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Theorem fourierdlem112 44449
Description: Here abbreviations (local definitions) are introduced to prove the fourier 44456 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 6842 . . . . . . . 8 (𝑛 = 𝑗 → (𝐴𝑛) = (𝐴𝑗))
3 oveq1 7364 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
43fveq2d 6846 . . . . . . . 8 (𝑛 = 𝑗 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑗 · 𝑋)))
52, 4oveq12d 7375 . . . . . . 7 (𝑛 = 𝑗 → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))))
6 fveq2 6842 . . . . . . . 8 (𝑛 = 𝑗 → (𝐵𝑛) = (𝐵𝑗))
73fveq2d 6846 . . . . . . . 8 (𝑛 = 𝑗 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑗 · 𝑋)))
86, 7oveq12d 7375 . . . . . . 7 (𝑛 = 𝑗 → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))
95, 8oveq12d 7375 . . . . . 6 (𝑛 = 𝑗 → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
109cbvmptv 5218 . . . . 5 (𝑛 ∈ ℕ ↦ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
111, 10eqtri 2764 . . . 4 𝑆 = (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))
12 seqeq3 13911 . . . 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 12806 . . . . 5 ℕ = (ℤ‘1)
15 1zzd 12534 . . . . 5 (𝜑 → 1 ∈ ℤ)
16 nfv 1917 . . . . . . 7 𝑛𝜑
17 nfcv 2907 . . . . . . . 8 𝑛
18 nfcv 2907 . . . . . . . . 9 𝑛(-π(,)0)
19 nfcv 2907 . . . . . . . . . 10 𝑛(𝐹‘(𝑋 + 𝑠))
20 nfcv 2907 . . . . . . . . . 10 𝑛 ·
21 nfcv 2907 . . . . . . . . . 10 𝑛((𝐷𝑚)‘𝑠)
2219, 20, 21nfov 7387 . . . . . . . . 9 𝑛((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠))
2318, 22nfitg 25139 . . . . . . . 8 𝑛∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠
2417, 23nfmpt 5212 . . . . . . 7 𝑛(𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)
25 nfcv 2907 . . . . . . . . 9 𝑛(0(,)π)
2625, 22nfitg 25139 . . . . . . . 8 𝑛∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠
2717, 26nfmpt 5212 . . . . . . 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 5213 . . . . . . . . . . . . 13 𝑛(𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
3129, 30nfcxfr 2905 . . . . . . . . . . . 12 𝑛𝐴
32 nfcv 2907 . . . . . . . . . . . 12 𝑛0
3331, 32nffv 6852 . . . . . . . . . . 11 𝑛(𝐴‘0)
34 nfcv 2907 . . . . . . . . . . 11 𝑛 /
35 nfcv 2907 . . . . . . . . . . 11 𝑛2
3633, 34, 35nfov 7387 . . . . . . . . . 10 𝑛((𝐴‘0) / 2)
37 nfcv 2907 . . . . . . . . . 10 𝑛 +
38 nfcv 2907 . . . . . . . . . . 11 𝑛(1...𝑚)
3938nfsum1 15574 . . . . . . . . . 10 𝑛Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))
4036, 37, 39nfov 7387 . . . . . . . . 9 𝑛(((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
4117, 40nfmpt 5212 . . . . . . . 8 𝑛(𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
4228, 41nfcxfr 2905 . . . . . . 7 𝑛𝑍
43 fourierdlem112.f . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
44 fourierdlem112.25 . . . . . . . 8 (𝜑𝑋 ∈ ℝ)
45 eqid 2736 . . . . . . . 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 25816 . . . . . . . . . . . . 13 π ∈ ℂ
47462timesi 12291 . . . . . . . . . . . 12 (2 · π) = (π + π)
48 fourierdlem112.t . . . . . . . . . . . 12 𝑇 = (2 · π)
4946, 46subnegi 11480 . . . . . . . . . . . 12 (π − -π) = (π + π)
5047, 48, 493eqtr4i 2774 . . . . . . . . . . 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 25815 . . . . . . . . . . . . . 14 π ∈ ℝ
5554a1i 11 . . . . . . . . . . . . 13 (𝜑 → π ∈ ℝ)
5655renegcld 11582 . . . . . . . . . . . 12 (𝜑 → -π ∈ ℝ)
5756, 44readdcld 11184 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) ∈ ℝ)
5855, 44readdcld 11184 . . . . . . . . . . 11 (𝜑 → (π + 𝑋) ∈ ℝ)
59 negpilt0 43504 . . . . . . . . . . . . . 14 -π < 0
60 pipos 25817 . . . . . . . . . . . . . 14 0 < π
6154renegcli 11462 . . . . . . . . . . . . . . 15 -π ∈ ℝ
62 0re 11157 . . . . . . . . . . . . . . 15 0 ∈ ℝ
6361, 62, 54lttri 11281 . . . . . . . . . . . . . 14 ((-π < 0 ∧ 0 < π) → -π < π)
6459, 60, 63mp2an 690 . . . . . . . . . . . . 13 -π < π
6564a1i 11 . . . . . . . . . . . 12 (𝜑 → -π < π)
6656, 55, 44, 65ltadd1dd 11766 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) < (π + 𝑋))
67 oveq1 7364 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑦 + (𝑘 · 𝑇)) = (𝑥 + (𝑘 · 𝑇)))
6867eleq1d 2822 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
6968rexbidv 3175 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
7069cbvrabv 3417 . . . . . . . . . . . 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 4120 . . . . . . . . . . 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 44391 . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑝𝑖) = (𝑝𝑗))
81 oveq1 7364 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
8281fveq2d 6846 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1)))
8380, 82breq12d 5118 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝𝑗) < (𝑝‘(𝑗 + 1))))
8483cbvralvw 3225 . . . . . . . . . . . . . 14 (∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))
8584anbi2i 623 . . . . . . . . . . . . 13 ((((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1))))
8685a1i 11 . . . . . . . . . . . 12 (𝑝 ∈ (ℝ ↑m (0...𝑛)) → ((((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑗 ∈ (0..^𝑛)(𝑝𝑗) < (𝑝‘(𝑗 + 1)))))
8786rabbiia 3411 . . . . . . . . . . 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 5210 . . . . . . . . . 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 2764 . . . . . . . . 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 713 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
94 eleq1w 2820 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
9594anbi2d 629 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑗 ∈ (0..^𝑀))))
96 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑄𝑖) = (𝑄𝑗))
9781fveq2d 6846 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
9896, 97oveq12d 7375 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1))))
9998reseq2d 5937 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))))
10098oveq1d 7372 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) = (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
10199, 100eleq12d 2832 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) ↔ (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ)))
10295, 101imbi12d 344 . . . . . . . . . . 11 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))))
103 fourierdlem112.fcn . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
104102, 103chvarvv 2002 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
105104adantlr 713 . . . . . . . . 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 11205 . . . . . . . . . . 11 (𝜑 → (-π + 𝑋) ∈ ℝ*)
108 pnfxr 11209 . . . . . . . . . . . 12 +∞ ∈ ℝ*
109108a1i 11 . . . . . . . . . . 11 (𝜑 → +∞ ∈ ℝ*)
11058ltpnfd 13042 . . . . . . . . . . 11 (𝜑 → (π + 𝑋) < +∞)
111107, 109, 58, 66, 110eliood 43726 . . . . . . . . . 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 7368 . . . . . . . . . . 11 (0..^𝑁) = (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))
115113, 114eleqtrdi 2848 . . . . . . . . . 10 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))
116115adantl 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 7368 . . . . . . . . . . . 12 (0...𝑁) = (0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))
118 isoeq4 7265 . . . . . . . . . . . 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 6481 . . . . . . . . . 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 2764 . . . . . . . . 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 44435 . . . . . . . 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 3267 . . . . . . . . . . 11 𝑡𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤
126 elioore 13294 . . . . . . . . . . . . 13 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
127 rspa 3231 . . . . . . . . . . . . 13 ((∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤𝑡 ∈ ℝ) → (abs‘(𝐹𝑡)) ≤ 𝑤)
128126, 127sylan2 593 . . . . . . . . . . . 12 ((∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(𝐹𝑡)) ≤ 𝑤)
129128ex 413 . . . . . . . . . . 11 (∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤 → (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → (abs‘(𝐹𝑡)) ≤ 𝑤))
130125, 129ralrimi 3240 . . . . . . . . . 10 (∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑤 → ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹𝑡)) ≤ 𝑤)
131130reximi 3087 . . . . . . . . 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 3966 . . . . . . . . . . . 12 ℝ ⊆ ℝ
134 dvfre 25315 . . . . . . . . . . . 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 2736 . . . . . . . . . . . . 13 (ℝ D 𝐹) = (ℝ D 𝐹)
13854a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → π ∈ ℝ)
13961a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → -π ∈ ℝ)
14098reseq2d 5937 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))))
141140, 100eleq12d 2832 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) ↔ ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ)))
14295, 141imbi12d 344 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))))
143 fourierdlem112.fdvcn . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
144142, 143chvarvv 2002 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) ∈ (((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))–cn→ℂ))
145144adantlr 713 . . . . . . . . . . . . 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 11184 . . . . . . . . . . . . . 14 (𝜑 → (-π + 𝑋) ∈ ℝ)
148147adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → (-π + 𝑋) ∈ ℝ)
149147rexrd 11205 . . . . . . . . . . . . . . 15 (𝜑 → (-π + 𝑋) ∈ ℝ*)
15055, 146readdcld 11184 . . . . . . . . . . . . . . 15 (𝜑 → (π + 𝑋) ∈ ℝ)
15156, 55, 146, 65ltadd1dd 11766 . . . . . . . . . . . . . . 15 (𝜑 → (-π + 𝑋) < (π + 𝑋))
152150ltpnfd 13042 . . . . . . . . . . . . . . 15 (𝜑 → (π + 𝑋) < +∞)
153149, 109, 150, 151, 152eliood 43726 . . . . . . . . . . . . . 14 (𝜑 → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
154153adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → (π + 𝑋) ∈ ((-π + 𝑋)(,)+∞))
155 oveq1 7364 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = → (𝑘 · 𝑇) = ( · 𝑇))
156155oveq2d 7373 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝑦 + (𝑘 · 𝑇)) = (𝑦 + ( · 𝑇)))
157156eleq1d 2822 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + ( · 𝑇)) ∈ ran 𝑄))
158157cbvrexvw 3226 . . . . . . . . . . . . . . . . . . 19 (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄)
159158rgenw 3068 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋))(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄)
160 rabbi 3432 . . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . . . 17 {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄}
162161uneq2i 4120 . . . . . . . . . . . . . . . 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 7266 . . . . . . . . . . . . . . . 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 6481 . . . . . . . . . . . . . 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 2764 . . . . . . . . . . . . 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 2820 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → (𝑣 ∈ dom (ℝ D 𝐹) ↔ 𝑢 ∈ dom (ℝ D 𝐹)))
168 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → ((ℝ D 𝐹)‘𝑣) = ((ℝ D 𝐹)‘𝑢))
169167, 168ifbieq1d 4510 . . . . . . . . . . . . . 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 5218 . . . . . . . . . . . . 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 44434 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
172 cncff 24256 . . . . . . . . . . . 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 6677 . . . . . . . . . . . 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 5960 . . . . . . . . . . 11 (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))
176174, 175sylibr 233 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹))
177136, 176fssresd 6709 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
178 ax-resscn 11108 . . . . . . . . . . 11 ℝ ⊆ ℂ
179178a1i 11 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → ℝ ⊆ ℂ)
180 cncfcdm 24261 . . . . . . . . . 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 256 . . . . . . . 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 1917 . . . . . . . . . . . . . 14 𝑡(𝜑𝑖 ∈ (0..^𝑁))
186 nfra1 3267 . . . . . . . . . . . . . 14 𝑡𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧
187185, 186nfan 1902 . . . . . . . . . . . . 13 𝑡((𝜑𝑖 ∈ (0..^𝑁)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
188 fvres 6861 . . . . . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → (abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡)))
191190adantlr 713 . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . 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 3944 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) → 𝑡 ∈ dom (ℝ D 𝐹))
194193adantlr 713 . . . . . . . . . . . . . . . 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 3231 . . . . . . . . . . . . . . . 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 5127 . . . . . . . . . . . . . 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 3240 . . . . . . . . . . . 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 3167 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧))
202184, 201mpd 15 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧)
203 nfra1 3267 . . . . . . . . . . . 12 𝑡𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧
204188eqcomd 2742 . . . . . . . . . . . . . . . 16 (𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))) → ((ℝ D 𝐹)‘𝑡) = (((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡))
205204fveq2d 6846 . . . . . . . . . . . . . . 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 3231 . . . . . . . . . . . . . 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 5127 . . . . . . . . . . . . 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 3240 . . . . . . . . . . 11 (∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 → ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
211210a1i 11 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 → ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧))
212211reximdv 3167 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1))))‘𝑡)) ≤ 𝑧 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧))
213202, 212mpd 15 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
214 nfv 1917 . . . . . . . . . . . 12 𝑖(𝜑𝑗 ∈ (0..^𝑀))
215 nfcsb1v 3880 . . . . . . . . . . . . 13 𝑖𝑗 / 𝑖𝐶
216215nfel1 2923 . . . . . . . . . . . 12 𝑖𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗))
217214, 216nfim 1899 . . . . . . . . . . 11 𝑖((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
218 csbeq1a 3869 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝐶 = 𝑗 / 𝑖𝐶)
21999, 96oveq12d 7375 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
220218, 219eleq12d 2832 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) ↔ 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗))))
22195, 220imbi12d 344 . . . . . . . . . . 11 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖))) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))))
222 fourierdlem112.c . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
223217, 221, 222chvarfv 2233 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝐶 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄𝑗)))
224223adantlr 713 . . . . . . . . 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 44433 . . . . . . . 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 3880 . . . . . . . . . . . . 13 𝑖𝑗 / 𝑖𝑈
227226nfel1 2923 . . . . . . . . . . . 12 𝑖𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1)))
228214, 227nfim 1899 . . . . . . . . . . 11 𝑖((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
229 csbeq1a 3869 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝑈 = 𝑗 / 𝑖𝑈)
23099, 97oveq12d 7375 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
231229, 230eleq12d 2832 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑈 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) ↔ 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1)))))
23295, 231imbi12d 344 . . . . . . . . . . 11 (𝑖 = 𝑗 → (((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1)))) ↔ ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))))
233 fourierdlem112.u . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
234228, 232, 233chvarfv 2233 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 / 𝑖𝑈 ∈ ((𝐹 ↾ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) lim (𝑄‘(𝑗 + 1))))
235234adantlr 713 . . . . . . . . 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 44436 . . . . . . . 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 2740 . . . . . . . . . 10 (𝑔 = 𝑠 → (𝑔 = 0 ↔ 𝑠 = 0))
238 oveq2 7365 . . . . . . . . . . . . 13 (𝑔 = 𝑠 → (𝑋 + 𝑔) = (𝑋 + 𝑠))
239238fveq2d 6846 . . . . . . . . . . . 12 (𝑔 = 𝑠 → (𝐹‘(𝑋 + 𝑔)) = (𝐹‘(𝑋 + 𝑠)))
240 breq2 5109 . . . . . . . . . . . . 13 (𝑔 = 𝑠 → (0 < 𝑔 ↔ 0 < 𝑠))
241240ifbid 4509 . . . . . . . . . . . 12 (𝑔 = 𝑠 → if(0 < 𝑔, 𝑅, 𝐿) = if(0 < 𝑠, 𝑅, 𝐿))
242239, 241oveq12d 7375 . . . . . . . . . . 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 7375 . . . . . . . . . 10 (𝑔 = 𝑠 → (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑅, 𝐿)) / 𝑠))
245237, 244ifbieq2d 4512 . . . . . . . . 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 5218 . . . . . . . 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 2740 . . . . . . . . . 10 (𝑜 = 𝑠 → (𝑜 = 0 ↔ 𝑠 = 0))
248 id 22 . . . . . . . . . . 11 (𝑜 = 𝑠𝑜 = 𝑠)
249 oveq1 7364 . . . . . . . . . . . . 13 (𝑜 = 𝑠 → (𝑜 / 2) = (𝑠 / 2))
250249fveq2d 6846 . . . . . . . . . . . 12 (𝑜 = 𝑠 → (sin‘(𝑜 / 2)) = (sin‘(𝑠 / 2)))
251250oveq2d 7373 . . . . . . . . . . 11 (𝑜 = 𝑠 → (2 · (sin‘(𝑜 / 2))) = (2 · (sin‘(𝑠 / 2))))
252248, 251oveq12d 7375 . . . . . . . . . 10 (𝑜 = 𝑠 → (𝑜 / (2 · (sin‘(𝑜 / 2)))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
253247, 252ifbieq2d 4512 . . . . . . . . 9 (𝑜 = 𝑠 → if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
254253cbvmptv 5218 . . . . . . . 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 6842 . . . . . . . . . 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 6842 . . . . . . . . . 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 7375 . . . . . . . . 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 5218 . . . . . . . 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 7365 . . . . . . . . . 10 (𝑑 = 𝑠 → ((𝑘 + (1 / 2)) · 𝑑) = ((𝑘 + (1 / 2)) · 𝑠))
260259fveq2d 6846 . . . . . . . . 9 (𝑑 = 𝑠 → (sin‘((𝑘 + (1 / 2)) · 𝑑)) = (sin‘((𝑘 + (1 / 2)) · 𝑠)))
261260cbvmptv 5218 . . . . . . . 8 (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑))) = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑠)))
262 fveq2 6842 . . . . . . . . . 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 6842 . . . . . . . . . 10 (𝑧 = 𝑠 → ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧) = ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑠))
264262, 263oveq12d 7375 . . . . . . . . 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 5218 . . . . . . . 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 6842 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝐷𝑚) = (𝐷𝑛))
267266fveq1d 6844 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((𝐷𝑚)‘𝑠) = ((𝐷𝑛)‘𝑠))
268267oveq2d 7373 . . . . . . . . . . 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 25146 . . . . . . . . 9 (𝑚 = 𝑛 → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
271270cbvmptv 5218 . . . . . . . 8 (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) = (𝑛 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
272 oveq1 7364 . . . . . . . . . . . . . . . . . . 19 (𝑐 = 𝑘 → (𝑐 + (1 / 2)) = (𝑘 + (1 / 2)))
273272oveq1d 7372 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝑘 → ((𝑐 + (1 / 2)) · 𝑑) = ((𝑘 + (1 / 2)) · 𝑑))
274273fveq2d 6846 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑘 → (sin‘((𝑐 + (1 / 2)) · 𝑑)) = (sin‘((𝑘 + (1 / 2)) · 𝑑)))
275274mpteq2dv 5207 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑘 → (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑))) = (𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑))))
276275fveq1d 6844 . . . . . . . . . . . . . . 15 (𝑐 = 𝑘 → ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑐 + (1 / 2)) · 𝑑)))‘𝑧) = ((𝑑 ∈ (-π[,]π) ↦ (sin‘((𝑘 + (1 / 2)) · 𝑑)))‘𝑧))
277276oveq2d 7373 . . . . . . . . . . . . . 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 5207 . . . . . . . . . . . . 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 6844 . . . . . . . . . . . 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 25146 . . . . . . . . . 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 7372 . . . . . . . . 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 5218 . . . . . . . 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 7364 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → (𝑦 mod (2 · π)) = (𝑠 mod (2 · π)))
290289eqeq1d 2738 . . . . . . . . . . . . 13 (𝑦 = 𝑠 → ((𝑦 mod (2 · π)) = 0 ↔ (𝑠 mod (2 · π)) = 0))
291 oveq2 7365 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → ((𝑚 + (1 / 2)) · 𝑦) = ((𝑚 + (1 / 2)) · 𝑠))
292291fveq2d 6846 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → (sin‘((𝑚 + (1 / 2)) · 𝑦)) = (sin‘((𝑚 + (1 / 2)) · 𝑠)))
293 oveq1 7364 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑠 → (𝑦 / 2) = (𝑠 / 2))
294293fveq2d 6846 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → (sin‘(𝑦 / 2)) = (sin‘(𝑠 / 2)))
295294oveq2d 7373 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → ((2 · π) · (sin‘(𝑦 / 2))) = ((2 · π) · (sin‘(𝑠 / 2))))
296292, 295oveq12d 7375 . . . . . . . . . . . . 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 4512 . . . . . . . . . . . 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 5218 . . . . . . . . . . 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 7373 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (2 · 𝑚) = (2 · 𝑘))
301300oveq1d 7372 . . . . . . . . . . . . . 14 ((𝑚 = 𝑘𝑠 ∈ ℝ) → ((2 · 𝑚) + 1) = ((2 · 𝑘) + 1))
302301oveq1d 7372 . . . . . . . . . . . . 13 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (((2 · 𝑚) + 1) / (2 · π)) = (((2 · 𝑘) + 1) / (2 · π)))
303299oveq1d 7372 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (𝑚 + (1 / 2)) = (𝑘 + (1 / 2)))
304303oveq1d 7372 . . . . . . . . . . . . . . 15 ((𝑚 = 𝑘𝑠 ∈ ℝ) → ((𝑚 + (1 / 2)) · 𝑠) = ((𝑘 + (1 / 2)) · 𝑠))
305304fveq2d 6846 . . . . . . . . . . . . . 14 ((𝑚 = 𝑘𝑠 ∈ ℝ) → (sin‘((𝑚 + (1 / 2)) · 𝑠)) = (sin‘((𝑘 + (1 / 2)) · 𝑠)))
306305oveq1d 7372 . . . . . . . . . . . . 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 4507 . . . . . . . . . . . 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 5205 . . . . . . . . . . 11 (𝑚 = 𝑘 → (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))
309298, 308eqtrid 2788 . . . . . . . . . 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 5218 . . . . . . . . 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 2764 . . . . . . . 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 2736 . . . . . . . 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 2736 . . . . . . . 8 ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙))) = ({-π, 𝑙} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (-π(,)𝑙)))
314 eqid 2736 . . . . . . . 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 7262 . . . . . . . . 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 6456 . . . . . . . 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 6842 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝑉𝑗) = (𝑉𝑖))
318317oveq1d 7372 . . . . . . . . 9 (𝑗 = 𝑖 → ((𝑉𝑗) − 𝑋) = ((𝑉𝑖) − 𝑋))
319318cbvmptv 5218 . . . . . . . 8 (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) = (𝑖 ∈ (0...𝑁) ↦ ((𝑉𝑖) − 𝑋))
320 eqid 2736 . . . . . . . 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 6842 . . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . . 15 (𝑎 = 𝑠 → ((𝑏 + (1 / 2)) · 𝑎) = ((𝑏 + (1 / 2)) · 𝑠))
323322fveq2d 6846 . . . . . . . . . . . . . 14 (𝑎 = 𝑠 → (sin‘((𝑏 + (1 / 2)) · 𝑎)) = (sin‘((𝑏 + (1 / 2)) · 𝑠)))
324321, 323oveq12d 7375 . . . . . . . . . . . . 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 25141 . . . . . . . . . . . 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 6845 . . . . . . . . . . 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 5112 . . . . . . . . . 10 ((abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑖 / 2) ↔ (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑖 / 2))
328327anbi2i 623 . . . . . . . . 9 (((((𝜑𝑖 ∈ ℝ+) ∧ 𝑙 ∈ (-π(,)0)) ∧ 𝑏 ∈ ℕ) ∧ (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑖 / 2)) ↔ ((((𝜑𝑖 ∈ ℝ+) ∧ 𝑙 ∈ (-π(,)0)) ∧ 𝑏 ∈ ℕ) ∧ (abs‘∫(𝑙(,)0)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑖 / 2)))
329324cbvitgv 25141 . . . . . . . . . . 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 6845 . . . . . . . . . 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 5112 . . . . . . . . 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 627 . . . . . . . 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 44440 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) ⇝ (𝐿 / 2))
334 nnex 12159 . . . . . . . . . 10 ℕ ∈ V
335334mptex 7173 . . . . . . . . 9 (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))) ∈ V
33628, 335eqeltri 2834 . . . . . . . 8 𝑍 ∈ V
337336a1i 11 . . . . . . 7 (𝜑𝑍 ∈ V)
338268adantr 481 . . . . . . . . . 10 ((𝑚 = 𝑛𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)))
339338itgeq2dv 25146 . . . . . . . . 9 (𝑚 = 𝑛 → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
340339cbvmptv 5218 . . . . . . . 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 25146 . . . . . . . . . 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 7372 . . . . . . . . 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 5218 . . . . . . . 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 2736 . . . . . . . 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 2736 . . . . . . . 8 ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π))) = ({𝑒, π} ∪ (ran (𝑗 ∈ (0...𝑁) ↦ ((𝑉𝑗) − 𝑋)) ∩ (𝑒(,)π)))
347 eqid 2736 . . . . . . . 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 7262 . . . . . . . . 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 6456 . . . . . . . 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 2736 . . . . . . . 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 25141 . . . . . . . . . . . 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 6845 . . . . . . . . . . 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 5112 . . . . . . . . . 10 ((abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑞 / 2) ↔ (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑞 / 2))
354353anbi2i 623 . . . . . . . . 9 (((((𝜑𝑞 ∈ ℝ+) ∧ 𝑒 ∈ (0(,)π)) ∧ 𝑏 ∈ ℕ) ∧ (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑎) · (sin‘((𝑏 + (1 / 2)) · 𝑎))) d𝑎) < (𝑞 / 2)) ↔ ((((𝜑𝑞 ∈ ℝ+) ∧ 𝑒 ∈ (0(,)π)) ∧ 𝑏 ∈ ℕ) ∧ (abs‘∫(0(,)𝑒)(((𝑟 ∈ (-π[,]π) ↦ (((𝑔 ∈ (-π[,]π) ↦ if(𝑔 = 0, 0, (((𝐹‘(𝑋 + 𝑔)) − if(0 < 𝑔, 𝑅, 𝐿)) / 𝑔)))‘𝑟) · ((𝑜 ∈ (-π[,]π) ↦ if(𝑜 = 0, 1, (𝑜 / (2 · (sin‘(𝑜 / 2))))))‘𝑟)))‘𝑠) · (sin‘((𝑏 + (1 / 2)) · 𝑠))) d𝑠) < (𝑞 / 2)))
355324cbvitgv 25141 . . . . . . . . . . 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 6845 . . . . . . . . . 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 5112 . . . . . . . . 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 627 . . . . . . . 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 44441 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) ⇝ (𝑅 / 2))
360 eqidd 2737 . . . . . . . . 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 13294 . . . . . . . . . . 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 11184 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ ℝ) → (𝑋 + 𝑠) ∈ ℝ)
368364, 367ffvelcdmd 7036 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ ℝ) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
369368adantlr 713 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
370288dirkerre 44326 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
371370adantll 712 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
372369, 371remulcld 11185 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
373363, 372sylan2 593 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
374 ioossicc 13350 . . . . . . . . . . . . 13 (-π(,)0) ⊆ (-π[,]0)
37561leidi 11689 . . . . . . . . . . . . . 14 -π ≤ -π
37662, 54, 60ltleii 11278 . . . . . . . . . . . . . 14 0 ≤ π
377 iccss 13332 . . . . . . . . . . . . . 14 (((-π ∈ ℝ ∧ π ∈ ℝ) ∧ (-π ≤ -π ∧ 0 ≤ π)) → (-π[,]0) ⊆ (-π[,]π))
37861, 54, 375, 376, 377mp4an 691 . . . . . . . . . . . . 13 (-π[,]0) ⊆ (-π[,]π)
379374, 378sstri 3953 . . . . . . . . . . . 12 (-π(,)0) ⊆ (-π[,]π)
380379a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (-π(,)0) ⊆ (-π[,]π))
381 ioombl 24929 . . . . . . . . . . . 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 13351 . . . . . . . . . . . . . . . 16 (𝜑 → (-π[,]π) ⊆ ℝ)
386385sselda 3944 . . . . . . . . . . . . . . 15 ((𝜑𝑠 ∈ (-π[,]π)) → 𝑠 ∈ ℝ)
387384, 386readdcld 11184 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (-π[,]π)) → (𝑋 + 𝑠) ∈ ℝ)
388383, 387ffvelcdmd 7036 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (-π[,]π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
389388adantlr 713 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
390 iccssre 13346 . . . . . . . . . . . . . . . 16 ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
39161, 54, 390mp2an 690 . . . . . . . . . . . . . . 15 (-π[,]π) ⊆ ℝ
392391sseli 3940 . . . . . . . . . . . . . 14 (𝑠 ∈ (-π[,]π) → 𝑠 ∈ ℝ)
393392, 370sylan2 593 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
394393adantll 712 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
395389, 394remulcld 11185 . . . . . . . . . . 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 713 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
403225adantlr 713 . . . . . . . . . . . 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 713 . . . . . . . . . . . 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 44338 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝐷𝑛) ∈ (ℝ–cn→ℝ))
406405adantl 482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) ∈ (ℝ–cn→ℝ))
407 eqid 2736 . . . . . . . . . . . 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 44421 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
409380, 382, 395, 408iblss 25169 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (-π(,)0) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
410373, 409itgcl 25148 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 ∈ ℂ)
411360, 361, 362, 410fvmptd 6955 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) = ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
412411, 410eqeltrd 2838 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) ∈ ℂ)
413 eqidd 2737 . . . . . . . . 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 13294 . . . . . . . . . . . . . . 15 (𝑠 ∈ (0(,)π) → 𝑠 ∈ ℝ)
418417adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (0(,)π)) → 𝑠 ∈ ℝ)
419416, 418readdcld 11184 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (0(,)π)) → (𝑋 + 𝑠) ∈ ℝ)
420415, 419ffvelcdmd 7036 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ (0(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
421420adantlr 713 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
422417, 370sylan2 593 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (0(,)π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
423422adantll 712 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐷𝑛)‘𝑠) ∈ ℝ)
424421, 423remulcld 11185 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) ∈ ℝ)
425 ioossicc 13350 . . . . . . . . . . . . 13 (0(,)π) ⊆ (0[,]π)
42661, 62, 59ltleii 11278 . . . . . . . . . . . . . 14 -π ≤ 0
42754leidi 11689 . . . . . . . . . . . . . 14 π ≤ π
428 iccss 13332 . . . . . . . . . . . . . 14 (((-π ∈ ℝ ∧ π ∈ ℝ) ∧ (-π ≤ 0 ∧ π ≤ π)) → (0[,]π) ⊆ (-π[,]π))
42961, 54, 426, 427, 428mp4an 691 . . . . . . . . . . . . 13 (0[,]π) ⊆ (-π[,]π)
430425, 429sstri 3953 . . . . . . . . . . . 12 (0(,)π) ⊆ (-π[,]π)
431430a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (0(,)π) ⊆ (-π[,]π))
432 ioombl 24929 . . . . . . . . . . . 12 (0(,)π) ∈ dom vol
433432a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (0(,)π) ∈ dom vol)
434431, 433, 395, 408iblss 25169 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑠 ∈ (0(,)π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠))) ∈ 𝐿1)
435424, 434itgcl 25148 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 ∈ ℂ)
436413, 414, 362, 435fvmptd 6955 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) = ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
437436, 435eqeltrd 2838 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)‘𝑛) ∈ ℂ)
438 eleq1w 2820 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑚 ∈ ℕ ↔ 𝑛 ∈ ℕ))
439438anbi2d 629 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝜑𝑚 ∈ ℕ) ↔ (𝜑𝑛 ∈ ℕ)))
440 fveq2 6842 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑍𝑚) = (𝑍𝑛))
441270, 339oveq12d 7375 . . . . . . . . . . 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 2752 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝑍𝑚) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠) ↔ (𝑍𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)))
443439, 442imbi12d 344 . . . . . . . . 9 (𝑚 = 𝑛 → (((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)) ↔ ((𝜑𝑛 ∈ ℕ) → (𝑍𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠))))
444 oveq1 7364 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝑛 · 𝑥) = (𝑚 · 𝑥))
445444fveq2d 6846 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (cos‘(𝑛 · 𝑥)) = (cos‘(𝑚 · 𝑥)))
446445oveq2d 7373 . . . . . . . . . . . . . . 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 25146 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 = ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥)
449448oveq1d 7372 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π) = (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π))
450449cbvmptv 5218 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) = (𝑚 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑚 · 𝑥))) d𝑥 / π))
45129, 450eqtri 2764 . . . . . . . . . 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 6846 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (sin‘(𝑛 · 𝑥)) = (sin‘(𝑚 · 𝑥)))
454453oveq2d 7373 . . . . . . . . . . . . . . 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 25146 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 = ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥)
457456oveq1d 7372 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π) = (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
458457cbvmptv 5218 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) = (𝑚 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
459452, 458eqtri 2764 . . . . . . . . . 10 𝐵 = (𝑚 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑚 · 𝑥))) d𝑥 / π))
460 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐴𝑛) = (𝐴𝑘))
461 oveq1 7364 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
462461fveq2d 6846 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
463460, 462oveq12d 7375 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))))
464 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐵𝑛) = (𝐵𝑘))
465461fveq2d 6846 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
466464, 465oveq12d 7375 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
467463, 466oveq12d 7375 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
468467cbvsumv 15581 . . . . . . . . . . . . 13 Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
469468oveq2i 7368 . . . . . . . . . . . 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 5210 . . . . . . . . . . 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 7365 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
472471sumeq1d 15586 . . . . . . . . . . . . . 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 7373 . . . . . . . . . . . . 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 5218 . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝐴𝑘) = (𝐴𝑚))
476 oveq1 7364 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → (𝑘 · 𝑋) = (𝑚 · 𝑋))
477476fveq2d 6846 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (cos‘(𝑘 · 𝑋)) = (cos‘(𝑚 · 𝑋)))
478475, 477oveq12d 7375 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) = ((𝐴𝑚) · (cos‘(𝑚 · 𝑋))))
479 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝐵𝑘) = (𝐵𝑚))
480476fveq2d 6846 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (sin‘(𝑘 · 𝑋)) = (sin‘(𝑚 · 𝑋)))
481479, 480oveq12d 7375 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) = ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))
482478, 481oveq12d 7375 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋)))))
483482cbvsumv 15581 . . . . . . . . . . . . . 14 Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))
484483oveq2i 7368 . . . . . . . . . . . . 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 5210 . . . . . . . . . . . 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 2764 . . . . . . . . . . 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 2768 . . . . . . . . . 10 𝑍 = (𝑛 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑚 ∈ (1...𝑛)(((𝐴𝑚) · (cos‘(𝑚 · 𝑋))) + ((𝐵𝑚) · (sin‘(𝑚 · 𝑋))))))
488 oveq2 7365 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑋 + 𝑦) = (𝑋 + 𝑥))
489488fveq2d 6846 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑥)))
490 fveq2 6842 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝐷𝑚)‘𝑦) = ((𝐷𝑚)‘𝑥))
491489, 490oveq12d 7375 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹‘(𝑋 + 𝑦)) · ((𝐷𝑚)‘𝑦)) = ((𝐹‘(𝑋 + 𝑥)) · ((𝐷𝑚)‘𝑥)))
492491cbvmptv 5218 . . . . . . . . . 10 (𝑦 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑦)) · ((𝐷𝑚)‘𝑦))) = (𝑥 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷𝑚)‘𝑥)))
493 eqid 2736 . . . . . . . . . 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 6842 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑄𝑗) = (𝑄𝑖))
495494oveq1d 7372 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝑄𝑗) − 𝑋) = ((𝑄𝑖) − 𝑋))
496495cbvmptv 5218 . . . . . . . . . 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 44448 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠))
498443, 497chvarvv 2002 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝑍𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠))
499411, 436oveq12d 7375 . . . . . . . 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 2779 . . . . . . 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 43846 . . . . . 6 (𝜑𝑍 ⇝ ((𝐿 / 2) + (𝑅 / 2)))
502 limccl 25239 . . . . . . . 8 ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋) ⊆ ℂ
503502, 285sselid 3942 . . . . . . 7 (𝜑𝐿 ∈ ℂ)
504 limccl 25239 . . . . . . . 8 ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋) ⊆ ℂ
505504, 284sselid 3942 . . . . . . 7 (𝜑𝑅 ∈ ℂ)
506 2cnd 12231 . . . . . . 7 (𝜑 → 2 ∈ ℂ)
507 2pos 12256 . . . . . . . . 9 0 < 2
508507a1i 11 . . . . . . . 8 (𝜑 → 0 < 2)
509508gt0ne0d 11719 . . . . . . 7 (𝜑 → 2 ≠ 0)
510503, 505, 506, 509divdird 11969 . . . . . 6 (𝜑 → ((𝐿 + 𝑅) / 2) = ((𝐿 / 2) + (𝑅 / 2)))
511501, 510breqtrrd 5133 . . . . 5 (𝜑𝑍 ⇝ ((𝐿 + 𝑅) / 2))
512 0nn0 12428 . . . . . . . 8 0 ∈ ℕ0
51343adantr 481 . . . . . . . . . 10 ((𝜑 ∧ 0 ∈ ℕ0) → 𝐹:ℝ⟶ℝ)
514 eqid 2736 . . . . . . . . . 10 (-π(,)π) = (-π(,)π)
515 ioossre 13325 . . . . . . . . . . . . . 14 (-π(,)π) ⊆ ℝ
516515a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ⊆ ℝ)
51743, 516feqresmpt 6911 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ (-π(,)π)) = (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)))
518 ioossicc 13350 . . . . . . . . . . . . . 14 (-π(,)π) ⊆ (-π[,]π)
519518a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ⊆ (-π[,]π))
520 ioombl 24929 . . . . . . . . . . . . . 14 (-π(,)π) ∈ dom vol
521520a1i 11 . . . . . . . . . . . . 13 (𝜑 → (-π(,)π) ∈ dom vol)
52243adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℝ)
523385sselda 3944 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (-π[,]π)) → 𝑥 ∈ ℝ)
524522, 523ffvelcdmd 7036 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (-π[,]π)) → (𝐹𝑥) ∈ ℝ)
52543, 385feqresmpt 6911 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (-π[,]π)) = (𝑥 ∈ (-π[,]π) ↦ (𝐹𝑥)))
526178a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ℝ ⊆ ℂ)
52743, 526fssd 6686 . . . . . . . . . . . . . . . 16 (𝜑𝐹:ℝ⟶ℂ)
528527, 385fssresd 6709 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 ↾ (-π[,]π)):(-π[,]π)⟶ℂ)
529 ioossicc 13350 . . . . . . . . . . . . . . . . . 18 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1)))
53061rexri 11213 . . . . . . . . . . . . . . . . . . . 20 -π ∈ ℝ*
531530a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ*)
53254rexri 11213 . . . . . . . . . . . . . . . . . . . 20 π ∈ ℝ*
533532a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑀)) → π ∈ ℝ*)
53451, 52, 53fourierdlem15 44353 . . . . . . . . . . . . . . . . . . . 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 44346 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
538529, 537sstrid 3955 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
539538resabs1d 5968 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
540539, 103eqeltrd 2838 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
541539eqcomd 2742 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
542541oveq1d 7372 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) = (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
543222, 542eleqtrd 2840 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
544541oveq1d 7372 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) = (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
545233, 544eleqtrd 2840 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
54651, 52, 53, 528, 540, 543, 545fourierdlem69 44406 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (-π[,]π)) ∈ 𝐿1)
547525, 546eqeltrrd 2839 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (-π[,]π) ↦ (𝐹𝑥)) ∈ 𝐿1)
548519, 521, 524, 547iblss 25169 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1)
549517, 548eqeltrd 2838 . . . . . . . . . . 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 44354 . . . . . . . . 9 ((𝜑 ∧ 0 ∈ ℕ0) → (((𝐴‘0) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (cos‘(0 · 𝑥))) d𝑥 ∈ ℝ))
553552simplld 766 . . . . . . . 8 ((𝜑 ∧ 0 ∈ ℕ0) → (𝐴‘0) ∈ ℝ)
554512, 553mpan2 689 . . . . . . 7 (𝜑 → (𝐴‘0) ∈ ℝ)
555554rehalfcld 12400 . . . . . 6 (𝜑 → ((𝐴‘0) / 2) ∈ ℝ)
556555recnd 11183 . . . . 5 (𝜑 → ((𝐴‘0) / 2) ∈ ℂ)
557334mptex 7173 . . . . . 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 13878 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (1...𝑚) ∈ Fin)
562 simpll 765 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → 𝜑)
563 elfznn 13470 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑚) → 𝑛 ∈ ℕ)
564563adantl 482 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑚)) → 𝑛 ∈ ℕ)
565 simpl 483 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝜑)
566362nnnn0d 12473 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
567 eleq1w 2820 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝑘 ∈ ℕ0𝑛 ∈ ℕ0))
568567anbi2d 629 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝜑𝑘 ∈ ℕ0) ↔ (𝜑𝑛 ∈ ℕ0)))
569 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → (𝐴𝑘) = (𝐴𝑛))
570569eleq1d 2822 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ((𝐴𝑘) ∈ ℝ ↔ (𝐴𝑛) ∈ ℝ))
571568, 570imbi12d 344 . . . . . . . . . . . . . . 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 44354 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ0) → (((𝐴𝑘) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ (𝐹𝑥)) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑘 · 𝑥))) d𝑥 ∈ ℝ))
576575simplld 766 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℝ)
577571, 576chvarvv 2002 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℝ)
578565, 566, 577syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ∈ ℝ)
579362nnred 12168 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
580579, 399remulcld 11185 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝑛 · 𝑋) ∈ ℝ)
581580recoscld 16026 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (cos‘(𝑛 · 𝑋)) ∈ ℝ)
582578, 581remulcld 11185 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) ∈ ℝ)
583 eleq1w 2820 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝑘 ∈ ℕ ↔ 𝑛 ∈ ℕ))
584583anbi2d 629 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → ((𝜑𝑘 ∈ ℕ) ↔ (𝜑𝑛 ∈ ℕ)))
585 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝐵𝑘) = (𝐵𝑛))
586585eleq1d 2822 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → ((𝐵𝑘) ∈ ℝ ↔ (𝐵𝑛) ∈ ℝ))
587584, 586imbi12d 344 . . . . . . . . . . . . . 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 44359 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (((𝐵𝑘) ∈ ℝ ∧ (𝑥 ∈ (-π(,)π) ↦ ((𝐹𝑥) · (sin‘(𝑘 · 𝑥)))) ∈ 𝐿1) ∧ ∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑘 · 𝑥))) d𝑥 ∈ ℝ))
592591simplld 766 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐵𝑘) ∈ ℝ)
593587, 592chvarvv 2002 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐵𝑛) ∈ ℝ)
594580resincld 16025 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (sin‘(𝑛 · 𝑋)) ∈ ℝ)
595593, 594remulcld 11185 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) ∈ ℝ)
596582, 595readdcld 11184 . . . . . . . . . . 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 15619 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℝ)
599560, 598readdcld 11184 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ ℝ)
60028fvmpt2 6959 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ ℝ) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
601559, 599, 600syl2anc 584 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
602601, 599eqeltrd 2838 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) ∈ ℝ)
603602recnd 11183 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑍𝑚) ∈ ℂ)
604 eqidd 2737 . . . . . . 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 7365 . . . . . . . . 9 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
606605sumeq1d 15586 . . . . . . . 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 15572 . . . . . . . 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 6955 . . . . . 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 11183 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ((𝐴‘0) / 2) ∈ ℂ)
612598recnd 11183 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℂ)
613611, 612pncan2d 11514 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → ((((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) − ((𝐴‘0) / 2)) = Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
614613, 468eqtr2di 2793 . . . . . 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 7390 . . . . . . . . 9 (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) ∈ V
61628fvmpt2 6959 . . . . . . . . 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 2742 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝑍𝑚))
619618oveq1d 7372 . . . . . 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 2780 . . . . 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 15520 . . . 4 (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
622 seqex 13908 . . . . . 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 2737 . . . . . . 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 7365 . . . . . . . . 9 (𝑛 = 𝑙 → (1...𝑛) = (1...𝑙))
626625sumeq1d 15586 . . . . . . . 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 13878 . . . . . . . 8 ((𝜑𝑙 ∈ ℕ) → (1...𝑙) ∈ Fin)
630 elfznn 13470 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ)
631630nnnn0d 12473 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℕ0)
632631, 576sylan2 593 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (𝐴𝑘) ∈ ℝ)
633630nnred 12168 . . . . . . . . . . . . . 14 (𝑘 ∈ (1...𝑙) → 𝑘 ∈ ℝ)
634633adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1...𝑙)) → 𝑘 ∈ ℝ)
635146adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1...𝑙)) → 𝑋 ∈ ℝ)
636634, 635remulcld 11185 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (1...𝑙)) → (𝑘 · 𝑋) ∈ ℝ)
637636recoscld 16026 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (cos‘(𝑘 · 𝑋)) ∈ ℝ)
638632, 637remulcld 11185 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1...𝑙)) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
639630, 592sylan2 593 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (𝐵𝑘) ∈ ℝ)
640636resincld 16025 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1...𝑙)) → (sin‘(𝑘 · 𝑋)) ∈ ℝ)
641639, 640remulcld 11185 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1...𝑙)) → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) ∈ ℝ)
642638, 641readdcld 11184 . . . . . . . . 9 ((𝜑𝑘 ∈ (1...𝑙)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
643642adantlr 713 . . . . . . . 8 (((𝜑𝑙 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑙)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
644629, 643fsumrecl 15619 . . . . . . 7 ((𝜑𝑙 ∈ ℕ) → Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℝ)
645624, 627, 628, 644fvmptd 6955 . . . . . 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 2820 . . . . . . . . 9 (𝑛 = 𝑙 → (𝑛 ∈ ℕ ↔ 𝑙 ∈ ℕ))
647646anbi2d 629 . . . . . . . 8 (𝑛 = 𝑙 → ((𝜑𝑛 ∈ ℕ) ↔ (𝜑𝑙 ∈ ℕ)))
648 fveq2 6842 . . . . . . . . 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 2752 . . . . . . . 8 (𝑛 = 𝑙 → (Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑛) ↔ Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑙)))
650647, 649imbi12d 344 . . . . . . 7 (𝑛 = 𝑙 → (((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑛)) ↔ ((𝜑𝑙 ∈ ℕ) → Σ𝑘 ∈ (1...𝑙)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = (seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))))))‘𝑙))))
651 eqidd 2737 . . . . . . . . 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 6842 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
653 oveq1 7364 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑗 · 𝑋) = (𝑘 · 𝑋))
654653fveq2d 6846 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (cos‘(𝑗 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
655652, 654oveq12d 7375 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) = ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))))
656 fveq2 6842 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐵𝑗) = (𝐵𝑘))
657653fveq2d 6846 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (sin‘(𝑗 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
658656, 657oveq12d 7375 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))) = ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
659655, 658oveq12d 7375 . . . . . . . . . 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 13470 . . . . . . . . . 10 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
662661adantl 482 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
663 simpll 765 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
664 nnnn0 12420 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
665 nn0re 12422 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
666665adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
667146adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → 𝑋 ∈ ℝ)
668666, 667remulcld 11185 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ0) → (𝑘 · 𝑋) ∈ ℝ)
669668recoscld 16026 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ0) → (cos‘(𝑘 · 𝑋)) ∈ ℝ)
670576, 669remulcld 11185 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
671664, 670sylan2 593 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) ∈ ℝ)
672664, 668sylan2 593 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (𝑘 · 𝑋) ∈ ℝ)
673672resincld 16025 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (sin‘(𝑘 · 𝑋)) ∈ ℝ)
674592, 673remulcld 11185 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))) ∈ ℝ)
675671, 674readdcld 11184 . . . . . . . . . 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 6955 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))‘𝑘) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
678362, 14eleqtrdi 2848 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
679676recnd 11183 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) ∈ ℂ)
680677, 678, 679fsumser 15615 . . . . . . 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 2002 . . . . . 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 2776 . . . . 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 15449 . . . 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 231 . . 3 (𝜑 → seq1( + , (𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
68513, 684eqbrtrd 5127 . 2 (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
686 eqidd 2737 . . . . . 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 6842 . . . . . . . . 9 (𝑗 = 𝑛 → (𝐴𝑗) = (𝐴𝑛))
688 oveq1 7364 . . . . . . . . . 10 (𝑗 = 𝑛 → (𝑗 · 𝑋) = (𝑛 · 𝑋))
689688fveq2d 6846 . . . . . . . . 9 (𝑗 = 𝑛 → (cos‘(𝑗 · 𝑋)) = (cos‘(𝑛 · 𝑋)))
690687, 689oveq12d 7375 . . . . . . . 8 (𝑗 = 𝑛 → ((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) = ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))))
691 fveq2 6842 . . . . . . . . 9 (𝑗 = 𝑛 → (𝐵𝑗) = (𝐵𝑛))
692688fveq2d 6846 . . . . . . . . 9 (𝑗 = 𝑛 → (sin‘(𝑗 · 𝑋)) = (sin‘(𝑛 · 𝑋)))
693691, 692oveq12d 7375 . . . . . . . 8 (𝑗 = 𝑛 → ((𝐵𝑗) · (sin‘(𝑗 · 𝑋))) = ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))
694690, 693oveq12d 7375 . . . . . . 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 6955 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (((𝐴𝑗) · (cos‘(𝑗 · 𝑋))) + ((𝐵𝑗) · (sin‘(𝑗 · 𝑋)))))‘𝑛) = (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
697596recnd 11183 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) ∈ ℂ)
69814, 15, 696, 697, 684isumclim 15642 . . . 4 (𝜑 → Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))
699698oveq2d 7373 . . 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 11174 . . . . 5 (𝜑 → (𝐿 + 𝑅) ∈ ℂ)
701700halfcld 12398 . . . 4 (𝜑 → ((𝐿 + 𝑅) / 2) ∈ ℂ)
702556, 701pncan3d 11515 . . 3 (𝜑 → (((𝐴‘0) / 2) + (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))) = ((𝐿 + 𝑅) / 2))
703699, 702eqtrd 2776 . 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 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  csb 3855  cun 3908  cin 3909  wss 3910  ifcif 4486  {cpr 4588   class class class wbr 5105  cmpt 5188  dom cdm 5633  ran crn 5634  cres 5635  cio 6446  wf 6492  cfv 6496   Isom wiso 6497  crio 7312  (class class class)co 7357  m cmap 8765  supcsup 9376  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  +∞cpnf 11186  -∞cmnf 11187  *cxr 11188   < clt 11189  cle 11190  cmin 11385  -cneg 11386   / cdiv 11812  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  +crp 12915  (,)cioo 13264  (,]cioc 13265  [,]cicc 13267  ...cfz 13424  ..^cfzo 13567  cfl 13695   mod cmo 13774  seqcseq 13906  chash 14230  abscabs 15119  cli 15366  Σcsu 15570  sincsin 15946  cosccos 15947  πcpi 15949  cnccncf 24239  volcvol 24827  𝐿1cibl 24981  citg 24982   lim climc 25226   D cdv 25227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-symdif 4202  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-t1 22665  df-haus 22666  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-ovol 24828  df-vol 24829  df-mbf 24983  df-itg1 24984  df-itg2 24985  df-ibl 24986  df-itg 24987  df-0p 25034  df-ditg 25211  df-limc 25230  df-dv 25231
This theorem is referenced by:  fourierdlem113  44450
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