Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fourierdlem112 Structured version   Visualization version   GIF version

Theorem fourierdlem112 46173
Description: Here abbreviations (local definitions) are introduced to prove the fourier 46180 theorem. (𝑍𝑚) is the mth partial sum of the fourier series. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem112.f (𝜑𝐹:ℝ⟶ℝ)
fourierdlem112.d 𝐷 = (𝑚 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))
fourierdlem112.p 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem112.m (𝜑𝑀 ∈ ℕ)
fourierdlem112.q (𝜑𝑄 ∈ (𝑃𝑀))
fourierdlem112.n 𝑁 = ((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)
fourierdlem112.v 𝑉 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))
fourierdlem112.x (𝜑𝑋 ∈ ℝ)
fourierdlem112.xran (𝜑𝑋 ∈ ran 𝑉)
fourierdlem112.t 𝑇 = (2 · π)
fourierdlem112.fper ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
fourierdlem112.fcn ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem112.c ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
fourierdlem112.u ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
fourierdlem112.fdvcn ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem112.e (𝜑𝐸 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim 𝑋))
fourierdlem112.i (𝜑𝐼 ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim 𝑋))
fourierdlem112.l (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋))
fourierdlem112.r (𝜑𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))
fourierdlem112.a 𝐴 = (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
fourierdlem112.b 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
fourierdlem112.z 𝑍 = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
fourierdlem112.23 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
fourierdlem112.fbd (𝜑 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤)
fourierdlem112.fdvbd (𝜑 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
fourierdlem112.25 (𝜑𝑋 ∈ ℝ)
Assertion
Ref Expression
fourierdlem112 (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))
Distinct variable groups:   𝐴,𝑘,𝑚,𝑛   𝐵,𝑘,𝑚,𝑛   𝑡,𝐶,𝑚   𝑥,𝐶,𝑚   𝐷,𝑖,𝑘,𝑚,𝑛,𝑥,𝑦   𝑖,𝐹,𝑡,𝑧   𝑦,𝐹,𝑡,𝑘,𝑚   𝑧,𝑘,𝑚   𝑛,𝐹   𝑤,𝐹,𝑖,𝑡,𝑧   𝑥,𝐹   𝑖,𝐿,𝑡,𝑧,𝑘,𝑚   𝑛,𝐿   𝑤,𝐿   𝑓,𝑀,𝑖,𝑡,𝑦,𝑚   𝑛,𝑀,𝑥   𝑀,𝑝,𝑖,𝑛,𝑦   𝑖,𝑁,𝑡,𝑤,𝑧   𝑓,𝑁,𝑦,𝑚   𝑛,𝑁,𝑝   𝑥,𝑁,𝑓   𝑄,𝑓,𝑖,𝑡,𝑦,𝑘,𝑚   𝑄,𝑛,𝑥   𝑄,𝑝,𝑘   𝑅,𝑖,𝑡,𝑧,𝑘,𝑚   𝑅,𝑛   𝑤,𝑅   𝑇,𝑓,𝑡,𝑦,𝑖,𝑘,𝑚   𝑇,𝑛,𝑥   𝑇,𝑝   𝑡,𝑈,𝑚   𝑥,𝑈   𝑖,𝑉,𝑡,𝑤,𝑧   𝑓,𝑉,𝑘,𝑚   𝑛,𝑉,𝑝   𝑥,𝑉   𝑖,𝑋,𝑡,𝑧   𝑓,𝑋,𝑦,𝑘,𝑚   𝑛,𝑋,𝑝   𝑤,𝑋   𝑥,𝑋   𝑚,𝑍   𝜑,𝑖,𝑡,𝑤,𝑧   𝜑,𝑓,𝑘,𝑚,𝑦   𝜑,𝑛   𝑤,𝑚   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑝)   𝐴(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑝)   𝐵(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑝)   𝐶(𝑦,𝑧,𝑤,𝑓,𝑖,𝑘,𝑛,𝑝)   𝐷(𝑧,𝑤,𝑡,𝑓,𝑝)   𝑃(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑘,𝑚,𝑛,𝑝)   𝑄(𝑧,𝑤)   𝑅(𝑥,𝑦,𝑓,𝑝)   𝑆(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑘,𝑚,𝑛,𝑝)   𝑇(𝑧,𝑤)   𝑈(𝑦,𝑧,𝑤,𝑓,𝑖,𝑘,𝑛,𝑝)   𝐸(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑘,𝑚,𝑛,𝑝)   𝐹(𝑓,𝑝)   𝐼(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑘,𝑚,𝑛,𝑝)   𝐿(𝑥,𝑦,𝑓,𝑝)   𝑀(𝑧,𝑤,𝑘)   𝑁(𝑘)   𝑉(𝑦)   𝑍(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑘,𝑛,𝑝)

Proof of Theorem fourierdlem112
Dummy variables 𝑗 𝑙 𝑎 𝑠 𝑏 𝑒 𝑔 𝑐 𝑢 𝑞 𝑟 𝑣 𝑑 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem112.23 . . . . 5 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
2 fveq2 6906 . . . . . . . 8 (𝑛 = 𝑗 → (𝐴𝑛) = (𝐴𝑗))
3 oveq1 7437 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
43fveq2d 6910 . . . . . . . 8 (𝑛 = 𝑗 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑗 · 𝑋)))
52, 4oveq12d 7448 . . . . . . 7 (𝑛 = 𝑗 → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))))
6 fveq2 6906 . . . . . . . 8 (𝑛 = 𝑗 → (𝐵𝑛) = (𝐵𝑗))
73fveq2d 6910 . . . . . . . 8 (𝑛 = 𝑗 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑗 · 𝑋)))
86, 7oveq12d 7448 . . . . . . 7 (𝑛 = 𝑗 → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))
95, 8oveq12d 7448 . . . . . 6 (𝑛 = 𝑗 → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
109cbvmptv 5260 . . . . 5 (𝑛 ∈ ℕ ↦ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
111, 10eqtri 2762 . . . 4 𝑆 = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
12 seqeq3 14043 . . . 4 (𝑆 = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))) → seq1( + , 𝑆) = seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))))
1311, 12mp1i 13 . . 3 (𝜑 → seq1( + , 𝑆) = seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))))
14 nnuz 12918 . . . . 5 ℕ = (ℤ‘1)
15 1zzd 12645 . . . . 5 (𝜑 → 1 ∈ ℤ)
16 nfv 1911 . . . . . . 7 𝑛𝜑
17 nfcv 2902 . . . . . . . 8 𝑛
18 nfcv 2902 . . . . . . . . 9 𝑛(-π(,)0)
19 nfcv 2902 . . . . . . . . . 10 𝑛(𝐹‘(𝑋 + 𝑠))
20 nfcv 2902 . . . . . . . . . 10 𝑛 ·
21 nfcv 2902 . . . . . . . . . 10 𝑛((𝐷𝑚)‘𝑠)
2219, 20, 21nfov 7460 . . . . . . . . 9 𝑛((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠))
2318, 22nfitg 25824 . . . . . . . 8 𝑛∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠
2417, 23nfmpt 5254 . . . . . . 7 𝑛(𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)
25 nfcv 2902 . . . . . . . . 9 𝑛(0(,)π)
2625, 22nfitg 25824 . . . . . . . 8 𝑛∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠
2717, 26nfmpt 5254 . . . . . . 7 𝑛(𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)
28 fourierdlem112.z . . . . . . . 8 𝑍 = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
29 fourierdlem112.a . . . . . . . . . . . . 13 𝐴 = (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
30 nfmpt1 5255 . . . . . . . . . . . . 13 𝑛(𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
3129, 30nfcxfr 2900 . . . . . . . . . . . 12 𝑛𝐴
32 nfcv 2902 . . . . . . . . . . . 12 𝑛0
3331, 32nffv 6916 . . . . . . . . . . 11 𝑛(𝐴‘0)
34 nfcv 2902 . . . . . . . . . . 11 𝑛 /
35 nfcv 2902 . . . . . . . . . . 11 𝑛2
3633, 34, 35nfov 7460 . . . . . . . . . 10 𝑛((𝐴‘0) / 2)
37 nfcv 2902 . . . . . . . . . 10 𝑛 +
38 nfcv 2902 . . . . . . . . . . 11 𝑛(1...𝑚)
3938nfsum1 15722 . . . . . . . . . 10 𝑛Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))
4036, 37, 39nfov 7460 . . . . . . . . 9 𝑛(((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
4117, 40nfmpt 5254 . . . . . . . 8 𝑛(𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
4228, 41nfcxfr 2900 . . . . . . 7 𝑛𝑍
43 fourierdlem112.f . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
44 fourierdlem112.25 . . . . . . . 8 (𝜑𝑋 ∈ ℝ)
45 eqid 2734 . . . . . . . 8 (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
46 picn 26515 . . . . . . . . . . . . 13 π ∈ ℂ
47462timesi 12401 . . . . . . . . . . . 12 (2 · π) = (π + π)
48 fourierdlem112.t . . . . . . . . . . . 12 𝑇 = (2 · π)
4946, 46subnegi 11585 . . . . . . . . . . . 12 (π − -π) = (π + π)
5047, 48, 493eqtr4i 2772 . . . . . . . . . . 11 𝑇 = (π − -π)
51 fourierdlem112.p . . . . . . . . . . 11 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
52 fourierdlem112.m . . . . . . . . . . 11 (𝜑𝑀 ∈ ℕ)
53 fourierdlem112.q . . . . . . . . . . 11 (𝜑𝑄 ∈ (𝑃𝑀))
54 pire 26514 . . . . . . . . . . . . . 14 π ∈ ℝ
5554a1i 11 . . . . . . . . . . . . 13 (𝜑 → π ∈ ℝ)
5655renegcld 11687 . . . . . . . . . . . 12 (𝜑 → -π ∈ ℝ)
5756, 44readdcld 11287 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) ∈ ℝ)
5855, 44readdcld 11287 . . . . . . . . . . 11 (𝜑 → (π + 𝑋) ∈ ℝ)
59 negpilt0 45230 . . . . . . . . . . . . . 14 -π < 0
60 pipos 26516 . . . . . . . . . . . . . 14 0 < π
6154renegcli 11567 . . . . . . . . . . . . . . 15 -π ∈ ℝ
62 0re 11260 . . . . . . . . . . . . . . 15 0 ∈ ℝ
6361, 62, 54lttri 11384 . . . . . . . . . . . . . 14 ((-π < 0 ∧ 0 < π) → -π < π)
6459, 60, 63mp2an 692 . . . . . . . . . . . . 13 -π < π
6564a1i 11 . . . . . . . . . . . 12 (𝜑 → -π < π)
6656, 55, 44, 65ltadd1dd 11871 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) < (π + 𝑋))
67 oveq1 7437 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑦 + (𝑘 · 𝑇)) = (𝑥 + (𝑘 · 𝑇)))
6867eleq1d 2823 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
6968rexbidv 3176 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
7069cbvrabv 3443 . . . . . . . . . . . 12 {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑥 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}
7170uneq2i 4174 . . . . . . . . . . 11 ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑥 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
72 fourierdlem112.n . . . . . . . . . . 11 𝑁 = ((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)
73 fourierdlem112.v . . . . . . . . . . 11 𝑉 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))
7450, 51, 52, 53, 57, 58, 66, 45, 71, 72, 73fourierdlem54 46115 . . . . . . . . . 10 (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑉 ∈ ((𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑁)) ∧ 𝑉 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))))
7574simpld 494 . . . . . . . . 9 (𝜑 → (𝑁 ∈ ℕ ∧ 𝑉 ∈ ((𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑁)))
7675simpld 494 . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
7775simprd 495 . . . . . . . 8 (𝜑𝑉 ∈ ((𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑁))
78 fourierdlem112.xran . . . . . . . 8 (𝜑𝑋 ∈ ran 𝑉)
7943adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℝ)
80 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑝𝑖) = (𝑝𝑗))
81 oveq1 7437 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
8281fveq2d 6910 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1)))
8380, 82breq12d 5160 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝𝑗) < (𝑝‘(𝑗 + 1))))
8483cbvralvw 3234 . . . . . . . . . . . . . 14 (∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))
8584anbi2i 623 . . . . . . . . . . . . 13 ((((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1))))
8685a1i 11 . . . . . . . . . . . 12 (𝑝 ∈ (ℝ ↑m (0...𝑛)) → ((((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))))
8786rabbiia 3436 . . . . . . . . . . 11 {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))} = {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))}
8887mpteq2i 5252 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))})
8951, 88eqtri 2762 . . . . . . . . 9 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))})
9052adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ)
9153adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑄 ∈ (𝑃𝑀))
92 fourierdlem112.fper . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
9392adantlr 715 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
94 eleq1w 2821 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
9594anbi2d 630 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑗 ∈ (0..^𝑀))))
96 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
9781fveq2d 6910 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
9896, 97oveq12d 7448 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1))))
9998reseq2d 5999 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))))
10098oveq1d 7445 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) = (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
10199, 100eleq12d 2832 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) ↔ (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ)))
10295, 101imbi12d 344 . . . . . . . . . . 11 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))))
103 fourierdlem112.fcn . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
104102, 103chvarvv 1995 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
105104adantlr 715 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
10657adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (-π + 𝑋) ∈ ℝ)
10757rexrd 11308 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) ∈ ℝ*)
108 pnfxr 11312 . . . . . . . . . . . 12 +∞ ∈ ℝ*
109108a1i 11 . . . . . . . . . . 11 (𝜑 → +∞ ∈ ℝ*)
11058ltpnfd 13160 . . . . . . . . . . 11 (𝜑 → (π + 𝑋) < +∞)
111107, 109, 58, 66, 110eliood 45450 . . . . . . . . . 10 (𝜑 → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
112111adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
113 id 22 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^𝑁))
11472oveq2i 7441 . . . . . . . . . . 11 (0..^𝑁) = (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))
115113, 114eleqtrdi 2848 . . . . . . . . . 10 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))
116115adantl 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))
11772oveq2i 7441 . . . . . . . . . . . 12 (0...𝑁) = (0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))
118 isoeq4 7339 . . . . . . . . . . . 12 ((0...𝑁) = (0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)) → (𝑓 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))))
119117, 118ax-mp 5 . . . . . . . . . . 11 (𝑓 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))
120119iotabii 6547 . . . . . . . . . 10 (℩𝑓𝑓 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) = (℩𝑓𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))
12173, 120eqtri 2762 . . . . . . . . 9 𝑉 = (℩𝑓𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))
12279, 89, 50, 90, 91, 93, 105, 106, 112, 116, 121fourierdlem98 46159 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
123 fourierdlem112.fbd . . . . . . . . . 10 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤)
124123adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤)
125 nfra1 3281 . . . . . . . . . . 11 𝑡𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤
126 elioore 13413 . . . . . . . . . . . . 13 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
127 rspa 3245 . . . . . . . . . . . . 13 ((∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤𝑡 ∈ ℝ) → (abs‘(𝐹𝑡)) ≤ 𝑤)
128126, 127sylan2 593 . . . . . . . . . . . 12 ((∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(𝐹𝑡)) ≤ 𝑤)
129128ex 412 . . . . . . . . . . 11 (∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤 → (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘(𝐹𝑡)) ≤ 𝑤))
130125, 129ralrimi 3254 . . . . . . . . . 10 (∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤 → ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹𝑡)) ≤ 𝑤)
131130reximi 3081 . . . . . . . . 9 (∃𝑤 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹𝑡)) ≤ 𝑤)
132124, 131syl 17 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹𝑡)) ≤ 𝑤)
133 ssid 4017 . . . . . . . . . . . 12 ℝ ⊆ ℝ
134 dvfre 26003 . . . . . . . . . . . 12 ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
13543, 133, 134sylancl 586 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
136135adantr 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
137 eqid 2734 . . . . . . . . . . . . 13 (ℝ D 𝐹) = (ℝ D 𝐹)
13854a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → π ∈ ℝ)
13961a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → -π ∈ ℝ)
14098reseq2d 5999 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))))
141140, 100eleq12d 2832 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) ↔ ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ)))
14295, 141imbi12d 344 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))))
143 fourierdlem112.fdvcn . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
144142, 143chvarvv 1995 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
145144adantlr 715 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
146 fourierdlem112.x . . . . . . . . . . . . . . 15 (𝜑𝑋 ∈ ℝ)
14756, 146readdcld 11287 . . . . . . . . . . . . . 14 (𝜑 → (-π + 𝑋) ∈ ℝ)
148147adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → (-π + 𝑋) ∈ ℝ)
149147rexrd 11308 . . . . . . . . . . . . . . 15 (𝜑 → (-π + 𝑋) ∈ ℝ*)
15055, 146readdcld 11287 . . . . . . . . . . . . . . 15 (𝜑 → (π + 𝑋) ∈ ℝ)
15156, 55, 146, 65ltadd1dd 11871 . . . . . . . . . . . . . . 15 (𝜑 → (-π + 𝑋) < (π + 𝑋))
152150ltpnfd 13160 . . . . . . . . . . . . . . 15 (𝜑 → (π + 𝑋) < +∞)
153149, 109, 150, 151, 152eliood 45450 . . . . . . . . . . . . . 14 (𝜑 → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
154153adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
155 oveq1 7437 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = → (𝑘 · 𝑇) = ( · 𝑇))
156155oveq2d 7446 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝑦 + (𝑘 · 𝑇)) = (𝑦 + ( · 𝑇)))
157156eleq1d 2823 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + ( · 𝑇)) ∈ ran 𝑄))
158157cbvrexvw 3235 . . . . . . . . . . . . . . . . . . 19 (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄)
159158rgenw 3062 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋))(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄)
160 rabbi 3464 . . . . . . . . . . . . . . . . . 18 (∀𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋))(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄) ↔ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄})
161159, 160mpbi 230 . . . . . . . . . . . . . . . . 17 {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄}
162161uneq2i 4174 . . . . . . . . . . . . . . . 16 ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄})
163 isoeq5 7340 . . . . . . . . . . . . . . . 16 (({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄}) → (𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄}))))
164162, 163ax-mp 5 . . . . . . . . . . . . . . 15 (𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄})))
165164iotabii 6547 . . . . . . . . . . . . . 14 (℩𝑓𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) = (℩𝑓𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄})))
166121, 165eqtri 2762 . . . . . . . . . . . . 13 𝑉 = (℩𝑓𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄})))
167 eleq1w 2821 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → (𝑣 ∈ dom (ℝ D 𝐹) ↔ 𝑢 ∈ dom (ℝ D 𝐹)))
168 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → ((ℝ D 𝐹)‘𝑣) = ((ℝ D 𝐹)‘𝑢))
169167, 168ifbieq1d 4554 . . . . . . . . . . . . . 14 (𝑣 = 𝑢 → if(𝑣 ∈ dom (ℝ D 𝐹), ((ℝ D 𝐹)‘𝑣), 0) = if(𝑢 ∈ dom (ℝ D 𝐹), ((ℝ D 𝐹)‘𝑢), 0))
170169cbvmptv 5260 . . . . . . . . . . . . 13 (𝑣 ∈ ℝ ↦ if(𝑣 ∈ dom (ℝ D 𝐹), ((ℝ D 𝐹)‘𝑣), 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ dom (ℝ D 𝐹), ((ℝ D 𝐹)‘𝑢), 0))
17179, 137, 89, 138, 139, 50, 90, 91, 93, 145, 148, 154, 116, 166, 170fourierdlem97 46158 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
172 cncff 24932 . . . . . . . . . . . 12 (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
173 fdm 6745 . . . . . . . . . . . 12 (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ → dom ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
174171, 172, 1733syl 18 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → dom ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
175 ssdmres 6032 . . . . . . . . . . 11 (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
176174, 175sylibr 234 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹))
177136, 176fssresd 6775 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
178 ax-resscn 11209 . . . . . . . . . . 11 ℝ ⊆ ℂ
179178a1i 11 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ℝ ⊆ ℂ)
180 cncfcdm 24937 . . . . . . . . . 10 ((ℝ ⊆ ℂ ∧ ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) → (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ) ↔ ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ))
181179, 171, 180syl2anc 584 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ) ↔ ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ))
182177, 181mpbird 257 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ))
183 fourierdlem112.fdvbd . . . . . . . . . . 11 (𝜑 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
184183adantr 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
185 nfv 1911 . . . . . . . . . . . . . 14 𝑡(𝜑𝑖 ∈ (0..^𝑁))
186 nfra1 3281 . . . . . . . . . . . . . 14 𝑡𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧
187185, 186nfan 1896 . . . . . . . . . . . . 13 𝑡((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
188 fvres 6925 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
189188adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
190189fveq2d 6910 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡)))
191190adantlr 715 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡)))
192 simplr 769 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
193176sselda 3994 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → 𝑡 ∈ dom (ℝ D 𝐹))
194193adantlr 715 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → 𝑡 ∈ dom (ℝ D 𝐹))
195 rspa 3245 . . . . . . . . . . . . . . . 16 ((∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧𝑡 ∈ dom (ℝ D 𝐹)) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
196192, 194, 195syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
197191, 196eqbrtrd 5169 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧)
198197ex 412 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) → (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧))
199187, 198ralrimi 3254 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) → ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧)
200199ex 412 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → (∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 → ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧))
201200reximdv 3167 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧))
202184, 201mpd 15 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧)
203 nfra1 3281 . . . . . . . . . . . 12 𝑡𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧
204188eqcomd 2740 . . . . . . . . . . . . . . . 16 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → ((ℝ D 𝐹)‘𝑡) = (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡))
205204fveq2d 6910 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘((ℝ D 𝐹)‘𝑡)) = (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)))
206205adantl 481 . . . . . . . . . . . . . 14 ((∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘((ℝ D 𝐹)‘𝑡)) = (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)))
207 rspa 3245 . . . . . . . . . . . . . 14 ((∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧)
208206, 207eqbrtrd 5169 . . . . . . . . . . . . 13 ((∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
209208ex 412 . . . . . . . . . . . 12 (∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 → (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧))
210203, 209ralrimi 3254 . . . . . . . . . . 11 (∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 → ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
211210a1i 11 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 → ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧))
212211reximdv 3167 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧))
213202, 212mpd 15 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
214 nfv 1911 . . . . . . . . . . . 12 𝑖(𝜑𝑗 ∈ (0..^𝑀))
215 nfcsb1v 3932 . . . . . . . . . . . . 13 𝑖𝑗 / 𝑖𝐶
216215nfel1 2919 . . . . . . . . . . . 12 𝑖𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗))
217214, 216nfim 1893 . . . . . . . . . . 11 𝑖((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
218 csbeq1a 3921 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝐶 = 𝑗 / 𝑖𝐶)
21999, 96oveq12d 7448 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
220218, 219eleq12d 2832 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) ↔ 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗))))
22195, 220imbi12d 344 . . . . . . . . . . 11 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖))) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))))
222 fourierdlem112.c . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
223217, 221, 222chvarfv 2237 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
224223adantlr 715 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
22579, 89, 50, 90, 91, 93, 105, 224, 106, 112, 116, 121fourierdlem96 46157 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → if(((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘(𝑉𝑖))) = (𝑄‘((𝑦 ∈ ℝ ↦ sup({𝑓 ∈ (0..^𝑀) ∣ (𝑄𝑓) ≤ ((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉𝑖))), ((𝑗 ∈ (0..^𝑀) ↦ 𝑗 / 𝑖𝐶)‘((𝑦 ∈ ℝ ↦ sup({𝑓 ∈ (0..^𝑀) ∣ (𝑄𝑓) ≤ ((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉𝑖))), (𝐹‘((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘(𝑉𝑖))))) ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))
226 nfcsb1v 3932 . . . . . . . . . . . . 13 𝑖𝑗 / 𝑖𝑈
227226nfel1 2919 . . . . . . . . . . . 12 𝑖𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1)))
228214, 227nfim 1893 . . . . . . . . . . 11 𝑖((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
229 csbeq1a 3921 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝑈 = 𝑗 / 𝑖𝑈)
23099, 97oveq12d 7448 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
231229, 230eleq12d 2832 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑈 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) ↔ 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1)))))
23295, 231imbi12d 344 . . . . . . . . . . 11 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1)))) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))))
233 fourierdlem112.u . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
234228, 232, 233chvarfv 2237 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
235234adantlr 715 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
23679, 89, 50, 90, 91, 93, 105, 235, 148, 154, 116, 121fourierdlem99 46160 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → if(((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘(𝑉‘(𝑖 + 1))) = (𝑄‘(((𝑦 ∈ ℝ ↦ sup({ ∈ (0..^𝑀) ∣ (𝑄) ≤ ((𝑔 ∈ (-π(,]π) ↦ if(𝑔 = π, -π, 𝑔))‘((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉𝑖)) + 1)), ((𝑗 ∈ (0..^𝑀) ↦ 𝑗 / 𝑖𝑈)‘((𝑦 ∈ ℝ ↦ sup({ ∈ (0..^𝑀) ∣ (𝑄) ≤ ((𝑔 ∈ (-π(,]π) ↦ if(𝑔 = π, -π, 𝑔))‘((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉𝑖))), (𝐹‘((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘(𝑉‘(𝑖 + 1))))) ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))
237 eqeq1 2738 . . . . . . . . . 10 (𝑔 = 𝑠 → (𝑔 = 0 ↔ 𝑠 = 0))
238 oveq2 7438 . . . . . . . . . . . . 13 (𝑔 = 𝑠 → (𝑋 + 𝑔) = (𝑋 + 𝑠))
239238fveq2d 6910 . . . . . . . . . . . 12 (𝑔 = 𝑠 → (𝐹‘(𝑋 + 𝑔)) = (𝐹‘(𝑋 + 𝑠)))
240 breq2 5151 . . . . . . . . . . . . 13 (𝑔 = 𝑠 → (0 < 𝑔 ↔ 0 < 𝑠))
241240ifbid 4553 . . . . . . . . . . . 12 (𝑔 = 𝑠 → if(0 < 𝑔, 𝑅, 𝐿) = if(0 < 𝑠, 𝑅, 𝐿))
242239, 241oveq12d 7448 . . . . . . . . . . 11 (𝑔 = 𝑠 → ((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) = ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑅, 𝐿)))
243 id 22 . . . . . . . . . . 11 (𝑔 = 𝑠𝑔 = 𝑠)
244242, 243oveq12d 7448 . . . . . . . . . 10 (𝑔 = 𝑠 → (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑅, 𝐿)) / 𝑠))
245237, 244ifbieq2d 4556 . . . . . . . . 9 (𝑔 = 𝑠 → if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑅, 𝐿)) / 𝑠)))
246245cbvmptv 5260 . . . . . . . 8 (𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔))) = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑅, 𝐿)) / 𝑠)))
247 eqeq1 2738 . . . . . . . . . 10 (𝑜 = 𝑠 → (𝑜 = 0 ↔ 𝑠 = 0))
248 id 22 . . . . . . . . . . 11 (𝑜 = 𝑠𝑜 = 𝑠)
249 oveq1 7437 . . . . . . . . . . . . 13 (𝑜 = 𝑠 → (𝑜 / 2) = (𝑠 / 2))
250249fveq2d 6910 . . . . . . . . . . . 12 (𝑜 = 𝑠 → (sin‘(𝑜 / 2)) = (sin‘(𝑠 / 2)))
251250oveq2d 7446 . . . . . . . . . . 11 (𝑜 = 𝑠 → (2 · (sin‘(𝑜 / 2))) = (2 · (sin‘(𝑠 / 2))))
252248, 251oveq12d 7448 . . . . . . . . . 10 (𝑜 = 𝑠 → (𝑜 / (2 · (sin‘(𝑜 / 2)))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
253247, 252ifbieq2d 4556 . . . . . . . . 9 (𝑜 = 𝑠 → if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
254253cbvmptv 5260 . . . . . . . 8 (𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2)))))) = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
255 fveq2 6906 . . . . . . . . . 10 (𝑟 = 𝑠 → ((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) = ((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑠))
256 fveq2 6906 . . . . . . . . . 10 (𝑟 = 𝑠 → ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟) = ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑠))
257255, 256oveq12d 7448 . . . . . . . . 9 (𝑟 = 𝑠 → (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)) = (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑠) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑠)))
258257cbvmptv 5260 . . . . . . . 8 (𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟))) = (𝑠 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑠) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑠)))
259 oveq2 7438 . . . . . . . . . 10 (𝑑 = 𝑠 → ((𝑘 + (1 / 2)) · 𝑑) = ((𝑘 + (1 / 2)) · 𝑠))
260259fveq2d 6910 . . . . . . . . 9 (𝑑 = 𝑠 → (sin‘((𝑘 + (1 / 2)) · 𝑑)) = (sin‘((𝑘 + (1 / 2)) · 𝑠)))
261260cbvmptv 5260 . . . . . . . 8 (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑))) = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑠)))
262 fveq2 6906 . . . . . . . . . 10 (𝑧 = 𝑠 → ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) = ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠))
263 fveq2 6906 . . . . . . . . . 10 (𝑧 = 𝑠 → ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧) = ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑠))
264262, 263oveq12d 7448 . . . . . . . . 9 (𝑧 = 𝑠 → (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)) = (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑠)))
265264cbvmptv 5260 . . . . . . . 8 (𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧))) = (𝑠 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑠)))
266 fveq2 6906 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝐷𝑚) = (𝐷𝑛))
267266fveq1d 6908 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((𝐷𝑚)‘𝑠) = ((𝐷𝑛)‘𝑠))
268267oveq2d 7446 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
269268adantr 480 . . . . . . . . . 10 ((𝑚 = 𝑛𝑠 ∈ (-π(,)0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
270269itgeq2dv 25831 . . . . . . . . 9 (𝑚 = 𝑛 → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
271270cbvmptv 5260 . . . . . . . 8 (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑛 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
272 oveq1 7437 . . . . . . . . . . . . . . . . . . 19 (𝑐 = 𝑘 → (𝑐 + (1 / 2)) = (𝑘 + (1 / 2)))
273272oveq1d 7445 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝑘 → ((𝑐 + (1 / 2)) · 𝑑) = ((𝑘 + (1 / 2)) · 𝑑))
274273fveq2d 6910 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑘 → (sin‘((𝑐 + (1 / 2)) · 𝑑)) = (sin‘((𝑘 + (1 / 2)) · 𝑑)))
275274mpteq2dv 5249 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑘 → (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑))) = (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑))))
276275fveq1d 6908 . . . . . . . . . . . . . . 15 (𝑐 = 𝑘 → ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧) = ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧))
277276oveq2d 7446 . . . . . . . . . . . . . 14 (𝑐 = 𝑘 → (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)) = (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))
278277mpteq2dv 5249 . . . . . . . . . . . . 13 (𝑐 = 𝑘 → (𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧))) = (𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧))))
279278fveq1d 6908 . . . . . . . . . . . 12 (𝑐 = 𝑘 → ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) = ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠))
280279adantr 480 . . . . . . . . . . 11 ((𝑐 = 𝑘𝑠 ∈ (-π(,)0)) → ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) = ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠))
281280itgeq2dv 25831 . . . . . . . . . 10 (𝑐 = 𝑘 → ∫(-π(,)0)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) d𝑠 = ∫(-π(,)0)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) d𝑠)
282281oveq1d 7445 . . . . . . . . 9 (𝑐 = 𝑘 → (∫(-π(,)0)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π) = (∫(-π(,)0)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π))
283282cbvmptv 5260 . . . . . . . 8 (𝑐 ∈ ℕ ↦ (∫(-π(,)0)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π)) = (𝑘 ∈ ℕ ↦ (∫(-π(,)0)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π))
284 fourierdlem112.r . . . . . . . 8 (𝜑𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))
285 fourierdlem112.l . . . . . . . 8 (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋))
286 fourierdlem112.e . . . . . . . 8 (𝜑𝐸 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim 𝑋))
287 fourierdlem112.i . . . . . . . 8 (𝜑𝐼 ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim 𝑋))
288 fourierdlem112.d . . . . . . . . 9 𝐷 = (𝑚 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))
289 oveq1 7437 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → (𝑦 mod (2 · π)) = (𝑠 mod (2 · π)))
290289eqeq1d 2736 . . . . . . . . . . . . 13 (𝑦 = 𝑠 → ((𝑦 mod (2 · π)) = 0 ↔ (𝑠 mod (2 · π)) = 0))
291 oveq2 7438 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → ((𝑚 + (1 / 2)) · 𝑦) = ((𝑚 + (1 / 2)) · 𝑠))
292291fveq2d 6910 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → (sin‘((𝑚 + (1 / 2)) · 𝑦)) = (sin‘((𝑚 + (1 / 2)) · 𝑠)))
293 oveq1 7437 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑠 → (𝑦 / 2) = (𝑠 / 2))
294293fveq2d 6910 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → (sin‘(𝑦 / 2)) = (sin‘(𝑠 / 2)))
295294oveq2d 7446 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → ((2 · π) · (sin‘(𝑦 / 2))) = ((2 · π) · (sin‘(𝑠 / 2))))
296292, 295oveq12d 7448 . . . . . . . . . . . . 13 (𝑦 = 𝑠 → ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))) = ((sin‘((𝑚 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))
297290, 296ifbieq2d 4556 . . . . . . . . . . . 12 (𝑦 = 𝑠 → if((𝑦 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2))))) = if((𝑠 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))
298297cbvmptv 5260 . . . . . . . . . . 11 (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))
299 simpl 482 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑘𝑠 ∈ ℝ) → 𝑚 = 𝑘)
300299oveq2d 7446 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (2 · 𝑚) = (2 · 𝑘))
301300oveq1d 7445 . . . . . . . . . . . . . 14 ((𝑚 = 𝑘𝑠 ∈ ℝ) → ((2 · 𝑚) + 1) = ((2 · 𝑘) + 1))
302301oveq1d 7445 . . . . . . . . . . . . 13 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (((2 · 𝑚) + 1) / (2 · π)) = (((2 · 𝑘) + 1) / (2 · π)))
303299oveq1d 7445 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (𝑚 + (1 / 2)) = (𝑘 + (1 / 2)))
304303oveq1d 7445 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑘𝑠 ∈ ℝ) → ((𝑚 + (1 / 2)) · 𝑠) = ((𝑘 + (1 / 2)) · 𝑠))
305304fveq2d 6910 . . . . . . . . . . . . . 14 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (sin‘((𝑚 + (1 / 2)) · 𝑠)) = (sin‘((𝑘 + (1 / 2)) · 𝑠)))
306305oveq1d 7445 . . . . . . . . . . . . 13 ((𝑚 = 𝑘𝑠 ∈ ℝ) → ((sin‘((𝑚 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))) = ((sin‘((𝑘 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))
307302, 306ifeq12d 4551 . . . . . . . . . . . 12 ((𝑚 = 𝑘𝑠 ∈ ℝ) → if((𝑠 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))) = if((𝑠 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))
308307mpteq2dva 5247 . . . . . . . . . . 11 (𝑚 = 𝑘 → (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))
309298, 308eqtrid 2786 . . . . . . . . . 10 (𝑚 = 𝑘 → (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))
310309cbvmptv 5260 . . . . . . . . 9 (𝑚 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2))))))) = (𝑘 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))
311288, 310eqtri 2762 . . . . . . . 8 𝐷 = (𝑘 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))
312 eqid 2734 . . . . . . . 8 ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟))) ↾ (-π[,]𝑙)) = ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟))) ↾ (-π[,]𝑙))
313 eqid 2734 . . . . . . . 8 ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙))) = ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))
314 eqid 2734 . . . . . . . 8 ((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1) = ((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)
315 isoeq1 7336 . . . . . . . . 9 (𝑢 = 𝑤 → (𝑢 Isom < , < ((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) ↔ 𝑤 Isom < , < ((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙))))))
316315cbviotavw 6523 . . . . . . . 8 (℩𝑢𝑢 Isom < , < ((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙))))) = (℩𝑤𝑤 Isom < , < ((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))))
317 fveq2 6906 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝑉𝑗) = (𝑉𝑖))
318317oveq1d 7445 . . . . . . . . 9 (𝑗 = 𝑖 → ((𝑉𝑗) − 𝑋) = ((𝑉𝑖) − 𝑋))
319318cbvmptv 5260 . . . . . . . 8 (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) = (𝑖 ∈ (0...𝑁) ↦ ((𝑉𝑖) − 𝑋))
320 eqid 2734 . . . . . . . 8 (𝑚 ∈ (0..^𝑁)(((℩𝑢𝑢 Isom < , < ((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))))‘𝑏)(,)((℩𝑢𝑢 Isom < , < ((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))))‘(𝑏 + 1))) ⊆ (((𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋))‘𝑚)(,)((𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋))‘(𝑚 + 1)))) = (𝑚 ∈ (0..^𝑁)(((℩𝑢𝑢 Isom < , < ((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))))‘𝑏)(,)((℩𝑢𝑢 Isom < , < ((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))))‘(𝑏 + 1))) ⊆ (((𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋))‘𝑚)(,)((𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋))‘(𝑚 + 1))))
321 fveq2 6906 . . . . . . . . . . . . . 14 (𝑎 = 𝑠 → ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) = ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠))
322 oveq2 7438 . . . . . . . . . . . . . . 15 (𝑎 = 𝑠 → ((𝑏 + (1 / 2)) · 𝑎) = ((𝑏 + (1 / 2)) · 𝑠))
323322fveq2d 6910 . . . . . . . . . . . . . 14 (𝑎 = 𝑠 → (sin‘((𝑏 + (1 / 2)) · 𝑎)) = (sin‘((𝑏 + (1 / 2)) · 𝑠)))
324321, 323oveq12d 7448 . . . . . . . . . . . . 13 (𝑎 = 𝑠 → (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) = (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))))
325324cbvitgv 25826 . . . . . . . . . . . 12 ∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎 = ∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠
326325fveq2i 6909 . . . . . . . . . . 11 (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) = (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠)
327326breq1i 5154 . . . . . . . . . 10 ((abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑖 / 2) ↔ (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑖 / 2))
328327anbi2i 623 . . . . . . . . 9 (((((𝜑𝑖 ∈ ℝ+) ∧ 𝑙 ∈ (-π(,)0)) ∧ 𝑏 ∈ ℕ) ∧ (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑖 / 2)) ↔ ((((𝜑𝑖 ∈ ℝ+) ∧ 𝑙 ∈ (-π(,)0)) ∧ 𝑏 ∈ ℕ) ∧ (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑖 / 2)))
329324cbvitgv 25826 . . . . . . . . . . 11 ∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎 = ∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠
330329fveq2i 6909 . . . . . . . . . 10 (abs‘∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) = (abs‘∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠)
331330breq1i 5154 . . . . . . . . 9 ((abs‘∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑖 / 2) ↔ (abs‘∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑖 / 2))
332328, 331anbi12i 628 . . . . . . . 8 ((((((𝜑𝑖 ∈ ℝ+) ∧ 𝑙 ∈ (-π(,)0)) ∧ 𝑏 ∈ ℕ) ∧ (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑖 / 2)) ∧ (abs‘∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑖 / 2)) ↔ (((((𝜑𝑖 ∈ ℝ+) ∧ 𝑙 ∈ (-π(,)0)) ∧ 𝑏 ∈ ℕ) ∧ (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑖 / 2)) ∧ (abs‘∫(-π(,)𝑙)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑖 / 2)))
33343, 44, 45, 76, 77, 78, 122, 132, 182, 213, 225, 236, 246, 254, 258, 261, 265, 271, 283, 284, 285, 286, 287, 311, 312, 313, 314, 316, 319, 320, 332fourierdlem103 46164 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) ⇝ (𝐿 / 2))
334 nnex 12269 . . . . . . . . . 10 ℕ ∈ V
335334mptex 7242 . . . . . . . . 9 (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))) ∈ V
33628, 335eqeltri 2834 . . . . . . . 8 𝑍 ∈ V
337336a1i 11 . . . . . . 7 (𝜑𝑍 ∈ V)
338268adantr 480 . . . . . . . . . 10 ((𝑚 = 𝑛𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
339338itgeq2dv 25831 . . . . . . . . 9 (𝑚 = 𝑛 → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
340339cbvmptv 5260 . . . . . . . 8 (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑛 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
341279adantr 480 . . . . . . . . . . 11 ((𝑐 = 𝑘𝑠 ∈ (0(,)π)) → ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) = ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠))
342341itgeq2dv 25831 . . . . . . . . . 10 (𝑐 = 𝑘 → ∫(0(,)π)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) d𝑠 = ∫(0(,)π)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) d𝑠)
343342oveq1d 7445 . . . . . . . . 9 (𝑐 = 𝑘 → (∫(0(,)π)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π) = (∫(0(,)π)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π))
344343cbvmptv 5260 . . . . . . . 8 (𝑐 ∈ ℕ ↦ (∫(0(,)π)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π)) = (𝑘 ∈ ℕ ↦ (∫(0(,)π)((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) d𝑠 / π))
345 eqid 2734 . . . . . . . 8 ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟))) ↾ (𝑒[,]π)) = ((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟))) ↾ (𝑒[,]π))
346 eqid 2734 . . . . . . . 8 ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π))) = ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))
347 eqid 2734 . . . . . . . 8 ((♯‘({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1) = ((♯‘({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)
348 isoeq1 7336 . . . . . . . . 9 (𝑢 = 𝑣 → (𝑢 Isom < , < ((0...((♯‘({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) ↔ 𝑣 Isom < , < ((0...((♯‘({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π))))))
349348cbviotavw 6523 . . . . . . . 8 (℩𝑢𝑢 Isom < , < ((0...((♯‘({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π))))) = (℩𝑣𝑣 Isom < , < ((0...((♯‘({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))))
350 eqid 2734 . . . . . . . 8 (𝑎 ∈ (0..^𝑁)(((℩𝑢𝑢 Isom < , < ((0...((♯‘({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))))‘𝑏)(,)((℩𝑢𝑢 Isom < , < ((0...((♯‘({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))))‘(𝑏 + 1))) ⊆ (((𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋))‘𝑎)(,)((𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋))‘(𝑎 + 1)))) = (𝑎 ∈ (0..^𝑁)(((℩𝑢𝑢 Isom < , < ((0...((♯‘({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))))‘𝑏)(,)((℩𝑢𝑢 Isom < , < ((0...((♯‘({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))))‘(𝑏 + 1))) ⊆ (((𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋))‘𝑎)(,)((𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋))‘(𝑎 + 1))))
351324cbvitgv 25826 . . . . . . . . . . . 12 ∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎 = ∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠
352351fveq2i 6909 . . . . . . . . . . 11 (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) = (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠)
353352breq1i 5154 . . . . . . . . . 10 ((abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑞 / 2) ↔ (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑞 / 2))
354353anbi2i 623 . . . . . . . . 9 (((((𝜑𝑞 ∈ ℝ+) ∧ 𝑒 ∈ (0(,)π)) ∧ 𝑏 ∈ ℕ) ∧ (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑞 / 2)) ↔ ((((𝜑𝑞 ∈ ℝ+) ∧ 𝑒 ∈ (0(,)π)) ∧ 𝑏 ∈ ℕ) ∧ (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑞 / 2)))
355324cbvitgv 25826 . . . . . . . . . . 11 ∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎 = ∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠
356355fveq2i 6909 . . . . . . . . . 10 (abs‘∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) = (abs‘∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠)
357356breq1i 5154 . . . . . . . . 9 ((abs‘∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑞 / 2) ↔ (abs‘∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑞 / 2))
358354, 357anbi12i 628 . . . . . . . 8 ((((((𝜑𝑞 ∈ ℝ+) ∧ 𝑒 ∈ (0(,)π)) ∧ 𝑏 ∈ ℕ) ∧ (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑞 / 2)) ∧ (abs‘∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑞 / 2)) ↔ (((((𝜑𝑞 ∈ ℝ+) ∧ 𝑒 ∈ (0(,)π)) ∧ 𝑏 ∈ ℕ) ∧ (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑞 / 2)) ∧ (abs‘∫(𝑒(,)π)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑞 / 2)))
35943, 44, 45, 76, 77, 78, 122, 132, 182, 213, 225, 236, 246, 254, 258, 261, 265, 340, 344, 284, 285, 286, 287, 311, 345, 346, 347, 349, 319, 350, 358fourierdlem104 46165 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) ⇝ (𝑅 / 2))
360 eqidd 2735 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
361270adantl 481 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 = 𝑛) → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
362 simpr 484 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
363 elioore 13413 . . . . . . . . . . 11 (𝑠 ∈ (-π(,)0) → 𝑠 ∈ ℝ)
36443adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ ℝ) → 𝐹:ℝ⟶ℝ)
36544adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ ℝ) → 𝑋 ∈ ℝ)
366 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ ℝ) → 𝑠 ∈ ℝ)
367365, 366readdcld 11287 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ ℝ) → (𝑋 + 𝑠) ∈ ℝ)
368364, 367ffvelcdmd 7104 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ ℝ) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
369368adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
370288dirkerre 46050 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
371370adantll 714 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
372369, 371remulcld 11288 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
373363, 372sylan2 593 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
374 ioossicc 13469 . . . . . . . . . . . . 13 (-π(,)0) ⊆ (-π[,]0)
37561leidi 11794 . . . . . . . . . . . . . 14 -π ≤ -π
37662, 54, 60ltleii 11381 . . . . . . . . . . . . . 14 0 ≤ π
377 iccss 13451 . . . . . . . . . . . . . 14 (((-π ∈ ℝ ∧ π ∈ ℝ) ∧ (-π ≤ -π ∧ 0 ≤ π)) → (-π[,]0) ⊆ (-π[,]π))
37861, 54, 375, 376, 377mp4an 693 . . . . . . . . . . . . 13 (-π[,]0) ⊆ (-π[,]π)
379374, 378sstri 4004 . . . . . . . . . . . 12 (-π(,)0) ⊆ (-π[,]π)
380379a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (-π(,)0) ⊆ (-π[,]π))
381 ioombl 25613 . . . . . . . . . . . 12 (-π(,)0) ∈ dom vol
382381a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (-π(,)0) ∈ dom vol)
38343adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℝ)
38444adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ (-π[,]π)) → 𝑋 ∈ ℝ)
38556, 55iccssred 13470 . . . . . . . . . . . . . . . 16 (𝜑 → (-π[,]π) ⊆ ℝ)
386385sselda 3994 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ (-π[,]π)) → 𝑠 ∈ ℝ)
387384, 386readdcld 11287 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (-π[,]π)) → (𝑋 + 𝑠) ∈ ℝ)
388383, 387ffvelcdmd 7104 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (-π[,]π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
389388adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
390 iccssre 13465 . . . . . . . . . . . . . . . 16 ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
39161, 54, 390mp2an 692 . . . . . . . . . . . . . . 15 (-π[,]π) ⊆ ℝ
392391sseli 3990 . . . . . . . . . . . . . 14 (𝑠 ∈ (-π[,]π) → 𝑠 ∈ ℝ)
393392, 370sylan2 593 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
394393adantll 714 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
395389, 394remulcld 11288 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
39661a1i 11 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → -π ∈ ℝ)
39754a1i 11 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → π ∈ ℝ)
39843adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
39944adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑋 ∈ ℝ)
40076adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑁 ∈ ℕ)
40177adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑉 ∈ ((𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑁))
402122adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
403225adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑁)) → if(((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘(𝑉𝑖))) = (𝑄‘((𝑦 ∈ ℝ ↦ sup({𝑓 ∈ (0..^𝑀) ∣ (𝑄𝑓) ≤ ((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉𝑖))), ((𝑗 ∈ (0..^𝑀) ↦ 𝑗 / 𝑖𝐶)‘((𝑦 ∈ ℝ ↦ sup({𝑓 ∈ (0..^𝑀) ∣ (𝑄𝑓) ≤ ((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉𝑖))), (𝐹‘((𝑑 ∈ (-π(,]π) ↦ if(𝑑 = π, -π, 𝑑))‘((𝑐 ∈ ℝ ↦ (𝑐 + ((⌊‘((π − 𝑐) / 𝑇)) · 𝑇)))‘(𝑉𝑖))))) ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))
404236adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑁)) → if(((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘(𝑉‘(𝑖 + 1))) = (𝑄‘(((𝑦 ∈ ℝ ↦ sup({ ∈ (0..^𝑀) ∣ (𝑄) ≤ ((𝑔 ∈ (-π(,]π) ↦ if(𝑔 = π, -π, 𝑔))‘((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉𝑖)) + 1)), ((𝑗 ∈ (0..^𝑀) ↦ 𝑗 / 𝑖𝑈)‘((𝑦 ∈ ℝ ↦ sup({ ∈ (0..^𝑀) ∣ (𝑄) ≤ ((𝑔 ∈ (-π(,]π) ↦ if(𝑔 = π, -π, 𝑔))‘((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉𝑖))), (𝐹‘((𝑒 ∈ ℝ ↦ (𝑒 + ((⌊‘((π − 𝑒) / 𝑇)) · 𝑇)))‘(𝑉‘(𝑖 + 1))))) ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))
405288dirkercncf 46062 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝐷𝑛) ∈ (ℝ–cn→ℝ))
406405adantl 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) ∈ (ℝ–cn→ℝ))
407 eqid 2734 . . . . . . . . . . . 12 (𝑠 ∈ (-π[,]π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) = (𝑠 ∈ (-π[,]π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
408396, 397, 398, 399, 45, 400, 401, 402, 403, 404, 319, 51, 406, 407fourierdlem84 46145 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
409380, 382, 395, 408iblss 25854 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (-π(,)0) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
410373, 409itgcl 25833 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 ∈ ℂ)
411360, 361, 362, 410fvmptd 7022 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
412411, 410eqeltrd 2838 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) ∈ ℂ)
413 eqidd 2735 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
414339adantl 481 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 = 𝑛) → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
41543adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (0(,)π)) → 𝐹:ℝ⟶ℝ)
41644adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (0(,)π)) → 𝑋 ∈ ℝ)
417 elioore 13413 . . . . . . . . . . . . . . 15 (𝑠 ∈ (0(,)π) → 𝑠 ∈ ℝ)
418417adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (0(,)π)) → 𝑠 ∈ ℝ)
419416, 418readdcld 11287 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (0(,)π)) → (𝑋 + 𝑠) ∈ ℝ)
420415, 419ffvelcdmd 7104 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
421420adantlr 715 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
422417, 370sylan2 593 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (0(,)π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
423422adantll 714 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
424421, 423remulcld 11288 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
425 ioossicc 13469 . . . . . . . . . . . . 13 (0(,)π) ⊆ (0[,]π)
42661, 62, 59ltleii 11381 . . . . . . . . . . . . . 14 -π ≤ 0
42754leidi 11794 . . . . . . . . . . . . . 14 π ≤ π
428 iccss 13451 . . . . . . . . . . . . . 14 (((-π ∈ ℝ ∧ π ∈ ℝ) ∧ (-π ≤ 0 ∧ π ≤ π)) → (0[,]π) ⊆ (-π[,]π))
42961, 54, 426, 427, 428mp4an 693 . . . . . . . . . . . . 13 (0[,]π) ⊆ (-π[,]π)
430425, 429sstri 4004 . . . . . . . . . . . 12 (0(,)π) ⊆ (-π[,]π)
431430a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (0(,)π) ⊆ (-π[,]π))
432 ioombl 25613 . . . . . . . . . . . 12 (0(,)π) ∈ dom vol
433432a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (0(,)π) ∈ dom vol)
434431, 433, 395, 408iblss 25854 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (0(,)π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
435424, 434itgcl 25833 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 ∈ ℂ)
436413, 414, 362, 435fvmptd 7022 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
437436, 435eqeltrd 2838 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) ∈ ℂ)
438 eleq1w 2821 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑚 ∈ ℕ ↔ 𝑛 ∈ ℕ))
439438anbi2d 630 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝜑𝑚 ∈ ℕ) ↔ (𝜑𝑛 ∈ ℕ)))
440 fveq2 6906 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑍𝑚) = (𝑍𝑛))
441270, 339oveq12d 7448 . . . . . . . . . . 11 (𝑚 = 𝑛 → (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠))
442440, 441eqeq12d 2750 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝑍𝑚) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) ↔ (𝑍𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)))
443439, 442imbi12d 344 . . . . . . . . 9 (𝑚 = 𝑛 → (((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)) ↔ ((𝜑𝑛 ∈ ℕ) → (𝑍𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠))))
444 oveq1 7437 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝑛 · 𝑥) = (𝑚 · 𝑥))
445444fveq2d 6910 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (cos‘(𝑛 · 𝑥)) = (cos‘(𝑚 · 𝑥)))
446445oveq2d 7446 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (cos‘(𝑚 · 𝑥))))
447446adantr 480 . . . . . . . . . . . . . 14 ((𝑛 = 𝑚𝑥 ∈ (-π(,)π)) → ((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (cos‘(𝑚 · 𝑥))))
448447itgeq2dv 25831 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 = ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥)
449448oveq1d 7445 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π) = (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π))
450449cbvmptv 5260 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) = (𝑚 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π))
45129, 450eqtri 2762 . . . . . . . . . 10 𝐴 = (𝑚 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π))
452 fourierdlem112.b . . . . . . . . . . 11 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
453444fveq2d 6910 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (sin‘(𝑛 · 𝑥)) = (sin‘(𝑚 · 𝑥)))
454453oveq2d 7446 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (sin‘(𝑚 · 𝑥))))
455454adantr 480 . . . . . . . . . . . . . 14 ((𝑛 = 𝑚𝑥 ∈ (-π(,)π)) → ((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (sin‘(𝑚 · 𝑥))))
456455itgeq2dv 25831 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 = ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥)
457456oveq1d 7445 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π) = (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
458457cbvmptv 5260 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) = (𝑚 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
459452, 458eqtri 2762 . . . . . . . . . 10 𝐵 = (𝑚 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
460 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐴𝑛) = (𝐴𝑘))
461 oveq1 7437 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
462461fveq2d 6910 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
463460, 462oveq12d 7448 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))))
464 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐵𝑛) = (𝐵𝑘))
465461fveq2d 6910 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
466464, 465oveq12d 7448 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
467463, 466oveq12d 7448 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
468467cbvsumv 15728 . . . . . . . . . . . . 13 Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
469468oveq2i 7441 . . . . . . . . . . . 12 (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
470469mpteq2i 5252 . . . . . . . . . . 11 (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))) = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))))
471 oveq2 7438 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
472471sumeq1d 15732 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
473472oveq2d 7446 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) = (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))))
474473cbvmptv 5260 . . . . . . . . . . . 12 (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))) = (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))))
475 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝐴𝑘) = (𝐴𝑚))
476 oveq1 7437 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → (𝑘 · 𝑋) = (𝑚 · 𝑋))
477476fveq2d 6910 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (cos‘(𝑘 · 𝑋)) = (cos‘(𝑚 · 𝑋)))
478475, 477oveq12d 7448 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) = ((𝐴𝑚) · (cos‘(𝑚 · 𝑋))))
479 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝐵𝑘) = (𝐵𝑚))
480476fveq2d 6910 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (sin‘(𝑘 · 𝑋)) = (sin‘(𝑚 · 𝑋)))
481479, 480oveq12d 7448 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) = ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))
482478, 481oveq12d 7448 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋)))))
483482cbvsumv 15728 . . . . . . . . . . . . . 14 Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))
484483oveq2i 7441 . . . . . . . . . . . . 13 (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) = (((𝐴‘0) / 2) + Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋)))))
485484mpteq2i 5252 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))) = (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))))
486474, 485eqtri 2762 . . . . . . . . . . 11 (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))) = (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))))
48728, 470, 4863eqtri 2766 . . . . . . . . . 10 𝑍 = (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))))
488 oveq2 7438 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑋 + 𝑦) = (𝑋 + 𝑥))
489488fveq2d 6910 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑥)))
490 fveq2 6906 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝐷𝑚)‘𝑦) = ((𝐷𝑚)‘𝑥))
491489, 490oveq12d 7448 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹‘(𝑋 + 𝑦)) · ((𝐷𝑚)‘𝑦)) = ((𝐹‘(𝑋 + 𝑥)) · ((𝐷𝑚)‘𝑥)))
492491cbvmptv 5260 . . . . . . . . . 10 (𝑦 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑦)) · ((𝐷𝑚)‘𝑦))) = (𝑥 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷𝑚)‘𝑥)))
493 eqid 2734 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = (-π − 𝑋) ∧ (𝑝𝑛) = (π − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = (-π − 𝑋) ∧ (𝑝𝑛) = (π − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
494 fveq2 6906 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑄𝑗) = (𝑄𝑖))
495494oveq1d 7445 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝑄𝑗) − 𝑋) = ((𝑄𝑖) − 𝑋))
496495cbvmptv 5260 . . . . . . . . . 10 (𝑗 ∈ (0...𝑀) ↦ ((𝑄𝑗) − 𝑋)) = (𝑖 ∈ (0...𝑀) ↦ ((𝑄𝑖) − 𝑋))
497451, 459, 487, 288, 51, 52, 53, 146, 43, 92, 492, 103, 222, 233, 48, 493, 496fourierdlem111 46172 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
498443, 497chvarvv 1995 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝑍𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠))
499411, 436oveq12d 7448 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) + ((𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛)) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠))
500498, 499eqtr4d 2777 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝑍𝑛) = (((𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) + ((𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛)))
50116, 24, 27, 42, 14, 15, 333, 337, 359, 412, 437, 500climaddf 45570 . . . . . 6 (𝜑𝑍 ⇝ ((𝐿 / 2) + (𝑅 / 2)))
502 limccl 25924 . . . . . . . 8 ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋) ⊆ ℂ
503502, 285sselid 3992 . . . . . . 7 (𝜑𝐿 ∈ ℂ)
504 limccl 25924 . . . . . . . 8 ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋) ⊆ ℂ
505504, 284sselid 3992 . . . . . . 7 (𝜑𝑅 ∈ ℂ)
506 2cnd 12341 . . . . . . 7 (𝜑 → 2 ∈ ℂ)
507 2pos 12366 . . . . . . . . 9 0 < 2
508507a1i 11 . . . . . . . 8 (𝜑 → 0 < 2)
509508gt0ne0d 11824 . . . . . . 7 (𝜑 → 2 ≠ 0)
510503, 505, 506, 509divdird 12078 . . . . . 6 (𝜑 → ((𝐿 + 𝑅) / 2) = ((𝐿 / 2) + (𝑅 / 2)))
511501, 510breqtrrd 5175 . . . . 5 (𝜑𝑍 ⇝ ((𝐿 + 𝑅) / 2))
512 0nn0 12538 . . . . . . . 8 0 ∈ ℕ0
51343adantr 480 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → 𝐹:ℝ⟶ℝ)
514 eqid 2734 . . . . . . . . . 10 (-π(,)π) = (-π(,)π)
515 ioossre 13444 . . . . . . . . . . . . . 14 (-π(,)π) ⊆ ℝ
516515a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ⊆ ℝ)
51743, 516feqresmpt 6977 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ (-π(,)π)) = (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)))
518 ioossicc 13469 . . . . . . . . . . . . . 14 (-π(,)π) ⊆ (-π[,]π)
519518a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ⊆ (-π[,]π))
520 ioombl 25613 . . . . . . . . . . . . . 14 (-π(,)π) ∈ dom vol
521520a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ∈ dom vol)
52243adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℝ)
523385sselda 3994 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (-π[,]π)) → 𝑥 ∈ ℝ)
524522, 523ffvelcdmd 7104 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (-π[,]π)) → (𝐹𝑥) ∈ ℝ)
52543, 385feqresmpt 6977 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (-π[,]π)) = (𝑥 ∈ (-π[,]π) ↦ (𝐹𝑥)))
526178a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ℝ ⊆ ℂ)
52743, 526fssd 6753 . . . . . . . . . . . . . . . 16 (𝜑𝐹:ℝ⟶ℂ)
528527, 385fssresd 6775 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 ↾ (-π[,]π)):(-π[,]π)⟶ℂ)
529 ioossicc 13469 . . . . . . . . . . . . . . . . . 18 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
53061rexri 11316 . . . . . . . . . . . . . . . . . . . 20 -π ∈ ℝ*
531530a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ*)
53254rexri 11316 . . . . . . . . . . . . . . . . . . . 20 π ∈ ℝ*
533532a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → π ∈ ℝ*)
53451, 52, 53fourierdlem15 46077 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑄:(0...𝑀)⟶(-π[,]π))
535534adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
536 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
537531, 533, 535, 536fourierdlem8 46070 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
538529, 537sstrid 4006 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
539538resabs1d 6027 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
540539, 103eqeltrd 2838 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
541539eqcomd 2740 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
542541oveq1d 7445 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
543222, 542eleqtrd 2840 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
544541oveq1d 7445 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
545233, 544eleqtrd 2840 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
54651, 52, 53, 528, 540, 543, 545fourierdlem69 46130 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (-π[,]π)) ∈ 𝐿1)
547525, 546eqeltrrd 2839 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (-π[,]π) ↦ (𝐹𝑥)) ∈ 𝐿1)
548519, 521, 524, 547iblss 25854 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1)
549517, 548eqeltrd 2838 . . . . . . . . . . 11 (𝜑 → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
550549adantr 480 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
551 simpr 484 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → 0 ∈ ℕ0)
552513, 514, 550, 29, 551fourierdlem16 46078 . . . . . . . . 9 ((𝜑 ∧ 0 ∈ ℕ0) → (((𝐴‘0) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (cos‘(0 · 𝑥))) d𝑥 ∈ ℝ))
553552simplld 768 . . . . . . . 8 ((𝜑 ∧ 0 ∈ ℕ0) → (𝐴‘0) ∈ ℝ)
554512, 553mpan2 691 . . . . . . 7 (𝜑 → (𝐴‘0) ∈ ℝ)
555554rehalfcld 12510 . . . . . 6 (𝜑 → ((𝐴‘0) / 2) ∈ ℝ)
556555recnd 11286 . . . . 5 (𝜑 → ((𝐴‘0) / 2) ∈ ℂ)
557334mptex 7242 . . . . . 6 (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) ∈ V
558557a1i 11 . . . . 5 (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) ∈ V)
559 simpr 484 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
560555adantr 480 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ((𝐴‘0) / 2) ∈ ℝ)
561 fzfid 14010 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (1...𝑚) ∈ Fin)
562 simpll 767 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → 𝜑)
563 elfznn 13589 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑚) → 𝑛 ∈ ℕ)
564563adantl 481 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → 𝑛 ∈ ℕ)
565 simpl 482 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝜑)
566362nnnn0d 12584 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
567 eleq1w 2821 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘 ∈ ℕ0𝑛 ∈ ℕ0))
568567anbi2d 630 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝜑𝑘 ∈ ℕ0) ↔ (𝜑𝑛 ∈ ℕ0)))
569 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝐴𝑘) = (𝐴𝑛))
570569eleq1d 2823 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝐴𝑘) ∈ ℝ ↔ (𝐴𝑛) ∈ ℝ))
571568, 570imbi12d 344 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℝ) ↔ ((𝜑𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℝ)))
57243adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → 𝐹:ℝ⟶ℝ)
573549adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
574 simpr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
575572, 514, 573, 29, 574fourierdlem16 46078 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ0) → (((𝐴𝑘) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑘 · 𝑥))) d𝑥 ∈ ℝ))
576575simplld 768 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℝ)
577571, 576chvarvv 1995 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℝ)
578565, 566, 577syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ∈ ℝ)
579362nnred 12278 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
580579, 399remulcld 11288 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝑛 · 𝑋) ∈ ℝ)
581580recoscld 16176 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (cos‘(𝑛 · 𝑋)) ∈ ℝ)
582578, 581remulcld 11288 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) ∈ ℝ)
583 eleq1w 2821 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝑘 ∈ ℕ ↔ 𝑛 ∈ ℕ))
584583anbi2d 630 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → ((𝜑𝑘 ∈ ℕ) ↔ (𝜑𝑛 ∈ ℕ)))
585 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝐵𝑘) = (𝐵𝑛))
586585eleq1d 2823 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → ((𝐵𝑘) ∈ ℝ ↔ (𝐵𝑛) ∈ ℝ))
587584, 586imbi12d 344 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → (((𝜑𝑘 ∈ ℕ) → (𝐵𝑘) ∈ ℝ) ↔ ((𝜑𝑛 ∈ ℕ) → (𝐵𝑛) ∈ ℝ)))
58843adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
589549adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
590 simpr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
591588, 514, 589, 452, 590fourierdlem21 46083 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (((𝐵𝑘) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ ((𝐹𝑥) · (sin‘(𝑘 · 𝑥)))) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑘 · 𝑥))) d𝑥 ∈ ℝ))
592591simplld 768 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐵𝑘) ∈ ℝ)
593587, 592chvarvv 1995 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐵𝑛) ∈ ℝ)
594580resincld 16175 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (sin‘(𝑛 · 𝑋)) ∈ ℝ)
595593, 594remulcld 11288 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) ∈ ℝ)
596582, 595readdcld 11287 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
597562, 564, 596syl2anc 584 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
598561, 597fsumrecl 15766 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
599560, 598readdcld 11287 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ ℝ)
60028fvmpt2 7026 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ ℝ) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
601559, 599, 600syl2anc 584 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
602601, 599eqeltrd 2838 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) ∈ ℝ)
603602recnd 11286 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) ∈ ℂ)
604 eqidd 2735 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))))
605 oveq2 7438 . . . . . . . . 9 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
606605sumeq1d 15732 . . . . . . . 8 (𝑛 = 𝑚 → Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
607606adantl 481 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 = 𝑚) → Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
608 sumex 15720 . . . . . . . 8 Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ V
609608a1i 11 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ V)
610604, 607, 559, 609fvmptd 7022 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))‘𝑚) = Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
611560recnd 11286 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝐴‘0) / 2) ∈ ℂ)
612598recnd 11286 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℂ)
613611, 612pncan2d 11619 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → ((((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) − ((𝐴‘0) / 2)) = Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
614613, 468eqtr2di 2791 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = ((((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) − ((𝐴‘0) / 2)))
615 ovex 7463 . . . . . . . . 9 (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ V
61628fvmpt2 7026 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ V) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
617559, 615, 616sylancl 586 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
618617eqcomd 2740 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑍𝑚))
619618oveq1d 7445 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → ((((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) − ((𝐴‘0) / 2)) = ((𝑍𝑚) − ((𝐴‘0) / 2)))
620610, 614, 6193eqtrd 2778 . . . . 5 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))‘𝑚) = ((𝑍𝑚) − ((𝐴‘0) / 2)))
62114, 15, 511, 556, 558, 603, 620climsubc1 15670 . . . 4 (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
622 seqex 14040 . . . . . 6 seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))) ∈ V
623622a1i 11 . . . . 5 (𝜑 → seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))) ∈ V)
624 eqidd 2735 . . . . . . 7 ((𝜑𝑙 ∈ ℕ) → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))))
625 oveq2 7438 . . . . . . . . 9 (𝑛 = 𝑙 → (1...𝑛) = (1...𝑙))
626625sumeq1d 15732 . . . . . . . 8 (𝑛 = 𝑙 → Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
627626adantl 481 . . . . . . 7 (((𝜑𝑙 ∈ ℕ) ∧ 𝑛 = 𝑙) → Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
628 simpr 484 . . . . . . 7 ((𝜑𝑙 ∈ ℕ) → 𝑙 ∈ ℕ)
629 fzfid 14010 . . . . . . . 8 ((𝜑𝑙 ∈ ℕ) → (1...𝑙) ∈ Fin)
630 elfznn 13589 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ)
631630nnnn0d 12584 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ0)
632631, 576sylan2 593 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (𝐴𝑘) ∈ ℝ)
633630nnred 12278 . . . . . . . . . . . . . 14 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℝ)
634633adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1...𝑙)) → 𝑘 ∈ ℝ)
635146adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1...𝑙)) → 𝑋 ∈ ℝ)
636634, 635remulcld 11288 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (1...𝑙)) → (𝑘 · 𝑋) ∈ ℝ)
637636recoscld 16176 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (cos‘(𝑘 · 𝑋)) ∈ ℝ)
638632, 637remulcld 11288 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1...𝑙)) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
639630, 592sylan2 593 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (𝐵𝑘) ∈ ℝ)
640636resincld 16175 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (sin‘(𝑘 · 𝑋)) ∈ ℝ)
641639, 640remulcld 11288 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1...𝑙)) → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) ∈ ℝ)
642638, 641readdcld 11287 . . . . . . . . 9 ((𝜑𝑘 ∈ (1...𝑙)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
643642adantlr 715 . . . . . . . 8 (((𝜑𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
644629, 643fsumrecl 15766 . . . . . . 7 ((𝜑𝑙 ∈ ℕ) → Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
645624, 627, 628, 644fvmptd 7022 . . . . . 6 ((𝜑𝑙 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))‘𝑙) = Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
646 eleq1w 2821 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑛 ∈ ℕ ↔ 𝑙 ∈ ℕ))
647646anbi2d 630 . . . . . . . 8 (𝑛 = 𝑙 → ((𝜑𝑛 ∈ ℕ) ↔ (𝜑𝑙 ∈ ℕ)))
648 fveq2 6906 . . . . . . . . 9 (𝑛 = 𝑙 → (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑛) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑙))
649626, 648eqeq12d 2750 . . . . . . . 8 (𝑛 = 𝑙 → (Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑛) ↔ Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑙)))
650647, 649imbi12d 344 . . . . . . 7 (𝑛 = 𝑙 → (((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑛)) ↔ ((𝜑𝑙 ∈ ℕ) → Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑙))))
651 eqidd 2735 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))) = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))
652 fveq2 6906 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
653 oveq1 7437 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑗 · 𝑋) = (𝑘 · 𝑋))
654653fveq2d 6910 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (cos‘(𝑗 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
655652, 654oveq12d 7448 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) = ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))))
656 fveq2 6906 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐵𝑗) = (𝐵𝑘))
657653fveq2d 6910 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (sin‘(𝑗 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
658656, 657oveq12d 7448 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))) = ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
659655, 658oveq12d 7448 . . . . . . . . . 10 (𝑗 = 𝑘 → (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
660659adantl 481 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑗 = 𝑘) → (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
661 elfznn 13589 . . . . . . . . . 10 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
662661adantl 481 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
663 simpll 767 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
664 nnnn0 12530 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
665 nn0re 12532 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
666665adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
667146adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → 𝑋 ∈ ℝ)
668666, 667remulcld 11288 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ0) → (𝑘 · 𝑋) ∈ ℝ)
669668recoscld 16176 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ0) → (cos‘(𝑘 · 𝑋)) ∈ ℝ)
670576, 669remulcld 11288 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
671664, 670sylan2 593 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
672664, 668sylan2 593 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝑘 · 𝑋) ∈ ℝ)
673672resincld 16175 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (sin‘(𝑘 · 𝑋)) ∈ ℝ)
674592, 673remulcld 11288 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) ∈ ℝ)
675671, 674readdcld 11287 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
676663, 662, 675syl2anc 584 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
677651, 660, 662, 676fvmptd 7022 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))‘𝑘) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
678362, 14eleqtrdi 2848 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
679676recnd 11286 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℂ)
680677, 678, 679fsumser 15762 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑛))
681650, 680chvarvv 1995 . . . . . 6 ((𝜑𝑙 ∈ ℕ) → Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑙))
682645, 681eqtrd 2774 . . . . 5 ((𝜑𝑙 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))‘𝑙) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑙))
68314, 558, 623, 15, 682climeq 15599 . . . 4 (𝜑 → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ↔ seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))))
684621, 683mpbid 232 . . 3 (𝜑 → seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
68513, 684eqbrtrd 5169 . 2 (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
686 eqidd 2735 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))) = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))
687 fveq2 6906 . . . . . . . . 9 (𝑗 = 𝑛 → (𝐴𝑗) = (𝐴𝑛))
688 oveq1 7437 . . . . . . . . . 10 (𝑗 = 𝑛 → (𝑗 · 𝑋) = (𝑛 · 𝑋))
689688fveq2d 6910 . . . . . . . . 9 (𝑗 = 𝑛 → (cos‘(𝑗 · 𝑋)) = (cos‘(𝑛 · 𝑋)))
690687, 689oveq12d 7448 . . . . . . . 8 (𝑗 = 𝑛 → ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) = ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))))
691 fveq2 6906 . . . . . . . . 9 (𝑗 = 𝑛 → (𝐵𝑗) = (𝐵𝑛))
692688fveq2d 6910 . . . . . . . . 9 (𝑗 = 𝑛 → (sin‘(𝑗 · 𝑋)) = (sin‘(𝑛 · 𝑋)))
693691, 692oveq12d 7448 . . . . . . . 8 (𝑗 = 𝑛 → ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))) = ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))
694690, 693oveq12d 7448 . . . . . . 7 (𝑗 = 𝑛 → (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
695694adantl 481 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 = 𝑛) → (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
696686, 695, 362, 596fvmptd 7022 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))‘𝑛) = (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
697596recnd 11286 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℂ)
69814, 15, 696, 697, 684isumclim 15789 . . . 4 (𝜑 → Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
699698oveq2d 7446 . . 3 (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = (((𝐴‘0) / 2) + (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))))
700503, 505addcld 11277 . . . . 5 (𝜑 → (𝐿 + 𝑅) ∈ ℂ)
701700halfcld 12508 . . . 4 (𝜑 → ((𝐿 + 𝑅) / 2) ∈ ℂ)
702556, 701pncan3d 11620 . . 3 (𝜑 → (((𝐴‘0) / 2) + (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) = ((𝐿 + 𝑅) / 2))
703699, 702eqtrd 2774 . 2 (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2))
704685, 703jca 511 1 (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  csb 3907  cun 3960  cin 3961  wss 3962  ifcif 4530  {cpr 4632   class class class wbr 5147  cmpt 5230  dom cdm 5688  ran crn 5689  cres 5690  cio 6513  wf 6558  cfv 6562   Isom wiso 6563  crio 7386  (class class class)co 7430  m cmap 8864  supcsup 9477  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157  +∞cpnf 11289  -∞cmnf 11290  *cxr 11291   < clt 11292  cle 11293  cmin 11489  -cneg 11490   / cdiv 11917  cn 12263  2c2 12318  0cn0 12523  cz 12610  cuz 12875  +crp 13031  (,)cioo 13383  (,]cioc 13384  [,]cicc 13386  ...cfz 13543  ..^cfzo 13690  cfl 13826   mod cmo 13905  seqcseq 14038  chash 14365  abscabs 15269  cli 15516  Σcsu 15718  sincsin 16095  cosccos 16096  πcpi 16098  cnccncf 24915  volcvol 25511  𝐿1cibl 25665  citg 25666   lim climc 25911   D cdv 25912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cc 10472  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-symdif 4258  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-acn 9979  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ioc 13388  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-fac 14309  df-bc 14338  df-hash 14366  df-shft 15102  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-limsup 15503  df-clim 15520  df-rlim 15521  df-sum 15719  df-ef 16099  df-sin 16101  df-cos 16102  df-pi 16104  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-fbas 21378  df-fg 21379  df-cnfld 21382  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-lp 23159  df-perf 23160  df-cn 23250  df-cnp 23251  df-t1 23337  df-haus 23338  df-cmp 23410  df-tx 23585  df-hmeo 23778  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-xms 24345  df-ms 24346  df-tms 24347  df-cncf 24917  df-ovol 25512  df-vol 25513  df-mbf 25667  df-itg1 25668  df-itg2 25669  df-ibl 25670  df-itg 25671  df-0p 25718  df-ditg 25896  df-limc 25915  df-dv 25916
This theorem is referenced by:  fourierdlem113  46174
  Copyright terms: Public domain W3C validator