∈ ℝ ∀ s ∈ dom ⁡ ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s ⁡ s ≤ b
620 366 mpteq2dva ⊢ φ ∧ d ∈ 0 π → s ∈ d π ⟼ F ⁡ X + s − Y s ⁢ s 2 ⁢ sin ⁡ s 2 = s ∈ d π ⟼ F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2
621 271 620 eqtrd ⊢ φ ∧ d ∈ 0 π → O = s ∈ d π ⟼ F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2
622 621 oveq2d ⊢ φ ∧ d ∈ 0 π → ℝ D O = ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s
623 622 dmeqd ⊢ φ ∧ d ∈ 0 π → dom ⁡ O ℝ ′ = dom ⁡ ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s
624 nfcv ⊢ Ⅎ _ s dom ⁡ O ℝ ′
625 nfcv ⊢ Ⅎ _ s ℝ
626 nfcv ⊢ Ⅎ _ s D
627 nfmpt1 ⊢ Ⅎ _ s s ∈ d π ⟼ F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2
628 625 626 627 nfov ⊢ Ⅎ _ s ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s
629 628 nfdm ⊢ Ⅎ _ s dom ⁡ ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s
630 624 629 raleqf ⊢ dom ⁡ O ℝ ′ = dom ⁡ ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s → ∀ s ∈ dom ⁡ O ℝ ′ O ℝ ′ ⁡ s ≤ b ↔ ∀ s ∈ dom ⁡ ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s O ℝ ′ ⁡ s ≤ b
631 623 630 syl ⊢ φ ∧ d ∈ 0 π → ∀ s ∈ dom ⁡ O ℝ ′ O ℝ ′ ⁡ s ≤ b ↔ ∀ s ∈ dom ⁡ ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s O ℝ ′ ⁡ s ≤ b
632 622 fveq1d ⊢ φ ∧ d ∈ 0 π → O ℝ ′ ⁡ s = ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s ⁡ s
633 632 fveq2d ⊢ φ ∧ d ∈ 0 π → O ℝ ′ ⁡ s = ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s ⁡ s
634 633 breq1d ⊢ φ ∧ d ∈ 0 π → O ℝ ′ ⁡ s ≤ b ↔ ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s ⁡ s ≤ b
635 634 ralbidv ⊢ φ ∧ d ∈ 0 π → ∀ s ∈ dom ⁡ ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s O ℝ ′ ⁡ s ≤ b ↔ ∀ s ∈ dom ⁡ ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s ⁡ s ≤ b
636 631 635 bitrd ⊢ φ ∧ d ∈ 0 π → ∀ s ∈ dom ⁡ O ℝ ′ O ℝ ′ ⁡ s ≤ b ↔ ∀ s ∈ dom ⁡ ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s ⁡ s ≤ b
637 636 rexbidv ⊢ φ ∧ d ∈ 0 π → ∃ b ∈ ℝ ∀ s ∈ dom ⁡ O ℝ ′ O ℝ ′ ⁡ s ≤ b ↔ ∃ b ∈ ℝ ∀ s ∈ dom ⁡ ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s ds ∈ d π F ⁡ X + s − Y 2 ⁢ sin ⁡ s 2 d ℝ s ⁡ s ≤ b
638 619 637 mpbird ⊢ φ ∧ d ∈ 0 π → ∃ b ∈ ℝ ∀ s ∈ dom ⁡ O ℝ ′ O ℝ ′ ⁡ s ≤ b
639 eqid ⊢ l ∈ ℝ + ⟼ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds = l ∈ ℝ + ⟼ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds
640 eqeq1 ⊢ t = s → t = J ⁡ k ↔ s = J ⁡ k
641 fveq2 ⊢ h = l → Q ⁡ h = Q ⁡ l
642 oveq1 ⊢ h = l → h + 1 = l + 1
643 642 fveq2d ⊢ h = l → Q ⁡ h + 1 = Q ⁡ l + 1
644 641 643 oveq12d ⊢ h = l → Q ⁡ h Q ⁡ h + 1 = Q ⁡ l Q ⁡ l + 1
645 644 sseq2d ⊢ h = l → J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 ↔ J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1
646 645 cbvriotavw ⊢ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 = ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1
647 646 fveq2i ⊢ Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1
648 647 eqeq2i ⊢ J ⁡ k = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 ↔ J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1
649 648 a1i ⊢ ⊤ → J ⁡ k = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 ↔ J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1
650 csbeq1 ⊢ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 = ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 → ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ R = ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R
651 646 650 mp1i ⊢ ⊤ → ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ R = ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R
652 649 651 ifbieq1d ⊢ ⊤ → if J ⁡ k = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ R F ⁡ X + J ⁡ k = if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k
653 652 mptru ⊢ if J ⁡ k = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ R F ⁡ X + J ⁡ k = if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k
654 653 oveq1i ⊢ if J ⁡ k = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ R F ⁡ X + J ⁡ k − Y = if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k − Y
655 654 oveq1i ⊢ if J ⁡ k = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k = if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k
656 655 oveq1i ⊢ if J ⁡ k = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2 = if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2
657 656 a1i ⊢ t = s → if J ⁡ k = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2 = if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2
658 eqeq1 ⊢ t = s → t = J ⁡ k + 1 ↔ s = J ⁡ k + 1
659 646 oveq1i ⊢ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 = ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1
660 659 fveq2i ⊢ Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1
661 660 eqeq2i ⊢ J ⁡ k + 1 = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 ↔ J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1
662 661 a1i ⊢ ⊤ → J ⁡ k + 1 = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 ↔ J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1
663 csbeq1 ⊢ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 = ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 → ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ L = ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L
664 646 663 mp1i ⊢ ⊤ → ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ L = ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L
665 662 664 ifbieq1d ⊢ ⊤ → if J ⁡ k + 1 = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 = if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1
666 665 mptru ⊢ if J ⁡ k + 1 = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 = if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1
667 666 oveq1i ⊢ if J ⁡ k + 1 = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y = if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y
668 667 oveq1i ⊢ if J ⁡ k + 1 = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 = if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1
669 668 oveq1i ⊢ if J ⁡ k + 1 = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2 = if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2
670 669 a1i ⊢ t = s → if J ⁡ k + 1 = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2 = if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2
671 fveq2 ⊢ t = s → O ⁡ t = O ⁡ s
672 658 670 671 ifbieq12d ⊢ t = s → if t = J ⁡ k + 1 if J ⁡ k + 1 = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2 O ⁡ t = if s = J ⁡ k + 1 if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2 O ⁡ s
673 640 657 672 ifbieq12d ⊢ t = s → if t = J ⁡ k if J ⁡ k = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2 if t = J ⁡ k + 1 if J ⁡ k + 1 = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2 O ⁡ t = if s = J ⁡ k if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2 if s = J ⁡ k + 1 if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2 O ⁡ s
674 673 cbvmptv ⊢ t ∈ J ⁡ k J ⁡ k + 1 ⟼ if t = J ⁡ k if J ⁡ k = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2 if t = J ⁡ k + 1 if J ⁡ k + 1 = Q ⁡ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 + 1 ⦋ ι h ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ h Q ⁡ h + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2 O ⁡ t = s ∈ J ⁡ k J ⁡ k + 1 ⟼ if s = J ⁡ k if J ⁡ k = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ R F ⁡ X + J ⁡ k − Y J ⁡ k ⁢ J ⁡ k 2 ⁢ sin ⁡ J ⁡ k 2 if s = J ⁡ k + 1 if J ⁡ k + 1 = Q ⁡ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 + 1 ⦋ ι l ∈ 0 ..^ M | J ⁡ k J ⁡ k + 1 ⊆ Q ⁡ l Q ⁡ l + 1 / i⦌ L F ⁡ X + J ⁡ k + 1 − Y J ⁡ k + 1 ⁢ J ⁡ k + 1 2 ⁢ sin ⁡ J ⁡ k + 1 2 O ⁡ s
675 41 43 89 143 167 168 169 191 301 306 309 310 432 638 639 674 fourierdlem73 ⊢ φ ∧ d ∈ 0 π → ∀ e ∈ ℝ + ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e
676 breq2 ⊢ e = a → ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e ↔ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < a
677 676 rexralbidv ⊢ e = a → ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e ↔ ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < a
678 677 cbvralvw ⊢ ∀ e ∈ ℝ + ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e ↔ ∀ a ∈ ℝ + ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < a
679 675 678 sylib ⊢ φ ∧ d ∈ 0 π → ∀ a ∈ ℝ + ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < a
680 679 adantlr ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π → ∀ a ∈ ℝ + ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < a
681 rphalfcl ⊢ e ∈ ℝ + → e 2 ∈ ℝ +
682 681 ad2antlr ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π → e 2 ∈ ℝ +
683 breq2 ⊢ a = e 2 → ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < a ↔ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
684 683 rexralbidv ⊢ a = e 2 → ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < a ↔ ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
685 684 rspccva ⊢ ∀ a ∈ ℝ + ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < a ∧ e 2 ∈ ℝ + → ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
686 680 682 685 syl2anc ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π → ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
687 156 a1i ⊢ φ ∧ d ∈ 0 π → d π ⊆ d π
688 687 sselda ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → s ∈ d π
689 688 355 syl ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → U ↾ d π ⁡ s = U ⁡ s
690 353 689 eqtr2id ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → U ⁡ s = O ⁡ s
691 690 oveq1d ⊢ φ ∧ d ∈ 0 π ∧ s ∈ d π → U ⁡ s ⁢ sin ⁡ l ⁢ s = O ⁡ s ⁢ sin ⁡ l ⁢ s
692 691 itgeq2dv ⊢ φ ∧ d ∈ 0 π → ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds = ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds
693 692 adantr ⊢ φ ∧ d ∈ 0 π ∧ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds = ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds
694 693 fveq2d ⊢ φ ∧ d ∈ 0 π ∧ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds = ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds
695 simpr ⊢ φ ∧ d ∈ 0 π ∧ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
696 694 695 eqbrtrd ⊢ φ ∧ d ∈ 0 π ∧ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
697 696 ex ⊢ φ ∧ d ∈ 0 π → ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
698 697 adantlr ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π → ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
699 698 ralimdv ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π → ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
700 699 reximdv ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π → ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π O ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
701 686 700 mpd ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π → ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
702 701 adantr ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
703 nfv ⊢ Ⅎ k φ ∧ e ∈ ℝ + ∧ d ∈ 0 π
704 nfra1 ⊢ Ⅎ k ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
705 703 704 nfan ⊢ Ⅎ k φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
706 nfv ⊢ Ⅎ k j ∈ ℕ
707 705 706 nfan ⊢ Ⅎ k φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ
708 nfv ⊢ Ⅎ k ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
709 707 708 nfan ⊢ Ⅎ k φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ ∧ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
710 simpll ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ j ∈ ℕ ∧ k ∈ ℤ ≥ j → φ ∧ e ∈ ℝ + ∧ d ∈ 0 π
711 eluznn ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → k ∈ ℕ
712 711 adantll ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ j ∈ ℕ ∧ k ∈ ℤ ≥ j → k ∈ ℕ
713 710 712 jca ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ j ∈ ℕ ∧ k ∈ ℤ ≥ j → φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ k ∈ ℕ
714 713 adantllr ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ ∧ k ∈ ℤ ≥ j → φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ k ∈ ℕ
715 simpllr ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ ∧ k ∈ ℤ ≥ j → ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
716 711 adantll ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ ∧ k ∈ ℤ ≥ j → k ∈ ℕ
717 rspa ⊢ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ k ∈ ℕ → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
718 715 716 717 syl2anc ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ ∧ k ∈ ℤ ≥ j → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
719 714 718 jca ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ ∧ k ∈ ℤ ≥ j → φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ k ∈ ℕ ∧ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
720 719 adantlr ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ ∧ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 ∧ k ∈ ℤ ≥ j → φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ k ∈ ℕ ∧ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
721 nnre ⊢ j ∈ ℕ → j ∈ ℝ
722 721 rexrd ⊢ j ∈ ℕ → j ∈ ℝ *
723 722 adantr ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → j ∈ ℝ *
724 50 a1i ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → +∞ ∈ ℝ *
725 eluzelre ⊢ k ∈ ℤ ≥ j → k ∈ ℝ
726 halfre ⊢ 1 2 ∈ ℝ
727 726 a1i ⊢ k ∈ ℤ ≥ j → 1 2 ∈ ℝ
728 725 727 readdcld ⊢ k ∈ ℤ ≥ j → k + 1 2 ∈ ℝ
729 728 adantl ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → k + 1 2 ∈ ℝ
730 721 adantr ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → j ∈ ℝ
731 725 adantl ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → k ∈ ℝ
732 eluzle ⊢ k ∈ ℤ ≥ j → j ≤ k
733 732 adantl ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → j ≤ k
734 halfgt0 ⊢ 0 < 1 2
735 734 a1i ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → 0 < 1 2
736 726 a1i ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → 1 2 ∈ ℝ
737 736 731 ltaddposd ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → 0 < 1 2 ↔ k < k + 1 2
738 735 737 mpbid ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → k < k + 1 2
739 730 731 729 733 738 lelttrd ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → j < k + 1 2
740 729 ltpnfd ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → k + 1 2 < +∞
741 723 724 729 739 740 eliood ⊢ j ∈ ℕ ∧ k ∈ ℤ ≥ j → k + 1 2 ∈ j +∞
742 741 adantlr ⊢ j ∈ ℕ ∧ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 ∧ k ∈ ℤ ≥ j → k + 1 2 ∈ j +∞
743 simplr ⊢ j ∈ ℕ ∧ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 ∧ k ∈ ℤ ≥ j → ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2
744 oveq1 ⊢ l = k + 1 2 → l ⁢ s = k + 1 2 ⁢ s
745 744 fveq2d ⊢ l = k + 1 2 → sin ⁡ l ⁢ s = sin ⁡ k + 1 2 ⁢ s
746 745 oveq2d ⊢ l = k + 1 2 → U ⁡ s ⁢ sin ⁡ l ⁢ s = U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s
747 746 adantr ⊢ l = k + 1 2 ∧ s ∈ d π → U ⁡ s ⁢ sin ⁡ l ⁢ s = U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s
748 747 itgeq2dv ⊢ l = k + 1 2 → ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds = ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds
749 748 fveq2d ⊢ l = k + 1 2 → ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds = ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds
750 749 breq1d ⊢ l = k + 1 2 → ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 ↔ ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
751 750 rspcv ⊢ k + 1 2 ∈ j +∞ → ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
752 742 743 751 sylc ⊢ j ∈ ℕ ∧ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 ∧ k ∈ ℤ ≥ j → ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
753 752 adantlll ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ ∧ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 ∧ k ∈ ℤ ≥ j → ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
754 720 753 31 sylanbrc ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ ∧ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 ∧ k ∈ ℤ ≥ j → χ
755 0red ⊢ χ → 0 ∈ ℝ
756 42 a1i ⊢ χ → π ∈ ℝ
757 ioossicc ⊢ 0 π ⊆ 0 π
758 31 biimpi ⊢ χ → φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ k ∈ ℕ ∧ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
759 simp-4r ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ k ∈ ℕ ∧ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → d ∈ 0 π
760 758 759 syl ⊢ χ → d ∈ 0 π
761 757 760 sselid ⊢ χ → d ∈ 0 π
762 simp-5l ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ k ∈ ℕ ∧ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → φ
763 758 762 syl ⊢ χ → φ
764 65 adantr ⊢ φ ∧ s ∈ 0 π → U : − π π ⟶ ℝ
765 70 rexri ⊢ − π ∈ ℝ *
766 0re ⊢ 0 ∈ ℝ
767 70 766 74 ltleii ⊢ − π ≤ 0
768 iooss1 ⊢ − π ∈ ℝ * ∧ − π ≤ 0 → 0 π ⊆ − π π
769 765 767 768 mp2an ⊢ 0 π ⊆ − π π
770 ioossicc ⊢ − π π ⊆ − π π
771 769 770 sstri ⊢ 0 π ⊆ − π π
772 771 sseli ⊢ s ∈ 0 π → s ∈ − π π
773 772 adantl ⊢ φ ∧ s ∈ 0 π → s ∈ − π π
774 764 773 ffvelcdmd ⊢ φ ∧ s ∈ 0 π → U ⁡ s ∈ ℝ
775 763 774 sylan ⊢ χ ∧ s ∈ 0 π → U ⁡ s ∈ ℝ
776 simpllr ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ k ∈ ℕ ∧ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → k ∈ ℕ
777 758 776 syl ⊢ χ → k ∈ ℕ
778 777 nnred ⊢ χ → k ∈ ℝ
779 726 a1i ⊢ χ → 1 2 ∈ ℝ
780 778 779 readdcld ⊢ χ → k + 1 2 ∈ ℝ
781 780 adantr ⊢ χ ∧ s ∈ 0 π → k + 1 2 ∈ ℝ
782 elioore ⊢ s ∈ 0 π → s ∈ ℝ
783 782 adantl ⊢ χ ∧ s ∈ 0 π → s ∈ ℝ
784 781 783 remulcld ⊢ χ ∧ s ∈ 0 π → k + 1 2 ⁢ s ∈ ℝ
785 784 resincld ⊢ χ ∧ s ∈ 0 π → sin ⁡ k + 1 2 ⁢ s ∈ ℝ
786 775 785 remulcld ⊢ χ ∧ s ∈ 0 π → U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ∈ ℝ
787 786 recnd ⊢ χ ∧ s ∈ 0 π → U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ∈ ℂ
788 76 a1i ⊢ χ → 0 ∈ ℝ *
789 77 a1i ⊢ χ → π ∈ ℝ *
790 755 leidd ⊢ χ → 0 ≤ 0
791 ioossre ⊢ 0 π ⊆ ℝ
792 791 760 sselid ⊢ χ → d ∈ ℝ
793 788 789 760 123 syl3anc ⊢ χ → d < π
794 792 756 793 ltled ⊢ χ → d ≤ π
795 ioossioo ⊢ 0 ∈ ℝ * ∧ π ∈ ℝ * ∧ 0 ≤ 0 ∧ d ≤ π → 0 d ⊆ 0 π
796 788 789 790 794 795 syl22anc ⊢ χ → 0 d ⊆ 0 π
797 ioombl ⊢ 0 d ∈ dom ⁡ vol
798 797 a1i ⊢ χ → 0 d ∈ dom ⁡ vol
799 eleq1 ⊢ n = k → n ∈ ℕ ↔ k ∈ ℕ
800 799 anbi2d ⊢ n = k → φ ∧ n ∈ ℕ ↔ φ ∧ k ∈ ℕ
801 simpl ⊢ n = k ∧ s ∈ 0 π → n = k
802 801 oveq1d ⊢ n = k ∧ s ∈ 0 π → n + 1 2 = k + 1 2
803 802 oveq1d ⊢ n = k ∧ s ∈ 0 π → n + 1 2 ⁢ s = k + 1 2 ⁢ s
804 803 fveq2d ⊢ n = k ∧ s ∈ 0 π → sin ⁡ n + 1 2 ⁢ s = sin ⁡ k + 1 2 ⁢ s
805 804 oveq2d ⊢ n = k ∧ s ∈ 0 π → U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s = U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s
806 805 mpteq2dva ⊢ n = k → s ∈ 0 π ⟼ U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s = s ∈ 0 π ⟼ U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s
807 806 eleq1d ⊢ n = k → s ∈ 0 π ⟼ U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s ∈ 𝐿 1 ↔ s ∈ 0 π ⟼ U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ∈ 𝐿 1
808 800 807 imbi12d ⊢ n = k → φ ∧ n ∈ ℕ → s ∈ 0 π ⟼ U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s ∈ 𝐿 1 ↔ φ ∧ k ∈ ℕ → s ∈ 0 π ⟼ U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ∈ 𝐿 1
809 771 a1i ⊢ φ ∧ n ∈ ℕ → 0 π ⊆ − π π
810 ioombl ⊢ 0 π ∈ dom ⁡ vol
811 810 a1i ⊢ φ ∧ n ∈ ℕ → 0 π ∈ dom ⁡ vol
812 65 ffvelcdmda ⊢ φ ∧ s ∈ − π π → U ⁡ s ∈ ℝ
813 812 adantlr ⊢ φ ∧ n ∈ ℕ ∧ s ∈ − π π → U ⁡ s ∈ ℝ
814 nnre ⊢ n ∈ ℕ → n ∈ ℝ
815 readdcl ⊢ n ∈ ℝ ∧ 1 2 ∈ ℝ → n + 1 2 ∈ ℝ
816 814 726 815 sylancl ⊢ n ∈ ℕ → n + 1 2 ∈ ℝ
817 816 adantr ⊢ n ∈ ℕ ∧ s ∈ − π π → n + 1 2 ∈ ℝ
818 simpr ⊢ n ∈ ℕ ∧ s ∈ − π π → s ∈ − π π
819 228 818 sselid ⊢ n ∈ ℕ ∧ s ∈ − π π → s ∈ ℝ
820 817 819 remulcld ⊢ n ∈ ℕ ∧ s ∈ − π π → n + 1 2 ⁢ s ∈ ℝ
821 820 resincld ⊢ n ∈ ℕ ∧ s ∈ − π π → sin ⁡ n + 1 2 ⁢ s ∈ ℝ
822 821 adantll ⊢ φ ∧ n ∈ ℕ ∧ s ∈ − π π → sin ⁡ n + 1 2 ⁢ s ∈ ℝ
823 813 822 remulcld ⊢ φ ∧ n ∈ ℕ ∧ s ∈ − π π → U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s ∈ ℝ
824 16 fvmpt2 ⊢ s ∈ − π π ∧ sin ⁡ n + 1 2 ⁢ s ∈ ℝ → S ⁡ s = sin ⁡ n + 1 2 ⁢ s
825 818 821 824 syl2anc ⊢ n ∈ ℕ ∧ s ∈ − π π → S ⁡ s = sin ⁡ n + 1 2 ⁢ s
826 825 adantll ⊢ φ ∧ n ∈ ℕ ∧ s ∈ − π π → S ⁡ s = sin ⁡ n + 1 2 ⁢ s
827 826 oveq2d ⊢ φ ∧ n ∈ ℕ ∧ s ∈ − π π → U ⁡ s ⁢ S ⁡ s = U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s
828 827 mpteq2dva ⊢ φ ∧ n ∈ ℕ → s ∈ − π π ⟼ U ⁡ s ⁢ S ⁡ s = s ∈ − π π ⟼ U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s
829 17 828 eqtr2id ⊢ φ ∧ n ∈ ℕ → s ∈ − π π ⟼ U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s = G
830 1 adantr ⊢ φ ∧ n ∈ ℕ → F : ℝ ⟶ ℝ
831 6 adantr ⊢ φ ∧ n ∈ ℕ → X ∈ ran ⁡ V
832 20 adantr ⊢ φ ∧ n ∈ ℕ → Y ∈ F ↾ X +∞ lim ℂ X
833 21 adantr ⊢ φ ∧ n ∈ ℕ → W ∈ F ↾ −∞ X lim ℂ X
834 814 adantl ⊢ φ ∧ n ∈ ℕ → n ∈ ℝ
835 4 adantr ⊢ φ ∧ n ∈ ℕ → M ∈ ℕ
836 5 adantr ⊢ φ ∧ n ∈ ℕ → V ∈ P ⁡ M
837 7 adantlr ⊢ φ ∧ n ∈ ℕ ∧ i ∈ 0 ..^ M → F ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶cn ℂ
838 11 adantlr ⊢ φ ∧ n ∈ ℕ ∧ i ∈ 0 ..^ M → R ∈ F ↾ V ⁡ i V ⁡ i + 1 lim ℂ V ⁡ i
839 12 adantlr ⊢ φ ∧ n ∈ ℕ ∧ i ∈ 0 ..^ M → L ∈ F ↾ V ⁡ i V ⁡ i + 1 lim ℂ V ⁡ i + 1
840 eqid ⊢ m ∈ ℕ ⟼ p ∈ ℝ 0 … m | p ⁡ 0 = − π ∧ p ⁡ m = π ∧ ∀ i ∈ 0 ..^ m p ⁡ i < p ⁡ i + 1 = m ∈ ℕ ⟼ p ∈ ℝ 0 … m | p ⁡ 0 = − π ∧ p ⁡ m = π ∧ ∀ i ∈ 0 ..^ m p ⁡ i < p ⁡ i + 1
841 eqid ⊢ ℝ D F = ℝ D F
842 601 adantlr ⊢ φ ∧ n ∈ ℕ ∧ i ∈ 0 ..^ M → F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶ ℝ
843 22 adantr ⊢ φ ∧ n ∈ ℕ → A ∈ F ℝ ′ ↾ −∞ X lim ℂ X
844 23 adantr ⊢ φ ∧ n ∈ ℕ → B ∈ F ℝ ′ ↾ X +∞ lim ℂ X
845 3 830 831 832 833 13 14 15 834 16 17 835 836 837 838 839 29 840 841 842 843 844 fourierdlem88 ⊢ φ ∧ n ∈ ℕ → G ∈ 𝐿 1
846 829 845 eqeltrd ⊢ φ ∧ n ∈ ℕ → s ∈ − π π ⟼ U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s ∈ 𝐿 1
847 809 811 823 846 iblss ⊢ φ ∧ n ∈ ℕ → s ∈ 0 π ⟼ U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s ∈ 𝐿 1
848 808 847 chvarvv ⊢ φ ∧ k ∈ ℕ → s ∈ 0 π ⟼ U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ∈ 𝐿 1
849 763 777 848 syl2anc ⊢ χ → s ∈ 0 π ⟼ U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ∈ 𝐿 1
850 796 798 786 849 iblss ⊢ χ → s ∈ 0 d ⟼ U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ∈ 𝐿 1
851 788 789 760 78 syl3anc ⊢ χ → 0 < d
852 755 792 851 ltled ⊢ χ → 0 ≤ d
853 756 leidd ⊢ χ → π ≤ π
854 ioossioo ⊢ 0 ∈ ℝ * ∧ π ∈ ℝ * ∧ 0 ≤ d ∧ π ≤ π → d π ⊆ 0 π
855 788 789 852 853 854 syl22anc ⊢ χ → d π ⊆ 0 π
856 ioombl ⊢ d π ∈ dom ⁡ vol
857 856 a1i ⊢ χ → d π ∈ dom ⁡ vol
858 855 857 786 849 iblss ⊢ χ → s ∈ d π ⟼ U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ∈ 𝐿 1
859 755 756 761 787 850 858 itgsplitioo ⊢ χ → ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds = ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds + ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds
860 859 fveq2d ⊢ χ → ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds = ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds + ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds
861 796 sselda ⊢ χ ∧ s ∈ 0 d → s ∈ 0 π
862 861 786 syldan ⊢ χ ∧ s ∈ 0 d → U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ∈ ℝ
863 862 850 itgcl ⊢ χ → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds ∈ ℂ
864 855 sselda ⊢ χ ∧ s ∈ d π → s ∈ 0 π
865 864 786 syldan ⊢ χ ∧ s ∈ d π → U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ∈ ℝ
866 865 858 itgcl ⊢ χ → ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds ∈ ℂ
867 863 866 addcld ⊢ χ → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds + ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds ∈ ℂ
868 867 abscld ⊢ χ → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds + ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds ∈ ℝ
869 863 abscld ⊢ χ → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds ∈ ℝ
870 866 abscld ⊢ χ → ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds ∈ ℝ
871 869 870 readdcld ⊢ χ → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds + ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds ∈ ℝ
872 simp-5r ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ k ∈ ℕ ∧ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → e ∈ ℝ +
873 758 872 syl ⊢ χ → e ∈ ℝ +
874 873 rpred ⊢ χ → e ∈ ℝ
875 863 866 abstrid ⊢ χ → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds + ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds ≤ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds + ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds
876 758 simplrd ⊢ χ → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
877 758 simprd ⊢ χ → ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
878 869 870 874 876 877 lt2halvesd ⊢ χ → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds + ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e
879 868 871 874 875 878 lelttrd ⊢ χ → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds + ∫ d π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e
880 860 879 eqbrtrd ⊢ χ → ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e
881 754 880 syl ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ ∧ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 ∧ k ∈ ℤ ≥ j → ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e
882 881 ex ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ ∧ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → k ∈ ℤ ≥ j → ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e
883 709 882 ralrimi ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ ∧ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → ∀ k ∈ ℤ ≥ j ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e
884 883 ex ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ∧ j ∈ ℕ → ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → ∀ k ∈ ℤ ≥ j ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e
885 884 reximdva ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → ∃ j ∈ ℕ ∀ l ∈ j +∞ ∫ d π U ⁡ s ⁢ sin ⁡ l ⁢ s ds < e 2 → ∃ j ∈ ℕ ∀ k ∈ ℤ ≥ j ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e
886 702 885 mpd ⊢ φ ∧ e ∈ ℝ + ∧ d ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → ∃ j ∈ ℕ ∀ k ∈ ℤ ≥ j ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e
887 pipos ⊢ 0 < π
888 70 766 42 lttri ⊢ − π < 0 ∧ 0 < π → − π < π
889 74 887 888 mp2an ⊢ − π < π
890 70 42 889 ltleii ⊢ − π ≤ π
891 890 a1i ⊢ φ → − π ≤ π
892 3 fourierdlem2 ⊢ M ∈ ℕ → V ∈ P ⁡ M ↔ V ∈ ℝ 0 … M ∧ V ⁡ 0 = - π + X ∧ V ⁡ M = π + X ∧ ∀ i ∈ 0 ..^ M V ⁡ i < V ⁡ i + 1
893 4 892 syl ⊢ φ → V ∈ P ⁡ M ↔ V ∈ ℝ 0 … M ∧ V ⁡ 0 = - π + X ∧ V ⁡ M = π + X ∧ ∀ i ∈ 0 ..^ M V ⁡ i < V ⁡ i + 1
894 5 893 mpbid ⊢ φ → V ∈ ℝ 0 … M ∧ V ⁡ 0 = - π + X ∧ V ⁡ M = π + X ∧ ∀ i ∈ 0 ..^ M V ⁡ i < V ⁡ i + 1
895 894 simpld ⊢ φ → V ∈ ℝ 0 … M
896 elmapi ⊢ V ∈ ℝ 0 … M → V : 0 … M ⟶ ℝ
897 895 896 syl ⊢ φ → V : 0 … M ⟶ ℝ
898 897 ffvelcdmda ⊢ φ ∧ i ∈ 0 … M → V ⁡ i ∈ ℝ
899 2 adantr ⊢ φ ∧ i ∈ 0 … M → X ∈ ℝ
900 898 899 resubcld ⊢ φ ∧ i ∈ 0 … M → V ⁡ i − X ∈ ℝ
901 900 29 fmptd ⊢ φ → Q : 0 … M ⟶ ℝ
902 29 a1i ⊢ φ → Q = i ∈ 0 … M ⟼ V ⁡ i − X
903 fveq2 ⊢ i = 0 → V ⁡ i = V ⁡ 0
904 903 oveq1d ⊢ i = 0 → V ⁡ i − X = V ⁡ 0 − X
905 904 adantl ⊢ φ ∧ i = 0 → V ⁡ i − X = V ⁡ 0 − X
906 4 nnnn0d ⊢ φ → M ∈ ℕ 0
907 nn0uz ⊢ ℕ 0 = ℤ ≥ 0
908 906 907 eleqtrdi ⊢ φ → M ∈ ℤ ≥ 0
909 eluzfz1 ⊢ M ∈ ℤ ≥ 0 → 0 ∈ 0 … M
910 908 909 syl ⊢ φ → 0 ∈ 0 … M
911 897 910 ffvelcdmd ⊢ φ → V ⁡ 0 ∈ ℝ
912 911 2 resubcld ⊢ φ → V ⁡ 0 − X ∈ ℝ
913 902 905 910 912 fvmptd ⊢ φ → Q ⁡ 0 = V ⁡ 0 − X
914 894 simprd ⊢ φ → V ⁡ 0 = - π + X ∧ V ⁡ M = π + X ∧ ∀ i ∈ 0 ..^ M V ⁡ i < V ⁡ i + 1
915 914 simplld ⊢ φ → V ⁡ 0 = - π + X
916 915 oveq1d ⊢ φ → V ⁡ 0 − X = − π + X - X
917 454 recnd ⊢ φ → − π ∈ ℂ
918 2 recnd ⊢ φ → X ∈ ℂ
919 917 918 pncand ⊢ φ → − π + X - X = − π
920 913 916 919 3eqtrd ⊢ φ → Q ⁡ 0 = − π
921 454 456 2 3 840 4 5 29 fourierdlem14 ⊢ φ → Q ∈ m ∈ ℕ ⟼ p ∈ ℝ 0 … m | p ⁡ 0 = − π ∧ p ⁡ m = π ∧ ∀ i ∈ 0 ..^ m p ⁡ i < p ⁡ i + 1 ⁡ M
922 840 fourierdlem2 ⊢ M ∈ ℕ → Q ∈ m ∈ ℕ ⟼ p ∈ ℝ 0 … m | p ⁡ 0 = − π ∧ p ⁡ m = π ∧ ∀ i ∈ 0 ..^ m p ⁡ i < p ⁡ i + 1 ⁡ M ↔ Q ∈ ℝ 0 … M ∧ Q ⁡ 0 = − π ∧ Q ⁡ M = π ∧ ∀ i ∈ 0 ..^ M Q ⁡ i < Q ⁡ i + 1
923 4 922 syl ⊢ φ → Q ∈ m ∈ ℕ ⟼ p ∈ ℝ 0 … m | p ⁡ 0 = − π ∧ p ⁡ m = π ∧ ∀ i ∈ 0 ..^ m p ⁡ i < p ⁡ i + 1 ⁡ M ↔ Q ∈ ℝ 0 … M ∧ Q ⁡ 0 = − π ∧ Q ⁡ M = π ∧ ∀ i ∈ 0 ..^ M Q ⁡ i < Q ⁡ i + 1
924 921 923 mpbid ⊢ φ → Q ∈ ℝ 0 … M ∧ Q ⁡ 0 = − π ∧ Q ⁡ M = π ∧ ∀ i ∈ 0 ..^ M Q ⁡ i < Q ⁡ i + 1
925 924 simprd ⊢ φ → Q ⁡ 0 = − π ∧ Q ⁡ M = π ∧ ∀ i ∈ 0 ..^ M Q ⁡ i < Q ⁡ i + 1
926 925 simplrd ⊢ φ → Q ⁡ M = π
927 925 simprd ⊢ φ → ∀ i ∈ 0 ..^ M Q ⁡ i < Q ⁡ i + 1
928 927 r19.21bi ⊢ φ ∧ i ∈ 0 ..^ M → Q ⁡ i < Q ⁡ i + 1
929 1 adantr ⊢ φ ∧ i ∈ 0 ..^ M → F : ℝ ⟶ ℝ
930 840 4 921 fourierdlem15 ⊢ φ → Q : 0 … M ⟶ − π π
931 930 adantr ⊢ φ ∧ i ∈ 0 ..^ M → Q : 0 … M ⟶ − π π
932 elfzofz ⊢ i ∈ 0 ..^ M → i ∈ 0 … M
933 932 adantl ⊢ φ ∧ i ∈ 0 ..^ M → i ∈ 0 … M
934 931 933 ffvelcdmd ⊢ φ ∧ i ∈ 0 ..^ M → Q ⁡ i ∈ − π π
935 fzofzp1 ⊢ i ∈ 0 ..^ M → i + 1 ∈ 0 … M
936 935 adantl ⊢ φ ∧ i ∈ 0 ..^ M → i + 1 ∈ 0 … M
937 931 936 ffvelcdmd ⊢ φ ∧ i ∈ 0 ..^ M → Q ⁡ i + 1 ∈ − π π
938 2 adantr ⊢ φ ∧ i ∈ 0 ..^ M → X ∈ ℝ
939 ffn ⊢ V : 0 … M ⟶ ℝ → V Fn 0 … M
940 895 896 939 3syl ⊢ φ → V Fn 0 … M
941 fvelrnb ⊢ V Fn 0 … M → X ∈ ran ⁡ V ↔ ∃ i ∈ 0 … M V ⁡ i = X
942 940 941 syl ⊢ φ → X ∈ ran ⁡ V ↔ ∃ i ∈ 0 … M V ⁡ i = X
943 6 942 mpbid ⊢ φ → ∃ i ∈ 0 … M V ⁡ i = X
944 oveq1 ⊢ V ⁡ i = X → V ⁡ i − X = X − X
945 944 adantl ⊢ φ ∧ i ∈ 0 … M ∧ V ⁡ i = X → V ⁡ i − X = X − X
946 918 subidd ⊢ φ → X − X = 0
947 946 ad2antrr ⊢ φ ∧ i ∈ 0 … M ∧ V ⁡ i = X → X − X = 0
948 945 947 eqtr2d ⊢ φ ∧ i ∈ 0 … M ∧ V ⁡ i = X → 0 = V ⁡ i − X
949 948 ex ⊢ φ ∧ i ∈ 0 … M → V ⁡ i = X → 0 = V ⁡ i − X
950 949 reximdva ⊢ φ → ∃ i ∈ 0 … M V ⁡ i = X → ∃ i ∈ 0 … M 0 = V ⁡ i − X
951 943 950 mpd ⊢ φ → ∃ i ∈ 0 … M 0 = V ⁡ i − X
952 29 elrnmpt ⊢ 0 ∈ ℝ → 0 ∈ ran ⁡ Q ↔ ∃ i ∈ 0 … M 0 = V ⁡ i − X
953 766 952 ax-mp ⊢ 0 ∈ ran ⁡ Q ↔ ∃ i ∈ 0 … M 0 = V ⁡ i − X
954 951 953 sylibr ⊢ φ → 0 ∈ ran ⁡ Q
955 840 4 921 954 fourierdlem12 ⊢ φ ∧ i ∈ 0 ..^ M → ¬ 0 ∈ Q ⁡ i Q ⁡ i + 1
956 897 adantr ⊢ φ ∧ i ∈ 0 ..^ M → V : 0 … M ⟶ ℝ
957 956 933 ffvelcdmd ⊢ φ ∧ i ∈ 0 ..^ M → V ⁡ i ∈ ℝ
958 957 938 resubcld ⊢ φ ∧ i ∈ 0 ..^ M → V ⁡ i − X ∈ ℝ
959 29 fvmpt2 ⊢ i ∈ 0 … M ∧ V ⁡ i − X ∈ ℝ → Q ⁡ i = V ⁡ i − X
960 933 958 959 syl2anc ⊢ φ ∧ i ∈ 0 ..^ M → Q ⁡ i = V ⁡ i − X
961 960 oveq1d ⊢ φ ∧ i ∈ 0 ..^ M → Q ⁡ i + X = V ⁡ i - X + X
962 957 recnd ⊢ φ ∧ i ∈ 0 ..^ M → V ⁡ i ∈ ℂ
963 918 adantr ⊢ φ ∧ i ∈ 0 ..^ M → X ∈ ℂ
964 962 963 npcand ⊢ φ ∧ i ∈ 0 ..^ M → V ⁡ i - X + X = V ⁡ i
965 961 964 eqtrd ⊢ φ ∧ i ∈ 0 ..^ M → Q ⁡ i + X = V ⁡ i
966 fveq2 ⊢ j = i → V ⁡ j = V ⁡ i
967 966 oveq1d ⊢ j = i → V ⁡ j − X = V ⁡ i − X
968 967 cbvmptv ⊢ j ∈ 0 … M ⟼ V ⁡ j − X = i ∈ 0 … M ⟼ V ⁡ i − X
969 29 968 eqtr4i ⊢ Q = j ∈ 0 … M ⟼ V ⁡ j − X
970 969 a1i ⊢ φ ∧ i ∈ 0 ..^ M → Q = j ∈ 0 … M ⟼ V ⁡ j − X
971 fveq2 ⊢ j = i + 1 → V ⁡ j = V ⁡ i + 1
972 971 oveq1d ⊢ j = i + 1 → V ⁡ j − X = V ⁡ i + 1 − X
973 972 adantl ⊢ φ ∧ i ∈ 0 ..^ M ∧ j = i + 1 → V ⁡ j − X = V ⁡ i + 1 − X
974 956 936 ffvelcdmd ⊢ φ ∧ i ∈ 0 ..^ M → V ⁡ i + 1 ∈ ℝ
975 974 938 resubcld ⊢ φ ∧ i ∈ 0 ..^ M → V ⁡ i + 1 − X ∈ ℝ
976 970 973 936 975 fvmptd ⊢ φ ∧ i ∈ 0 ..^ M → Q ⁡ i + 1 = V ⁡ i + 1 − X
977 976 oveq1d ⊢ φ ∧ i ∈ 0 ..^ M → Q ⁡ i + 1 + X = V ⁡ i + 1 - X + X
978 974 recnd ⊢ φ ∧ i ∈ 0 ..^ M → V ⁡ i + 1 ∈ ℂ
979 978 963 npcand ⊢ φ ∧ i ∈ 0 ..^ M → V ⁡ i + 1 - X + X = V ⁡ i + 1
980 977 979 eqtrd ⊢ φ ∧ i ∈ 0 ..^ M → Q ⁡ i + 1 + X = V ⁡ i + 1
981 965 980 oveq12d ⊢ φ ∧ i ∈ 0 ..^ M → Q ⁡ i + X Q ⁡ i + 1 + X = V ⁡ i V ⁡ i + 1
982 981 reseq2d ⊢ φ ∧ i ∈ 0 ..^ M → F ↾ Q ⁡ i + X Q ⁡ i + 1 + X = F ↾ V ⁡ i V ⁡ i + 1
983 981 oveq1d ⊢ φ ∧ i ∈ 0 ..^ M → Q ⁡ i + X Q ⁡ i + 1 + X ⟶cn ℂ = V ⁡ i V ⁡ i + 1 ⟶cn ℂ
984 7 982 983 3eltr4d ⊢ φ ∧ i ∈ 0 ..^ M → F ↾ Q ⁡ i + X Q ⁡ i + 1 + X : Q ⁡ i + X Q ⁡ i + 1 + X ⟶cn ℂ
985 54 adantr ⊢ φ ∧ i ∈ 0 ..^ M → Y ∈ ℝ
986 64 adantr ⊢ φ ∧ i ∈ 0 ..^ M → W ∈ ℝ
987 929 934 937 938 955 984 985 986 13 fourierdlem40 ⊢ φ ∧ i ∈ 0 ..^ M → H ↾ Q ⁡ i Q ⁡ i + 1 : Q ⁡ i Q ⁡ i + 1 ⟶cn ℂ
988 id ⊢ F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶ ℝ → F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶ ℝ
989 66 a1i ⊢ F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶ ℝ → ℝ ⊆ ℂ
990 988 989 fssd ⊢ F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶ ℝ → F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶ ℂ
991 9 600 990 3syl ⊢ φ ∧ i ∈ 0 ..^ M → F ℝ ′ ↾ V ⁡ i V ⁡ i + 1 : V ⁡ i V ⁡ i + 1 ⟶ ℂ
992 eqid ⊢ if V ⁡ i = X B R − if V ⁡ i < X W Y Q ⁡ i = if V ⁡ i = X B R − if V ⁡ i < X W Y Q ⁡ i
993 2 3 1 6 20 64 13 4 5 11 29 840 841 991 23 992 fourierdlem75 ⊢ φ ∧ i ∈ 0 ..^ M → if V ⁡ i = X B R − if V ⁡ i < X W Y Q ⁡ i ∈ H ↾ Q ⁡ i Q ⁡ i + 1 lim ℂ Q ⁡ i
994 eqid ⊢ if V ⁡ i + 1 = X A L − if V ⁡ i + 1 < X W Y Q ⁡ i + 1 = if V ⁡ i + 1 = X A L − if V ⁡ i + 1 < X W Y Q ⁡ i + 1
995 2 3 1 6 54 21 13 4 5 12 29 840 841 601 22 994 fourierdlem74 ⊢ φ ∧ i ∈ 0 ..^ M → if V ⁡ i + 1 = X A L − if V ⁡ i + 1 < X W Y Q ⁡ i + 1 ∈ H ↾ Q ⁡ i Q ⁡ i + 1 lim ℂ Q ⁡ i + 1
996 fveq2 ⊢ j = i → Q ⁡ j = Q ⁡ i
997 oveq1 ⊢ j = i → j + 1 = i + 1
998 997 fveq2d ⊢ j = i → Q ⁡ j + 1 = Q ⁡ i + 1
999 996 998 oveq12d ⊢ j = i → Q ⁡ j Q ⁡ j + 1 = Q ⁡ i Q ⁡ i + 1
1000 999 cbvmptv ⊢ j ∈ 0 ..^ M ⟼ Q ⁡ j Q ⁡ j + 1 = i ∈ 0 ..^ M ⟼ Q ⁡ i Q ⁡ i + 1
1001 454 456 891 195 4 901 920 926 928 987 993 995 1000 fourierdlem70 ⊢ φ → ∃ x ∈ ℝ ∀ s ∈ − π π H ⁡ s ≤ x
1002 eqid ⊢ e 3 y = e 3 y
1003 fveq2 ⊢ t = s → G ⁡ t = G ⁡ s
1004 1003 fveq2d ⊢ t = s → G ⁡ t = G ⁡ s
1005 1004 breq1d ⊢ t = s → G ⁡ t ≤ y ↔ G ⁡ s ≤ y
1006 1005 cbvralvw ⊢ ∀ t ∈ − π π G ⁡ t ≤ y ↔ ∀ s ∈ − π π G ⁡ s ≤ y
1007 1006 ralbii ⊢ ∀ n ∈ ℕ ∀ t ∈ − π π G ⁡ t ≤ y ↔ ∀ n ∈ ℕ ∀ s ∈ − π π G ⁡ s ≤ y
1008 1007 3anbi3i ⊢ φ ∧ e ∈ ℝ + ∧ y ∈ ℝ + ∧ ∀ n ∈ ℕ ∀ t ∈ − π π G ⁡ t ≤ y ↔ φ ∧ e ∈ ℝ + ∧ y ∈ ℝ + ∧ ∀ n ∈ ℕ ∀ s ∈ − π π G ⁡ s ≤ y
1009 1008 anbi1i ⊢ φ ∧ e ∈ ℝ + ∧ y ∈ ℝ + ∧ ∀ n ∈ ℕ ∀ t ∈ − π π G ⁡ t ≤ y ∧ u ∈ dom ⁡ vol ↔ φ ∧ e ∈ ℝ + ∧ y ∈ ℝ + ∧ ∀ n ∈ ℕ ∀ s ∈ − π π G ⁡ s ≤ y ∧ u ∈ dom ⁡ vol
1010 1009 anbi1i ⊢ φ ∧ e ∈ ℝ + ∧ y ∈ ℝ + ∧ ∀ n ∈ ℕ ∀ t ∈ − π π G ⁡ t ≤ y ∧ u ∈ dom ⁡ vol ∧ u ⊆ − π π ∧ vol ⁡ u ≤ e 3 y ↔ φ ∧ e ∈ ℝ + ∧ y ∈ ℝ + ∧ ∀ n ∈ ℕ ∀ s ∈ − π π G ⁡ s ≤ y ∧ u ∈ dom ⁡ vol ∧ u ⊆ − π π ∧ vol ⁡ u ≤ e 3 y
1011 1010 anbi1i ⊢ φ ∧ e ∈ ℝ + ∧ y ∈ ℝ + ∧ ∀ n ∈ ℕ ∀ t ∈ − π π G ⁡ t ≤ y ∧ u ∈ dom ⁡ vol ∧ u ⊆ − π π ∧ vol ⁡ u ≤ e 3 y ∧ n ∈ ℕ ↔ φ ∧ e ∈ ℝ + ∧ y ∈ ℝ + ∧ ∀ n ∈ ℕ ∀ s ∈ − π π G ⁡ s ≤ y ∧ u ∈ dom ⁡ vol ∧ u ⊆ − π π ∧ vol ⁡ u ≤ e 3 y ∧ n ∈ ℕ
1012 1 2 54 64 13 14 15 16 17 1001 845 1002 1011 fourierdlem87 ⊢ φ ∧ e ∈ ℝ + → ∃ c ∈ ℝ + ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1013 iftrue ⊢ c ≤ π 2 → if c ≤ π 2 c π 2 = c
1014 1013 adantl ⊢ c ∈ ℝ + ∧ c ≤ π 2 → if c ≤ π 2 c π 2 = c
1015 76 a1i ⊢ c ∈ ℝ + ∧ c ≤ π 2 → 0 ∈ ℝ *
1016 77 a1i ⊢ c ∈ ℝ + ∧ c ≤ π 2 → π ∈ ℝ *
1017 rpre ⊢ c ∈ ℝ + → c ∈ ℝ
1018 1017 adantr ⊢ c ∈ ℝ + ∧ c ≤ π 2 → c ∈ ℝ
1019 rpgt0 ⊢ c ∈ ℝ + → 0 < c
1020 1019 adantr ⊢ c ∈ ℝ + ∧ c ≤ π 2 → 0 < c
1021 42 rehalfcli ⊢ π 2 ∈ ℝ
1022 1021 a1i ⊢ c ∈ ℝ + ∧ c ≤ π 2 → π 2 ∈ ℝ
1023 42 a1i ⊢ c ∈ ℝ + ∧ c ≤ π 2 → π ∈ ℝ
1024 simpr ⊢ c ∈ ℝ + ∧ c ≤ π 2 → c ≤ π 2
1025 halfpos ⊢ π ∈ ℝ → 0 < π ↔ π 2 < π
1026 42 1025 ax-mp ⊢ 0 < π ↔ π 2 < π
1027 887 1026 mpbi ⊢ π 2 < π
1028 1027 a1i ⊢ c ∈ ℝ + ∧ c ≤ π 2 → π 2 < π
1029 1018 1022 1023 1024 1028 lelttrd ⊢ c ∈ ℝ + ∧ c ≤ π 2 → c < π
1030 1015 1016 1018 1020 1029 eliood ⊢ c ∈ ℝ + ∧ c ≤ π 2 → c ∈ 0 π
1031 1014 1030 eqeltrd ⊢ c ∈ ℝ + ∧ c ≤ π 2 → if c ≤ π 2 c π 2 ∈ 0 π
1032 iffalse ⊢ ¬ c ≤ π 2 → if c ≤ π 2 c π 2 = π 2
1033 2pos ⊢ 0 < 2
1034 42 119 887 1033 divgt0ii ⊢ 0 < π 2
1035 elioo2 ⊢ 0 ∈ ℝ * ∧ π ∈ ℝ * → π 2 ∈ 0 π ↔ π 2 ∈ ℝ ∧ 0 < π 2 ∧ π 2 < π
1036 76 77 1035 mp2an ⊢ π 2 ∈ 0 π ↔ π 2 ∈ ℝ ∧ 0 < π 2 ∧ π 2 < π
1037 1021 1034 1027 1036 mpbir3an ⊢ π 2 ∈ 0 π
1038 1037 a1i ⊢ ¬ c ≤ π 2 → π 2 ∈ 0 π
1039 1032 1038 eqeltrd ⊢ ¬ c ≤ π 2 → if c ≤ π 2 c π 2 ∈ 0 π
1040 1039 adantl ⊢ c ∈ ℝ + ∧ ¬ c ≤ π 2 → if c ≤ π 2 c π 2 ∈ 0 π
1041 1031 1040 pm2.61dan ⊢ c ∈ ℝ + → if c ≤ π 2 c π 2 ∈ 0 π
1042 1041 3ad2ant2 ⊢ φ ∧ e ∈ ℝ + ∧ c ∈ ℝ + ∧ ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → if c ≤ π 2 c π 2 ∈ 0 π
1043 ioombl ⊢ 0 if c ≤ π 2 c π 2 ∈ dom ⁡ vol
1044 1043 a1i ⊢ c ∈ ℝ + ∧ ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → 0 if c ≤ π 2 c π 2 ∈ dom ⁡ vol
1045 simpr ⊢ c ∈ ℝ + ∧ ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1046 1044 1045 jca ⊢ c ∈ ℝ + ∧ ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → 0 if c ≤ π 2 c π 2 ∈ dom ⁡ vol ∧ ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1047 ioossicc ⊢ 0 if c ≤ π 2 c π 2 ⊆ 0 if c ≤ π 2 c π 2
1048 70 a1i ⊢ c ∈ ℝ + → − π ∈ ℝ
1049 42 a1i ⊢ c ∈ ℝ + → π ∈ ℝ
1050 767 a1i ⊢ c ∈ ℝ + → − π ≤ 0
1051 791 1041 sselid ⊢ c ∈ ℝ + → if c ≤ π 2 c π 2 ∈ ℝ
1052 1021 a1i ⊢ c ∈ ℝ + → π 2 ∈ ℝ
1053 min2 ⊢ c ∈ ℝ ∧ π 2 ∈ ℝ → if c ≤ π 2 c π 2 ≤ π 2
1054 1017 1021 1053 sylancl ⊢ c ∈ ℝ + → if c ≤ π 2 c π 2 ≤ π 2
1055 1027 a1i ⊢ c ∈ ℝ + → π 2 < π
1056 1051 1052 1049 1054 1055 lelttrd ⊢ c ∈ ℝ + → if c ≤ π 2 c π 2 < π
1057 1051 1049 1056 ltled ⊢ c ∈ ℝ + → if c ≤ π 2 c π 2 ≤ π
1058 iccss ⊢ − π ∈ ℝ ∧ π ∈ ℝ ∧ − π ≤ 0 ∧ if c ≤ π 2 c π 2 ≤ π → 0 if c ≤ π 2 c π 2 ⊆ − π π
1059 1048 1049 1050 1057 1058 syl22anc ⊢ c ∈ ℝ + → 0 if c ≤ π 2 c π 2 ⊆ − π π
1060 1047 1059 sstrid ⊢ c ∈ ℝ + → 0 if c ≤ π 2 c π 2 ⊆ − π π
1061 0red ⊢ c ∈ ℝ + → 0 ∈ ℝ
1062 1020 1014 breqtrrd ⊢ c ∈ ℝ + ∧ c ≤ π 2 → 0 < if c ≤ π 2 c π 2
1063 1034 1032 breqtrrid ⊢ ¬ c ≤ π 2 → 0 < if c ≤ π 2 c π 2
1064 1063 adantl ⊢ c ∈ ℝ + ∧ ¬ c ≤ π 2 → 0 < if c ≤ π 2 c π 2
1065 1062 1064 pm2.61dan ⊢ c ∈ ℝ + → 0 < if c ≤ π 2 c π 2
1066 1061 1051 1065 ltled ⊢ c ∈ ℝ + → 0 ≤ if c ≤ π 2 c π 2
1067 volioo ⊢ 0 ∈ ℝ ∧ if c ≤ π 2 c π 2 ∈ ℝ ∧ 0 ≤ if c ≤ π 2 c π 2 → vol ⁡ 0 if c ≤ π 2 c π 2 = if c ≤ π 2 c π 2 − 0
1068 1061 1051 1066 1067 syl3anc ⊢ c ∈ ℝ + → vol ⁡ 0 if c ≤ π 2 c π 2 = if c ≤ π 2 c π 2 − 0
1069 1051 recnd ⊢ c ∈ ℝ + → if c ≤ π 2 c π 2 ∈ ℂ
1070 1069 subid1d ⊢ c ∈ ℝ + → if c ≤ π 2 c π 2 − 0 = if c ≤ π 2 c π 2
1071 1068 1070 eqtrd ⊢ c ∈ ℝ + → vol ⁡ 0 if c ≤ π 2 c π 2 = if c ≤ π 2 c π 2
1072 min1 ⊢ c ∈ ℝ ∧ π 2 ∈ ℝ → if c ≤ π 2 c π 2 ≤ c
1073 1017 1021 1072 sylancl ⊢ c ∈ ℝ + → if c ≤ π 2 c π 2 ≤ c
1074 1071 1073 eqbrtrd ⊢ c ∈ ℝ + → vol ⁡ 0 if c ≤ π 2 c π 2 ≤ c
1075 1060 1074 jca ⊢ c ∈ ℝ + → 0 if c ≤ π 2 c π 2 ⊆ − π π ∧ vol ⁡ 0 if c ≤ π 2 c π 2 ≤ c
1076 1075 adantr ⊢ c ∈ ℝ + ∧ ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → 0 if c ≤ π 2 c π 2 ⊆ − π π ∧ vol ⁡ 0 if c ≤ π 2 c π 2 ≤ c
1077 sseq1 ⊢ u = 0 if c ≤ π 2 c π 2 → u ⊆ − π π ↔ 0 if c ≤ π 2 c π 2 ⊆ − π π
1078 fveq2 ⊢ u = 0 if c ≤ π 2 c π 2 → vol ⁡ u = vol ⁡ 0 if c ≤ π 2 c π 2
1079 1078 breq1d ⊢ u = 0 if c ≤ π 2 c π 2 → vol ⁡ u ≤ c ↔ vol ⁡ 0 if c ≤ π 2 c π 2 ≤ c
1080 1077 1079 anbi12d ⊢ u = 0 if c ≤ π 2 c π 2 → u ⊆ − π π ∧ vol ⁡ u ≤ c ↔ 0 if c ≤ π 2 c π 2 ⊆ − π π ∧ vol ⁡ 0 if c ≤ π 2 c π 2 ≤ c
1081 itgeq1 ⊢ u = 0 if c ≤ π 2 c π 2 → ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds = ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds
1082 1081 fveq2d ⊢ u = 0 if c ≤ π 2 c π 2 → ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds = ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds
1083 1082 breq1d ⊢ u = 0 if c ≤ π 2 c π 2 → ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ↔ ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1084 1083 ralbidv ⊢ u = 0 if c ≤ π 2 c π 2 → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ↔ ∀ k ∈ ℕ ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1085 1080 1084 imbi12d ⊢ u = 0 if c ≤ π 2 c π 2 → u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ↔ 0 if c ≤ π 2 c π 2 ⊆ − π π ∧ vol ⁡ 0 if c ≤ π 2 c π 2 ≤ c → ∀ k ∈ ℕ ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1086 1085 rspcva ⊢ 0 if c ≤ π 2 c π 2 ∈ dom ⁡ vol ∧ ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → 0 if c ≤ π 2 c π 2 ⊆ − π π ∧ vol ⁡ 0 if c ≤ π 2 c π 2 ≤ c → ∀ k ∈ ℕ ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1087 1046 1076 1086 sylc ⊢ c ∈ ℝ + ∧ ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → ∀ k ∈ ℕ ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1088 1087 3adant1 ⊢ φ ∧ e ∈ ℝ + ∧ c ∈ ℝ + ∧ ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → ∀ k ∈ ℕ ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1089 oveq2 ⊢ d = if c ≤ π 2 c π 2 → 0 d = 0 if c ≤ π 2 c π 2
1090 1089 itgeq1d ⊢ d = if c ≤ π 2 c π 2 → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds = ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds
1091 1090 fveq2d ⊢ d = if c ≤ π 2 c π 2 → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds = ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds
1092 1091 breq1d ⊢ d = if c ≤ π 2 c π 2 → ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ↔ ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1093 1092 ralbidv ⊢ d = if c ≤ π 2 c π 2 → ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 ↔ ∀ k ∈ ℕ ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1094 1093 rspcev ⊢ if c ≤ π 2 c π 2 ∈ 0 π ∧ ∀ k ∈ ℕ ∫ 0 if c ≤ π 2 c π 2 U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → ∃ d ∈ 0 π ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1095 1042 1088 1094 syl2anc ⊢ φ ∧ e ∈ ℝ + ∧ c ∈ ℝ + ∧ ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → ∃ d ∈ 0 π ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1096 1095 rexlimdv3a ⊢ φ ∧ e ∈ ℝ + → ∃ c ∈ ℝ + ∀ u ∈ dom ⁡ vol u ⊆ − π π ∧ vol ⁡ u ≤ c → ∀ k ∈ ℕ ∫ u U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2 → ∃ d ∈ 0 π ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1097 1012 1096 mpd ⊢ φ ∧ e ∈ ℝ + → ∃ d ∈ 0 π ∀ k ∈ ℕ ∫ 0 d U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e 2
1098 886 1097 r19.29a ⊢ φ ∧ e ∈ ℝ + → ∃ j ∈ ℕ ∀ k ∈ ℤ ≥ j ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e
1099 1098 ralrimiva ⊢ φ → ∀ e ∈ ℝ + ∃ j ∈ ℕ ∀ k ∈ ℤ ≥ j ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e
1100 nnex ⊢ ℕ ∈ V
1101 1100 mptex ⊢ n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds ∈ V
1102 1101 a1i ⊢ φ → n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds ∈ V
1103 eqidd ⊢ φ ∧ k ∈ ℕ → n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds = n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds
1104 772 adantl ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → s ∈ − π π
1105 774 ad4ant14 ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → U ⁡ s ∈ ℝ
1106 772 adantl ⊢ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → s ∈ − π π
1107 simpr ⊢ k ∈ ℕ ∧ n = k → n = k
1108 simpl ⊢ k ∈ ℕ ∧ n = k → k ∈ ℕ
1109 1107 1108 eqeltrd ⊢ k ∈ ℕ ∧ n = k → n ∈ ℕ
1110 1109 nnred ⊢ k ∈ ℕ ∧ n = k → n ∈ ℝ
1111 726 a1i ⊢ k ∈ ℕ ∧ n = k → 1 2 ∈ ℝ
1112 1110 1111 readdcld ⊢ k ∈ ℕ ∧ n = k → n + 1 2 ∈ ℝ
1113 1112 adantr ⊢ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → n + 1 2 ∈ ℝ
1114 228 1106 sselid ⊢ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → s ∈ ℝ
1115 1113 1114 remulcld ⊢ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → n + 1 2 ⁢ s ∈ ℝ
1116 1115 resincld ⊢ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → sin ⁡ n + 1 2 ⁢ s ∈ ℝ
1117 1106 1116 824 syl2anc ⊢ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → S ⁡ s = sin ⁡ n + 1 2 ⁢ s
1118 1117 adantlll ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → S ⁡ s = sin ⁡ n + 1 2 ⁢ s
1119 1110 adantll ⊢ φ ∧ k ∈ ℕ ∧ n = k → n ∈ ℝ
1120 1119 adantr ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → n ∈ ℝ
1121 1red ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → 1 ∈ ℝ
1122 1121 rehalfcld ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → 1 2 ∈ ℝ
1123 1120 1122 readdcld ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → n + 1 2 ∈ ℝ
1124 228 1104 sselid ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → s ∈ ℝ
1125 1123 1124 remulcld ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → n + 1 2 ⁢ s ∈ ℝ
1126 1125 resincld ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → sin ⁡ n + 1 2 ⁢ s ∈ ℝ
1127 1118 1126 eqeltrd ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → S ⁡ s ∈ ℝ
1128 1105 1127 remulcld ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → U ⁡ s ⁢ S ⁡ s ∈ ℝ
1129 17 fvmpt2 ⊢ s ∈ − π π ∧ U ⁡ s ⁢ S ⁡ s ∈ ℝ → G ⁡ s = U ⁡ s ⁢ S ⁡ s
1130 1104 1128 1129 syl2anc ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → G ⁡ s = U ⁡ s ⁢ S ⁡ s
1131 oveq1 ⊢ n = k → n + 1 2 = k + 1 2
1132 1131 oveq1d ⊢ n = k → n + 1 2 ⁢ s = k + 1 2 ⁢ s
1133 1132 fveq2d ⊢ n = k → sin ⁡ n + 1 2 ⁢ s = sin ⁡ k + 1 2 ⁢ s
1134 1133 ad2antlr ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → sin ⁡ n + 1 2 ⁢ s = sin ⁡ k + 1 2 ⁢ s
1135 1118 1134 eqtrd ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → S ⁡ s = sin ⁡ k + 1 2 ⁢ s
1136 1135 oveq2d ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → U ⁡ s ⁢ S ⁡ s = U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s
1137 1130 1136 eqtrd ⊢ φ ∧ k ∈ ℕ ∧ n = k ∧ s ∈ 0 π → G ⁡ s = U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s
1138 1137 itgeq2dv ⊢ φ ∧ k ∈ ℕ ∧ n = k → ∫ 0 π G ⁡ s ds = ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds
1139 simpr ⊢ φ ∧ k ∈ ℕ → k ∈ ℕ
1140 805 itgeq2dv ⊢ n = k → ∫ 0 π U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s ds = ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds
1141 1140 eleq1d ⊢ n = k → ∫ 0 π U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s ds ∈ ℂ ↔ ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds ∈ ℂ
1142 800 1141 imbi12d ⊢ n = k → φ ∧ n ∈ ℕ → ∫ 0 π U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s ds ∈ ℂ ↔ φ ∧ k ∈ ℕ → ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds ∈ ℂ
1143 774 adantlr ⊢ φ ∧ n ∈ ℕ ∧ s ∈ 0 π → U ⁡ s ∈ ℝ
1144 simpr ⊢ φ ∧ n ∈ ℕ → n ∈ ℕ
1145 1144 772 821 syl2an ⊢ φ ∧ n ∈ ℕ ∧ s ∈ 0 π → sin ⁡ n + 1 2 ⁢ s ∈ ℝ
1146 1143 1145 remulcld ⊢ φ ∧ n ∈ ℕ ∧ s ∈ 0 π → U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s ∈ ℝ
1147 1146 847 itgcl ⊢ φ ∧ n ∈ ℕ → ∫ 0 π U ⁡ s ⁢ sin ⁡ n + 1 2 ⁢ s ds ∈ ℂ
1148 1142 1147 chvarvv ⊢ φ ∧ k ∈ ℕ → ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds ∈ ℂ
1149 1103 1138 1139 1148 fvmptd ⊢ φ ∧ k ∈ ℕ → n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds ⁡ k = ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds
1150 39 33 1102 1149 1148 clim0c ⊢ φ → n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds ⇝ 0 ↔ ∀ e ∈ ℝ + ∃ j ∈ ℕ ∀ k ∈ ℤ ≥ j ∫ 0 π U ⁡ s ⁢ sin ⁡ k + 1 2 ⁢ s ds < e
1151 1099 1150 mpbird ⊢ φ → n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds ⇝ 0
1152 1100 mptex ⊢ n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds π ∈ V
1153 19 1152 eqeltri ⊢ E ∈ V
1154 1153 a1i ⊢ φ → E ∈ V
1155 1100 mptex ⊢ n ∈ ℕ ⟼ π ∈ V
1156 1155 a1i ⊢ φ → n ∈ ℕ ⟼ π ∈ V
1157 42 recni ⊢ π ∈ ℂ
1158 1157 a1i ⊢ φ → π ∈ ℂ
1159 eqidd ⊢ m ∈ ℕ → n ∈ ℕ ⟼ π = n ∈ ℕ ⟼ π
1160 eqidd ⊢ m ∈ ℕ ∧ n = m → π = π
1161 id ⊢ m ∈ ℕ → m ∈ ℕ
1162 42 a1i ⊢ m ∈ ℕ → π ∈ ℝ
1163 1159 1160 1161 1162 fvmptd ⊢ m ∈ ℕ → n ∈ ℕ ⟼ π ⁡ m = π
1164 1163 adantl ⊢ φ ∧ m ∈ ℕ → n ∈ ℕ ⟼ π ⁡ m = π
1165 39 33 1156 1158 1164 climconst ⊢ φ → n ∈ ℕ ⟼ π ⇝ π
1166 766 887 gtneii ⊢ π ≠ 0
1167 1166 a1i ⊢ φ → π ≠ 0
1168 2 adantr ⊢ φ ∧ n ∈ ℕ → X ∈ ℝ
1169 54 adantr ⊢ φ ∧ n ∈ ℕ → Y ∈ ℝ
1170 64 adantr ⊢ φ ∧ n ∈ ℕ → W ∈ ℝ
1171 830 1168 1169 1170 13 14 15 834 16 17 fourierdlem67 ⊢ φ ∧ n ∈ ℕ → G : − π π ⟶ ℝ
1172 1171 adantr ⊢ φ ∧ n ∈ ℕ ∧ s ∈ 0 π → G : − π π ⟶ ℝ
1173 809 sselda ⊢ φ ∧ n ∈ ℕ ∧ s ∈ 0 π → s ∈ − π π
1174 1172 1173 ffvelcdmd ⊢ φ ∧ n ∈ ℕ ∧ s ∈ 0 π → G ⁡ s ∈ ℝ
1175 1171 ffvelcdmda ⊢ φ ∧ n ∈ ℕ ∧ s ∈ − π π → G ⁡ s ∈ ℝ
1176 1171 feqmptd ⊢ φ ∧ n ∈ ℕ → G = s ∈ − π π ⟼ G ⁡ s
1177 1176 845 eqeltrrd ⊢ φ ∧ n ∈ ℕ → s ∈ − π π ⟼ G ⁡ s ∈ 𝐿 1
1178 809 811 1175 1177 iblss ⊢ φ ∧ n ∈ ℕ → s ∈ 0 π ⟼ G ⁡ s ∈ 𝐿 1
1179 1174 1178 itgcl ⊢ φ ∧ n ∈ ℕ → ∫ 0 π G ⁡ s ds ∈ ℂ
1180 eqid ⊢ n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds = n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds
1181 1180 fvmpt2 ⊢ n ∈ ℕ ∧ ∫ 0 π G ⁡ s ds ∈ ℂ → n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds ⁡ n = ∫ 0 π G ⁡ s ds
1182 1144 1179 1181 syl2anc ⊢ φ ∧ n ∈ ℕ → n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds ⁡ n = ∫ 0 π G ⁡ s ds
1183 1182 1179 eqeltrd ⊢ φ ∧ n ∈ ℕ → n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds ⁡ n ∈ ℂ
1184 eqid ⊢ n ∈ ℕ ⟼ π = n ∈ ℕ ⟼ π
1185 1184 fvmpt2 ⊢ n ∈ ℕ ∧ π ∈ ℝ → n ∈ ℕ ⟼ π ⁡ n = π
1186 42 1185 mpan2 ⊢ n ∈ ℕ → n ∈ ℕ ⟼ π ⁡ n = π
1187 1157 a1i ⊢ n ∈ ℕ → π ∈ ℂ
1188 1166 a1i ⊢ n ∈ ℕ → π ≠ 0
1189 eldifsn ⊢ π ∈ ℂ ∖ 0 ↔ π ∈ ℂ ∧ π ≠ 0
1190 1187 1188 1189 sylanbrc ⊢ n ∈ ℕ → π ∈ ℂ ∖ 0
1191 1186 1190 eqeltrd ⊢ n ∈ ℕ → n ∈ ℕ ⟼ π ⁡ n ∈ ℂ ∖ 0
1192 1191 adantl ⊢ φ ∧ n ∈ ℕ → n ∈ ℕ ⟼ π ⁡ n ∈ ℂ ∖ 0
1193 1157 a1i ⊢ φ ∧ n ∈ ℕ → π ∈ ℂ
1194 1166 a1i ⊢ φ ∧ n ∈ ℕ → π ≠ 0
1195 1179 1193 1194 divcld ⊢ φ ∧ n ∈ ℕ → ∫ 0 π G ⁡ s ds π ∈ ℂ
1196 19 fvmpt2 ⊢ n ∈ ℕ ∧ ∫ 0 π G ⁡ s ds π ∈ ℂ → E ⁡ n = ∫ 0 π G ⁡ s ds π
1197 1144 1195 1196 syl2anc ⊢ φ ∧ n ∈ ℕ → E ⁡ n = ∫ 0 π G ⁡ s ds π
1198 1182 eqcomd ⊢ φ ∧ n ∈ ℕ → ∫ 0 π G ⁡ s ds = n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds ⁡ n
1199 1186 eqcomd ⊢ n ∈ ℕ → π = n ∈ ℕ ⟼ π ⁡ n
1200 1199 adantl ⊢ φ ∧ n ∈ ℕ → π = n ∈ ℕ ⟼ π ⁡ n
1201 1198 1200 oveq12d ⊢ φ ∧ n ∈ ℕ → ∫ 0 π G ⁡ s ds π = n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds ⁡ n n ∈ ℕ ⟼ π ⁡ n
1202 1197 1201 eqtrd ⊢ φ ∧ n ∈ ℕ → E ⁡ n = n ∈ ℕ ⟼ ∫ 0 π G ⁡ s ds ⁡ n n ∈ ℕ ⟼ π ⁡ n
1203 34 35 36 38 39 33 1151 1154 1165 1167 1183 1192 1202 climdivf ⊢ φ → E ⇝ 0 π
1204 1157 1166 div0i ⊢ 0 π = 0
1205 1204 a1i ⊢ φ → 0 π = 0
1206 1203 1205 breqtrd ⊢ φ → E ⇝ 0
1207 1100 mptex ⊢ m ∈ ℕ ⟼ ∫ 0 π F ⁡ X + s ⁢ D ⁡ m ⁡ s ds ∈ V
1208 18 1207 eqeltri ⊢ Z ∈ V
1209 1208 a1i ⊢ φ → Z ∈ V
1210 1100 mptex ⊢ m ∈ ℕ ⟼ Y 2 ∈ V
1211 1210 a1i ⊢ φ → m ∈ ℕ ⟼ Y 2 ∈ V
1212 limccl ⊢ F ↾ X +∞ lim ℂ X ⊆ ℂ
1213 1212 20 sselid ⊢ φ → Y ∈ ℂ
1214 1213 halfcld ⊢ φ → Y 2 ∈ ℂ
1215 eqidd ⊢ φ ∧ n ∈ ℤ ≥ 1 → m ∈ ℕ ⟼ Y 2 = m ∈ ℕ ⟼ Y 2
1216 eqidd ⊢ φ ∧ n ∈ ℤ ≥ 1 ∧ m = n → Y 2 = Y 2
1217 39 eqcomi ⊢ ℤ ≥ 1 = ℕ
1218 1217 eleq2i ⊢ n ∈ ℤ ≥ 1 ↔ n ∈ ℕ
1219 1218 biimpi ⊢ n ∈ ℤ ≥ 1 → n ∈ ℕ
1220 1219 adantl ⊢ φ ∧ n ∈ ℤ ≥ 1 → n ∈ ℕ
1221 1214 adantr ⊢ φ ∧ n ∈ ℤ ≥ 1 → Y 2 ∈ ℂ
1222 1215 1216 1220 1221 fvmptd ⊢ φ ∧ n ∈ ℤ ≥ 1 → m ∈ ℕ ⟼ Y 2 ⁡ n = Y 2
1223 32 33 1211 1214 1222 climconst ⊢ φ → m ∈ ℕ ⟼ Y 2 ⇝ Y 2
1224 1195 19 fmptd ⊢ φ → E : ℕ ⟶ ℂ
1225 1224 adantr ⊢ φ ∧ n ∈ ℤ ≥ 1 → E : ℕ ⟶ ℂ
1226 1225 1220 ffvelcdmd ⊢ φ ∧ n ∈ ℤ ≥ 1 → E ⁡ n ∈ ℂ
1227 1222 1221 eqeltrd ⊢ φ ∧ n ∈ ℤ ≥ 1 → m ∈ ℕ ⟼ Y 2 ⁡ n ∈ ℂ
1228 1222 oveq2d ⊢ φ ∧ n ∈ ℤ ≥ 1 → E ⁡ n + m ∈ ℕ ⟼ Y 2 ⁡ n = E ⁡ n + Y 2
1229 810 a1i ⊢ φ → 0 π ∈ dom ⁡ vol
1230 0red ⊢ s ∈ 0 π → 0 ∈ ℝ
1231 1230 rexrd ⊢ s ∈ 0 π → 0 ∈ ℝ *
1232 77 a1i ⊢ s ∈ 0 π → π ∈ ℝ *
1233 id ⊢ s ∈ 0 π → s ∈ 0 π
1234 ioogtlb ⊢ 0 ∈ ℝ * ∧ π ∈ ℝ * ∧ s ∈ 0 π → 0 < s
1235 1231 1232 1233 1234 syl3anc ⊢ s ∈ 0 π → 0 < s
1236 1235 gt0ne0d ⊢ s ∈ 0 π → s ≠ 0
1237 1236 neneqd ⊢ s ∈ 0 π → ¬ s = 0
1238 velsn ⊢ s ∈ 0 ↔ s = 0
1239 1237 1238 sylnibr ⊢ s ∈ 0 π → ¬ s ∈ 0
1240 772 1239 eldifd ⊢ s ∈ 0 π → s ∈ − π π ∖ 0
1241 1240 ssriv ⊢ 0 π ⊆ − π π ∖ 0
1242 1241 a1i ⊢ φ → 0 π ⊆ − π π ∖ 0
1243 1235 adantl ⊢ φ ∧ s ∈ 0 π → 0 < s
1244 1243 iftrued ⊢ φ ∧ s ∈ 0 π → if 0 < s Y W = Y
1245 eqid ⊢ D ⁡ n = D ⁡ n
1246 0red ⊢ φ ∧ n ∈ ℕ → 0 ∈ ℝ
1247 42 a1i ⊢ φ ∧ n ∈ ℕ → π ∈ ℝ
1248 766 42 887 ltleii ⊢ 0 ≤ π
1249 1248 a1i ⊢ φ ∧ n ∈ ℕ → 0 ≤ π
1250 eqid ⊢ s ∈ 0 π ⟼ s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π = s ∈ 0 π ⟼ s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π
1251 24 1144 1245 1246 1247 1249 1250 dirkeritg ⊢ φ ∧ n ∈ ℕ → ∫ 0 π D ⁡ n ⁡ s ds = s ∈ 0 π ⟼ s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π ⁡ π − s ∈ 0 π ⟼ s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π ⁡ 0
1252 ubicc2 ⊢ 0 ∈ ℝ * ∧ π ∈ ℝ * ∧ 0 ≤ π → π ∈ 0 π
1253 76 77 1248 1252 mp3an ⊢ π ∈ 0 π
1254 oveq1 ⊢ s = π → s 2 = π 2
1255 oveq2 ⊢ s = π → k ⁢ s = k ⁢ π
1256 1255 fveq2d ⊢ s = π → sin ⁡ k ⁢ s = sin ⁡ k ⁢ π
1257 1256 oveq1d ⊢ s = π → sin ⁡ k ⁢ s k = sin ⁡ k ⁢ π k
1258 elfzelz ⊢ k ∈ 1 … n → k ∈ ℤ
1259 1258 zcnd ⊢ k ∈ 1 … n → k ∈ ℂ
1260 1157 a1i ⊢ k ∈ 1 … n → π ∈ ℂ
1261 1166 a1i ⊢ k ∈ 1 … n → π ≠ 0
1262 1259 1260 1261 divcan4d ⊢ k ∈ 1 … n → k ⁢ π π = k
1263 1262 1258 eqeltrd ⊢ k ∈ 1 … n → k ⁢ π π ∈ ℤ
1264 1259 1260 mulcld ⊢ k ∈ 1 … n → k ⁢ π ∈ ℂ
1265 sineq0 ⊢ k ⁢ π ∈ ℂ → sin ⁡ k ⁢ π = 0 ↔ k ⁢ π π ∈ ℤ
1266 1264 1265 syl ⊢ k ∈ 1 … n → sin ⁡ k ⁢ π = 0 ↔ k ⁢ π π ∈ ℤ
1267 1263 1266 mpbird ⊢ k ∈ 1 … n → sin ⁡ k ⁢ π = 0
1268 1267 oveq1d ⊢ k ∈ 1 … n → sin ⁡ k ⁢ π k = 0 k
1269 0red ⊢ k ∈ 1 … n → 0 ∈ ℝ
1270 1red ⊢ k ∈ 1 … n → 1 ∈ ℝ
1271 1258 zred ⊢ k ∈ 1 … n → k ∈ ℝ
1272 117 a1i ⊢ k ∈ 1 … n → 0 < 1
1273 elfzle1 ⊢ k ∈ 1 … n → 1 ≤ k
1274 1269 1270 1271 1272 1273 ltletrd ⊢ k ∈ 1 … n → 0 < k
1275 1274 gt0ne0d ⊢ k ∈ 1 … n → k ≠ 0
1276 1259 1275 div0d ⊢ k ∈ 1 … n → 0 k = 0
1277 1268 1276 eqtrd ⊢ k ∈ 1 … n → sin ⁡ k ⁢ π k = 0
1278 1257 1277 sylan9eq ⊢ s = π ∧ k ∈ 1 … n → sin ⁡ k ⁢ s k = 0
1279 1278 sumeq2dv ⊢ s = π → ∑ k = 1 n sin ⁡ k ⁢ s k = ∑ k = 1 n 0
1280 fzfi ⊢ 1 … n ∈ Fin
1281 1280 olci ⊢ 1 … n ⊆ ℤ ≥ ∥ ˙ ∨ 1 … n ∈ Fin
1282 sumz ⊢ 1 … n ⊆ ℤ ≥ ∥ ˙ ∨ 1 … n ∈ Fin → ∑ k = 1 n 0 = 0
1283 1281 1282 ax-mp ⊢ ∑ k = 1 n 0 = 0
1284 1279 1283 eqtrdi ⊢ s = π → ∑ k = 1 n sin ⁡ k ⁢ s k = 0
1285 1254 1284 oveq12d ⊢ s = π → s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k = π 2 + 0
1286 1285 oveq1d ⊢ s = π → s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π = π 2 + 0 π
1287 ovex ⊢ π 2 + 0 π ∈ V
1288 1286 1250 1287 fvmpt ⊢ π ∈ 0 π → s ∈ 0 π ⟼ s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π ⁡ π = π 2 + 0 π
1289 1253 1288 ax-mp ⊢ s ∈ 0 π ⟼ s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π ⁡ π = π 2 + 0 π
1290 lbicc2 ⊢ 0 ∈ ℝ * ∧ π ∈ ℝ * ∧ 0 ≤ π → 0 ∈ 0 π
1291 76 77 1248 1290 mp3an ⊢ 0 ∈ 0 π
1292 oveq1 ⊢ s = 0 → s 2 = 0 2
1293 2cn ⊢ 2 ∈ ℂ
1294 1293 256 div0i ⊢ 0 2 = 0
1295 1292 1294 eqtrdi ⊢ s = 0 → s 2 = 0
1296 oveq2 ⊢ s = 0 → k ⁢ s = k ⋅ 0
1297 1259 mul01d ⊢ k ∈ 1 … n → k ⋅ 0 = 0
1298 1296 1297 sylan9eq ⊢ s = 0 ∧ k ∈ 1 … n → k ⁢ s = 0
1299 1298 fveq2d ⊢ s = 0 ∧ k ∈ 1 … n → sin ⁡ k ⁢ s = sin ⁡ 0
1300 sin0 ⊢ sin ⁡ 0 = 0
1301 1299 1300 eqtrdi ⊢ s = 0 ∧ k ∈ 1 … n → sin ⁡ k ⁢ s = 0
1302 1301 oveq1d ⊢ s = 0 ∧ k ∈ 1 … n → sin ⁡ k ⁢ s k = 0 k
1303 1276 adantl ⊢ s = 0 ∧ k ∈ 1 … n → 0 k = 0
1304 1302 1303 eqtrd ⊢ s = 0 ∧ k ∈ 1 … n → sin ⁡ k ⁢ s k = 0
1305 1304 sumeq2dv ⊢ s = 0 → ∑ k = 1 n sin ⁡ k ⁢ s k = ∑ k = 1 n 0
1306 1305 1283 eqtrdi ⊢ s = 0 → ∑ k = 1 n sin ⁡ k ⁢ s k = 0
1307 1295 1306 oveq12d ⊢ s = 0 → s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k = 0 + 0
1308 00id ⊢ 0 + 0 = 0
1309 1307 1308 eqtrdi ⊢ s = 0 → s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k = 0
1310 1309 oveq1d ⊢ s = 0 → s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π = 0 π
1311 1310 1204 eqtrdi ⊢ s = 0 → s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π = 0
1312 c0ex ⊢ 0 ∈ V
1313 1311 1250 1312 fvmpt ⊢ 0 ∈ 0 π → s ∈ 0 π ⟼ s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π ⁡ 0 = 0
1314 1291 1313 ax-mp ⊢ s ∈ 0 π ⟼ s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π ⁡ 0 = 0
1315 1289 1314 oveq12i ⊢ s ∈ 0 π ⟼ s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π ⁡ π − s ∈ 0 π ⟼ s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π ⁡ 0 = π 2 + 0 π − 0
1316 1315 a1i ⊢ φ ∧ n ∈ ℕ → s ∈ 0 π ⟼ s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π ⁡ π − s ∈ 0 π ⟼ s 2 + ∑ k = 1 n sin ⁡ k ⁢ s k π ⁡ 0 = π 2 + 0 π − 0
1317 1021 recni ⊢ π 2 ∈ ℂ
1318 1317 addridi ⊢ π 2 + 0 = π 2
1319 1318 oveq1i ⊢ π 2 + 0 π = π 2 π
1320 1157 1293 1157 256 1166 divdiv32i ⊢ π 2 π = π π 2
1321 1157 1166 dividi ⊢ π π = 1
1322 1321 oveq1i ⊢ π π 2 = 1 2
1323 1319 1320 1322 3eqtri ⊢ π 2 + 0 π = 1 2
1324 1323 oveq1i ⊢ π 2 + 0 π − 0 = 1 2 − 0
1325 halfcn ⊢ 1 2 ∈ ℂ
1326 1325 subid1i ⊢ 1 2 − 0 = 1 2
1327 1324 1326 eqtri ⊢ π 2 + 0 π − 0 = 1 2
1328 1327 a1i ⊢ φ ∧ n ∈ ℕ → π 2 + 0 π − 0 = 1 2
1329 1251 1316 1328 3eqtrd ⊢ φ ∧ n ∈ ℕ → ∫ 0 π D ⁡ n ⁡ s ds = 1 2
1330 1 2 3 4 5 6 7 11 12 13 14 15 16 17 841 601 22 23 20 21 1229 1242 19 24 54 1244 1329 fourierdlem95 ⊢ φ ∧ n ∈ ℕ → E ⁡ n + Y 2 = ∫ 0 π F ⁡ X + s ⁢ D ⁡ n ⁡ s ds
1331 1220 1330 syldan ⊢ φ ∧ n ∈ ℤ ≥ 1 → E ⁡ n + Y 2 = ∫ 0 π F ⁡ X + s ⁢ D ⁡ n ⁡ s ds
1332 18 a1i ⊢ φ ∧ n ∈ ℕ → Z = m ∈ ℕ ⟼ ∫ 0 π F ⁡ X + s ⁢ D ⁡ m ⁡ s ds
1333 fveq2 ⊢ m = n → D ⁡ m = D ⁡ n
1334 1333 fveq1d ⊢ m = n → D ⁡ m ⁡ s = D ⁡ n ⁡ s
1335 1334 oveq2d ⊢ m = n → F ⁡ X + s ⁢ D ⁡ m ⁡ s = F ⁡ X + s ⁢ D ⁡ n ⁡ s
1336 1335 adantr ⊢ m = n ∧ s ∈ 0 π → F ⁡ X + s ⁢ D ⁡ m ⁡ s = F ⁡ X + s ⁢ D ⁡ n ⁡ s
1337 1336 itgeq2dv ⊢ m = n → ∫ 0 π F ⁡ X + s ⁢ D ⁡ m ⁡ s ds = ∫ 0 π F ⁡ X + s ⁢ D ⁡ n ⁡ s ds
1338 1337 adantl ⊢ φ ∧ n ∈ ℕ ∧ m = n → ∫ 0 π F ⁡ X + s ⁢ D ⁡ m ⁡ s ds = ∫ 0 π F ⁡ X + s ⁢ D ⁡ n ⁡ s ds
1339 1 adantr ⊢ φ ∧ s ∈ 0 π → F : ℝ ⟶ ℝ
1340 2 adantr ⊢ φ ∧ s ∈ 0 π → X ∈ ℝ
1341 782 adantl ⊢ φ ∧ s ∈ 0 π → s ∈ ℝ
1342 1340 1341 readdcld ⊢ φ ∧ s ∈ 0 π → X + s ∈ ℝ
1343 1339 1342 ffvelcdmd ⊢ φ ∧ s ∈ 0 π → F ⁡ X + s ∈ ℝ
1344 1343 adantlr ⊢ φ ∧ n ∈ ℕ ∧ s ∈ 0 π → F ⁡ X + s ∈ ℝ
1345 24 dirkerf ⊢ n ∈ ℕ → D ⁡ n : ℝ ⟶ ℝ
1346 1345 ad2antlr ⊢ φ ∧ n ∈ ℕ ∧ s ∈ 0 π → D ⁡ n : ℝ ⟶ ℝ
1347 782 adantl ⊢ φ ∧ n ∈ ℕ ∧ s ∈ 0 π → s ∈ ℝ
1348 1346 1347 ffvelcdmd ⊢ φ ∧ n ∈ ℕ ∧ s ∈ 0 π → D ⁡ n ⁡ s ∈ ℝ
1349 1344 1348 remulcld ⊢ φ ∧ n ∈ ℕ ∧ s ∈ 0 π → F ⁡ X + s ⁢ D ⁡ n ⁡ s ∈ ℝ
1350 1 adantr ⊢ φ ∧ s ∈ − π π → F : ℝ ⟶ ℝ
1351 2 adantr ⊢ φ ∧ s ∈ − π π → X ∈ ℝ
1352 228 sseli ⊢ s ∈ − π π → s ∈ ℝ
1353 1352 adantl ⊢ φ ∧ s ∈ − π π → s ∈ ℝ
1354 1351 1353 readdcld ⊢ φ ∧ s ∈ − π π → X + s ∈ ℝ
1355 1350 1354 ffvelcdmd ⊢ φ ∧ s ∈ − π π → F ⁡ X + s ∈ ℝ
1356 1355 adantlr ⊢ φ ∧ n ∈ ℕ ∧ s ∈ − π π → F ⁡ X + s ∈ ℝ
1357 1345 ad2antlr ⊢ φ ∧ n ∈ ℕ ∧ s ∈ − π π → D ⁡ n : ℝ ⟶ ℝ
1358 1352 adantl ⊢ φ ∧ n ∈ ℕ ∧ s ∈ − π π → s ∈ ℝ
1359 1357 1358 ffvelcdmd ⊢ φ ∧ n ∈ ℕ ∧ s ∈ − π π → D ⁡ n ⁡ s ∈ ℝ
1360 1356 1359 remulcld ⊢ φ ∧ n ∈ ℕ ∧ s ∈ − π π → F ⁡ X + s ⁢ D ⁡ n ⁡ s ∈ ℝ
1361 70 a1i ⊢ φ ∧ n ∈ ℕ → − π ∈ ℝ
1362 24 dirkercncf ⊢ n ∈ ℕ → D ⁡ n : ℝ ⟶cn ℝ
1363 1362 adantl ⊢ φ ∧ n ∈ ℕ → D ⁡ n : ℝ ⟶cn ℝ
1364 eqid ⊢ s ∈ − π π ⟼ F ⁡ X + s ⁢ D ⁡ n ⁡ s = s ∈ − π π ⟼ F ⁡ X + s ⁢ D ⁡ n ⁡ s
1365 1361 1247 830 1168 3 835 836 837 838 839 29 840 1363 1364 fourierdlem84 ⊢ φ ∧ n ∈ ℕ → s ∈ − π π ⟼ F ⁡ X + s ⁢ D ⁡ n ⁡ s ∈ 𝐿 1
1366 809 811 1360 1365 iblss ⊢ φ ∧ n ∈ ℕ → s ∈ 0 π ⟼ F ⁡ X + s ⁢ D ⁡ n ⁡ s ∈ 𝐿 1
1367 1349 1366 itgrecl ⊢ φ ∧ n ∈ ℕ → ∫ 0 π F ⁡ X + s ⁢ D ⁡ n ⁡ s ds ∈ ℝ
1368 1332 1338 1144 1367 fvmptd ⊢ φ ∧ n ∈ ℕ → Z ⁡ n = ∫ 0 π F ⁡ X + s ⁢ D ⁡ n ⁡ s ds
1369 1368 eqcomd ⊢ φ ∧ n ∈ ℕ → ∫ 0 π F ⁡ X + s ⁢ D ⁡ n ⁡ s ds = Z ⁡ n
1370 1220 1369 syldan ⊢ φ ∧ n ∈ ℤ ≥ 1 → ∫ 0 π F ⁡ X + s ⁢ D ⁡ n ⁡ s ds = Z ⁡ n
1371 1228 1331 1370 3eqtrrd ⊢ φ ∧ n ∈ ℤ ≥ 1 → Z ⁡ n = E ⁡ n + m ∈ ℕ ⟼ Y 2 ⁡ n
1372 32 33 1206 1209 1223 1226 1227 1371 climadd ⊢ φ → Z ⇝ 0 + Y 2
1373 1214 addlidd ⊢ φ → 0 + Y 2 = Y 2
1374 1372 1373 breqtrd ⊢ φ → Z ⇝ Y 2