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Theorem fourierdlem112 42380
Description: Here abbreviations (local definitions) are introduced to prove the fourier 42387 theorem. (𝑍𝑚) is the mth partial sum of the fourier series. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem112.f (𝜑𝐹:ℝ⟶ℝ)
fourierdlem112.d 𝐷 = (𝑚 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))
fourierdlem112.p 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem112.m (𝜑𝑀 ∈ ℕ)
fourierdlem112.q (𝜑𝑄 ∈ (𝑃𝑀))
fourierdlem112.n 𝑁 = ((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)
fourierdlem112.v 𝑉 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))
fourierdlem112.x (𝜑𝑋 ∈ ℝ)
fourierdlem112.xran (𝜑𝑋 ∈ ran 𝑉)
fourierdlem112.t 𝑇 = (2 · π)
fourierdlem112.fper ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
fourierdlem112.fcn ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem112.c ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
fourierdlem112.u ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
fourierdlem112.fdvcn ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem112.e (𝜑𝐸 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim 𝑋))
fourierdlem112.i (𝜑𝐼 ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim 𝑋))
fourierdlem112.l (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋))
fourierdlem112.r (𝜑𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))
fourierdlem112.a 𝐴 = (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
fourierdlem112.b 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
fourierdlem112.z 𝑍 = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
fourierdlem112.23 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
fourierdlem112.fbd (𝜑 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤)
fourierdlem112.fdvbd (𝜑 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
fourierdlem112.25 (𝜑𝑋 ∈ ℝ)
Assertion
Ref Expression
fourierdlem112 (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))
Distinct variable groups:   𝐴,𝑘,𝑚,𝑛   𝐵,𝑘,𝑚,𝑛   𝑡,𝐶,𝑚   𝑥,𝐶,𝑚   𝐷,𝑖,𝑘,𝑚,𝑛,𝑥,𝑦   𝑖,𝐹,𝑡,𝑧   𝑦,𝐹,𝑡,𝑘,𝑚   𝑧,𝑘,𝑚   𝑛,𝐹   𝑤,𝐹,𝑖,𝑡,𝑧   𝑥,𝐹   𝑖,𝐿,𝑡,𝑧,𝑘,𝑚   𝑛,𝐿   𝑤,𝐿   𝑓,𝑀,𝑖,𝑡,𝑦,𝑚   𝑛,𝑀,𝑥   𝑀,𝑝,𝑖,𝑛,𝑦   𝑖,𝑁,𝑡,𝑤,𝑧   𝑓,𝑁,𝑦,𝑚   𝑛,𝑁,𝑝   𝑥,𝑁,𝑓   𝑄,𝑓,𝑖,𝑡,𝑦,𝑘,𝑚   𝑄,𝑛,𝑥   𝑄,𝑝,𝑘   𝑅,𝑖,𝑡,𝑧,𝑘,𝑚   𝑅,𝑛   𝑤,𝑅   𝑇,𝑓,𝑡,𝑦,𝑖,𝑘,𝑚   𝑇,𝑛,𝑥   𝑇,𝑝   𝑡,𝑈,𝑚   𝑥,𝑈   𝑖,𝑉,𝑡,𝑤,𝑧   𝑓,𝑉,𝑘,𝑚   𝑛,𝑉,𝑝   𝑥,𝑉   𝑖,𝑋,𝑡,𝑧   𝑓,𝑋,𝑦,𝑘,𝑚   𝑛,𝑋,𝑝   𝑤,𝑋   𝑥,𝑋   𝑚,𝑍   𝜑,𝑖,𝑡,𝑤,𝑧   𝜑,𝑓,𝑘,𝑚,𝑦   𝜑,𝑛   𝑤,𝑚   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑝)   𝐴(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑝)   𝐵(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑝)   𝐶(𝑦,𝑧,𝑤,𝑓,𝑖,𝑘,𝑛,𝑝)   𝐷(𝑧,𝑤,𝑡,𝑓,𝑝)   𝑃(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑘,𝑚,𝑛,𝑝)   𝑄(𝑧,𝑤)   𝑅(𝑥,𝑦,𝑓,𝑝)   𝑆(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑘,𝑚,𝑛,𝑝)   𝑇(𝑧,𝑤)   𝑈(𝑦,𝑧,𝑤,𝑓,𝑖,𝑘,𝑛,𝑝)   𝐸(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑘,𝑚,𝑛,𝑝)   𝐹(𝑓,𝑝)   𝐼(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑘,𝑚,𝑛,𝑝)   𝐿(𝑥,𝑦,𝑓,𝑝)   𝑀(𝑧,𝑤,𝑘)   𝑁(𝑘)   𝑉(𝑦)   𝑍(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑖,𝑘,𝑛,𝑝)

Proof of Theorem fourierdlem112
Dummy variables 𝑗 𝑙 𝑎 𝑠 𝑏 𝑒 𝑔 𝑐 𝑢 𝑞 𝑟 𝑣 𝑑 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem112.23 . . . . 5 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
2 fveq2 6663 . . . . . . . 8 (𝑛 = 𝑗 → (𝐴𝑛) = (𝐴𝑗))
3 oveq1 7152 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
43fveq2d 6667 . . . . . . . 8 (𝑛 = 𝑗 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑗 · 𝑋)))
52, 4oveq12d 7163 . . . . . . 7 (𝑛 = 𝑗 → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))))
6 fveq2 6663 . . . . . . . 8 (𝑛 = 𝑗 → (𝐵𝑛) = (𝐵𝑗))
73fveq2d 6667 . . . . . . . 8 (𝑛 = 𝑗 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑗 · 𝑋)))
86, 7oveq12d 7163 . . . . . . 7 (𝑛 = 𝑗 → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))
95, 8oveq12d 7163 . . . . . 6 (𝑛 = 𝑗 → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
109cbvmptv 5160 . . . . 5 (𝑛 ∈ ℕ ↦ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
111, 10eqtri 2841 . . . 4 𝑆 = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
12 seqeq3 13362 . . . 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 12269 . . . . 5 ℕ = (ℤ‘1)
15 1zzd 12001 . . . . 5 (𝜑 → 1 ∈ ℤ)
16 nfv 1906 . . . . . . 7 𝑛𝜑
17 nfcv 2974 . . . . . . . 8 𝑛
18 nfcv 2974 . . . . . . . . 9 𝑛(-π(,)0)
19 nfcv 2974 . . . . . . . . . 10 𝑛(𝐹‘(𝑋 + 𝑠))
20 nfcv 2974 . . . . . . . . . 10 𝑛 ·
21 nfcv 2974 . . . . . . . . . 10 𝑛((𝐷𝑚)‘𝑠)
2219, 20, 21nfov 7175 . . . . . . . . 9 𝑛((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠))
2318, 22nfitg 24302 . . . . . . . 8 𝑛∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠
2417, 23nfmpt 5154 . . . . . . 7 𝑛(𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)
25 nfcv 2974 . . . . . . . . 9 𝑛(0(,)π)
2625, 22nfitg 24302 . . . . . . . 8 𝑛∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠
2717, 26nfmpt 5154 . . . . . . 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 5155 . . . . . . . . . . . . 13 𝑛(𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
3129, 30nfcxfr 2972 . . . . . . . . . . . 12 𝑛𝐴
32 nfcv 2974 . . . . . . . . . . . 12 𝑛0
3331, 32nffv 6673 . . . . . . . . . . 11 𝑛(𝐴‘0)
34 nfcv 2974 . . . . . . . . . . 11 𝑛 /
35 nfcv 2974 . . . . . . . . . . 11 𝑛2
3633, 34, 35nfov 7175 . . . . . . . . . 10 𝑛((𝐴‘0) / 2)
37 nfcv 2974 . . . . . . . . . 10 𝑛 +
38 nfcv 2974 . . . . . . . . . . 11 𝑛(1...𝑚)
3938nfsum1 15034 . . . . . . . . . 10 𝑛Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))
4036, 37, 39nfov 7175 . . . . . . . . 9 𝑛(((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
4117, 40nfmpt 5154 . . . . . . . 8 𝑛(𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
4228, 41nfcxfr 2972 . . . . . . 7 𝑛𝑍
43 fourierdlem112.f . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
44 fourierdlem112.25 . . . . . . . 8 (𝜑𝑋 ∈ ℝ)
45 eqid 2818 . . . . . . . 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 24972 . . . . . . . . . . . . 13 π ∈ ℂ
47462timesi 11763 . . . . . . . . . . . 12 (2 · π) = (π + π)
48 fourierdlem112.t . . . . . . . . . . . 12 𝑇 = (2 · π)
4946, 46subnegi 10953 . . . . . . . . . . . 12 (π − -π) = (π + π)
5047, 48, 493eqtr4i 2851 . . . . . . . . . . 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 24971 . . . . . . . . . . . . . 14 π ∈ ℝ
5554a1i 11 . . . . . . . . . . . . 13 (𝜑 → π ∈ ℝ)
5655renegcld 11055 . . . . . . . . . . . 12 (𝜑 → -π ∈ ℝ)
5756, 44readdcld 10658 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) ∈ ℝ)
5855, 44readdcld 10658 . . . . . . . . . . 11 (𝜑 → (π + 𝑋) ∈ ℝ)
59 negpilt0 41422 . . . . . . . . . . . . . 14 -π < 0
60 pipos 24973 . . . . . . . . . . . . . 14 0 < π
6154renegcli 10935 . . . . . . . . . . . . . . 15 -π ∈ ℝ
62 0re 10631 . . . . . . . . . . . . . . 15 0 ∈ ℝ
6361, 62, 54lttri 10754 . . . . . . . . . . . . . 14 ((-π < 0 ∧ 0 < π) → -π < π)
6459, 60, 63mp2an 688 . . . . . . . . . . . . 13 -π < π
6564a1i 11 . . . . . . . . . . . 12 (𝜑 → -π < π)
6656, 55, 44, 65ltadd1dd 11239 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) < (π + 𝑋))
67 oveq1 7152 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑦 + (𝑘 · 𝑇)) = (𝑥 + (𝑘 · 𝑇)))
6867eleq1d 2894 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
6968rexbidv 3294 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
7069cbvrabv 3489 . . . . . . . . . . . 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 4133 . . . . . . . . . . 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 42322 . . . . . . . . . 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 495 . . . . . . . . 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 495 . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
7775simprd 496 . . . . . . . 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 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℝ)
80 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑝𝑖) = (𝑝𝑗))
81 oveq1 7152 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
8281fveq2d 6667 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1)))
8380, 82breq12d 5070 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝𝑗) < (𝑝‘(𝑗 + 1))))
8483cbvralvw 3447 . . . . . . . . . . . . . 14 (∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))
8584anbi2i 622 . . . . . . . . . . . . 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 3470 . . . . . . . . . . 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 5149 . . . . . . . . . 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 2841 . . . . . . . . 9 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))})
9052adantr 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ)
9153adantr 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑄 ∈ (𝑃𝑀))
92 fourierdlem112.fper . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
9392adantlr 711 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
94 eleq1w 2892 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
9594anbi2d 628 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑗 ∈ (0..^𝑀))))
96 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
9781fveq2d 6667 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
9896, 97oveq12d 7163 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1))))
9998reseq2d 5846 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))))
10098oveq1d 7160 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) = (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
10199, 100eleq12d 2904 . . . . . . . . . . . 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 346 . . . . . . . . . . 11 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))))
103 fourierdlem112.fcn . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
104102, 103chvarvv 1996 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
105104adantlr 711 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
10657adantr 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (-π + 𝑋) ∈ ℝ)
10757rexrd 10679 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) ∈ ℝ*)
108 pnfxr 10683 . . . . . . . . . . . 12 +∞ ∈ ℝ*
109108a1i 11 . . . . . . . . . . 11 (𝜑 → +∞ ∈ ℝ*)
11058ltpnfd 12504 . . . . . . . . . . 11 (𝜑 → (π + 𝑋) < +∞)
111107, 109, 58, 66, 110eliood 41649 . . . . . . . . . 10 (𝜑 → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
112111adantr 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
113 id 22 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^𝑁))
11472oveq2i 7156 . . . . . . . . . . 11 (0..^𝑁) = (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))
115113, 114eleqtrdi 2920 . . . . . . . . . 10 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))
116115adantl 482 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))
11772oveq2i 7156 . . . . . . . . . . . 12 (0...𝑁) = (0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))
118 isoeq4 7062 . . . . . . . . . . . 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 6333 . . . . . . . . . 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 2841 . . . . . . . . 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 42366 . . . . . . . 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 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤)
125 nfra1 3216 . . . . . . . . . . 11 𝑡𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤
126 elioore 12756 . . . . . . . . . . . . 13 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
127 rspa 3203 . . . . . . . . . . . . 13 ((∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤𝑡 ∈ ℝ) → (abs‘(𝐹𝑡)) ≤ 𝑤)
128126, 127sylan2 592 . . . . . . . . . . . 12 ((∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(𝐹𝑡)) ≤ 𝑤)
129128ex 413 . . . . . . . . . . 11 (∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤 → (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘(𝐹𝑡)) ≤ 𝑤))
130125, 129ralrimi 3213 . . . . . . . . . 10 (∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤 → ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹𝑡)) ≤ 𝑤)
131130reximi 3240 . . . . . . . . 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 3986 . . . . . . . . . . . 12 ℝ ⊆ ℝ
134 dvfre 24475 . . . . . . . . . . . 12 ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
13543, 133, 134sylancl 586 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
136135adantr 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
137 eqid 2818 . . . . . . . . . . . . 13 (ℝ D 𝐹) = (ℝ D 𝐹)
13854a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → π ∈ ℝ)
13961a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → -π ∈ ℝ)
14098reseq2d 5846 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))))
141140, 100eleq12d 2904 . . . . . . . . . . . . . . . 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 346 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))))
143 fourierdlem112.fdvcn . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
144142, 143chvarvv 1996 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
145144adantlr 711 . . . . . . . . . . . . 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 10658 . . . . . . . . . . . . . 14 (𝜑 → (-π + 𝑋) ∈ ℝ)
148147adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → (-π + 𝑋) ∈ ℝ)
149147rexrd 10679 . . . . . . . . . . . . . . 15 (𝜑 → (-π + 𝑋) ∈ ℝ*)
15055, 146readdcld 10658 . . . . . . . . . . . . . . 15 (𝜑 → (π + 𝑋) ∈ ℝ)
15156, 55, 146, 65ltadd1dd 11239 . . . . . . . . . . . . . . 15 (𝜑 → (-π + 𝑋) < (π + 𝑋))
152150ltpnfd 12504 . . . . . . . . . . . . . . 15 (𝜑 → (π + 𝑋) < +∞)
153149, 109, 150, 151, 152eliood 41649 . . . . . . . . . . . . . 14 (𝜑 → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
154153adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
155 oveq1 7152 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = → (𝑘 · 𝑇) = ( · 𝑇))
156155oveq2d 7161 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝑦 + (𝑘 · 𝑇)) = (𝑦 + ( · 𝑇)))
157156eleq1d 2894 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + ( · 𝑇)) ∈ ran 𝑄))
158157cbvrexvw 3448 . . . . . . . . . . . . . . . . . . 19 (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄)
159158rgenw 3147 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋))(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄)
160 rabbi 3381 . . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . . 17 {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄}
162161uneq2i 4133 . . . . . . . . . . . . . . . 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 7063 . . . . . . . . . . . . . . . 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 6333 . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . 13 𝑉 = (℩𝑓𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄})))
167 eleq1w 2892 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → (𝑣 ∈ dom (ℝ D 𝐹) ↔ 𝑢 ∈ dom (ℝ D 𝐹)))
168 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → ((ℝ D 𝐹)‘𝑣) = ((ℝ D 𝐹)‘𝑢))
169167, 168ifbieq1d 4486 . . . . . . . . . . . . . 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 5160 . . . . . . . . . . . . 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 42365 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
172 cncff 23428 . . . . . . . . . . . 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 6515 . . . . . . . . . . . 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 5869 . . . . . . . . . . 11 (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
176174, 175sylibr 235 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹))
177136, 176fssresd 6538 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
178 ax-resscn 10582 . . . . . . . . . . 11 ℝ ⊆ ℂ
179178a1i 11 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ℝ ⊆ ℂ)
180 cncffvrn 23433 . . . . . . . . . 10 ((ℝ ⊆ ℂ ∧ ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) → (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ) ↔ ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ))
181179, 171, 180syl2anc 584 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ) ↔ ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ))
182177, 181mpbird 258 . . . . . . . 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 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
185 nfv 1906 . . . . . . . . . . . . . 14 𝑡(𝜑𝑖 ∈ (0..^𝑁))
186 nfra1 3216 . . . . . . . . . . . . . 14 𝑡𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧
187185, 186nfan 1891 . . . . . . . . . . . . 13 𝑡((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
188 fvres 6682 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
189188adantl 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
190189fveq2d 6667 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡)))
191190adantlr 711 . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . 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 3964 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → 𝑡 ∈ dom (ℝ D 𝐹))
194193adantlr 711 . . . . . . . . . . . . . . . 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 3203 . . . . . . . . . . . . . . . 16 ((∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧𝑡 ∈ dom (ℝ D 𝐹)) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
196192, 194, 195syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
197191, 196eqbrtrd 5079 . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . 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 3213 . . . . . . . . . . . 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 413 . . . . . . . . . . 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 3270 . . . . . . . . . 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 3216 . . . . . . . . . . . 12 𝑡𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧
204188eqcomd 2824 . . . . . . . . . . . . . . . 16 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → ((ℝ D 𝐹)‘𝑡) = (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡))
205204fveq2d 6667 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘((ℝ D 𝐹)‘𝑡)) = (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)))
206205adantl 482 . . . . . . . . . . . . . 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 3203 . . . . . . . . . . . . . 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 5079 . . . . . . . . . . . . 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 413 . . . . . . . . . . . 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 3213 . . . . . . . . . . 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 3270 . . . . . . . . 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 1906 . . . . . . . . . . . 12 𝑖(𝜑𝑗 ∈ (0..^𝑀))
215 nfcsb1v 3904 . . . . . . . . . . . . 13 𝑖𝑗 / 𝑖𝐶
216215nfel1 2991 . . . . . . . . . . . 12 𝑖𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗))
217214, 216nfim 1888 . . . . . . . . . . 11 𝑖((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
218 csbeq1a 3894 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝐶 = 𝑗 / 𝑖𝐶)
21999, 96oveq12d 7163 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
220218, 219eleq12d 2904 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) ↔ 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗))))
22195, 220imbi12d 346 . . . . . . . . . . 11 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖))) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))))
222 fourierdlem112.c . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
223217, 221, 222chvarfv 2232 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
224223adantlr 711 . . . . . . . . 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 42364 . . . . . . . 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 3904 . . . . . . . . . . . . 13 𝑖𝑗 / 𝑖𝑈
227226nfel1 2991 . . . . . . . . . . . 12 𝑖𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1)))
228214, 227nfim 1888 . . . . . . . . . . 11 𝑖((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
229 csbeq1a 3894 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝑈 = 𝑗 / 𝑖𝑈)
23099, 97oveq12d 7163 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
231229, 230eleq12d 2904 . . . . . . . . . . . 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 346 . . . . . . . . . . 11 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1)))) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))))
233 fourierdlem112.u . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
234228, 232, 233chvarfv 2232 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
235234adantlr 711 . . . . . . . . 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 42367 . . . . . . . 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 2822 . . . . . . . . . 10 (𝑔 = 𝑠 → (𝑔 = 0 ↔ 𝑠 = 0))
238 oveq2 7153 . . . . . . . . . . . . 13 (𝑔 = 𝑠 → (𝑋 + 𝑔) = (𝑋 + 𝑠))
239238fveq2d 6667 . . . . . . . . . . . 12 (𝑔 = 𝑠 → (𝐹‘(𝑋 + 𝑔)) = (𝐹‘(𝑋 + 𝑠)))
240 breq2 5061 . . . . . . . . . . . . 13 (𝑔 = 𝑠 → (0 < 𝑔 ↔ 0 < 𝑠))
241240ifbid 4485 . . . . . . . . . . . 12 (𝑔 = 𝑠 → if(0 < 𝑔, 𝑅, 𝐿) = if(0 < 𝑠, 𝑅, 𝐿))
242239, 241oveq12d 7163 . . . . . . . . . . 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 7163 . . . . . . . . . 10 (𝑔 = 𝑠 → (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑅, 𝐿)) / 𝑠))
245237, 244ifbieq2d 4488 . . . . . . . . 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 5160 . . . . . . . 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 2822 . . . . . . . . . 10 (𝑜 = 𝑠 → (𝑜 = 0 ↔ 𝑠 = 0))
248 id 22 . . . . . . . . . . 11 (𝑜 = 𝑠𝑜 = 𝑠)
249 oveq1 7152 . . . . . . . . . . . . 13 (𝑜 = 𝑠 → (𝑜 / 2) = (𝑠 / 2))
250249fveq2d 6667 . . . . . . . . . . . 12 (𝑜 = 𝑠 → (sin‘(𝑜 / 2)) = (sin‘(𝑠 / 2)))
251250oveq2d 7161 . . . . . . . . . . 11 (𝑜 = 𝑠 → (2 · (sin‘(𝑜 / 2))) = (2 · (sin‘(𝑠 / 2))))
252248, 251oveq12d 7163 . . . . . . . . . 10 (𝑜 = 𝑠 → (𝑜 / (2 · (sin‘(𝑜 / 2)))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
253247, 252ifbieq2d 4488 . . . . . . . . 9 (𝑜 = 𝑠 → if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
254253cbvmptv 5160 . . . . . . . 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 6663 . . . . . . . . . 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 6663 . . . . . . . . . 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 7163 . . . . . . . . 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 5160 . . . . . . . 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 7153 . . . . . . . . . 10 (𝑑 = 𝑠 → ((𝑘 + (1 / 2)) · 𝑑) = ((𝑘 + (1 / 2)) · 𝑠))
260259fveq2d 6667 . . . . . . . . 9 (𝑑 = 𝑠 → (sin‘((𝑘 + (1 / 2)) · 𝑑)) = (sin‘((𝑘 + (1 / 2)) · 𝑠)))
261260cbvmptv 5160 . . . . . . . 8 (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑))) = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑠)))
262 fveq2 6663 . . . . . . . . . 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 6663 . . . . . . . . . 10 (𝑧 = 𝑠 → ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧) = ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑠))
264262, 263oveq12d 7163 . . . . . . . . 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 5160 . . . . . . . 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 6663 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝐷𝑚) = (𝐷𝑛))
267266fveq1d 6665 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((𝐷𝑚)‘𝑠) = ((𝐷𝑛)‘𝑠))
268267oveq2d 7161 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
269268adantr 481 . . . . . . . . . 10 ((𝑚 = 𝑛𝑠 ∈ (-π(,)0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
270269itgeq2dv 24309 . . . . . . . . 9 (𝑚 = 𝑛 → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
271270cbvmptv 5160 . . . . . . . 8 (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑛 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
272 oveq1 7152 . . . . . . . . . . . . . . . . . . 19 (𝑐 = 𝑘 → (𝑐 + (1 / 2)) = (𝑘 + (1 / 2)))
273272oveq1d 7160 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝑘 → ((𝑐 + (1 / 2)) · 𝑑) = ((𝑘 + (1 / 2)) · 𝑑))
274273fveq2d 6667 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑘 → (sin‘((𝑐 + (1 / 2)) · 𝑑)) = (sin‘((𝑘 + (1 / 2)) · 𝑑)))
275274mpteq2dv 5153 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑘 → (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑))) = (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑))))
276275fveq1d 6665 . . . . . . . . . . . . . . 15 (𝑐 = 𝑘 → ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧) = ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧))
277276oveq2d 7161 . . . . . . . . . . . . . 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 5153 . . . . . . . . . . . . 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 6665 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 24309 . . . . . . . . . 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 7160 . . . . . . . . 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 5160 . . . . . . . 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 7152 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → (𝑦 mod (2 · π)) = (𝑠 mod (2 · π)))
290289eqeq1d 2820 . . . . . . . . . . . . 13 (𝑦 = 𝑠 → ((𝑦 mod (2 · π)) = 0 ↔ (𝑠 mod (2 · π)) = 0))
291 oveq2 7153 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → ((𝑚 + (1 / 2)) · 𝑦) = ((𝑚 + (1 / 2)) · 𝑠))
292291fveq2d 6667 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → (sin‘((𝑚 + (1 / 2)) · 𝑦)) = (sin‘((𝑚 + (1 / 2)) · 𝑠)))
293 oveq1 7152 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑠 → (𝑦 / 2) = (𝑠 / 2))
294293fveq2d 6667 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → (sin‘(𝑦 / 2)) = (sin‘(𝑠 / 2)))
295294oveq2d 7161 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → ((2 · π) · (sin‘(𝑦 / 2))) = ((2 · π) · (sin‘(𝑠 / 2))))
296292, 295oveq12d 7163 . . . . . . . . . . . . 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 4488 . . . . . . . . . . . 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 5160 . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑘𝑠 ∈ ℝ) → 𝑚 = 𝑘)
300299oveq2d 7161 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (2 · 𝑚) = (2 · 𝑘))
301300oveq1d 7160 . . . . . . . . . . . . . 14 ((𝑚 = 𝑘𝑠 ∈ ℝ) → ((2 · 𝑚) + 1) = ((2 · 𝑘) + 1))
302301oveq1d 7160 . . . . . . . . . . . . 13 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (((2 · 𝑚) + 1) / (2 · π)) = (((2 · 𝑘) + 1) / (2 · π)))
303299oveq1d 7160 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (𝑚 + (1 / 2)) = (𝑘 + (1 / 2)))
304303oveq1d 7160 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑘𝑠 ∈ ℝ) → ((𝑚 + (1 / 2)) · 𝑠) = ((𝑘 + (1 / 2)) · 𝑠))
305304fveq2d 6667 . . . . . . . . . . . . . 14 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (sin‘((𝑚 + (1 / 2)) · 𝑠)) = (sin‘((𝑘 + (1 / 2)) · 𝑠)))
306305oveq1d 7160 . . . . . . . . . . . . 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 4483 . . . . . . . . . . . 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 5152 . . . . . . . . . . 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 2865 . . . . . . . . . 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 5160 . . . . . . . . 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 2841 . . . . . . . 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 2818 . . . . . . . 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 2818 . . . . . . . 8 ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙))) = ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))
314 eqid 2818 . . . . . . . 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 7059 . . . . . . . . 9 (𝑢 = 𝑤 → (𝑢 Isom < , < ((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) ↔ 𝑤 Isom < , < ((0...((♯‘({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))) − 1)), ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙))))))
316315cbviotavw 6315 . . . . . . . 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 6663 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝑉𝑗) = (𝑉𝑖))
318317oveq1d 7160 . . . . . . . . 9 (𝑗 = 𝑖 → ((𝑉𝑗) − 𝑋) = ((𝑉𝑖) − 𝑋))
319318cbvmptv 5160 . . . . . . . 8 (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) = (𝑖 ∈ (0...𝑁) ↦ ((𝑉𝑖) − 𝑋))
320 eqid 2818 . . . . . . . 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 6663 . . . . . . . . . . . . . 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 7153 . . . . . . . . . . . . . . 15 (𝑎 = 𝑠 → ((𝑏 + (1 / 2)) · 𝑎) = ((𝑏 + (1 / 2)) · 𝑠))
323322fveq2d 6667 . . . . . . . . . . . . . 14 (𝑎 = 𝑠 → (sin‘((𝑏 + (1 / 2)) · 𝑎)) = (sin‘((𝑏 + (1 / 2)) · 𝑠)))
324321, 323oveq12d 7163 . . . . . . . . . . . . 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 24304 . . . . . . . . . . . 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 6666 . . . . . . . . . . 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 5064 . . . . . . . . . 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 622 . . . . . . . . 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 24304 . . . . . . . . . . 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 6666 . . . . . . . . . 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 5064 . . . . . . . . 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 626 . . . . . . . 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 42371 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) ⇝ (𝐿 / 2))
334 nnex 11632 . . . . . . . . . 10 ℕ ∈ V
335334mptex 6977 . . . . . . . . 9 (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))) ∈ V
33628, 335eqeltri 2906 . . . . . . . 8 𝑍 ∈ V
337336a1i 11 . . . . . . 7 (𝜑𝑍 ∈ V)
338268adantr 481 . . . . . . . . . 10 ((𝑚 = 𝑛𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
339338itgeq2dv 24309 . . . . . . . . 9 (𝑚 = 𝑛 → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
340339cbvmptv 5160 . . . . . . . 8 (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑛 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
341279adantr 481 . . . . . . . . . . 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 24309 . . . . . . . . . 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 7160 . . . . . . . . 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 5160 . . . . . . . 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 2818 . . . . . . . 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 2818 . . . . . . . 8 ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π))) = ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))
347 eqid 2818 . . . . . . . 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 7059 . . . . . . . . 9 (𝑢 = 𝑣 → (𝑢 Isom < , < ((0...((♯‘({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) ↔ 𝑣 Isom < , < ((0...((♯‘({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))) − 1)), ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π))))))
349348cbviotavw 6315 . . . . . . . 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 2818 . . . . . . . 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 24304 . . . . . . . . . . . 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 6666 . . . . . . . . . . 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 5064 . . . . . . . . . 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 622 . . . . . . . . 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 24304 . . . . . . . . . . 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 6666 . . . . . . . . . 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 5064 . . . . . . . . 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 626 . . . . . . . 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 42372 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) ⇝ (𝑅 / 2))
360 eqidd 2819 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
361270adantl 482 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 = 𝑛) → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
362 simpr 485 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
363 elioore 12756 . . . . . . . . . . 11 (𝑠 ∈ (-π(,)0) → 𝑠 ∈ ℝ)
36443adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ ℝ) → 𝐹:ℝ⟶ℝ)
36544adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ ℝ) → 𝑋 ∈ ℝ)
366 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ ℝ) → 𝑠 ∈ ℝ)
367365, 366readdcld 10658 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ ℝ) → (𝑋 + 𝑠) ∈ ℝ)
368364, 367ffvelrnd 6844 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ ℝ) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
369368adantlr 711 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
370288dirkerre 42257 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
371370adantll 710 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
372369, 371remulcld 10659 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
373363, 372sylan2 592 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
374 ioossicc 12810 . . . . . . . . . . . . 13 (-π(,)0) ⊆ (-π[,]0)
37561leidi 11162 . . . . . . . . . . . . . 14 -π ≤ -π
37662, 54, 60ltleii 10751 . . . . . . . . . . . . . 14 0 ≤ π
377 iccss 12792 . . . . . . . . . . . . . 14 (((-π ∈ ℝ ∧ π ∈ ℝ) ∧ (-π ≤ -π ∧ 0 ≤ π)) → (-π[,]0) ⊆ (-π[,]π))
37861, 54, 375, 376, 377mp4an 689 . . . . . . . . . . . . 13 (-π[,]0) ⊆ (-π[,]π)
379374, 378sstri 3973 . . . . . . . . . . . 12 (-π(,)0) ⊆ (-π[,]π)
380379a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (-π(,)0) ⊆ (-π[,]π))
381 ioombl 24093 . . . . . . . . . . . 12 (-π(,)0) ∈ dom vol
382381a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (-π(,)0) ∈ dom vol)
38343adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℝ)
38444adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ (-π[,]π)) → 𝑋 ∈ ℝ)
38556, 55iccssred 41656 . . . . . . . . . . . . . . . 16 (𝜑 → (-π[,]π) ⊆ ℝ)
386385sselda 3964 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ (-π[,]π)) → 𝑠 ∈ ℝ)
387384, 386readdcld 10658 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (-π[,]π)) → (𝑋 + 𝑠) ∈ ℝ)
388383, 387ffvelrnd 6844 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (-π[,]π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
389388adantlr 711 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
390 iccssre 12806 . . . . . . . . . . . . . . . 16 ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
39161, 54, 390mp2an 688 . . . . . . . . . . . . . . 15 (-π[,]π) ⊆ ℝ
392391sseli 3960 . . . . . . . . . . . . . 14 (𝑠 ∈ (-π[,]π) → 𝑠 ∈ ℝ)
393392, 370sylan2 592 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
394393adantll 710 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
395389, 394remulcld 10659 . . . . . . . . . . 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 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
39944adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑋 ∈ ℝ)
40076adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑁 ∈ ℕ)
40177adantr 481 . . . . . . . . . . . 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 711 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
403225adantlr 711 . . . . . . . . . . . 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 711 . . . . . . . . . . . 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 42269 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝐷𝑛) ∈ (ℝ–cn→ℝ))
406405adantl 482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) ∈ (ℝ–cn→ℝ))
407 eqid 2818 . . . . . . . . . . . 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 42352 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
409380, 382, 395, 408iblss 24332 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (-π(,)0) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
410373, 409itgcl 24311 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 ∈ ℂ)
411360, 361, 362, 410fvmptd 6767 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
412411, 410eqeltrd 2910 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) ∈ ℂ)
413 eqidd 2819 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
414339adantl 482 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 = 𝑛) → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
41543adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (0(,)π)) → 𝐹:ℝ⟶ℝ)
41644adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (0(,)π)) → 𝑋 ∈ ℝ)
417 elioore 12756 . . . . . . . . . . . . . . 15 (𝑠 ∈ (0(,)π) → 𝑠 ∈ ℝ)
418417adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (0(,)π)) → 𝑠 ∈ ℝ)
419416, 418readdcld 10658 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (0(,)π)) → (𝑋 + 𝑠) ∈ ℝ)
420415, 419ffvelrnd 6844 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
421420adantlr 711 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
422417, 370sylan2 592 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (0(,)π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
423422adantll 710 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
424421, 423remulcld 10659 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
425 ioossicc 12810 . . . . . . . . . . . . 13 (0(,)π) ⊆ (0[,]π)
42661, 62, 59ltleii 10751 . . . . . . . . . . . . . 14 -π ≤ 0
42754leidi 11162 . . . . . . . . . . . . . 14 π ≤ π
428 iccss 12792 . . . . . . . . . . . . . 14 (((-π ∈ ℝ ∧ π ∈ ℝ) ∧ (-π ≤ 0 ∧ π ≤ π)) → (0[,]π) ⊆ (-π[,]π))
42961, 54, 426, 427, 428mp4an 689 . . . . . . . . . . . . 13 (0[,]π) ⊆ (-π[,]π)
430425, 429sstri 3973 . . . . . . . . . . . 12 (0(,)π) ⊆ (-π[,]π)
431430a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (0(,)π) ⊆ (-π[,]π))
432 ioombl 24093 . . . . . . . . . . . 12 (0(,)π) ∈ dom vol
433432a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (0(,)π) ∈ dom vol)
434431, 433, 395, 408iblss 24332 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (0(,)π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
435424, 434itgcl 24311 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 ∈ ℂ)
436413, 414, 362, 435fvmptd 6767 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
437436, 435eqeltrd 2910 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) ∈ ℂ)
438 eleq1w 2892 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑚 ∈ ℕ ↔ 𝑛 ∈ ℕ))
439438anbi2d 628 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝜑𝑚 ∈ ℕ) ↔ (𝜑𝑛 ∈ ℕ)))
440 fveq2 6663 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑍𝑚) = (𝑍𝑛))
441270, 339oveq12d 7163 . . . . . . . . . . 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 2834 . . . . . . . . . 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 346 . . . . . . . . 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 7152 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝑛 · 𝑥) = (𝑚 · 𝑥))
445444fveq2d 6667 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (cos‘(𝑛 · 𝑥)) = (cos‘(𝑚 · 𝑥)))
446445oveq2d 7161 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (cos‘(𝑚 · 𝑥))))
447446adantr 481 . . . . . . . . . . . . . 14 ((𝑛 = 𝑚𝑥 ∈ (-π(,)π)) → ((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (cos‘(𝑚 · 𝑥))))
448447itgeq2dv 24309 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 = ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥)
449448oveq1d 7160 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π) = (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π))
450449cbvmptv 5160 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) = (𝑚 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π))
45129, 450eqtri 2841 . . . . . . . . . 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 6667 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (sin‘(𝑛 · 𝑥)) = (sin‘(𝑚 · 𝑥)))
454453oveq2d 7161 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (sin‘(𝑚 · 𝑥))))
455454adantr 481 . . . . . . . . . . . . . 14 ((𝑛 = 𝑚𝑥 ∈ (-π(,)π)) → ((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) = ((𝐹𝑥) · (sin‘(𝑚 · 𝑥))))
456455itgeq2dv 24309 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 = ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥)
457456oveq1d 7160 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π) = (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
458457cbvmptv 5160 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) = (𝑚 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
459452, 458eqtri 2841 . . . . . . . . . 10 𝐵 = (𝑚 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
460 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐴𝑛) = (𝐴𝑘))
461 oveq1 7152 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
462461fveq2d 6667 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
463460, 462oveq12d 7163 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))))
464 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐵𝑛) = (𝐵𝑘))
465461fveq2d 6667 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
466464, 465oveq12d 7163 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
467463, 466oveq12d 7163 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
468467cbvsumv 15041 . . . . . . . . . . . . 13 Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
469468oveq2i 7156 . . . . . . . . . . . 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 5149 . . . . . . . . . . 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 7153 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
472471sumeq1d 15046 . . . . . . . . . . . . . 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 7161 . . . . . . . . . . . . 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 5160 . . . . . . . . . . . 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 6663 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝐴𝑘) = (𝐴𝑚))
476 oveq1 7152 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → (𝑘 · 𝑋) = (𝑚 · 𝑋))
477476fveq2d 6667 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (cos‘(𝑘 · 𝑋)) = (cos‘(𝑚 · 𝑋)))
478475, 477oveq12d 7163 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) = ((𝐴𝑚) · (cos‘(𝑚 · 𝑋))))
479 fveq2 6663 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝐵𝑘) = (𝐵𝑚))
480476fveq2d 6667 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (sin‘(𝑘 · 𝑋)) = (sin‘(𝑚 · 𝑋)))
481479, 480oveq12d 7163 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) = ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))
482478, 481oveq12d 7163 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋)))))
483482cbvsumv 15041 . . . . . . . . . . . . . 14 Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))
484483oveq2i 7156 . . . . . . . . . . . . 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 5149 . . . . . . . . . . . 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 2841 . . . . . . . . . . 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 2845 . . . . . . . . . 10 𝑍 = (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))))
488 oveq2 7153 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑋 + 𝑦) = (𝑋 + 𝑥))
489488fveq2d 6667 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑥)))
490 fveq2 6663 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝐷𝑚)‘𝑦) = ((𝐷𝑚)‘𝑥))
491489, 490oveq12d 7163 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹‘(𝑋 + 𝑦)) · ((𝐷𝑚)‘𝑦)) = ((𝐹‘(𝑋 + 𝑥)) · ((𝐷𝑚)‘𝑥)))
492491cbvmptv 5160 . . . . . . . . . 10 (𝑦 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑦)) · ((𝐷𝑚)‘𝑦))) = (𝑥 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷𝑚)‘𝑥)))
493 eqid 2818 . . . . . . . . . 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 6663 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑄𝑗) = (𝑄𝑖))
495494oveq1d 7160 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝑄𝑗) − 𝑋) = ((𝑄𝑖) − 𝑋))
496495cbvmptv 5160 . . . . . . . . . 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 42379 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
498443, 497chvarvv 1996 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝑍𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠))
499411, 436oveq12d 7163 . . . . . . . 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 2856 . . . . . . 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 41772 . . . . . 6 (𝜑𝑍 ⇝ ((𝐿 / 2) + (𝑅 / 2)))
502 limccl 24400 . . . . . . . 8 ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋) ⊆ ℂ
503502, 285sseldi 3962 . . . . . . 7 (𝜑𝐿 ∈ ℂ)
504 limccl 24400 . . . . . . . 8 ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋) ⊆ ℂ
505504, 284sseldi 3962 . . . . . . 7 (𝜑𝑅 ∈ ℂ)
506 2cnd 11703 . . . . . . 7 (𝜑 → 2 ∈ ℂ)
507 2pos 11728 . . . . . . . . 9 0 < 2
508507a1i 11 . . . . . . . 8 (𝜑 → 0 < 2)
509508gt0ne0d 11192 . . . . . . 7 (𝜑 → 2 ≠ 0)
510503, 505, 506, 509divdird 11442 . . . . . 6 (𝜑 → ((𝐿 + 𝑅) / 2) = ((𝐿 / 2) + (𝑅 / 2)))
511501, 510breqtrrd 5085 . . . . 5 (𝜑𝑍 ⇝ ((𝐿 + 𝑅) / 2))
512 0nn0 11900 . . . . . . . 8 0 ∈ ℕ0
51343adantr 481 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → 𝐹:ℝ⟶ℝ)
514 eqid 2818 . . . . . . . . . 10 (-π(,)π) = (-π(,)π)
515 ioossre 12786 . . . . . . . . . . . . . 14 (-π(,)π) ⊆ ℝ
516515a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ⊆ ℝ)
51743, 516feqresmpt 6727 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ (-π(,)π)) = (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)))
518 ioossicc 12810 . . . . . . . . . . . . . 14 (-π(,)π) ⊆ (-π[,]π)
519518a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ⊆ (-π[,]π))
520 ioombl 24093 . . . . . . . . . . . . . 14 (-π(,)π) ∈ dom vol
521520a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ∈ dom vol)
52243adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℝ)
523385sselda 3964 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (-π[,]π)) → 𝑥 ∈ ℝ)
524522, 523ffvelrnd 6844 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (-π[,]π)) → (𝐹𝑥) ∈ ℝ)
52543, 385feqresmpt 6727 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (-π[,]π)) = (𝑥 ∈ (-π[,]π) ↦ (𝐹𝑥)))
526178a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ℝ ⊆ ℂ)
52743, 526fssd 6521 . . . . . . . . . . . . . . . 16 (𝜑𝐹:ℝ⟶ℂ)
528527, 385fssresd 6538 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 ↾ (-π[,]π)):(-π[,]π)⟶ℂ)
529 ioossicc 12810 . . . . . . . . . . . . . . . . . 18 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
53061rexri 10687 . . . . . . . . . . . . . . . . . . . 20 -π ∈ ℝ*
531530a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ*)
53254rexri 10687 . . . . . . . . . . . . . . . . . . . 20 π ∈ ℝ*
533532a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → π ∈ ℝ*)
53451, 52, 53fourierdlem15 42284 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑄:(0...𝑀)⟶(-π[,]π))
535534adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
536 simpr 485 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
537531, 533, 535, 536fourierdlem8 42277 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
538529, 537sstrid 3975 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
539538resabs1d 5877 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
540539, 103eqeltrd 2910 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
541539eqcomd 2824 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
542541oveq1d 7160 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
543222, 542eleqtrd 2912 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
544541oveq1d 7160 . . . . . . . . . . . . . . . 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 2912 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
54651, 52, 53, 528, 540, 543, 545fourierdlem69 42337 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (-π[,]π)) ∈ 𝐿1)
547525, 546eqeltrrd 2911 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (-π[,]π) ↦ (𝐹𝑥)) ∈ 𝐿1)
548519, 521, 524, 547iblss 24332 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1)
549517, 548eqeltrd 2910 . . . . . . . . . . 11 (𝜑 → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
550549adantr 481 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
551 simpr 485 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → 0 ∈ ℕ0)
552513, 514, 550, 29, 551fourierdlem16 42285 . . . . . . . . 9 ((𝜑 ∧ 0 ∈ ℕ0) → (((𝐴‘0) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (cos‘(0 · 𝑥))) d𝑥 ∈ ℝ))
553552simplld 764 . . . . . . . 8 ((𝜑 ∧ 0 ∈ ℕ0) → (𝐴‘0) ∈ ℝ)
554512, 553mpan2 687 . . . . . . 7 (𝜑 → (𝐴‘0) ∈ ℝ)
555554rehalfcld 11872 . . . . . 6 (𝜑 → ((𝐴‘0) / 2) ∈ ℝ)
556555recnd 10657 . . . . 5 (𝜑 → ((𝐴‘0) / 2) ∈ ℂ)
557334mptex 6977 . . . . . 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 485 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
560555adantr 481 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ((𝐴‘0) / 2) ∈ ℝ)
561 fzfid 13329 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (1...𝑚) ∈ Fin)
562 simpll 763 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → 𝜑)
563 elfznn 12924 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑚) → 𝑛 ∈ ℕ)
564563adantl 482 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → 𝑛 ∈ ℕ)
565 simpl 483 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝜑)
566362nnnn0d 11943 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
567 eleq1w 2892 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘 ∈ ℕ0𝑛 ∈ ℕ0))
568567anbi2d 628 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝜑𝑘 ∈ ℕ0) ↔ (𝜑𝑛 ∈ ℕ0)))
569 fveq2 6663 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝐴𝑘) = (𝐴𝑛))
570569eleq1d 2894 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝐴𝑘) ∈ ℝ ↔ (𝐴𝑛) ∈ ℝ))
571568, 570imbi12d 346 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℝ) ↔ ((𝜑𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℝ)))
57243adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → 𝐹:ℝ⟶ℝ)
573549adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
574 simpr 485 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
575572, 514, 573, 29, 574fourierdlem16 42285 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ0) → (((𝐴𝑘) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑘 · 𝑥))) d𝑥 ∈ ℝ))
576575simplld 764 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℝ)
577571, 576chvarvv 1996 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℝ)
578565, 566, 577syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ∈ ℝ)
579362nnred 11641 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
580579, 399remulcld 10659 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝑛 · 𝑋) ∈ ℝ)
581580recoscld 15485 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (cos‘(𝑛 · 𝑋)) ∈ ℝ)
582578, 581remulcld 10659 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) ∈ ℝ)
583 eleq1w 2892 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝑘 ∈ ℕ ↔ 𝑛 ∈ ℕ))
584583anbi2d 628 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → ((𝜑𝑘 ∈ ℕ) ↔ (𝜑𝑛 ∈ ℕ)))
585 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝐵𝑘) = (𝐵𝑛))
586585eleq1d 2894 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → ((𝐵𝑘) ∈ ℝ ↔ (𝐵𝑛) ∈ ℝ))
587584, 586imbi12d 346 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → (((𝜑𝑘 ∈ ℕ) → (𝐵𝑘) ∈ ℝ) ↔ ((𝜑𝑛 ∈ ℕ) → (𝐵𝑛) ∈ ℝ)))
58843adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
589549adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿1)
590 simpr 485 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
591588, 514, 589, 452, 590fourierdlem21 42290 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (((𝐵𝑘) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ ((𝐹𝑥) · (sin‘(𝑘 · 𝑥)))) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑘 · 𝑥))) d𝑥 ∈ ℝ))
592591simplld 764 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐵𝑘) ∈ ℝ)
593587, 592chvarvv 1996 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐵𝑛) ∈ ℝ)
594580resincld 15484 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (sin‘(𝑛 · 𝑋)) ∈ ℝ)
595593, 594remulcld 10659 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) ∈ ℝ)
596582, 595readdcld 10658 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
597562, 564, 596syl2anc 584 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
598561, 597fsumrecl 15079 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
599560, 598readdcld 10658 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ ℝ)
60028fvmpt2 6771 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ ℝ) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
601559, 599, 600syl2anc 584 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
602601, 599eqeltrd 2910 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) ∈ ℝ)
603602recnd 10657 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) ∈ ℂ)
604 eqidd 2819 . . . . . . 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 7153 . . . . . . . . 9 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
606605sumeq1d 15046 . . . . . . . 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 482 . . . . . . 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 15032 . . . . . . . 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 6767 . . . . . 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 10657 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝐴‘0) / 2) ∈ ℂ)
612598recnd 10657 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℂ)
613611, 612pncan2d 10987 . . . . . . 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 2870 . . . . . 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 7178 . . . . . . . . 9 (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ V
61628fvmpt2 6771 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ V) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
617559, 615, 616sylancl 586 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
618617eqcomd 2824 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑍𝑚))
619618oveq1d 7160 . . . . . 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 2857 . . . . 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 14982 . . . 4 (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
622 seqex 13359 . . . . . 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 2819 . . . . . . 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 7153 . . . . . . . . 9 (𝑛 = 𝑙 → (1...𝑛) = (1...𝑙))
626625sumeq1d 15046 . . . . . . . 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 482 . . . . . . 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 485 . . . . . . 7 ((𝜑𝑙 ∈ ℕ) → 𝑙 ∈ ℕ)
629 fzfid 13329 . . . . . . . 8 ((𝜑𝑙 ∈ ℕ) → (1...𝑙) ∈ Fin)
630 elfznn 12924 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ)
631630nnnn0d 11943 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ0)
632631, 576sylan2 592 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (𝐴𝑘) ∈ ℝ)
633630nnred 11641 . . . . . . . . . . . . . 14 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℝ)
634633adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1...𝑙)) → 𝑘 ∈ ℝ)
635146adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1...𝑙)) → 𝑋 ∈ ℝ)
636634, 635remulcld 10659 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (1...𝑙)) → (𝑘 · 𝑋) ∈ ℝ)
637636recoscld 15485 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (cos‘(𝑘 · 𝑋)) ∈ ℝ)
638632, 637remulcld 10659 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1...𝑙)) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
639630, 592sylan2 592 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (𝐵𝑘) ∈ ℝ)
640636resincld 15484 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (sin‘(𝑘 · 𝑋)) ∈ ℝ)
641639, 640remulcld 10659 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1...𝑙)) → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) ∈ ℝ)
642638, 641readdcld 10658 . . . . . . . . 9 ((𝜑𝑘 ∈ (1...𝑙)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
643642adantlr 711 . . . . . . . 8 (((𝜑𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
644629, 643fsumrecl 15079 . . . . . . 7 ((𝜑𝑙 ∈ ℕ) → Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
645624, 627, 628, 644fvmptd 6767 . . . . . 6 ((𝜑𝑙 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))‘𝑙) = Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
646 eleq1w 2892 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑛 ∈ ℕ ↔ 𝑙 ∈ ℕ))
647646anbi2d 628 . . . . . . . 8 (𝑛 = 𝑙 → ((𝜑𝑛 ∈ ℕ) ↔ (𝜑𝑙 ∈ ℕ)))
648 fveq2 6663 . . . . . . . . 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 2834 . . . . . . . 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 346 . . . . . . 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 2819 . . . . . . . . 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 6663 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
653 oveq1 7152 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑗 · 𝑋) = (𝑘 · 𝑋))
654653fveq2d 6667 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (cos‘(𝑗 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
655652, 654oveq12d 7163 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) = ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))))
656 fveq2 6663 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐵𝑗) = (𝐵𝑘))
657653fveq2d 6667 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (sin‘(𝑗 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
658656, 657oveq12d 7163 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))) = ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
659655, 658oveq12d 7163 . . . . . . . . . 10 (𝑗 = 𝑘 → (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
660659adantl 482 . . . . . . . . 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 12924 . . . . . . . . . 10 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
662661adantl 482 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
663 simpll 763 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
664 nnnn0 11892 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
665 nn0re 11894 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
666665adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
667146adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → 𝑋 ∈ ℝ)
668666, 667remulcld 10659 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ0) → (𝑘 · 𝑋) ∈ ℝ)
669668recoscld 15485 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ0) → (cos‘(𝑘 · 𝑋)) ∈ ℝ)
670576, 669remulcld 10659 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
671664, 670sylan2 592 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
672664, 668sylan2 592 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝑘 · 𝑋) ∈ ℝ)
673672resincld 15484 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (sin‘(𝑘 · 𝑋)) ∈ ℝ)
674592, 673remulcld 10659 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) ∈ ℝ)
675671, 674readdcld 10658 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
676663, 662, 675syl2anc 584 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
677651, 660, 662, 676fvmptd 6767 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))‘𝑘) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
678362, 14eleqtrdi 2920 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
679676recnd 10657 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℂ)
680677, 678, 679fsumser 15075 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑛))
681650, 680chvarvv 1996 . . . . . 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 2853 . . . . 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 14912 . . . 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 233 . . 3 (𝜑 → seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
68513, 684eqbrtrd 5079 . 2 (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
686 eqidd 2819 . . . . . 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 6663 . . . . . . . . 9 (𝑗 = 𝑛 → (𝐴𝑗) = (𝐴𝑛))
688 oveq1 7152 . . . . . . . . . 10 (𝑗 = 𝑛 → (𝑗 · 𝑋) = (𝑛 · 𝑋))
689688fveq2d 6667 . . . . . . . . 9 (𝑗 = 𝑛 → (cos‘(𝑗 · 𝑋)) = (cos‘(𝑛 · 𝑋)))
690687, 689oveq12d 7163 . . . . . . . 8 (𝑗 = 𝑛 → ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) = ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))))
691 fveq2 6663 . . . . . . . . 9 (𝑗 = 𝑛 → (𝐵𝑗) = (𝐵𝑛))
692688fveq2d 6667 . . . . . . . . 9 (𝑗 = 𝑛 → (sin‘(𝑗 · 𝑋)) = (sin‘(𝑛 · 𝑋)))
693691, 692oveq12d 7163 . . . . . . . 8 (𝑗 = 𝑛 → ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))) = ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))
694690, 693oveq12d 7163 . . . . . . 7 (𝑗 = 𝑛 → (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))) = (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
695694adantl 482 . . . . . 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 6767 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))‘𝑛) = (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
697596recnd 10657 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℂ)
69814, 15, 696, 697, 684isumclim 15100 . . . 4 (𝜑 → Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
699698oveq2d 7161 . . 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 10648 . . . . 5 (𝜑 → (𝐿 + 𝑅) ∈ ℂ)
701700halfcld 11870 . . . 4 (𝜑 → ((𝐿 + 𝑅) / 2) ∈ ℂ)
702556, 701pncan3d 10988 . . 3 (𝜑 → (((𝐴‘0) / 2) + (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) = ((𝐿 + 𝑅) / 2))
703699, 702eqtrd 2853 . 2 (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2))
704685, 703jca 512 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 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  csb 3880  cun 3931  cin 3932  wss 3933  ifcif 4463  {cpr 4559   class class class wbr 5057  cmpt 5137  dom cdm 5548  ran crn 5549  cres 5550  cio 6305  wf 6344  cfv 6348   Isom wiso 6349  crio 7102  (class class class)co 7145  m cmap 8395  supcsup 8892  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530  +∞cpnf 10660  -∞cmnf 10661  *cxr 10662   < clt 10663  cle 10664  cmin 10858  -cneg 10859   / cdiv 11285  cn 11626  2c2 11680  0cn0 11885  cz 11969  cuz 12231  +crp 12377  (,)cioo 12726  (,]cioc 12727  [,]cicc 12729  ...cfz 12880  ..^cfzo 13021  cfl 13148   mod cmo 13225  seqcseq 13357  chash 13678  abscabs 14581  cli 14829  Σcsu 15030  sincsin 15405  cosccos 15406  πcpi 15408  cnccncf 23411  volcvol 23991  𝐿1cibl 24145  citg 24146   lim climc 24387   D cdv 24388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cc 9845  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-symdif 4216  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-ofr 7399  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-xnn0 11956  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-fac 13622  df-bc 13651  df-hash 13679  df-shft 14414  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-limsup 14816  df-clim 14833  df-rlim 14834  df-sum 15031  df-ef 15409  df-sin 15411  df-cos 15412  df-pi 15414  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-rest 16684  df-topn 16685  df-0g 16703  df-gsum 16704  df-topgen 16705  df-pt 16706  df-prds 16709  df-xrs 16763  df-qtop 16768  df-imas 16769  df-xps 16771  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-mulg 18163  df-cntz 18385  df-cmn 18837  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-fbas 20470  df-fg 20471  df-cnfld 20474  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-lp 21672  df-perf 21673  df-cn 21763  df-cnp 21764  df-t1 21850  df-haus 21851  df-cmp 21923  df-tx 22098  df-hmeo 22291  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-xms 22857  df-ms 22858  df-tms 22859  df-cncf 23413  df-ovol 23992  df-vol 23993  df-mbf 24147  df-itg1 24148  df-itg2 24149  df-ibl 24150  df-itg 24151  df-0p 24198  df-ditg 24372  df-limc 24391  df-dv 24392
This theorem is referenced by:  fourierdlem113  42381
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