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Theorem fourierdlem112 39111
 Description: Here abbreviations (local definitions) are introduced to prove the fourier 39118 theorem. (𝑍‘𝑚) 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 6103 . . . . . . . 8 (𝑛 = 𝑗 → (𝐴𝑛) = (𝐴𝑗))
3 oveq1 6556 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
43fveq2d 6107 . . . . . . . 8 (𝑛 = 𝑗 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑗 · 𝑋)))
52, 4oveq12d 6567 . . . . . . 7 (𝑛 = 𝑗 → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))))
6 fveq2 6103 . . . . . . . 8 (𝑛 = 𝑗 → (𝐵𝑛) = (𝐵𝑗))
73fveq2d 6107 . . . . . . . 8 (𝑛 = 𝑗 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑗 · 𝑋)))
86, 7oveq12d 6567 . . . . . . 7 (𝑛 = 𝑗 → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))
95, 8oveq12d 6567 . . . . . 6 (𝑛 = 𝑗 → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
109cbvmptv 4678 . . . . 5 (𝑛 ∈ ℕ ↦ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
111, 10eqtri 2632 . . . 4 𝑆 = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
12 seqeq3 12668 . . . 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 11599 . . . . 5 ℕ = (ℤ‘1)
15 1zzd 11285 . . . . 5 (𝜑 → 1 ∈ ℤ)
16 nfv 1830 . . . . . . 7 𝑛𝜑
17 nfcv 2751 . . . . . . . 8 𝑛
18 nfcv 2751 . . . . . . . . 9 𝑛(-π(,)0)
19 nfcv 2751 . . . . . . . . . 10 𝑛(𝐹‘(𝑋 + 𝑠))
20 nfcv 2751 . . . . . . . . . 10 𝑛 ·
21 nfcv 2751 . . . . . . . . . 10 𝑛((𝐷𝑚)‘𝑠)
2219, 20, 21nfov 6575 . . . . . . . . 9 𝑛((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠))
2318, 22nfitg 23347 . . . . . . . 8 𝑛∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠
2417, 23nfmpt 4674 . . . . . . 7 𝑛(𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)
25 nfcv 2751 . . . . . . . . 9 𝑛(0(,)π)
2625, 22nfitg 23347 . . . . . . . 8 𝑛∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠
2717, 26nfmpt 4674 . . . . . . 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 4675 . . . . . . . . . . . . 13 𝑛(𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
3129, 30nfcxfr 2749 . . . . . . . . . . . 12 𝑛𝐴
32 nfcv 2751 . . . . . . . . . . . 12 𝑛0
3331, 32nffv 6110 . . . . . . . . . . 11 𝑛(𝐴‘0)
34 nfcv 2751 . . . . . . . . . . 11 𝑛 /
35 nfcv 2751 . . . . . . . . . . 11 𝑛2
3633, 34, 35nfov 6575 . . . . . . . . . 10 𝑛((𝐴‘0) / 2)
37 nfcv 2751 . . . . . . . . . 10 𝑛 +
38 nfcv 2751 . . . . . . . . . . 11 𝑛(1...𝑚)
3938nfsum1 14268 . . . . . . . . . 10 𝑛Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))
4036, 37, 39nfov 6575 . . . . . . . . 9 𝑛(((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
4117, 40nfmpt 4674 . . . . . . . 8 𝑛(𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
4228, 41nfcxfr 2749 . . . . . . 7 𝑛𝑍
43 fourierdlem112.f . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
44 fourierdlem112.25 . . . . . . . 8 (𝜑𝑋 ∈ ℝ)
45 eqid 2610 . . . . . . . 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 24015 . . . . . . . . . . . . 13 π ∈ ℂ
47462timesi 11024 . . . . . . . . . . . 12 (2 · π) = (π + π)
48 fourierdlem112.t . . . . . . . . . . . 12 𝑇 = (2 · π)
4946, 46subnegi 10239 . . . . . . . . . . . 12 (π − -π) = (π + π)
5047, 48, 493eqtr4i 2642 . . . . . . . . . . 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 24014 . . . . . . . . . . . . . 14 π ∈ ℝ
5554a1i 11 . . . . . . . . . . . . 13 (𝜑 → π ∈ ℝ)
5655renegcld 10336 . . . . . . . . . . . 12 (𝜑 → -π ∈ ℝ)
5756, 44readdcld 9948 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) ∈ ℝ)
5855, 44readdcld 9948 . . . . . . . . . . 11 (𝜑 → (π + 𝑋) ∈ ℝ)
59 negpilt0 38433 . . . . . . . . . . . . . 14 -π < 0
60 pipos 24016 . . . . . . . . . . . . . 14 0 < π
6154renegcli 10221 . . . . . . . . . . . . . . 15 -π ∈ ℝ
62 0re 9919 . . . . . . . . . . . . . . 15 0 ∈ ℝ
6361, 62, 54lttri 10042 . . . . . . . . . . . . . 14 ((-π < 0 ∧ 0 < π) → -π < π)
6459, 60, 63mp2an 704 . . . . . . . . . . . . 13 -π < π
6564a1i 11 . . . . . . . . . . . 12 (𝜑 → -π < π)
6656, 55, 44, 65ltadd1dd 10517 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) < (π + 𝑋))
67 oveq1 6556 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑦 + (𝑘 · 𝑇)) = (𝑥 + (𝑘 · 𝑇)))
6867eleq1d 2672 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
6968rexbidv 3034 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
7069cbvrabv 3172 . . . . . . . . . . . 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 3726 . . . . . . . . . . 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 39053 . . . . . . . . . 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 474 . . . . . . . . 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 474 . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
7775simprd 478 . . . . . . . 8 (𝜑𝑉 ∈ ((𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑁))
78 fourierdlem112.xran . . . . . . . 8 (𝜑𝑋 ∈ ran 𝑉)
7943adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℝ)
80 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑝𝑖) = (𝑝𝑗))
81 oveq1 6556 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
8281fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1)))
8380, 82breq12d 4596 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝𝑗) < (𝑝‘(𝑗 + 1))))
8483cbvralv 3147 . . . . . . . . . . . . . 14 (∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))
8584anbi2i 726 . . . . . . . . . . . . 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 3161 . . . . . . . . . . 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 4669 . . . . . . . . . 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 2632 . . . . . . . . 9 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))})
9052adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ)
9153adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑄 ∈ (𝑃𝑀))
92 fourierdlem112.fper . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
9392adantlr 747 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
94 eleq1 2676 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
9594anbi2d 736 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑗 ∈ (0..^𝑀))))
96 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
9781fveq2d 6107 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
9896, 97oveq12d 6567 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1))))
9998reseq2d 5317 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))))
10098oveq1d 6564 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) = (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
10199, 100eleq12d 2682 . . . . . . . . . . . 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 333 . . . . . . . . . . 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 2251 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
105104adantlr 747 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
10657adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (-π + 𝑋) ∈ ℝ)
10757rexrd 9968 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) ∈ ℝ*)
108 pnfxr 9971 . . . . . . . . . . . 12 +∞ ∈ ℝ*
109108a1i 11 . . . . . . . . . . 11 (𝜑 → +∞ ∈ ℝ*)
11058ltpnfd 11831 . . . . . . . . . . 11 (𝜑 → (π + 𝑋) < +∞)
111107, 109, 58, 66, 110eliood 38567 . . . . . . . . . 10 (𝜑 → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
112111adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
113 id 22 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^𝑁))
11472oveq2i 6560 . . . . . . . . . . 11 (0..^𝑁) = (0..^((#‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))
115113, 114syl6eleq 2698 . . . . . . . . . 10 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^((#‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))
116115adantl 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^((#‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))
11772oveq2i 6560 . . . . . . . . . . . 12 (0...𝑁) = (0...((#‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))
118 isoeq4 6470 . . . . . . . . . . . 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 5790 . . . . . . . . . 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 2632 . . . . . . . . 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 39097 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
123 fourierdlem112.fbd . . . . . . . . . 10 (𝜑 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤)
124123adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤)
125 nfra1 2925 . . . . . . . . . . 11 𝑡𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤
126 elioore 12076 . . . . . . . . . . . . 13 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
127 rspa 2914 . . . . . . . . . . . . 13 ((∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤𝑡 ∈ ℝ) → (abs‘(𝐹𝑡)) ≤ 𝑤)
128126, 127sylan2 490 . . . . . . . . . . . 12 ((∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(𝐹𝑡)) ≤ 𝑤)
129128ex 449 . . . . . . . . . . 11 (∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤 → (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘(𝐹𝑡)) ≤ 𝑤))
130125, 129ralrimi 2940 . . . . . . . . . 10 (∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤 → ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹𝑡)) ≤ 𝑤)
131130reximi 2994 . . . . . . . . 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 3587 . . . . . . . . . . . 12 ℝ ⊆ ℝ
134 dvfre 23520 . . . . . . . . . . . 12 ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
13543, 133, 134sylancl 693 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
136135adantr 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
137 eqid 2610 . . . . . . . . . . . . 13 (ℝ D 𝐹) = (ℝ D 𝐹)
13854a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → π ∈ ℝ)
13961a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → -π ∈ ℝ)
14098reseq2d 5317 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))))
141140, 100eleq12d 2682 . . . . . . . . . . . . . . . 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 333 . . . . . . . . . . . . . . 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 2251 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
145144adantlr 747 . . . . . . . . . . . . 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 9948 . . . . . . . . . . . . . 14 (𝜑 → (-π + 𝑋) ∈ ℝ)
148147adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → (-π + 𝑋) ∈ ℝ)
149147rexrd 9968 . . . . . . . . . . . . . . 15 (𝜑 → (-π + 𝑋) ∈ ℝ*)
15055, 146readdcld 9948 . . . . . . . . . . . . . . 15 (𝜑 → (π + 𝑋) ∈ ℝ)
15156, 55, 146, 65ltadd1dd 10517 . . . . . . . . . . . . . . 15 (𝜑 → (-π + 𝑋) < (π + 𝑋))
152150ltpnfd 11831 . . . . . . . . . . . . . . 15 (𝜑 → (π + 𝑋) < +∞)
153149, 109, 150, 151, 152eliood 38567 . . . . . . . . . . . . . 14 (𝜑 → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
154153adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
155 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = → (𝑘 · 𝑇) = ( · 𝑇))
156155oveq2d 6565 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝑦 + (𝑘 · 𝑇)) = (𝑦 + ( · 𝑇)))
157156eleq1d 2672 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + ( · 𝑇)) ∈ ran 𝑄))
158157cbvrexv 3148 . . . . . . . . . . . . . . . . . . 19 (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄)
159158rgenw 2908 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋))(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄)
160 rabbi 3097 . . . . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . . . . 17 {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄}
162161uneq2i 3726 . . . . . . . . . . . . . . . 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 6471 . . . . . . . . . . . . . . . 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 5790 . . . . . . . . . . . . . 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 2632 . . . . . . . . . . . . 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 eleq1 2676 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → (𝑣 ∈ dom (ℝ D 𝐹) ↔ 𝑢 ∈ dom (ℝ D 𝐹)))
168 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → ((ℝ D 𝐹)‘𝑣) = ((ℝ D 𝐹)‘𝑢))
169167, 168ifbieq1d 4059 . . . . . . . . . . . . . 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 4678 . . . . . . . . . . . . 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 39096 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
172 cncff 22504 . . . . . . . . . . . 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 5964 . . . . . . . . . . . 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 5340 . . . . . . . . . . 11 (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
176174, 175sylibr 223 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹))
177136, 176fssresd 5984 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
178 ax-resscn 9872 . . . . . . . . . . 11 ℝ ⊆ ℂ
179178a1i 11 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ℝ ⊆ ℂ)
180 cncffvrn 22509 . . . . . . . . . 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 691 . . . . . . . . 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 246 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ))
183 fourierdlem112.fdvbd . . . . . . . . . . 11 (𝜑 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
184183adantr 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
185 nfv 1830 . . . . . . . . . . . . . 14 𝑡(𝜑𝑖 ∈ (0..^𝑁))
186 nfra1 2925 . . . . . . . . . . . . . 14 𝑡𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧
187185, 186nfan 1816 . . . . . . . . . . . . 13 𝑡((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
188 fvres 6117 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
189188adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
190189fveq2d 6107 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡)))
191190adantlr 747 . . . . . . . . . . . . . . 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 788 . . . . . . . . . . . . . . . 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 3568 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → 𝑡 ∈ dom (ℝ D 𝐹))
194193adantlr 747 . . . . . . . . . . . . . . . 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 2914 . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . 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 4605 . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . 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 2940 . . . . . . . . . . . 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 449 . . . . . . . . . . 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 2999 . . . . . . . . . 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 2925 . . . . . . . . . . . 12 𝑡𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧
204188eqcomd 2616 . . . . . . . . . . . . . . . 16 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → ((ℝ D 𝐹)‘𝑡) = (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡))
205204fveq2d 6107 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘((ℝ D 𝐹)‘𝑡)) = (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)))
206205adantl 481 . . . . . . . . . . . . . 14 ((∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘((ℝ D 𝐹)‘𝑡)) = (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)))
207 rspa 2914 . . . . . . . . . . . . . 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 4605 . . . . . . . . . . . . 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 449 . . . . . . . . . . . 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 2940 . . . . . . . . . . 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 2999 . . . . . . . . 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 1830 . . . . . . . . . . . 12 𝑖(𝜑𝑗 ∈ (0..^𝑀))
215 nfcsb1v 3515 . . . . . . . . . . . . 13 𝑖𝑗 / 𝑖𝐶
216215nfel1 2765 . . . . . . . . . . . 12 𝑖𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗))
217214, 216nfim 1813 . . . . . . . . . . 11 𝑖((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
218 csbeq1a 3508 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝐶 = 𝑗 / 𝑖𝐶)
21999, 96oveq12d 6567 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
220218, 219eleq12d 2682 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) ↔ 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗))))
22195, 220imbi12d 333 . . . . . . . . . . 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 2250 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
224223adantlr 747 . . . . . . . . 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 39095 . . . . . . . 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 3515 . . . . . . . . . . . . 13 𝑖𝑗 / 𝑖𝑈
227226nfel1 2765 . . . . . . . . . . . 12 𝑖𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1)))
228214, 227nfim 1813 . . . . . . . . . . 11 𝑖((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
229 csbeq1a 3508 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝑈 = 𝑗 / 𝑖𝑈)
23099, 97oveq12d 6567 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
231229, 230eleq12d 2682 . . . . . . . . . . . 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 333 . . . . . . . . . . 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 2250 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
235234adantlr 747 . . . . . . . . 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 39098 . . . . . . . 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 2614 . . . . . . . . . 10 (𝑔 = 𝑠 → (𝑔 = 0 ↔ 𝑠 = 0))
238 oveq2 6557 . . . . . . . . . . . . 13 (𝑔 = 𝑠 → (𝑋 + 𝑔) = (𝑋 + 𝑠))
239238fveq2d 6107 . . . . . . . . . . . 12 (𝑔 = 𝑠 → (𝐹‘(𝑋 + 𝑔)) = (𝐹‘(𝑋 + 𝑠)))
240 breq2 4587 . . . . . . . . . . . . 13 (𝑔 = 𝑠 → (0 < 𝑔 ↔ 0 < 𝑠))
241240ifbid 4058 . . . . . . . . . . . 12 (𝑔 = 𝑠 → if(0 < 𝑔, 𝑅, 𝐿) = if(0 < 𝑠, 𝑅, 𝐿))
242239, 241oveq12d 6567 . . . . . . . . . . 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 6567 . . . . . . . . . 10 (𝑔 = 𝑠 → (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑅, 𝐿)) / 𝑠))
245237, 244ifbieq2d 4061 . . . . . . . . 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 4678 . . . . . . . 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 2614 . . . . . . . . . 10 (𝑜 = 𝑠 → (𝑜 = 0 ↔ 𝑠 = 0))
248 id 22 . . . . . . . . . . 11 (𝑜 = 𝑠𝑜 = 𝑠)
249 oveq1 6556 . . . . . . . . . . . . 13 (𝑜 = 𝑠 → (𝑜 / 2) = (𝑠 / 2))
250249fveq2d 6107 . . . . . . . . . . . 12 (𝑜 = 𝑠 → (sin‘(𝑜 / 2)) = (sin‘(𝑠 / 2)))
251250oveq2d 6565 . . . . . . . . . . 11 (𝑜 = 𝑠 → (2 · (sin‘(𝑜 / 2))) = (2 · (sin‘(𝑠 / 2))))
252248, 251oveq12d 6567 . . . . . . . . . 10 (𝑜 = 𝑠 → (𝑜 / (2 · (sin‘(𝑜 / 2)))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
253247, 252ifbieq2d 4061 . . . . . . . . 9 (𝑜 = 𝑠 → if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
254253cbvmptv 4678 . . . . . . . 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 6103 . . . . . . . . . 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 6103 . . . . . . . . . 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 6567 . . . . . . . . 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 4678 . . . . . . . 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 6557 . . . . . . . . . 10 (𝑑 = 𝑠 → ((𝑘 + (1 / 2)) · 𝑑) = ((𝑘 + (1 / 2)) · 𝑠))
260259fveq2d 6107 . . . . . . . . 9 (𝑑 = 𝑠 → (sin‘((𝑘 + (1 / 2)) · 𝑑)) = (sin‘((𝑘 + (1 / 2)) · 𝑠)))
261260cbvmptv 4678 . . . . . . . 8 (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑))) = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑠)))
262 fveq2 6103 . . . . . . . . . 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 6103 . . . . . . . . . 10 (𝑧 = 𝑠 → ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧) = ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑠))
264262, 263oveq12d 6567 . . . . . . . . 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 4678 . . . . . . . 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 6103 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝐷𝑚) = (𝐷𝑛))
267266fveq1d 6105 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((𝐷𝑚)‘𝑠) = ((𝐷𝑛)‘𝑠))
268267oveq2d 6565 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
269268adantr 480 . . . . . . . . . 10 ((𝑚 = 𝑛𝑠 ∈ (-π(,)0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
270269itgeq2dv 23354 . . . . . . . . 9 (𝑚 = 𝑛 → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
271270cbvmptv 4678 . . . . . . . 8 (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑛 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
272 oveq1 6556 . . . . . . . . . . . . . . . . . . 19 (𝑐 = 𝑘 → (𝑐 + (1 / 2)) = (𝑘 + (1 / 2)))
273272oveq1d 6564 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝑘 → ((𝑐 + (1 / 2)) · 𝑑) = ((𝑘 + (1 / 2)) · 𝑑))
274273fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑘 → (sin‘((𝑐 + (1 / 2)) · 𝑑)) = (sin‘((𝑘 + (1 / 2)) · 𝑑)))
275274mpteq2dv 4673 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑘 → (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑))) = (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑))))
276275fveq1d 6105 . . . . . . . . . . . . . . 15 (𝑐 = 𝑘 → ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧) = ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧))
277276oveq2d 6565 . . . . . . . . . . . . . 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 4673 . . . . . . . . . . . . 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 6105 . . . . . . . . . . . 12 (𝑐 = 𝑘 → ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) = ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠))
280279adantr 480 . . . . . . . . . . 11 ((𝑐 = 𝑘𝑠 ∈ (-π(,)0)) → ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) = ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠))
281280itgeq2dv 23354 . . . . . . . . . 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 6564 . . . . . . . . 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 4678 . . . . . . . 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 6556 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → (𝑦 mod (2 · π)) = (𝑠 mod (2 · π)))
290289eqeq1d 2612 . . . . . . . . . . . . 13 (𝑦 = 𝑠 → ((𝑦 mod (2 · π)) = 0 ↔ (𝑠 mod (2 · π)) = 0))
291 oveq2 6557 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → ((𝑚 + (1 / 2)) · 𝑦) = ((𝑚 + (1 / 2)) · 𝑠))
292291fveq2d 6107 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → (sin‘((𝑚 + (1 / 2)) · 𝑦)) = (sin‘((𝑚 + (1 / 2)) · 𝑠)))
293 oveq1 6556 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑠 → (𝑦 / 2) = (𝑠 / 2))
294293fveq2d 6107 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → (sin‘(𝑦 / 2)) = (sin‘(𝑠 / 2)))
295294oveq2d 6565 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → ((2 · π) · (sin‘(𝑦 / 2))) = ((2 · π) · (sin‘(𝑠 / 2))))
296292, 295oveq12d 6567 . . . . . . . . . . . . 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 4061 . . . . . . . . . . . 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 4678 . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑘𝑠 ∈ ℝ) → 𝑚 = 𝑘)
300299oveq2d 6565 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (2 · 𝑚) = (2 · 𝑘))
301300oveq1d 6564 . . . . . . . . . . . . . 14 ((𝑚 = 𝑘𝑠 ∈ ℝ) → ((2 · 𝑚) + 1) = ((2 · 𝑘) + 1))
302301oveq1d 6564 . . . . . . . . . . . . 13 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (((2 · 𝑚) + 1) / (2 · π)) = (((2 · 𝑘) + 1) / (2 · π)))
303299oveq1d 6564 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (𝑚 + (1 / 2)) = (𝑘 + (1 / 2)))
304303oveq1d 6564 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑘𝑠 ∈ ℝ) → ((𝑚 + (1 / 2)) · 𝑠) = ((𝑘 + (1 / 2)) · 𝑠))
305304fveq2d 6107 . . . . . . . . . . . . . 14 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (sin‘((𝑚 + (1 / 2)) · 𝑠)) = (sin‘((𝑘 + (1 / 2)) · 𝑠)))
306305oveq1d 6564 . . . . . . . . . . . . 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 4056 . . . . . . . . . . . 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 4672 . . . . . . . . . . 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 2656 . . . . . . . . . 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 4678 . . . . . . . . 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 2632 . . . . . . . 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 2610 . . . . . . . 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 2610 . . . . . . . 8 ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙))) = ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))
314 eqid 2610 . . . . . . . 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 6467 . . . . . . . . 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 5774 . . . . . . . 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 6103 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝑉𝑗) = (𝑉𝑖))
318317oveq1d 6564 . . . . . . . . 9 (𝑗 = 𝑖 → ((𝑉𝑗) − 𝑋) = ((𝑉𝑖) − 𝑋))
319318cbvmptv 4678 . . . . . . . 8 (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) = (𝑖 ∈ (0...𝑁) ↦ ((𝑉𝑖) − 𝑋))
320 eqid 2610 . . . . . . . 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 6103 . . . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . . 15 (𝑎 = 𝑠 → ((𝑏 + (1 / 2)) · 𝑎) = ((𝑏 + (1 / 2)) · 𝑠))
323322fveq2d 6107 . . . . . . . . . . . . . 14 (𝑎 = 𝑠 → (sin‘((𝑏 + (1 / 2)) · 𝑎)) = (sin‘((𝑏 + (1 / 2)) · 𝑠)))
324321, 323oveq12d 6567 . . . . . . . . . . . . 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 23349 . . . . . . . . . . . 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 6106 . . . . . . . . . . 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 4590 . . . . . . . . . 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 726 . . . . . . . . 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 23349 . . . . . . . . . . 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 6106 . . . . . . . . . 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 4590 . . . . . . . . 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 729 . . . . . . . 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 39102 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) ⇝ (𝐿 / 2))
334 nnex 10903 . . . . . . . . . 10 ℕ ∈ V
335334mptex 6390 . . . . . . . . 9 (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))) ∈ V
33628, 335eqeltri 2684 . . . . . . . 8 𝑍 ∈ V
337336a1i 11 . . . . . . 7 (𝜑𝑍 ∈ V)
338268adantr 480 . . . . . . . . . 10 ((𝑚 = 𝑛𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
339338itgeq2dv 23354 . . . . . . . . 9 (𝑚 = 𝑛 → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
340339cbvmptv 4678 . . . . . . . 8 (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑛 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
341279adantr 480 . . . . . . . . . . 11 ((𝑐 = 𝑘𝑠 ∈ (0(,)π)) → ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠) = ((𝑧 ∈ (-π[,]π) ↦ (((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑧) · ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧)))‘𝑠))
342341itgeq2dv 23354 . . . . . . . . . 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 6564 . . . . . . . . 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 4678 . . . . . . . 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 2610 . . . . . . . 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 2610 . . . . . . . 8 ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π))) = ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))
347 eqid 2610 . . . . . . . 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 6467 . . . . . . . . 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 5774 . . . . . . . 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 2610 . . . . . . . 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 23349 . . . . . . . . . . . 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 6106 . . . . . . . . . . 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 4590 . . . . . . . . . 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 726 . . . . . . . . 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 23349 . . . . . . . . . . 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 6106 . . . . . . . . . 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 4590 . . . . . . . . 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 729 . . . . . . . 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 39103 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) ⇝ (𝑅 / 2))
360 eqidd 2611 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
361270adantl 481 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 = 𝑛) → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
362 simpr 476 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
363 elioore 12076 . . . . . . . . . . 11 (𝑠 ∈ (-π(,)0) → 𝑠 ∈ ℝ)
36443adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ ℝ) → 𝐹:ℝ⟶ℝ)
36544adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ ℝ) → 𝑋 ∈ ℝ)
366 simpr 476 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ ℝ) → 𝑠 ∈ ℝ)
367365, 366readdcld 9948 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ ℝ) → (𝑋 + 𝑠) ∈ ℝ)
368364, 367ffvelrnd 6268 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ ℝ) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
369368adantlr 747 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
370288dirkerre 38988 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
371370adantll 746 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
372369, 371remulcld 9949 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
373363, 372sylan2 490 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
374 ioossicc 12130 . . . . . . . . . . . . 13 (-π(,)0) ⊆ (-π[,]0)
37561leidi 10441 . . . . . . . . . . . . . 14 -π ≤ -π
37662, 54, 60ltleii 10039 . . . . . . . . . . . . . 14 0 ≤ π
377 iccss 12112 . . . . . . . . . . . . . 14 (((-π ∈ ℝ ∧ π ∈ ℝ) ∧ (-π ≤ -π ∧ 0 ≤ π)) → (-π[,]0) ⊆ (-π[,]π))
37861, 54, 375, 376, 377mp4an 705 . . . . . . . . . . . . 13 (-π[,]0) ⊆ (-π[,]π)
379374, 378sstri 3577 . . . . . . . . . . . 12 (-π(,)0) ⊆ (-π[,]π)
380379a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (-π(,)0) ⊆ (-π[,]π))
381 ioombl 23140 . . . . . . . . . . . 12 (-π(,)0) ∈ dom vol
382381a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (-π(,)0) ∈ dom vol)
38343adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℝ)
38444adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ (-π[,]π)) → 𝑋 ∈ ℝ)
38556, 55iccssred 38574 . . . . . . . . . . . . . . . 16 (𝜑 → (-π[,]π) ⊆ ℝ)
386385sselda 3568 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ (-π[,]π)) → 𝑠 ∈ ℝ)
387384, 386readdcld 9948 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (-π[,]π)) → (𝑋 + 𝑠) ∈ ℝ)
388383, 387ffvelrnd 6268 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (-π[,]π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
389388adantlr 747 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
390 iccssre 12126 . . . . . . . . . . . . . . . 16 ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
39161, 54, 390mp2an 704 . . . . . . . . . . . . . . 15 (-π[,]π) ⊆ ℝ
392391sseli 3564 . . . . . . . . . . . . . 14 (𝑠 ∈ (-π[,]π) → 𝑠 ∈ ℝ)
393392, 370sylan2 490 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
394393adantll 746 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
395389, 394remulcld 9949 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
39661a1i 11 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → -π ∈ ℝ)
39754a1i 11 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → π ∈ ℝ)
39843adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
39944adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑋 ∈ ℝ)
40076adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑁 ∈ ℕ)
40177adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑉 ∈ ((𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑛)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑛) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑁))
402122adantlr 747 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
403225adantlr 747 . . . . . . . . . . . 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 747 . . . . . . . . . . . 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 39000 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝐷𝑛) ∈ (ℝ–cn→ℝ))
406405adantl 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) ∈ (ℝ–cn→ℝ))
407 eqid 2610 . . . . . . . . . . . 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 39083 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
409380, 382, 395, 408iblss 23377 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (-π(,)0) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
410373, 409itgcl 23356 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 ∈ ℂ)
411360, 361, 362, 410fvmptd 6197 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
412411, 410eqeltrd 2688 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) ∈ ℂ)
413 eqidd 2611 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
414339adantl 481 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 = 𝑛) → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
41543adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (0(,)π)) → 𝐹:ℝ⟶ℝ)
41644adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (0(,)π)) → 𝑋 ∈ ℝ)
417 elioore 12076 . . . . . . . . . . . . . . 15 (𝑠 ∈ (0(,)π) → 𝑠 ∈ ℝ)
418417adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (0(,)π)) → 𝑠 ∈ ℝ)
419416, 418readdcld 9948 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (0(,)π)) → (𝑋 + 𝑠) ∈ ℝ)
420415, 419ffvelrnd 6268 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
421420adantlr 747 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
422417, 370sylan2 490 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (0(,)π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
423422adantll 746 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
424421, 423remulcld 9949 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
425 ioossicc 12130 . . . . . . . . . . . . 13 (0(,)π) ⊆ (0[,]π)
42661, 62, 59ltleii 10039 . . . . . . . . . . . . . 14 -π ≤ 0
42754leidi 10441 . . . . . . . . . . . . . 14 π ≤ π
428 iccss 12112 . . . . . . . . . . . . . 14 (((-π ∈ ℝ ∧ π ∈ ℝ) ∧ (-π ≤ 0 ∧ π ≤ π)) → (0[,]π) ⊆ (-π[,]π))
42961, 54, 426, 427, 428mp4an 705 . . . . . . . . . . . . 13 (0[,]π) ⊆ (-π[,]π)
430425, 429sstri 3577 . . . . . . . . . . . 12 (0(,)π) ⊆ (-π[,]π)
431430a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (0(,)π) ⊆ (-π[,]π))
432 ioombl 23140 . . . . . . . . . . . 12 (0(,)π) ∈ dom vol
433432a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (0(,)π) ∈ dom vol)
434431, 433, 395, 408iblss 23377 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (0(,)π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
435424, 434itgcl 23356 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 ∈ ℂ)
436413, 414, 362, 435fvmptd 6197 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
437436, 435eqeltrd 2688 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) ∈ ℂ)
438 eleq1 2676 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑚 ∈ ℕ ↔ 𝑛 ∈ ℕ))
439438anbi2d 736 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝜑𝑚 ∈ ℕ) ↔ (𝜑𝑛 ∈ ℕ)))
440 fveq2 6103 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑍𝑚) = (𝑍𝑛))
441270, 339oveq12d 6567 . . . . . . . . . . 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 2625 . . . . . . . . . 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 333 . . . . . . . . 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 6556 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝑛 · 𝑥) = (𝑚 · 𝑥))
445444fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (cos‘(𝑛 · 𝑥)) = (cos‘(𝑚 · 𝑥)))
446445oveq2d 6565 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (cos‘(𝑚 · 𝑥))))
447446adantr 480 . . . . . . . . . . . . . 14 ((𝑛 = 𝑚𝑥 ∈ (-π(,)π)) → ((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (cos‘(𝑚 · 𝑥))))
448447itgeq2dv 23354 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 = ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥)
449448oveq1d 6564 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π) = (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π))
450449cbvmptv 4678 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) = (𝑚 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π))
45129, 450eqtri 2632 . . . . . . . . . 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 6107 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (sin‘(𝑛 · 𝑥)) = (sin‘(𝑚 · 𝑥)))
454453oveq2d 6565 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (sin‘(𝑚 · 𝑥))))
455454adantr 480 . . . . . . . . . . . . . 14 ((𝑛 = 𝑚𝑥 ∈ (-π(,)π)) → ((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (sin‘(𝑚 · 𝑥))))
456455itgeq2dv 23354 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 = ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥)
457456oveq1d 6564 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π) = (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
458457cbvmptv 4678 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) = (𝑚 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
459452, 458eqtri 2632 . . . . . . . . . 10 𝐵 = (𝑚 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
460 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐴𝑛) = (𝐴𝑘))
461 oveq1 6556 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
462461fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
463460, 462oveq12d 6567 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))))
464 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐵𝑛) = (𝐵𝑘))
465461fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
466464, 465oveq12d 6567 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
467463, 466oveq12d 6567 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
468467cbvsumv 14274 . . . . . . . . . . . . 13 Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
469468oveq2i 6560 . . . . . . . . . . . 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 4669 . . . . . . . . . . 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 6557 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
472471sumeq1d 14279 . . . . . . . . . . . . . 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 6565 . . . . . . . . . . . . 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 4678 . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝐴𝑘) = (𝐴𝑚))
476 oveq1 6556 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → (𝑘 · 𝑋) = (𝑚 · 𝑋))
477476fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (cos‘(𝑘 · 𝑋)) = (cos‘(𝑚 · 𝑋)))
478475, 477oveq12d 6567 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) = ((𝐴𝑚) · (cos‘(𝑚 · 𝑋))))
479 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝐵𝑘) = (𝐵𝑚))
480476fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (sin‘(𝑘 · 𝑋)) = (sin‘(𝑚 · 𝑋)))
481479, 480oveq12d 6567 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) = ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))
482478, 481oveq12d 6567 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋)))))
483482cbvsumv 14274 . . . . . . . . . . . . . 14 Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))
484483oveq2i 6560 . . . . . . . . . . . . 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 4669 . . . . . . . . . . . 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 2632 . . . . . . . . . . 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 2636 . . . . . . . . . 10 𝑍 = (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))))
488 oveq2 6557 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑋 + 𝑦) = (𝑋 + 𝑥))
489488fveq2d 6107 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑥)))
490 fveq2 6103 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝐷𝑚)‘𝑦) = ((𝐷𝑚)‘𝑥))
491489, 490oveq12d 6567 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹‘(𝑋 + 𝑦)) · ((𝐷𝑚)‘𝑦)) = ((𝐹‘(𝑋 + 𝑥)) · ((𝐷𝑚)‘𝑥)))
492491cbvmptv 4678 . . . . . . . . . 10 (𝑦 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑦)) · ((𝐷𝑚)‘𝑦))) = (𝑥 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷𝑚)‘𝑥)))
493 eqid 2610 . . . . . . . . . 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 6103 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑄𝑗) = (𝑄𝑖))
495494oveq1d 6564 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝑄𝑗) − 𝑋) = ((𝑄𝑖) − 𝑋))
496495cbvmptv 4678 . . . . . . . . . 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 39110 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
498443, 497chvarv 2251 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝑍𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠))
499411, 436oveq12d 6567 . . . . . . . 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 2647 . . . . . . 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 38682 . . . . . 6 (𝜑𝑍 ⇝ ((𝐿 / 2) + (𝑅 / 2)))
502 limccl 23445 . . . . . . . 8 ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋) ⊆ ℂ
503502, 285sseldi 3566 . . . . . . 7 (𝜑𝐿 ∈ ℂ)
504 limccl 23445 . . . . . . . 8 ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋) ⊆ ℂ
505504, 284sseldi 3566 . . . . . . 7 (𝜑𝑅 ∈ ℂ)
506 2cnd 10970 . . . . . . 7 (𝜑 → 2 ∈ ℂ)
507 2pos 10989 . . . . . . . . 9 0 < 2
508507a1i 11 . . . . . . . 8 (𝜑 → 0 < 2)
509508gt0ne0d 10471 . . . . . . 7 (𝜑 → 2 ≠ 0)
510503, 505, 506, 509divdird 10718 . . . . . 6 (𝜑 → ((𝐿 + 𝑅) / 2) = ((𝐿 / 2) + (𝑅 / 2)))
511501, 510breqtrrd 4611 . . . . 5 (𝜑𝑍 ⇝ ((𝐿 + 𝑅) / 2))
512 0nn0 11184 . . . . . . . 8 0 ∈ ℕ0
51343adantr 480 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → 𝐹:ℝ⟶ℝ)
514 eqid 2610 . . . . . . . . . 10 (-π(,)π) = (-π(,)π)
515 ioossre 12106 . . . . . . . . . . . . . 14 (-π(,)π) ⊆ ℝ
516515a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ⊆ ℝ)
51743, 516feqresmpt 6160 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ (-π(,)π)) = (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)))
518 ioossicc 12130 . . . . . . . . . . . . . 14 (-π(,)π) ⊆ (-π[,]π)
519518a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ⊆ (-π[,]π))
520 ioombl 23140 . . . . . . . . . . . . . 14 (-π(,)π) ∈ dom vol
521520a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ∈ dom vol)
52243adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℝ)
523385sselda 3568 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (-π[,]π)) → 𝑥 ∈ ℝ)
524522, 523ffvelrnd 6268 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (-π[,]π)) → (𝐹𝑥) ∈ ℝ)
52543, 385feqresmpt 6160 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (-π[,]π)) = (𝑥 ∈ (-π[,]π) ↦ (𝐹𝑥)))
526178a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ℝ ⊆ ℂ)
52743, 526fssd 5970 . . . . . . . . . . . . . . . 16 (𝜑𝐹:ℝ⟶ℂ)
528527, 385fssresd 5984 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 ↾ (-π[,]π)):(-π[,]π)⟶ℂ)
529 ioossicc 12130 . . . . . . . . . . . . . . . . . 18 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
53061rexri 9976 . . . . . . . . . . . . . . . . . . . 20 -π ∈ ℝ*
531530a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ*)
53254rexri 9976 . . . . . . . . . . . . . . . . . . . 20 π ∈ ℝ*
533532a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → π ∈ ℝ*)
53451, 52, 53fourierdlem15 39015 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑄:(0...𝑀)⟶(-π[,]π))
535534adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
536 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
537531, 533, 535, 536fourierdlem8 39008 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
538529, 537syl5ss 3579 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
539538resabs1d 5348 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
540539, 103eqeltrd 2688 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
541539eqcomd 2616 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
542541oveq1d 6564 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
543222, 542eleqtrd 2690 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
544541oveq1d 6564 . . . . . . . . . . . . . . . 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 2690 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
54651, 52, 53, 528, 540, 543, 545fourierdlem69 39068 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (-π[,]π)) ∈ 𝐿1)
547525, 546eqeltrrd 2689 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (-π[,]π) ↦ (𝐹𝑥)) ∈ 𝐿1)
548519, 521, 524, 547iblss 23377 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1)
549517, 548eqeltrd 2688 . . . . . . . . . . 11 (𝜑 → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
550549adantr 480 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
551 simpr 476 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → 0 ∈ ℕ0)
552513, 514, 550, 29, 551fourierdlem16 39016 . . . . . . . . 9 ((𝜑 ∧ 0 ∈ ℕ0) → (((𝐴‘0) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (cos‘(0 · 𝑥))) d𝑥 ∈ ℝ))
553552simplld 787 . . . . . . . 8 ((𝜑 ∧ 0 ∈ ℕ0) → (𝐴‘0) ∈ ℝ)
554512, 553mpan2 703 . . . . . . 7 (𝜑 → (𝐴‘0) ∈ ℝ)
555554rehalfcld 11156 . . . . . 6 (𝜑 → ((𝐴‘0) / 2) ∈ ℝ)
556555recnd 9947 . . . . 5 (𝜑 → ((𝐴‘0) / 2) ∈ ℂ)
557334mptex 6390 . . . . . 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 476 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
560555adantr 480 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ((𝐴‘0) / 2) ∈ ℝ)
561 fzfid 12634 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (1...𝑚) ∈ Fin)
562 simpll 786 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → 𝜑)
563 elfznn 12241 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑚) → 𝑛 ∈ ℕ)
564563adantl 481 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → 𝑛 ∈ ℕ)
565 simpl 472 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝜑)
566362nnnn0d 11228 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
567 eleq1 2676 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘 ∈ ℕ0𝑛 ∈ ℕ0))
568567anbi2d 736 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝜑𝑘 ∈ ℕ0) ↔ (𝜑𝑛 ∈ ℕ0)))
569 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝐴𝑘) = (𝐴𝑛))
570569eleq1d 2672 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝐴𝑘) ∈ ℝ ↔ (𝐴𝑛) ∈ ℝ))
571568, 570imbi12d 333 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℝ) ↔ ((𝜑𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℝ)))
57243adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → 𝐹:ℝ⟶ℝ)
573549adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
574 simpr 476 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
575572, 514, 573, 29, 574fourierdlem16 39016 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ0) → (((𝐴𝑘) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑘 · 𝑥))) d𝑥 ∈ ℝ))
576575simplld 787 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℝ)
577571, 576chvarv 2251 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℝ)
578565, 566, 577syl2anc 691 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ∈ ℝ)
579362nnred 10912 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
580579, 399remulcld 9949 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝑛 · 𝑋) ∈ ℝ)
581580recoscld 14713 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (cos‘(𝑛 · 𝑋)) ∈ ℝ)
582578, 581remulcld 9949 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) ∈ ℝ)
583 eleq1 2676 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝑘 ∈ ℕ ↔ 𝑛 ∈ ℕ))
584583anbi2d 736 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → ((𝜑𝑘 ∈ ℕ) ↔ (𝜑𝑛 ∈ ℕ)))
585 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝐵𝑘) = (𝐵𝑛))
586585eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → ((𝐵𝑘) ∈ ℝ ↔ (𝐵𝑛) ∈ ℝ))
587584, 586imbi12d 333 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → (((𝜑𝑘 ∈ ℕ) → (𝐵𝑘) ∈ ℝ) ↔ ((𝜑𝑛 ∈ ℕ) → (𝐵𝑛) ∈ ℝ)))
58843adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
589549adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
590 simpr 476 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
591588, 514, 589, 452, 590fourierdlem21 39021 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (((𝐵𝑘) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ ((𝐹𝑥) · (sin‘(𝑘 · 𝑥)))) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑘 · 𝑥))) d𝑥 ∈ ℝ))
592591simplld 787 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐵𝑘) ∈ ℝ)
593587, 592chvarv 2251 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐵𝑛) ∈ ℝ)
594580resincld 14712 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (sin‘(𝑛 · 𝑋)) ∈ ℝ)
595593, 594remulcld 9949 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) ∈ ℝ)
596582, 595readdcld 9948 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
597562, 564, 596syl2anc 691 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
598561, 597fsumrecl 14312 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
599560, 598readdcld 9948 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ ℝ)
60028fvmpt2 6200 . . . . . . . 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 691 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
602601, 599eqeltrd 2688 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) ∈ ℝ)
603602recnd 9947 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) ∈ ℂ)
604 eqidd 2611 . . . . . . 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 6557 . . . . . . . . 9 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
606605sumeq1d 14279 . . . . . . . 8 (𝑛 = 𝑚 → Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
607606adantl 481 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 = 𝑚) → Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
608 sumex 14266 . . . . . . . 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 6197 . . . . . 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 9947 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝐴‘0) / 2) ∈ ℂ)
612598recnd 9947 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℂ)
613611, 612pncan2d 10273 . . . . . . 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 2661 . . . . . 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 6577 . . . . . . . . 9 (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ V
61628fvmpt2 6200 . . . . . . . . 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 693 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
618617eqcomd 2616 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑍𝑚))
619618oveq1d 6564 . . . . . 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 2648 . . . . 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 14216 . . . 4 (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
622 seqex 12665 . . . . . 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 2611 . . . . . . 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 6557 . . . . . . . . 9 (𝑛 = 𝑙 → (1...𝑛) = (1...𝑙))
626625sumeq1d 14279 . . . . . . . 8 (𝑛 = 𝑙 → Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
627626adantl 481 . . . . . . 7 (((𝜑𝑙 ∈ ℕ) ∧ 𝑛 = 𝑙) → Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
628 simpr 476 . . . . . . 7 ((𝜑𝑙 ∈ ℕ) → 𝑙 ∈ ℕ)
629 fzfid 12634 . . . . . . . 8 ((𝜑𝑙 ∈ ℕ) → (1...𝑙) ∈ Fin)
630 elfznn 12241 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ)
631630nnnn0d 11228 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ0)
632631, 576sylan2 490 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (𝐴𝑘) ∈ ℝ)
633630nnred 10912 . . . . . . . . . . . . . 14 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℝ)
634633adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1...𝑙)) → 𝑘 ∈ ℝ)
635146adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1...𝑙)) → 𝑋 ∈ ℝ)
636634, 635remulcld 9949 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (1...𝑙)) → (𝑘 · 𝑋) ∈ ℝ)
637636recoscld 14713 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (cos‘(𝑘 · 𝑋)) ∈ ℝ)
638632, 637remulcld 9949 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1...𝑙)) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
639630, 592sylan2 490 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (𝐵𝑘) ∈ ℝ)
640636resincld 14712 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (sin‘(𝑘 · 𝑋)) ∈ ℝ)
641639, 640remulcld 9949 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1...𝑙)) → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) ∈ ℝ)
642638, 641readdcld 9948 . . . . . . . . 9 ((𝜑𝑘 ∈ (1...𝑙)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
643642adantlr 747 . . . . . . . 8 (((𝜑𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
644629, 643fsumrecl 14312 . . . . . . 7 ((𝜑𝑙 ∈ ℕ) → Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
645624, 627, 628, 644fvmptd 6197 . . . . . 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 eleq1 2676 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑛 ∈ ℕ ↔ 𝑙 ∈ ℕ))
647646anbi2d 736 . . . . . . . 8 (𝑛 = 𝑙 → ((𝜑𝑛 ∈ ℕ) ↔ (𝜑𝑙 ∈ ℕ)))
648 fveq2 6103 . . . . . . . . 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 2625 . . . . . . . 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 333 . . . . . . 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 2611 . . . . . . . . 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 6103 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
653 oveq1 6556 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑗 · 𝑋) = (𝑘 · 𝑋))
654653fveq2d 6107 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (cos‘(𝑗 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
655652, 654oveq12d 6567 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) = ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))))
656 fveq2 6103 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐵𝑗) = (𝐵𝑘))
657653fveq2d 6107 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (sin‘(𝑗 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
658656, 657oveq12d 6567 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))) = ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
659655, 658oveq12d 6567 . . . . . . . . . 10 (𝑗 = 𝑘 → (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
660659adantl 481 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑗 = 𝑘) → (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
661 elfznn 12241 . . . . . . . . . 10 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
662661adantl 481 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
663 simpll 786 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
664 nnnn0 11176 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
665 nn0re 11178 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
666665adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
667146adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → 𝑋 ∈ ℝ)
668666, 667remulcld 9949 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ0) → (𝑘 · 𝑋) ∈ ℝ)
669668recoscld 14713 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ0) → (cos‘(𝑘 · 𝑋)) ∈ ℝ)
670576, 669remulcld 9949 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
671664, 670sylan2 490 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
672664, 668sylan2 490 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝑘 · 𝑋) ∈ ℝ)
673672resincld 14712 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (sin‘(𝑘 · 𝑋)) ∈ ℝ)
674592, 673remulcld 9949 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) ∈ ℝ)
675671, 674readdcld 9948 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
676663, 662, 675syl2anc 691 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
677651, 660, 662, 676fvmptd 6197 . . . . . . . 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 2698 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
679676recnd 9947 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℂ)
680677, 678, 679fsumser 14308 . . . . . . 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 2251 . . . . . 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 2644 . . . . 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 14146 . . . 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 221 . . 3 (𝜑 → seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
68513, 684eqbrtrd 4605 . 2 (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
686 eqidd 2611 . . . . . 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 6103 . . . . . . . . 9 (𝑗 = 𝑛 → (𝐴𝑗) = (𝐴𝑛))
688 oveq1 6556 . . . . . . . . . 10 (𝑗 = 𝑛 → (𝑗 · 𝑋) = (𝑛 · 𝑋))
689688fveq2d 6107 . . . . . . . . 9 (𝑗 = 𝑛 → (cos‘(𝑗 · 𝑋)) = (cos‘(𝑛 · 𝑋)))
690687, 689oveq12d 6567 . . . . . . . 8 (𝑗 = 𝑛 → ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) = ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))))
691 fveq2 6103 . . . . . . . . 9 (𝑗 = 𝑛 → (𝐵𝑗) = (𝐵𝑛))
692688fveq2d 6107 . . . . . . . . 9 (𝑗 = 𝑛 → (sin‘(𝑗 · 𝑋)) = (sin‘(𝑛 · 𝑋)))
693691, 692oveq12d 6567 . . . . . . . 8 (𝑗 = 𝑛 → ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))) = ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))
694690, 693oveq12d 6567 . . . . . . 7 (𝑗 = 𝑛 → (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
695694adantl 481 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 = 𝑛) → (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
696686, 695, 362, 596fvmptd 6197 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))‘𝑛) = (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
697596recnd 9947 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℂ)
69814, 15, 696, 697, 684isumclim 14330 . . . 4 (𝜑 → Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
699698oveq2d 6565 . . 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 9938 . . . . 5 (𝜑 → (𝐿 + 𝑅) ∈ ℂ)
701700halfcld 11154 . . . 4 (𝜑 → ((𝐿 + 𝑅) / 2) ∈ ℂ)
702556, 701pncan3d 10274 . . 3 (𝜑 → (((𝐴‘0) / 2) + (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) = ((𝐿 + 𝑅) / 2))
703699, 702eqtrd 2644 . 2 (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2))
704685, 703jca 553 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ⦋csb 3499   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ifcif 4036  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   ↾ cres 5040  ℩cio 5766  ⟶wf 5800  ‘cfv 5804   Isom wiso 5805  ℩crio 6510  (class class class)co 6549   ↑𝑚 cmap 7744  supcsup 8229  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  +∞cpnf 9950  -∞cmnf 9951  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  -cneg 10146   / cdiv 10563  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ℝ+crp 11708  (,)cioo 12046  (,]cioc 12047  [,]cicc 12049  ...cfz 12197  ..^cfzo 12334  ⌊cfl 12453   mod cmo 12530  seqcseq 12663  #chash 12979  abscabs 13822   ⇝ cli 14063  Σcsu 14264  sincsin 14633  cosccos 14634  πcpi 14636  –cn→ccncf 22487  volcvol 23039  𝐿1cibl 23192  ∫citg 23193   limℂ climc 23432   D cdv 23433 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-pi 14642  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-t1 20928  df-haus 20929  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195  df-itg2 23196  df-ibl 23197  df-itg 23198  df-0p 23243  df-ditg 23417  df-limc 23436  df-dv 23437 This theorem is referenced by:  fourierdlem113  39112
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