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Theorem fourierdlem112 41096
Description: Here abbreviations (local definitions) are introduced to prove the fourier 41103 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 6379 . . . . . . . 8 (𝑛 = 𝑗 → (𝐴𝑛) = (𝐴𝑗))
3 oveq1 6853 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
43fveq2d 6383 . . . . . . . 8 (𝑛 = 𝑗 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑗 · 𝑋)))
52, 4oveq12d 6864 . . . . . . 7 (𝑛 = 𝑗 → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))))
6 fveq2 6379 . . . . . . . 8 (𝑛 = 𝑗 → (𝐵𝑛) = (𝐵𝑗))
73fveq2d 6383 . . . . . . . 8 (𝑛 = 𝑗 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑗 · 𝑋)))
86, 7oveq12d 6864 . . . . . . 7 (𝑛 = 𝑗 → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))
95, 8oveq12d 6864 . . . . . 6 (𝑛 = 𝑗 → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
109cbvmptv 4911 . . . . 5 (𝑛 ∈ ℕ ↦ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
111, 10eqtri 2787 . . . 4 𝑆 = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
12 seqeq3 13018 . . . 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 11928 . . . . 5 ℕ = (ℤ‘1)
15 1zzd 11660 . . . . 5 (𝜑 → 1 ∈ ℤ)
16 nfv 2009 . . . . . . 7 𝑛𝜑
17 nfcv 2907 . . . . . . . 8 𝑛
18 nfcv 2907 . . . . . . . . 9 𝑛(-π(,)0)
19 nfcv 2907 . . . . . . . . . 10 𝑛(𝐹‘(𝑋 + 𝑠))
20 nfcv 2907 . . . . . . . . . 10 𝑛 ·
21 nfcv 2907 . . . . . . . . . 10 𝑛((𝐷𝑚)‘𝑠)
2219, 20, 21nfov 6876 . . . . . . . . 9 𝑛((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠))
2318, 22nfitg 23846 . . . . . . . 8 𝑛∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠
2417, 23nfmpt 4907 . . . . . . 7 𝑛(𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)
25 nfcv 2907 . . . . . . . . 9 𝑛(0(,)π)
2625, 22nfitg 23846 . . . . . . . 8 𝑛∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠
2717, 26nfmpt 4907 . . . . . . 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 4908 . . . . . . . . . . . . 13 𝑛(𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
3129, 30nfcxfr 2905 . . . . . . . . . . . 12 𝑛𝐴
32 nfcv 2907 . . . . . . . . . . . 12 𝑛0
3331, 32nffv 6389 . . . . . . . . . . 11 𝑛(𝐴‘0)
34 nfcv 2907 . . . . . . . . . . 11 𝑛 /
35 nfcv 2907 . . . . . . . . . . 11 𝑛2
3633, 34, 35nfov 6876 . . . . . . . . . 10 𝑛((𝐴‘0) / 2)
37 nfcv 2907 . . . . . . . . . 10 𝑛 +
38 nfcv 2907 . . . . . . . . . . 11 𝑛(1...𝑚)
3938nfsum1 14719 . . . . . . . . . 10 𝑛Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))
4036, 37, 39nfov 6876 . . . . . . . . 9 𝑛(((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
4117, 40nfmpt 4907 . . . . . . . 8 𝑛(𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
4228, 41nfcxfr 2905 . . . . . . 7 𝑛𝑍
43 fourierdlem112.f . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
44 fourierdlem112.25 . . . . . . . 8 (𝜑𝑋 ∈ ℝ)
45 eqid 2765 . . . . . . . 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 24517 . . . . . . . . . . . . 13 π ∈ ℂ
47462timesi 11421 . . . . . . . . . . . 12 (2 · π) = (π + π)
48 fourierdlem112.t . . . . . . . . . . . 12 𝑇 = (2 · π)
4946, 46subnegi 10618 . . . . . . . . . . . 12 (π − -π) = (π + π)
5047, 48, 493eqtr4i 2797 . . . . . . . . . . 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 24516 . . . . . . . . . . . . . 14 π ∈ ℝ
5554a1i 11 . . . . . . . . . . . . 13 (𝜑 → π ∈ ℝ)
5655renegcld 10715 . . . . . . . . . . . 12 (𝜑 → -π ∈ ℝ)
5756, 44readdcld 10327 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) ∈ ℝ)
5855, 44readdcld 10327 . . . . . . . . . . 11 (𝜑 → (π + 𝑋) ∈ ℝ)
59 negpilt0 40156 . . . . . . . . . . . . . 14 -π < 0
60 pipos 24518 . . . . . . . . . . . . . 14 0 < π
6154renegcli 10600 . . . . . . . . . . . . . . 15 -π ∈ ℝ
62 0re 10299 . . . . . . . . . . . . . . 15 0 ∈ ℝ
6361, 62, 54lttri 10421 . . . . . . . . . . . . . 14 ((-π < 0 ∧ 0 < π) → -π < π)
6459, 60, 63mp2an 683 . . . . . . . . . . . . 13 -π < π
6564a1i 11 . . . . . . . . . . . 12 (𝜑 → -π < π)
6656, 55, 44, 65ltadd1dd 10896 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) < (π + 𝑋))
67 oveq1 6853 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑦 + (𝑘 · 𝑇)) = (𝑥 + (𝑘 · 𝑇)))
6867eleq1d 2829 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
6968rexbidv 3199 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
7069cbvrabv 3348 . . . . . . . . . . . 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 3928 . . . . . . . . . . 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 41038 . . . . . . . . . 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 488 . . . . . . . . 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 488 . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
7775simprd 489 . . . . . . . 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 472 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℝ)
80 fveq2 6379 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑝𝑖) = (𝑝𝑗))
81 oveq1 6853 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
8281fveq2d 6383 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1)))
8380, 82breq12d 4824 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝𝑗) < (𝑝‘(𝑗 + 1))))
8483cbvralv 3319 . . . . . . . . . . . . . 14 (∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))
8584anbi2i 616 . . . . . . . . . . . . 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 3333 . . . . . . . . . . 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 4902 . . . . . . . . . 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 2787 . . . . . . . . 9 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))})
9052adantr 472 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ)
9153adantr 472 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑄 ∈ (𝑃𝑀))
92 fourierdlem112.fper . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
9392adantlr 706 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
94 eleq1w 2827 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
9594anbi2d 622 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑗 ∈ (0..^𝑀))))
96 fveq2 6379 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
9781fveq2d 6383 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
9896, 97oveq12d 6864 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1))))
9998reseq2d 5567 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))))
10098oveq1d 6861 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) = (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
10199, 100eleq12d 2838 . . . . . . . . . . . 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 335 . . . . . . . . . . 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, 103chvarv 2369 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
105104adantlr 706 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
10657adantr 472 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (-π + 𝑋) ∈ ℝ)
10757rexrd 10347 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) ∈ ℝ*)
108 pnfxr 10350 . . . . . . . . . . . 12 +∞ ∈ ℝ*
109108a1i 11 . . . . . . . . . . 11 (𝜑 → +∞ ∈ ℝ*)
11058ltpnfd 12160 . . . . . . . . . . 11 (𝜑 → (π + 𝑋) < +∞)
111107, 109, 58, 66, 110eliood 40386 . . . . . . . . . 10 (𝜑 → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
112111adantr 472 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
113 id 22 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^𝑁))
11472oveq2i 6857 . . . . . . . . . . 11 (0..^𝑁) = (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))
115113, 114syl6eleq 2854 . . . . . . . . . 10 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))
116115adantl 473 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))
11772oveq2i 6857 . . . . . . . . . . . 12 (0...𝑁) = (0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))
118 isoeq4 6766 . . . . . . . . . . . 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 6055 . . . . . . . . . 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 2787 . . . . . . . . 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 41082 . . . . . . . 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 472 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤)
125 nfra1 3088 . . . . . . . . . . 11 𝑡𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤
126 elioore 12412 . . . . . . . . . . . . 13 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
127 rspa 3077 . . . . . . . . . . . . 13 ((∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤𝑡 ∈ ℝ) → (abs‘(𝐹𝑡)) ≤ 𝑤)
128126, 127sylan2 586 . . . . . . . . . . . 12 ((∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(𝐹𝑡)) ≤ 𝑤)
129128ex 401 . . . . . . . . . . 11 (∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤 → (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘(𝐹𝑡)) ≤ 𝑤))
130125, 129ralrimi 3104 . . . . . . . . . 10 (∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤 → ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹𝑡)) ≤ 𝑤)
131130reximi 3157 . . . . . . . . 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 3785 . . . . . . . . . . . 12 ℝ ⊆ ℝ
134 dvfre 24019 . . . . . . . . . . . 12 ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
13543, 133, 134sylancl 580 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
136135adantr 472 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
137 eqid 2765 . . . . . . . . . . . . 13 (ℝ D 𝐹) = (ℝ D 𝐹)
13854a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → π ∈ ℝ)
13961a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → -π ∈ ℝ)
14098reseq2d 5567 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))))
141140, 100eleq12d 2838 . . . . . . . . . . . . . . . 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 335 . . . . . . . . . . . . . . 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, 143chvarv 2369 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
145144adantlr 706 . . . . . . . . . . . . 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 10327 . . . . . . . . . . . . . 14 (𝜑 → (-π + 𝑋) ∈ ℝ)
148147adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → (-π + 𝑋) ∈ ℝ)
149147rexrd 10347 . . . . . . . . . . . . . . 15 (𝜑 → (-π + 𝑋) ∈ ℝ*)
15055, 146readdcld 10327 . . . . . . . . . . . . . . 15 (𝜑 → (π + 𝑋) ∈ ℝ)
15156, 55, 146, 65ltadd1dd 10896 . . . . . . . . . . . . . . 15 (𝜑 → (-π + 𝑋) < (π + 𝑋))
152150ltpnfd 12160 . . . . . . . . . . . . . . 15 (𝜑 → (π + 𝑋) < +∞)
153149, 109, 150, 151, 152eliood 40386 . . . . . . . . . . . . . 14 (𝜑 → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
154153adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
155 oveq1 6853 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = → (𝑘 · 𝑇) = ( · 𝑇))
156155oveq2d 6862 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝑦 + (𝑘 · 𝑇)) = (𝑦 + ( · 𝑇)))
157156eleq1d 2829 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + ( · 𝑇)) ∈ ran 𝑄))
158157cbvrexv 3320 . . . . . . . . . . . . . . . . . . 19 (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄)
159158rgenw 3071 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋))(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄)
160 rabbi 3268 . . . . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . . . . . 17 {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄}
162161uneq2i 3928 . . . . . . . . . . . . . . . 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 6767 . . . . . . . . . . . . . . . 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 6055 . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → (𝑣 ∈ dom (ℝ D 𝐹) ↔ 𝑢 ∈ dom (ℝ D 𝐹)))
168 fveq2 6379 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → ((ℝ D 𝐹)‘𝑣) = ((ℝ D 𝐹)‘𝑢))
169167, 168ifbieq1d 4268 . . . . . . . . . . . . . 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 4911 . . . . . . . . . . . . 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 41081 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
172 cncff 22989 . . . . . . . . . . . 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 6233 . . . . . . . . . . . 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 5597 . . . . . . . . . . 11 (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
176174, 175sylibr 225 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹))
177136, 176fssresd 6255 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
178 ax-resscn 10250 . . . . . . . . . . 11 ℝ ⊆ ℂ
179178a1i 11 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ℝ ⊆ ℂ)
180 cncffvrn 22994 . . . . . . . . . 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 579 . . . . . . . . 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 248 . . . . . . . 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 472 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
185 nfv 2009 . . . . . . . . . . . . . 14 𝑡(𝜑𝑖 ∈ (0..^𝑁))
186 nfra1 3088 . . . . . . . . . . . . . 14 𝑡𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧
187185, 186nfan 1998 . . . . . . . . . . . . 13 𝑡((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
188 fvres 6398 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
189188adantl 473 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
190189fveq2d 6383 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡)))
191190adantlr 706 . . . . . . . . . . . . . . 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 785 . . . . . . . . . . . . . . . 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 3763 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → 𝑡 ∈ dom (ℝ D 𝐹))
194193adantlr 706 . . . . . . . . . . . . . . . 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 3077 . . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . . 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 4833 . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . 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 3104 . . . . . . . . . . . 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 401 . . . . . . . . . . 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 3162 . . . . . . . . . 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 3088 . . . . . . . . . . . 12 𝑡𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧
204188eqcomd 2771 . . . . . . . . . . . . . . . 16 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → ((ℝ D 𝐹)‘𝑡) = (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡))
205204fveq2d 6383 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘((ℝ D 𝐹)‘𝑡)) = (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)))
206205adantl 473 . . . . . . . . . . . . . 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 3077 . . . . . . . . . . . . . 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 4833 . . . . . . . . . . . . 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 401 . . . . . . . . . . . 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 3104 . . . . . . . . . . 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 3162 . . . . . . . . 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 2009 . . . . . . . . . . . 12 𝑖(𝜑𝑗 ∈ (0..^𝑀))
215 nfcsb1v 3709 . . . . . . . . . . . . 13 𝑖𝑗 / 𝑖𝐶
216215nfel1 2922 . . . . . . . . . . . 12 𝑖𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗))
217214, 216nfim 1995 . . . . . . . . . . 11 𝑖((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
218 csbeq1a 3702 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝐶 = 𝑗 / 𝑖𝐶)
21999, 96oveq12d 6864 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
220218, 219eleq12d 2838 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) ↔ 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗))))
22195, 220imbi12d 335 . . . . . . . . . . 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, 222chvar 2368 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
224223adantlr 706 . . . . . . . . 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 41080 . . . . . . . 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 3709 . . . . . . . . . . . . 13 𝑖𝑗 / 𝑖𝑈
227226nfel1 2922 . . . . . . . . . . . 12 𝑖𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1)))
228214, 227nfim 1995 . . . . . . . . . . 11 𝑖((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
229 csbeq1a 3702 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝑈 = 𝑗 / 𝑖𝑈)
23099, 97oveq12d 6864 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
231229, 230eleq12d 2838 . . . . . . . . . . . 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 335 . . . . . . . . . . 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, 233chvar 2368 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
235234adantlr 706 . . . . . . . . 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 41083 . . . . . . . 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 2769 . . . . . . . . . 10 (𝑔 = 𝑠 → (𝑔 = 0 ↔ 𝑠 = 0))
238 oveq2 6854 . . . . . . . . . . . . 13 (𝑔 = 𝑠 → (𝑋 + 𝑔) = (𝑋 + 𝑠))
239238fveq2d 6383 . . . . . . . . . . . 12 (𝑔 = 𝑠 → (𝐹‘(𝑋 + 𝑔)) = (𝐹‘(𝑋 + 𝑠)))
240 breq2 4815 . . . . . . . . . . . . 13 (𝑔 = 𝑠 → (0 < 𝑔 ↔ 0 < 𝑠))
241240ifbid 4267 . . . . . . . . . . . 12 (𝑔 = 𝑠 → if(0 < 𝑔, 𝑅, 𝐿) = if(0 < 𝑠, 𝑅, 𝐿))
242239, 241oveq12d 6864 . . . . . . . . . . 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 6864 . . . . . . . . . 10 (𝑔 = 𝑠 → (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑅, 𝐿)) / 𝑠))
245237, 244ifbieq2d 4270 . . . . . . . . 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 4911 . . . . . . . 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 2769 . . . . . . . . . 10 (𝑜 = 𝑠 → (𝑜 = 0 ↔ 𝑠 = 0))
248 id 22 . . . . . . . . . . 11 (𝑜 = 𝑠𝑜 = 𝑠)
249 oveq1 6853 . . . . . . . . . . . . 13 (𝑜 = 𝑠 → (𝑜 / 2) = (𝑠 / 2))
250249fveq2d 6383 . . . . . . . . . . . 12 (𝑜 = 𝑠 → (sin‘(𝑜 / 2)) = (sin‘(𝑠 / 2)))
251250oveq2d 6862 . . . . . . . . . . 11 (𝑜 = 𝑠 → (2 · (sin‘(𝑜 / 2))) = (2 · (sin‘(𝑠 / 2))))
252248, 251oveq12d 6864 . . . . . . . . . 10 (𝑜 = 𝑠 → (𝑜 / (2 · (sin‘(𝑜 / 2)))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
253247, 252ifbieq2d 4270 . . . . . . . . 9 (𝑜 = 𝑠 → if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
254253cbvmptv 4911 . . . . . . . 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 6379 . . . . . . . . . 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 6379 . . . . . . . . . 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 6864 . . . . . . . . 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 4911 . . . . . . . 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 6854 . . . . . . . . . 10 (𝑑 = 𝑠 → ((𝑘 + (1 / 2)) · 𝑑) = ((𝑘 + (1 / 2)) · 𝑠))
260259fveq2d 6383 . . . . . . . . 9 (𝑑 = 𝑠 → (sin‘((𝑘 + (1 / 2)) · 𝑑)) = (sin‘((𝑘 + (1 / 2)) · 𝑠)))
261260cbvmptv 4911 . . . . . . . 8 (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑))) = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑠)))
262 fveq2 6379 . . . . . . . . . 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 6379 . . . . . . . . . 10 (𝑧 = 𝑠 → ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧) = ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑠))
264262, 263oveq12d 6864 . . . . . . . . 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 4911 . . . . . . . 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 6379 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝐷𝑚) = (𝐷𝑛))
267266fveq1d 6381 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((𝐷𝑚)‘𝑠) = ((𝐷𝑛)‘𝑠))
268267oveq2d 6862 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
269268adantr 472 . . . . . . . . . 10 ((𝑚 = 𝑛𝑠 ∈ (-π(,)0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
270269itgeq2dv 23853 . . . . . . . . 9 (𝑚 = 𝑛 → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
271270cbvmptv 4911 . . . . . . . 8 (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑛 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
272 oveq1 6853 . . . . . . . . . . . . . . . . . . 19 (𝑐 = 𝑘 → (𝑐 + (1 / 2)) = (𝑘 + (1 / 2)))
273272oveq1d 6861 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝑘 → ((𝑐 + (1 / 2)) · 𝑑) = ((𝑘 + (1 / 2)) · 𝑑))
274273fveq2d 6383 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑘 → (sin‘((𝑐 + (1 / 2)) · 𝑑)) = (sin‘((𝑘 + (1 / 2)) · 𝑑)))
275274mpteq2dv 4906 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑘 → (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑))) = (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑))))
276275fveq1d 6381 . . . . . . . . . . . . . . 15 (𝑐 = 𝑘 → ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧) = ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧))
277276oveq2d 6862 . . . . . . . . . . . . . 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 4906 . . . . . . . . . . . . 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 6381 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 23853 . . . . . . . . . 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 6861 . . . . . . . . 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 4911 . . . . . . . 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 6853 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → (𝑦 mod (2 · π)) = (𝑠 mod (2 · π)))
290289eqeq1d 2767 . . . . . . . . . . . . 13 (𝑦 = 𝑠 → ((𝑦 mod (2 · π)) = 0 ↔ (𝑠 mod (2 · π)) = 0))
291 oveq2 6854 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → ((𝑚 + (1 / 2)) · 𝑦) = ((𝑚 + (1 / 2)) · 𝑠))
292291fveq2d 6383 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → (sin‘((𝑚 + (1 / 2)) · 𝑦)) = (sin‘((𝑚 + (1 / 2)) · 𝑠)))
293 oveq1 6853 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑠 → (𝑦 / 2) = (𝑠 / 2))
294293fveq2d 6383 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → (sin‘(𝑦 / 2)) = (sin‘(𝑠 / 2)))
295294oveq2d 6862 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → ((2 · π) · (sin‘(𝑦 / 2))) = ((2 · π) · (sin‘(𝑠 / 2))))
296292, 295oveq12d 6864 . . . . . . . . . . . . 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 4270 . . . . . . . . . . . 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 4911 . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑘𝑠 ∈ ℝ) → 𝑚 = 𝑘)
300299oveq2d 6862 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (2 · 𝑚) = (2 · 𝑘))
301300oveq1d 6861 . . . . . . . . . . . . . 14 ((𝑚 = 𝑘𝑠 ∈ ℝ) → ((2 · 𝑚) + 1) = ((2 · 𝑘) + 1))
302301oveq1d 6861 . . . . . . . . . . . . 13 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (((2 · 𝑚) + 1) / (2 · π)) = (((2 · 𝑘) + 1) / (2 · π)))
303299oveq1d 6861 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (𝑚 + (1 / 2)) = (𝑘 + (1 / 2)))
304303oveq1d 6861 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑘𝑠 ∈ ℝ) → ((𝑚 + (1 / 2)) · 𝑠) = ((𝑘 + (1 / 2)) · 𝑠))
305304fveq2d 6383 . . . . . . . . . . . . . 14 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (sin‘((𝑚 + (1 / 2)) · 𝑠)) = (sin‘((𝑘 + (1 / 2)) · 𝑠)))
306305oveq1d 6861 . . . . . . . . . . . . 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 4265 . . . . . . . . . . . 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 4905 . . . . . . . . . . 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, 308syl5eq 2811 . . . . . . . . . 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 4911 . . . . . . . . 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 2787 . . . . . . . 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 2765 . . . . . . . 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 2765 . . . . . . . 8 ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙))) = ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))
314 eqid 2765 . . . . . . . 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 6763 . . . . . . . . 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))))))
316315cbviotav 6039 . . . . . . . 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 6379 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝑉𝑗) = (𝑉𝑖))
318317oveq1d 6861 . . . . . . . . 9 (𝑗 = 𝑖 → ((𝑉𝑗) − 𝑋) = ((𝑉𝑖) − 𝑋))
319318cbvmptv 4911 . . . . . . . 8 (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) = (𝑖 ∈ (0...𝑁) ↦ ((𝑉𝑖) − 𝑋))
320 eqid 2765 . . . . . . . 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 6379 . . . . . . . . . . . . . 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 6854 . . . . . . . . . . . . . . 15 (𝑎 = 𝑠 → ((𝑏 + (1 / 2)) · 𝑎) = ((𝑏 + (1 / 2)) · 𝑠))
323322fveq2d 6383 . . . . . . . . . . . . . 14 (𝑎 = 𝑠 → (sin‘((𝑏 + (1 / 2)) · 𝑎)) = (sin‘((𝑏 + (1 / 2)) · 𝑠)))
324321, 323oveq12d 6864 . . . . . . . . . . . . 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 23848 . . . . . . . . . . . 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 6382 . . . . . . . . . . 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 4818 . . . . . . . . . 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 616 . . . . . . . . 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 23848 . . . . . . . . . . 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 6382 . . . . . . . . . 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 4818 . . . . . . . . 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 620 . . . . . . . 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 41087 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) ⇝ (𝐿 / 2))
334 nnex 11285 . . . . . . . . . 10 ℕ ∈ V
335334mptex 6683 . . . . . . . . 9 (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))) ∈ V
33628, 335eqeltri 2840 . . . . . . . 8 𝑍 ∈ V
337336a1i 11 . . . . . . 7 (𝜑𝑍 ∈ V)
338268adantr 472 . . . . . . . . . 10 ((𝑚 = 𝑛𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
339338itgeq2dv 23853 . . . . . . . . 9 (𝑚 = 𝑛 → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
340339cbvmptv 4911 . . . . . . . 8 (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑛 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
341279adantr 472 . . . . . . . . . . 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 23853 . . . . . . . . . 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 6861 . . . . . . . . 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 4911 . . . . . . . 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 2765 . . . . . . . 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 2765 . . . . . . . 8 ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π))) = ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))
347 eqid 2765 . . . . . . . 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 6763 . . . . . . . . 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))))))
349348cbviotav 6039 . . . . . . . 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 2765 . . . . . . . 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 23848 . . . . . . . . . . . 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 6382 . . . . . . . . . . 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 4818 . . . . . . . . . 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 616 . . . . . . . . 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 23848 . . . . . . . . . . 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 6382 . . . . . . . . . 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 4818 . . . . . . . . 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 620 . . . . . . . 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 41088 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) ⇝ (𝑅 / 2))
360 eqidd 2766 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
361270adantl 473 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 = 𝑛) → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
362 simpr 477 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
363 elioore 12412 . . . . . . . . . . 11 (𝑠 ∈ (-π(,)0) → 𝑠 ∈ ℝ)
36443adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ ℝ) → 𝐹:ℝ⟶ℝ)
36544adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ ℝ) → 𝑋 ∈ ℝ)
366 simpr 477 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ ℝ) → 𝑠 ∈ ℝ)
367365, 366readdcld 10327 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ ℝ) → (𝑋 + 𝑠) ∈ ℝ)
368364, 367ffvelrnd 6554 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ ℝ) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
369368adantlr 706 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
370288dirkerre 40973 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
371370adantll 705 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
372369, 371remulcld 10328 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
373363, 372sylan2 586 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
374 ioossicc 12466 . . . . . . . . . . . . 13 (-π(,)0) ⊆ (-π[,]0)
37561leidi 10820 . . . . . . . . . . . . . 14 -π ≤ -π
37662, 54, 60ltleii 10418 . . . . . . . . . . . . . 14 0 ≤ π
377 iccss 12448 . . . . . . . . . . . . . 14 (((-π ∈ ℝ ∧ π ∈ ℝ) ∧ (-π ≤ -π ∧ 0 ≤ π)) → (-π[,]0) ⊆ (-π[,]π))
37861, 54, 375, 376, 377mp4an 684 . . . . . . . . . . . . 13 (-π[,]0) ⊆ (-π[,]π)
379374, 378sstri 3772 . . . . . . . . . . . 12 (-π(,)0) ⊆ (-π[,]π)
380379a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (-π(,)0) ⊆ (-π[,]π))
381 ioombl 23637 . . . . . . . . . . . 12 (-π(,)0) ∈ dom vol
382381a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (-π(,)0) ∈ dom vol)
38343adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℝ)
38444adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ (-π[,]π)) → 𝑋 ∈ ℝ)
38556, 55iccssred 40393 . . . . . . . . . . . . . . . 16 (𝜑 → (-π[,]π) ⊆ ℝ)
386385sselda 3763 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ (-π[,]π)) → 𝑠 ∈ ℝ)
387384, 386readdcld 10327 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (-π[,]π)) → (𝑋 + 𝑠) ∈ ℝ)
388383, 387ffvelrnd 6554 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (-π[,]π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
389388adantlr 706 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
390 iccssre 12462 . . . . . . . . . . . . . . . 16 ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
39161, 54, 390mp2an 683 . . . . . . . . . . . . . . 15 (-π[,]π) ⊆ ℝ
392391sseli 3759 . . . . . . . . . . . . . 14 (𝑠 ∈ (-π[,]π) → 𝑠 ∈ ℝ)
393392, 370sylan2 586 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
394393adantll 705 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
395389, 394remulcld 10328 . . . . . . . . . . 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 472 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
39944adantr 472 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑋 ∈ ℝ)
40076adantr 472 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑁 ∈ ℕ)
40177adantr 472 . . . . . . . . . . . 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 706 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
403225adantlr 706 . . . . . . . . . . . 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 706 . . . . . . . . . . . 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 40985 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝐷𝑛) ∈ (ℝ–cn→ℝ))
406405adantl 473 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) ∈ (ℝ–cn→ℝ))
407 eqid 2765 . . . . . . . . . . . 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 41068 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
409380, 382, 395, 408iblss 23876 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (-π(,)0) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
410373, 409itgcl 23855 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 ∈ ℂ)
411360, 361, 362, 410fvmptd 6481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
412411, 410eqeltrd 2844 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) ∈ ℂ)
413 eqidd 2766 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
414339adantl 473 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 = 𝑛) → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
41543adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (0(,)π)) → 𝐹:ℝ⟶ℝ)
41644adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (0(,)π)) → 𝑋 ∈ ℝ)
417 elioore 12412 . . . . . . . . . . . . . . 15 (𝑠 ∈ (0(,)π) → 𝑠 ∈ ℝ)
418417adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (0(,)π)) → 𝑠 ∈ ℝ)
419416, 418readdcld 10327 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (0(,)π)) → (𝑋 + 𝑠) ∈ ℝ)
420415, 419ffvelrnd 6554 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
421420adantlr 706 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
422417, 370sylan2 586 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (0(,)π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
423422adantll 705 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
424421, 423remulcld 10328 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
425 ioossicc 12466 . . . . . . . . . . . . 13 (0(,)π) ⊆ (0[,]π)
42661, 62, 59ltleii 10418 . . . . . . . . . . . . . 14 -π ≤ 0
42754leidi 10820 . . . . . . . . . . . . . 14 π ≤ π
428 iccss 12448 . . . . . . . . . . . . . 14 (((-π ∈ ℝ ∧ π ∈ ℝ) ∧ (-π ≤ 0 ∧ π ≤ π)) → (0[,]π) ⊆ (-π[,]π))
42961, 54, 426, 427, 428mp4an 684 . . . . . . . . . . . . 13 (0[,]π) ⊆ (-π[,]π)
430425, 429sstri 3772 . . . . . . . . . . . 12 (0(,)π) ⊆ (-π[,]π)
431430a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (0(,)π) ⊆ (-π[,]π))
432 ioombl 23637 . . . . . . . . . . . 12 (0(,)π) ∈ dom vol
433432a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (0(,)π) ∈ dom vol)
434431, 433, 395, 408iblss 23876 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (0(,)π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
435424, 434itgcl 23855 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 ∈ ℂ)
436413, 414, 362, 435fvmptd 6481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
437436, 435eqeltrd 2844 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) ∈ ℂ)
438 eleq1w 2827 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑚 ∈ ℕ ↔ 𝑛 ∈ ℕ))
439438anbi2d 622 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝜑𝑚 ∈ ℕ) ↔ (𝜑𝑛 ∈ ℕ)))
440 fveq2 6379 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑍𝑚) = (𝑍𝑛))
441270, 339oveq12d 6864 . . . . . . . . . . 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 2780 . . . . . . . . . 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 335 . . . . . . . . 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 6853 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝑛 · 𝑥) = (𝑚 · 𝑥))
445444fveq2d 6383 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (cos‘(𝑛 · 𝑥)) = (cos‘(𝑚 · 𝑥)))
446445oveq2d 6862 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (cos‘(𝑚 · 𝑥))))
447446adantr 472 . . . . . . . . . . . . . 14 ((𝑛 = 𝑚𝑥 ∈ (-π(,)π)) → ((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (cos‘(𝑚 · 𝑥))))
448447itgeq2dv 23853 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 = ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥)
449448oveq1d 6861 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π) = (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π))
450449cbvmptv 4911 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) = (𝑚 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π))
45129, 450eqtri 2787 . . . . . . . . . 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 6383 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (sin‘(𝑛 · 𝑥)) = (sin‘(𝑚 · 𝑥)))
454453oveq2d 6862 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (sin‘(𝑚 · 𝑥))))
455454adantr 472 . . . . . . . . . . . . . 14 ((𝑛 = 𝑚𝑥 ∈ (-π(,)π)) → ((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (sin‘(𝑚 · 𝑥))))
456455itgeq2dv 23853 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 = ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥)
457456oveq1d 6861 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π) = (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
458457cbvmptv 4911 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) = (𝑚 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
459452, 458eqtri 2787 . . . . . . . . . 10 𝐵 = (𝑚 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
460 fveq2 6379 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐴𝑛) = (𝐴𝑘))
461 oveq1 6853 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
462461fveq2d 6383 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
463460, 462oveq12d 6864 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))))
464 fveq2 6379 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐵𝑛) = (𝐵𝑘))
465461fveq2d 6383 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
466464, 465oveq12d 6864 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
467463, 466oveq12d 6864 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
468467cbvsumv 14725 . . . . . . . . . . . . 13 Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
469468oveq2i 6857 . . . . . . . . . . . 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 4902 . . . . . . . . . . 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 6854 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
472471sumeq1d 14730 . . . . . . . . . . . . . 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 6862 . . . . . . . . . . . . 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 4911 . . . . . . . . . . . 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 6379 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝐴𝑘) = (𝐴𝑚))
476 oveq1 6853 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → (𝑘 · 𝑋) = (𝑚 · 𝑋))
477476fveq2d 6383 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (cos‘(𝑘 · 𝑋)) = (cos‘(𝑚 · 𝑋)))
478475, 477oveq12d 6864 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) = ((𝐴𝑚) · (cos‘(𝑚 · 𝑋))))
479 fveq2 6379 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝐵𝑘) = (𝐵𝑚))
480476fveq2d 6383 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (sin‘(𝑘 · 𝑋)) = (sin‘(𝑚 · 𝑋)))
481479, 480oveq12d 6864 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) = ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))
482478, 481oveq12d 6864 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋)))))
483482cbvsumv 14725 . . . . . . . . . . . . . 14 Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))
484483oveq2i 6857 . . . . . . . . . . . . 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 4902 . . . . . . . . . . . 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 2787 . . . . . . . . . . 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 2791 . . . . . . . . . 10 𝑍 = (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))))
488 oveq2 6854 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑋 + 𝑦) = (𝑋 + 𝑥))
489488fveq2d 6383 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑥)))
490 fveq2 6379 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝐷𝑚)‘𝑦) = ((𝐷𝑚)‘𝑥))
491489, 490oveq12d 6864 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹‘(𝑋 + 𝑦)) · ((𝐷𝑚)‘𝑦)) = ((𝐹‘(𝑋 + 𝑥)) · ((𝐷𝑚)‘𝑥)))
492491cbvmptv 4911 . . . . . . . . . 10 (𝑦 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑦)) · ((𝐷𝑚)‘𝑦))) = (𝑥 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷𝑚)‘𝑥)))
493 eqid 2765 . . . . . . . . . 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 6379 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑄𝑗) = (𝑄𝑖))
495494oveq1d 6861 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝑄𝑗) − 𝑋) = ((𝑄𝑖) − 𝑋))
496495cbvmptv 4911 . . . . . . . . . 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 41095 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
498443, 497chvarv 2369 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝑍𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠))
499411, 436oveq12d 6864 . . . . . . . 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 2802 . . . . . . 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 40509 . . . . . 6 (𝜑𝑍 ⇝ ((𝐿 / 2) + (𝑅 / 2)))
502 limccl 23944 . . . . . . . 8 ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋) ⊆ ℂ
503502, 285sseldi 3761 . . . . . . 7 (𝜑𝐿 ∈ ℂ)
504 limccl 23944 . . . . . . . 8 ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋) ⊆ ℂ
505504, 284sseldi 3761 . . . . . . 7 (𝜑𝑅 ∈ ℂ)
506 2cnd 11354 . . . . . . 7 (𝜑 → 2 ∈ ℂ)
507 2pos 11386 . . . . . . . . 9 0 < 2
508507a1i 11 . . . . . . . 8 (𝜑 → 0 < 2)
509508gt0ne0d 10850 . . . . . . 7 (𝜑 → 2 ≠ 0)
510503, 505, 506, 509divdird 11097 . . . . . 6 (𝜑 → ((𝐿 + 𝑅) / 2) = ((𝐿 / 2) + (𝑅 / 2)))
511501, 510breqtrrd 4839 . . . . 5 (𝜑𝑍 ⇝ ((𝐿 + 𝑅) / 2))
512 0nn0 11559 . . . . . . . 8 0 ∈ ℕ0
51343adantr 472 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → 𝐹:ℝ⟶ℝ)
514 eqid 2765 . . . . . . . . . 10 (-π(,)π) = (-π(,)π)
515 ioossre 12442 . . . . . . . . . . . . . 14 (-π(,)π) ⊆ ℝ
516515a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ⊆ ℝ)
51743, 516feqresmpt 6443 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ (-π(,)π)) = (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)))
518 ioossicc 12466 . . . . . . . . . . . . . 14 (-π(,)π) ⊆ (-π[,]π)
519518a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ⊆ (-π[,]π))
520 ioombl 23637 . . . . . . . . . . . . . 14 (-π(,)π) ∈ dom vol
521520a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ∈ dom vol)
52243adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℝ)
523385sselda 3763 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (-π[,]π)) → 𝑥 ∈ ℝ)
524522, 523ffvelrnd 6554 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (-π[,]π)) → (𝐹𝑥) ∈ ℝ)
52543, 385feqresmpt 6443 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (-π[,]π)) = (𝑥 ∈ (-π[,]π) ↦ (𝐹𝑥)))
526178a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ℝ ⊆ ℂ)
52743, 526fssd 6239 . . . . . . . . . . . . . . . 16 (𝜑𝐹:ℝ⟶ℂ)
528527, 385fssresd 6255 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 ↾ (-π[,]π)):(-π[,]π)⟶ℂ)
529 ioossicc 12466 . . . . . . . . . . . . . . . . . 18 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
53061rexri 10355 . . . . . . . . . . . . . . . . . . . 20 -π ∈ ℝ*
531530a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ*)
53254rexri 10355 . . . . . . . . . . . . . . . . . . . 20 π ∈ ℝ*
533532a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → π ∈ ℝ*)
53451, 52, 53fourierdlem15 41000 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑄:(0...𝑀)⟶(-π[,]π))
535534adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
536 simpr 477 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
537531, 533, 535, 536fourierdlem8 40993 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
538529, 537syl5ss 3774 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
539538resabs1d 5605 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
540539, 103eqeltrd 2844 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
541539eqcomd 2771 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
542541oveq1d 6861 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
543222, 542eleqtrd 2846 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
544541oveq1d 6861 . . . . . . . . . . . . . . . 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 2846 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
54651, 52, 53, 528, 540, 543, 545fourierdlem69 41053 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (-π[,]π)) ∈ 𝐿1)
547525, 546eqeltrrd 2845 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (-π[,]π) ↦ (𝐹𝑥)) ∈ 𝐿1)
548519, 521, 524, 547iblss 23876 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1)
549517, 548eqeltrd 2844 . . . . . . . . . . 11 (𝜑 → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
550549adantr 472 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
551 simpr 477 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → 0 ∈ ℕ0)
552513, 514, 550, 29, 551fourierdlem16 41001 . . . . . . . . 9 ((𝜑 ∧ 0 ∈ ℕ0) → (((𝐴‘0) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (cos‘(0 · 𝑥))) d𝑥 ∈ ℝ))
553552simplld 784 . . . . . . . 8 ((𝜑 ∧ 0 ∈ ℕ0) → (𝐴‘0) ∈ ℝ)
554512, 553mpan2 682 . . . . . . 7 (𝜑 → (𝐴‘0) ∈ ℝ)
555554rehalfcld 11529 . . . . . 6 (𝜑 → ((𝐴‘0) / 2) ∈ ℝ)
556555recnd 10326 . . . . 5 (𝜑 → ((𝐴‘0) / 2) ∈ ℂ)
557334mptex 6683 . . . . . 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 477 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
560555adantr 472 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ((𝐴‘0) / 2) ∈ ℝ)
561 fzfid 12985 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (1...𝑚) ∈ Fin)
562 simpll 783 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → 𝜑)
563 elfznn 12582 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑚) → 𝑛 ∈ ℕ)
564563adantl 473 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → 𝑛 ∈ ℕ)
565 simpl 474 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝜑)
566362nnnn0d 11602 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
567 eleq1w 2827 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘 ∈ ℕ0𝑛 ∈ ℕ0))
568567anbi2d 622 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝜑𝑘 ∈ ℕ0) ↔ (𝜑𝑛 ∈ ℕ0)))
569 fveq2 6379 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝐴𝑘) = (𝐴𝑛))
570569eleq1d 2829 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝐴𝑘) ∈ ℝ ↔ (𝐴𝑛) ∈ ℝ))
571568, 570imbi12d 335 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℝ) ↔ ((𝜑𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℝ)))
57243adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → 𝐹:ℝ⟶ℝ)
573549adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
574 simpr 477 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
575572, 514, 573, 29, 574fourierdlem16 41001 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ0) → (((𝐴𝑘) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑘 · 𝑥))) d𝑥 ∈ ℝ))
576575simplld 784 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℝ)
577571, 576chvarv 2369 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℝ)
578565, 566, 577syl2anc 579 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ∈ ℝ)
579362nnred 11295 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
580579, 399remulcld 10328 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝑛 · 𝑋) ∈ ℝ)
581580recoscld 15170 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (cos‘(𝑛 · 𝑋)) ∈ ℝ)
582578, 581remulcld 10328 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) ∈ ℝ)
583 eleq1w 2827 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝑘 ∈ ℕ ↔ 𝑛 ∈ ℕ))
584583anbi2d 622 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → ((𝜑𝑘 ∈ ℕ) ↔ (𝜑𝑛 ∈ ℕ)))
585 fveq2 6379 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝐵𝑘) = (𝐵𝑛))
586585eleq1d 2829 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → ((𝐵𝑘) ∈ ℝ ↔ (𝐵𝑛) ∈ ℝ))
587584, 586imbi12d 335 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → (((𝜑𝑘 ∈ ℕ) → (𝐵𝑘) ∈ ℝ) ↔ ((𝜑𝑛 ∈ ℕ) → (𝐵𝑛) ∈ ℝ)))
58843adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
589549adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
590 simpr 477 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
591588, 514, 589, 452, 590fourierdlem21 41006 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (((𝐵𝑘) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ ((𝐹𝑥) · (sin‘(𝑘 · 𝑥)))) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑘 · 𝑥))) d𝑥 ∈ ℝ))
592591simplld 784 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐵𝑘) ∈ ℝ)
593587, 592chvarv 2369 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐵𝑛) ∈ ℝ)
594580resincld 15169 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (sin‘(𝑛 · 𝑋)) ∈ ℝ)
595593, 594remulcld 10328 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) ∈ ℝ)
596582, 595readdcld 10327 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
597562, 564, 596syl2anc 579 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
598561, 597fsumrecl 14764 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
599560, 598readdcld 10327 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ ℝ)
60028fvmpt2 6484 . . . . . . . 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 579 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
602601, 599eqeltrd 2844 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) ∈ ℝ)
603602recnd 10326 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) ∈ ℂ)
604 eqidd 2766 . . . . . . 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 6854 . . . . . . . . 9 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
606605sumeq1d 14730 . . . . . . . 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 473 . . . . . . 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 14717 . . . . . . . 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 6481 . . . . . 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 10326 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝐴‘0) / 2) ∈ ℂ)
612598recnd 10326 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℂ)
613611, 612pncan2d 10652 . . . . . . 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, 468syl6req 2816 . . . . . 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 6878 . . . . . . . . 9 (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ V
61628fvmpt2 6484 . . . . . . . . 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 580 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
618617eqcomd 2771 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑍𝑚))
619618oveq1d 6861 . . . . . 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 2803 . . . . 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 14667 . . . 4 (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
622 seqex 13015 . . . . . 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 2766 . . . . . . 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 6854 . . . . . . . . 9 (𝑛 = 𝑙 → (1...𝑛) = (1...𝑙))
626625sumeq1d 14730 . . . . . . . 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 473 . . . . . . 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 477 . . . . . . 7 ((𝜑𝑙 ∈ ℕ) → 𝑙 ∈ ℕ)
629 fzfid 12985 . . . . . . . 8 ((𝜑𝑙 ∈ ℕ) → (1...𝑙) ∈ Fin)
630 elfznn 12582 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ)
631630nnnn0d 11602 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ0)
632631, 576sylan2 586 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (𝐴𝑘) ∈ ℝ)
633630nnred 11295 . . . . . . . . . . . . . 14 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℝ)
634633adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1...𝑙)) → 𝑘 ∈ ℝ)
635146adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1...𝑙)) → 𝑋 ∈ ℝ)
636634, 635remulcld 10328 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (1...𝑙)) → (𝑘 · 𝑋) ∈ ℝ)
637636recoscld 15170 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (cos‘(𝑘 · 𝑋)) ∈ ℝ)
638632, 637remulcld 10328 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1...𝑙)) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
639630, 592sylan2 586 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (𝐵𝑘) ∈ ℝ)
640636resincld 15169 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (sin‘(𝑘 · 𝑋)) ∈ ℝ)
641639, 640remulcld 10328 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1...𝑙)) → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) ∈ ℝ)
642638, 641readdcld 10327 . . . . . . . . 9 ((𝜑𝑘 ∈ (1...𝑙)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
643642adantlr 706 . . . . . . . 8 (((𝜑𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
644629, 643fsumrecl 14764 . . . . . . 7 ((𝜑𝑙 ∈ ℕ) → Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
645624, 627, 628, 644fvmptd 6481 . . . . . 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 2827 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑛 ∈ ℕ ↔ 𝑙 ∈ ℕ))
647646anbi2d 622 . . . . . . . 8 (𝑛 = 𝑙 → ((𝜑𝑛 ∈ ℕ) ↔ (𝜑𝑙 ∈ ℕ)))
648 fveq2 6379 . . . . . . . . 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 2780 . . . . . . . 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 335 . . . . . . 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 2766 . . . . . . . . 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 6379 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
653 oveq1 6853 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑗 · 𝑋) = (𝑘 · 𝑋))
654653fveq2d 6383 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (cos‘(𝑗 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
655652, 654oveq12d 6864 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) = ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))))
656 fveq2 6379 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐵𝑗) = (𝐵𝑘))
657653fveq2d 6383 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (sin‘(𝑗 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
658656, 657oveq12d 6864 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))) = ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
659655, 658oveq12d 6864 . . . . . . . . . 10 (𝑗 = 𝑘 → (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
660659adantl 473 . . . . . . . . 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 12582 . . . . . . . . . 10 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
662661adantl 473 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
663 simpll 783 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
664 nnnn0 11550 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
665 nn0re 11552 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
666665adantl 473 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
667146adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → 𝑋 ∈ ℝ)
668666, 667remulcld 10328 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ0) → (𝑘 · 𝑋) ∈ ℝ)
669668recoscld 15170 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ0) → (cos‘(𝑘 · 𝑋)) ∈ ℝ)
670576, 669remulcld 10328 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
671664, 670sylan2 586 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
672664, 668sylan2 586 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝑘 · 𝑋) ∈ ℝ)
673672resincld 15169 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (sin‘(𝑘 · 𝑋)) ∈ ℝ)
674592, 673remulcld 10328 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) ∈ ℝ)
675671, 674readdcld 10327 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
676663, 662, 675syl2anc 579 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
677651, 660, 662, 676fvmptd 6481 . . . . . . . 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, 14syl6eleq 2854 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
679676recnd 10326 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℂ)
680677, 678, 679fsumser 14760 . . . . . . 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, 680chvarv 2369 . . . . . 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 2799 . . . . 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 14597 . . . 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 223 . . 3 (𝜑 → seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
68513, 684eqbrtrd 4833 . 2 (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
686 eqidd 2766 . . . . . 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 6379 . . . . . . . . 9 (𝑗 = 𝑛 → (𝐴𝑗) = (𝐴𝑛))
688 oveq1 6853 . . . . . . . . . 10 (𝑗 = 𝑛 → (𝑗 · 𝑋) = (𝑛 · 𝑋))
689688fveq2d 6383 . . . . . . . . 9 (𝑗 = 𝑛 → (cos‘(𝑗 · 𝑋)) = (cos‘(𝑛 · 𝑋)))
690687, 689oveq12d 6864 . . . . . . . 8 (𝑗 = 𝑛 → ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) = ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))))
691 fveq2 6379 . . . . . . . . 9 (𝑗 = 𝑛 → (𝐵𝑗) = (𝐵𝑛))
692688fveq2d 6383 . . . . . . . . 9 (𝑗 = 𝑛 → (sin‘(𝑗 · 𝑋)) = (sin‘(𝑛 · 𝑋)))
693691, 692oveq12d 6864 . . . . . . . 8 (𝑗 = 𝑛 → ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))) = ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))
694690, 693oveq12d 6864 . . . . . . 7 (𝑗 = 𝑛 → (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
695694adantl 473 . . . . . 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 6481 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))‘𝑛) = (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
697596recnd 10326 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℂ)
69814, 15, 696, 697, 684isumclim 14787 . . . 4 (𝜑 → Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
699698oveq2d 6862 . . 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 10317 . . . . 5 (𝜑 → (𝐿 + 𝑅) ∈ ℂ)
701700halfcld 11527 . . . 4 (𝜑 → ((𝐿 + 𝑅) / 2) ∈ ℂ)
702556, 701pncan3d 10653 . . 3 (𝜑 → (((𝐴‘0) / 2) + (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) = ((𝐿 + 𝑅) / 2))
703699, 702eqtrd 2799 . 2 (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2))
704685, 703jca 507 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 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  csb 3693  cun 3732  cin 3733  wss 3734  ifcif 4245  {cpr 4338   class class class wbr 4811  cmpt 4890  dom cdm 5279  ran crn 5280  cres 5281  cio 6031  wf 6066  cfv 6070   Isom wiso 6071  crio 6806  (class class class)co 6846  𝑚 cmap 8064  supcsup 8557  cc 10191  cr 10192  0cc0 10193  1c1 10194   + caddc 10196   · cmul 10198  +∞cpnf 10329  -∞cmnf 10330  *cxr 10331   < clt 10332  cle 10333  cmin 10524  -cneg 10525   / cdiv 10942  cn 11278  2c2 11331  0cn0 11542  cz 11628  cuz 11891  +crp 12033  (,)cioo 12382  (,]cioc 12383  [,]cicc 12385  ...cfz 12538  ..^cfzo 12678  cfl 12804   mod cmo 12881  seqcseq 13013  chash 13326  abscabs 14273  cli 14514  Σcsu 14715  sincsin 15090  cosccos 15091  πcpi 15093  cnccncf 22972  volcvol 23535  𝐿1cibl 23689  citg 23690   lim climc 23931   D cdv 23932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cc 9514  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271  ax-addf 10272  ax-mulf 10273
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-symdif 4007  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-disj 4780  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-ofr 7100  df-om 7268  df-1st 7370  df-2nd 7371  df-supp 7502  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-omul 7773  df-er 7951  df-map 8066  df-pm 8067  df-ixp 8118  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-fsupp 8487  df-fi 8528  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-acn 9023  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-4 11341  df-5 11342  df-6 11343  df-7 11344  df-8 11345  df-9 11346  df-n0 11543  df-xnn0 11615  df-z 11629  df-dec 11746  df-uz 11892  df-q 11995  df-rp 12034  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12386  df-ioc 12387  df-ico 12388  df-icc 12389  df-fz 12539  df-fzo 12679  df-fl 12806  df-mod 12882  df-seq 13014  df-exp 13073  df-fac 13270  df-bc 13299  df-hash 13327  df-shft 14106  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-limsup 14501  df-clim 14518  df-rlim 14519  df-sum 14716  df-ef 15094  df-sin 15096  df-cos 15097  df-pi 15099  df-struct 16146  df-ndx 16147  df-slot 16148  df-base 16150  df-sets 16151  df-ress 16152  df-plusg 16241  df-mulr 16242  df-starv 16243  df-sca 16244  df-vsca 16245  df-ip 16246  df-tset 16247  df-ple 16248  df-ds 16250  df-unif 16251  df-hom 16252  df-cco 16253  df-rest 16363  df-topn 16364  df-0g 16382  df-gsum 16383  df-topgen 16384  df-pt 16385  df-prds 16388  df-xrs 16442  df-qtop 16447  df-imas 16448  df-xps 16450  df-mre 16526  df-mrc 16527  df-acs 16529  df-mgm 17522  df-sgrp 17564  df-mnd 17575  df-submnd 17616  df-mulg 17822  df-cntz 18027  df-cmn 18475  df-psmet 20025  df-xmet 20026  df-met 20027  df-bl 20028  df-mopn 20029  df-fbas 20030  df-fg 20031  df-cnfld 20034  df-top 20992  df-topon 21009  df-topsp 21031  df-bases 21044  df-cld 21117  df-ntr 21118  df-cls 21119  df-nei 21196  df-lp 21234  df-perf 21235  df-cn 21325  df-cnp 21326  df-t1 21412  df-haus 21413  df-cmp 21484  df-tx 21659  df-hmeo 21852  df-fil 21943  df-fm 22035  df-flim 22036  df-flf 22037  df-xms 22418  df-ms 22419  df-tms 22420  df-cncf 22974  df-ovol 23536  df-vol 23537  df-mbf 23691  df-itg1 23692  df-itg2 23693  df-ibl 23694  df-itg 23695  df-0p 23742  df-ditg 23916  df-limc 23935  df-dv 23936
This theorem is referenced by:  fourierdlem113  41097
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