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Theorem fourierdlem113 40957
Description: Fourier series convergence for periodic, piecewise smooth functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem113.f (𝜑𝐹:ℝ⟶ℝ)
fourierdlem113.t 𝑇 = (2 · π)
fourierdlem113.per ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
fourierdlem113.x (𝜑𝑋 ∈ ℝ)
fourierdlem113.l (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋))
fourierdlem113.r (𝜑𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))
fourierdlem113.p 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem113.m (𝜑𝑀 ∈ ℕ)
fourierdlem113.q (𝜑𝑄 ∈ (𝑃𝑀))
fourierdlem113.dvcn ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem113.dvlb ((𝜑𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) ≠ ∅)
fourierdlem113.dvub ((𝜑𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) ≠ ∅)
fourierdlem113.a 𝐴 = (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
fourierdlem113.b 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
fourierdlem113.15 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
fourierdlem113.e 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇)))
fourierdlem113.exq (𝜑 → (𝐸𝑋) ∈ ran 𝑄)
Assertion
Ref Expression
fourierdlem113 (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,𝑛   𝐵,𝑛   𝑥,𝐸   𝑖,𝐹,𝑛,𝑥   𝑖,𝐿,𝑛   𝑖,𝑀,𝑥,𝑛   𝑀,𝑝,𝑖,𝑛   𝑄,𝑖,𝑥,𝑛   𝑄,𝑝   𝑅,𝑖,𝑛   𝑇,𝑖,𝑥,𝑛   𝑇,𝑝   𝑖,𝑋,𝑥,𝑛   𝑋,𝑝   𝜑,𝑖,𝑥,𝑛
Allowed substitution hints:   𝜑(𝑝)   𝐴(𝑥,𝑖,𝑝)   𝐵(𝑥,𝑖,𝑝)   𝑃(𝑥,𝑖,𝑛,𝑝)   𝑅(𝑥,𝑝)   𝑆(𝑥,𝑖,𝑛,𝑝)   𝐸(𝑖,𝑛,𝑝)   𝐹(𝑝)   𝐿(𝑥,𝑝)

Proof of Theorem fourierdlem113
Dummy variables 𝑗 𝑘 𝑚 𝑤 𝑦 𝑡 𝑢 𝑧 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem113.f . 2 (𝜑𝐹:ℝ⟶ℝ)
2 oveq1 6821 . . . . . . 7 (𝑤 = 𝑦 → (𝑤 mod (2 · π)) = (𝑦 mod (2 · π)))
32eqeq1d 2762 . . . . . 6 (𝑤 = 𝑦 → ((𝑤 mod (2 · π)) = 0 ↔ (𝑦 mod (2 · π)) = 0))
4 oveq2 6822 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑘 + (1 / 2)) · 𝑤) = ((𝑘 + (1 / 2)) · 𝑦))
54fveq2d 6357 . . . . . . 7 (𝑤 = 𝑦 → (sin‘((𝑘 + (1 / 2)) · 𝑤)) = (sin‘((𝑘 + (1 / 2)) · 𝑦)))
6 oveq1 6821 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤 / 2) = (𝑦 / 2))
76fveq2d 6357 . . . . . . . 8 (𝑤 = 𝑦 → (sin‘(𝑤 / 2)) = (sin‘(𝑦 / 2)))
87oveq2d 6830 . . . . . . 7 (𝑤 = 𝑦 → ((2 · π) · (sin‘(𝑤 / 2))) = ((2 · π) · (sin‘(𝑦 / 2))))
95, 8oveq12d 6832 . . . . . 6 (𝑤 = 𝑦 → ((sin‘((𝑘 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2)))) = ((sin‘((𝑘 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))
103, 9ifbieq2d 4255 . . . . 5 (𝑤 = 𝑦 → if((𝑤 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2))))) = if((𝑦 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2))))))
1110cbvmptv 4902 . . . 4 (𝑤 ∈ ℝ ↦ if((𝑤 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2)))))) = (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2))))))
12 oveq2 6822 . . . . . . . 8 (𝑘 = 𝑚 → (2 · 𝑘) = (2 · 𝑚))
1312oveq1d 6829 . . . . . . 7 (𝑘 = 𝑚 → ((2 · 𝑘) + 1) = ((2 · 𝑚) + 1))
1413oveq1d 6829 . . . . . 6 (𝑘 = 𝑚 → (((2 · 𝑘) + 1) / (2 · π)) = (((2 · 𝑚) + 1) / (2 · π)))
15 oveq1 6821 . . . . . . . . 9 (𝑘 = 𝑚 → (𝑘 + (1 / 2)) = (𝑚 + (1 / 2)))
1615oveq1d 6829 . . . . . . . 8 (𝑘 = 𝑚 → ((𝑘 + (1 / 2)) · 𝑦) = ((𝑚 + (1 / 2)) · 𝑦))
1716fveq2d 6357 . . . . . . 7 (𝑘 = 𝑚 → (sin‘((𝑘 + (1 / 2)) · 𝑦)) = (sin‘((𝑚 + (1 / 2)) · 𝑦)))
1817oveq1d 6829 . . . . . 6 (𝑘 = 𝑚 → ((sin‘((𝑘 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))) = ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))
1914, 18ifeq12d 4250 . . . . 5 (𝑘 = 𝑚 → if((𝑦 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2))))) = if((𝑦 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2))))))
2019mpteq2dv 4897 . . . 4 (𝑘 = 𝑚 → (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))) = (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))
2111, 20syl5eq 2806 . . 3 (𝑘 = 𝑚 → (𝑤 ∈ ℝ ↦ if((𝑤 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2)))))) = (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))
2221cbvmptv 4902 . 2 (𝑘 ∈ ℕ ↦ (𝑤 ∈ ℝ ↦ if((𝑤 mod (2 · π)) = 0, (((2 · 𝑘) + 1) / (2 · π)), ((sin‘((𝑘 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2))))))) = (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)))))))
23 fourierdlem113.p . 2 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
24 fourierdlem113.m . 2 (𝜑𝑀 ∈ ℕ)
25 fourierdlem113.q . 2 (𝜑𝑄 ∈ (𝑃𝑀))
26 oveq1 6821 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤 + (𝑗 · 𝑇)) = (𝑦 + (𝑗 · 𝑇)))
2726eleq1d 2824 . . . . . . 7 (𝑤 = 𝑦 → ((𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
2827rexbidv 3190 . . . . . 6 (𝑤 = 𝑦 → (∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
2928cbvrabv 3339 . . . . 5 {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄}
3029uneq2i 3907 . . . 4 ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄}) = ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄})
3130fveq2i 6356 . . 3 (♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})) = (♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄}))
3231oveq1i 6824 . 2 ((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})) − 1) = ((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄})) − 1)
33 oveq1 6821 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑘 · 𝑇) = (𝑗 · 𝑇))
3433oveq2d 6830 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑗 · 𝑇)))
3534eleq1d 2824 . . . . . . . . . 10 (𝑘 = 𝑗 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
3635cbvrexv 3311 . . . . . . . . 9 (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄)
3736a1i 11 . . . . . . . 8 (𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
3837rabbiia 3324 . . . . . . 7 {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄}
3938uneq2i 3907 . . . . . 6 ({(-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)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄})
40 isoeq5 6735 . . . . . 6 (({(-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)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄}) → (𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄}))))
4139, 40ax-mp 5 . . . . 5 (𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄})))
4241a1i 11 . . . 4 (𝑔 = 𝑓 → (𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄}))))
4333oveq2d 6830 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑤 + (𝑘 · 𝑇)) = (𝑤 + (𝑗 · 𝑇)))
4443eleq1d 2824 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → ((𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄))
4544cbvrexv 3311 . . . . . . . . . . . 12 (∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄)
4645a1i 11 . . . . . . . . . . 11 (𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) → (∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄))
4746rabbiia 3324 . . . . . . . . . 10 {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄}
4847uneq2i 3907 . . . . . . . . 9 ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})
4948fveq2i 6356 . . . . . . . 8 (♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) = (♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄}))
5049oveq1i 6824 . . . . . . 7 ((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1) = ((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})) − 1)
5150oveq2i 6825 . . . . . 6 (0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)) = (0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})) − 1))
52 isoeq4 6734 . . . . . 6 ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)) = (0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})) − 1)) → (𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄}))))
5351, 52ax-mp 5 . . . . 5 (𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄})))
5453a1i 11 . . . 4 (𝑔 = 𝑓 → (𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄}))))
55 isoeq1 6731 . . . 4 (𝑔 = 𝑓 → (𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄}))))
5642, 54, 553bitrd 294 . . 3 (𝑔 = 𝑓 → (𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (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)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄}))))
5756cbviotav 6018 . 2 (℩𝑔𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (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)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄})))
58 fourierdlem113.x . 2 (𝜑𝑋 ∈ ℝ)
59 pire 24430 . . . . 5 π ∈ ℝ
6059renegcli 10554 . . . 4 -π ∈ ℝ
6160a1i 11 . . 3 (𝜑 → -π ∈ ℝ)
6259a1i 11 . . 3 (𝜑 → π ∈ ℝ)
63 negpilt0 40009 . . . 4 -π < 0
6463a1i 11 . . 3 (𝜑 → -π < 0)
65 pipos 24432 . . . 4 0 < π
6665a1i 11 . . 3 (𝜑 → 0 < π)
67 picn 24431 . . . . 5 π ∈ ℂ
68672timesi 11359 . . . 4 (2 · π) = (π + π)
69 fourierdlem113.t . . . 4 𝑇 = (2 · π)
7067, 67subnegi 10572 . . . 4 (π − -π) = (π + π)
7168, 69, 703eqtr4i 2792 . . 3 𝑇 = (π − -π)
7223fourierdlem2 40847 . . . . . . . 8 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
7324, 72syl 17 . . . . . . 7 (𝜑 → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
7425, 73mpbid 222 . . . . . 6 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
7574simpld 477 . . . . 5 (𝜑𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)))
76 elmapi 8047 . . . . 5 (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
7775, 76syl 17 . . . 4 (𝜑𝑄:(0...𝑀)⟶ℝ)
78 fzfid 12986 . . . 4 (𝜑 → (0...𝑀) ∈ Fin)
79 rnffi 39873 . . . 4 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
8077, 78, 79syl2anc 696 . . 3 (𝜑 → ran 𝑄 ∈ Fin)
8123, 24, 25fourierdlem15 40860 . . . 4 (𝜑𝑄:(0...𝑀)⟶(-π[,]π))
82 frn 6214 . . . 4 (𝑄:(0...𝑀)⟶(-π[,]π) → ran 𝑄 ⊆ (-π[,]π))
8381, 82syl 17 . . 3 (𝜑 → ran 𝑄 ⊆ (-π[,]π))
8474simprd 482 . . . . 5 (𝜑 → (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
8584simplrd 810 . . . 4 (𝜑 → (𝑄𝑀) = π)
86 ffun 6209 . . . . . 6 (𝑄:(0...𝑀)⟶(-π[,]π) → Fun 𝑄)
8781, 86syl 17 . . . . 5 (𝜑 → Fun 𝑄)
8824nnnn0d 11563 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
89 nn0uz 11935 . . . . . . . 8 0 = (ℤ‘0)
9088, 89syl6eleq 2849 . . . . . . 7 (𝜑𝑀 ∈ (ℤ‘0))
91 eluzfz2 12562 . . . . . . 7 (𝑀 ∈ (ℤ‘0) → 𝑀 ∈ (0...𝑀))
9290, 91syl 17 . . . . . 6 (𝜑𝑀 ∈ (0...𝑀))
93 fdm 6212 . . . . . . . 8 (𝑄:(0...𝑀)⟶(-π[,]π) → dom 𝑄 = (0...𝑀))
9481, 93syl 17 . . . . . . 7 (𝜑 → dom 𝑄 = (0...𝑀))
9594eqcomd 2766 . . . . . 6 (𝜑 → (0...𝑀) = dom 𝑄)
9692, 95eleqtrd 2841 . . . . 5 (𝜑𝑀 ∈ dom 𝑄)
97 fvelrn 6516 . . . . 5 ((Fun 𝑄𝑀 ∈ dom 𝑄) → (𝑄𝑀) ∈ ran 𝑄)
9887, 96, 97syl2anc 696 . . . 4 (𝜑 → (𝑄𝑀) ∈ ran 𝑄)
9985, 98eqeltrrd 2840 . . 3 (𝜑 → π ∈ ran 𝑄)
100 fourierdlem113.e . . 3 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇)))
101 fourierdlem113.exq . . 3 (𝜑 → (𝐸𝑋) ∈ ran 𝑄)
102 eqid 2760 . . 3 ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})
103 isoeq1 6731 . . . . 5 (𝑔 = 𝑓 → (𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (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)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))))
10430, 48, 393eqtr4ri 2793 . . . . . 6 ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})
105 isoeq5 6735 . . . . . 6 (({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄}) → (𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (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)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄}))))
106104, 105ax-mp 5 . . . . 5 (𝑓 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (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)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})))
107103, 106syl6bb 276 . . . 4 (𝑔 = 𝑓 → (𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (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)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄}))))
108107cbviotav 6018 . . 3 (℩𝑔𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (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)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})))
109 eqid 2760 . . 3 {𝑤 ∈ ((-π + 𝑋)(,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑤 ∈ ((-π + 𝑋)(,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄}
11061, 62, 64, 66, 71, 80, 83, 99, 100, 58, 101, 102, 108, 109fourierdlem51 40895 . 2 (𝜑𝑋 ∈ ran (℩𝑔𝑔 Isom < , < ((0...((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))))
111 fourierdlem113.per . 2 ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
112 ax-resscn 10205 . . . 4 ℝ ⊆ ℂ
113112a1i 11 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → ℝ ⊆ ℂ)
114 ioossre 12448 . . . . . . 7 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
115114a1i 11 . . . . . 6 (𝜑 → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
1161, 115fssresd 6232 . . . . 5 (𝜑 → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℝ)
117112a1i 11 . . . . 5 (𝜑 → ℝ ⊆ ℂ)
118116, 117fssd 6218 . . . 4 (𝜑 → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
119118adantr 472 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
120114a1i 11 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
1211, 117fssd 6218 . . . . . . 7 (𝜑𝐹:ℝ⟶ℂ)
122121adantr 472 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℂ)
123 ssid 3765 . . . . . . 7 ℝ ⊆ ℝ
124123a1i 11 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ℝ ⊆ ℝ)
125 eqid 2760 . . . . . . 7 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
126125tgioo2 22827 . . . . . . 7 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
127125, 126dvres 23894 . . . . . 6 (((ℝ ⊆ ℂ ∧ 𝐹:ℝ⟶ℂ) ∧ (ℝ ⊆ ℝ ∧ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)) → (ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))))
128113, 122, 124, 120, 127syl22anc 1478 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))))
129128dmeqd 5481 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → dom (ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = dom ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))))
130 ioontr 40257 . . . . . . 7 ((int‘(topGen‘ran (,)))‘((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))
131130reseq2i 5548 . . . . . 6 ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
132131dmeqi 5480 . . . . 5 dom ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = dom ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
133132a1i 11 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → dom ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = dom ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
134 fourierdlem113.dvcn . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
135 cncff 22917 . . . . 5 (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
136 fdm 6212 . . . . 5 (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ → dom ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
137134, 135, 1363syl 18 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → dom ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
138129, 133, 1373eqtrd 2798 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → dom (ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
139 dvcn 23903 . . 3 (((ℝ ⊆ ℂ ∧ (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ ∧ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ) ∧ dom (ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
140113, 119, 120, 138, 139syl31anc 1480 . 2 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
141120, 113sstrd 3754 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
14277adantr 472 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
143 fzofzp1 12779 . . . . . . 7 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
144143adantl 473 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
145142, 144ffvelrnd 6524 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
146145rexrd 10301 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
147 elfzofz 12699 . . . . . 6 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
148147adantl 473 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
149142, 148ffvelrnd 6524 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
15074simprrd 814 . . . . 5 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
151150r19.21bi 3070 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
152125, 146, 149, 151lptioo1cn 40399 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ((limPt‘(TopOpen‘ℂfld))‘((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
153116adantr 472 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℝ)
154123a1i 11 . . . . . . . 8 (𝜑 → ℝ ⊆ ℝ)
155117, 121, 154dvbss 23884 . . . . . . 7 (𝜑 → dom (ℝ D 𝐹) ⊆ ℝ)
156 dvfre 23933 . . . . . . . 8 ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
1571, 154, 156syl2anc 696 . . . . . . 7 (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
158 0re 10252 . . . . . . . . . 10 0 ∈ ℝ
15960, 158, 59lttri 10375 . . . . . . . . 9 ((-π < 0 ∧ 0 < π) → -π < π)
16063, 65, 159mp2an 710 . . . . . . . 8 -π < π
161160a1i 11 . . . . . . 7 (𝜑 → -π < π)
16284simplld 808 . . . . . . 7 (𝜑 → (𝑄‘0) = -π)
163134, 135syl 17 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
164 fourierdlem113.dvlb . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) ≠ ∅)
165163, 141, 152, 164, 125ellimciota 40367 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (℩𝑥𝑥 ∈ (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖))) ∈ (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
166149rexrd 10301 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ*)
167125, 166, 145, 151lptioo2cn 40398 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ((limPt‘(TopOpen‘ℂfld))‘((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
168 fourierdlem113.dvub . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) ≠ ∅)
169163, 141, 167, 168, 125ellimciota 40367 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (℩𝑥𝑥 ∈ (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1)))) ∈ (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
170121adantr 472 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℤ) → 𝐹:ℝ⟶ℂ)
171 zre 11593 . . . . . . . . . . . 12 (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
172171adantl 473 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℤ) → 𝑘 ∈ ℝ)
173 2re 11302 . . . . . . . . . . . . . . 15 2 ∈ ℝ
174173, 59remulcli 10266 . . . . . . . . . . . . . 14 (2 · π) ∈ ℝ
175174a1i 11 . . . . . . . . . . . . 13 (𝜑 → (2 · π) ∈ ℝ)
17669, 175syl5eqel 2843 . . . . . . . . . . . 12 (𝜑𝑇 ∈ ℝ)
177176adantr 472 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℤ) → 𝑇 ∈ ℝ)
178172, 177remulcld 10282 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℤ) → (𝑘 · 𝑇) ∈ ℝ)
179170adantr 472 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝐹:ℝ⟶ℂ)
180177adantr 472 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑇 ∈ ℝ)
181 simplr 809 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑘 ∈ ℤ)
182 simpr 479 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
183111ad4ant14 1209 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
184179, 180, 181, 182, 183fperiodmul 40035 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → (𝐹‘(𝑡 + (𝑘 · 𝑇))) = (𝐹𝑡))
185 eqid 2760 . . . . . . . . . 10 (ℝ D 𝐹) = (ℝ D 𝐹)
186170, 178, 184, 185fperdvper 40654 . . . . . . . . 9 (((𝜑𝑘 ∈ ℤ) ∧ 𝑡 ∈ dom (ℝ D 𝐹)) → ((𝑡 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹) ∧ ((ℝ D 𝐹)‘(𝑡 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑡)))
187186an32s 881 . . . . . . . 8 (((𝜑𝑡 ∈ dom (ℝ D 𝐹)) ∧ 𝑘 ∈ ℤ) → ((𝑡 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹) ∧ ((ℝ D 𝐹)‘(𝑡 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑡)))
188187simpld 477 . . . . . . 7 (((𝜑𝑡 ∈ dom (ℝ D 𝐹)) ∧ 𝑘 ∈ ℤ) → (𝑡 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹))
189187simprd 482 . . . . . . 7 (((𝜑𝑡 ∈ dom (ℝ D 𝐹)) ∧ 𝑘 ∈ ℤ) → ((ℝ D 𝐹)‘(𝑡 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑡))
190 fveq2 6353 . . . . . . . . 9 (𝑗 = 𝑖 → (𝑄𝑗) = (𝑄𝑖))
191 oveq1 6821 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝑗 + 1) = (𝑖 + 1))
192191fveq2d 6357 . . . . . . . . 9 (𝑗 = 𝑖 → (𝑄‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)))
193190, 192oveq12d 6832 . . . . . . . 8 (𝑗 = 𝑖 → ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
194193cbvmptv 4902 . . . . . . 7 (𝑗 ∈ (0..^𝑀) ↦ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
195 eqid 2760 . . . . . . 7 (𝑡 ∈ ℝ ↦ (𝑡 + ((⌊‘((π − 𝑡) / 𝑇)) · 𝑇))) = (𝑡 ∈ ℝ ↦ (𝑡 + ((⌊‘((π − 𝑡) / 𝑇)) · 𝑇)))
196155, 157, 61, 62, 161, 71, 24, 77, 162, 85, 134, 165, 169, 188, 189, 194, 195fourierdlem71 40915 . . . . . 6 (𝜑 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
197196adantr 472 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
198 nfv 1992 . . . . . . . . 9 𝑡(𝜑𝑖 ∈ (0..^𝑀))
199 nfra1 3079 . . . . . . . . 9 𝑡𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧
200198, 199nfan 1977 . . . . . . . 8 𝑡((𝜑𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
201128, 131syl6eq 2810 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
202201fveq1d 6355 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡) = (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡))
203 fvres 6369 . . . . . . . . . . . . 13 (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
204202, 203sylan9eq 2814 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
205204fveq2d 6357 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (abs‘((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡)))
206205adantlr 753 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (abs‘((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡)))
207 simplr 809 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
208 ssdmres 5578 . . . . . . . . . . . . . 14 (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
209137, 208sylibr 224 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹))
210209ad2antrr 764 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹))
211 simpr 479 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
212210, 211sseldd 3745 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑡 ∈ dom (ℝ D 𝐹))
213 rspa 3068 . . . . . . . . . . 11 ((∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧𝑡 ∈ dom (ℝ D 𝐹)) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
214207, 212, 213syl2anc 696 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
215206, 214eqbrtrd 4826 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (abs‘((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧)
216215ex 449 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) → (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧))
217200, 216ralrimi 3095 . . . . . . 7 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) → ∀𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧)
218217ex 449 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 → ∀𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧))
219218reximdv 3154 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧))
220197, 219mpd 15 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧)
221149, 145, 153, 138, 220ioodvbdlimc1 40669 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) ≠ ∅)
222119, 141, 152, 221, 125ellimciota 40367 . 2 ((𝜑𝑖 ∈ (0..^𝑀)) → (℩𝑦𝑦 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖))) ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
223149, 145, 153, 138, 220ioodvbdlimc2 40671 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) ≠ ∅)
224119, 141, 167, 223, 125ellimciota 40367 . 2 ((𝜑𝑖 ∈ (0..^𝑀)) → (℩𝑦𝑦 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1)))) ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
225 frel 6211 . . . . . . 7 ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ → Rel (ℝ D 𝐹))
226157, 225syl 17 . . . . . 6 (𝜑 → Rel (ℝ D 𝐹))
227 resindm 5602 . . . . . 6 (Rel (ℝ D 𝐹) → ((ℝ D 𝐹) ↾ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))) = ((ℝ D 𝐹) ↾ (-∞(,)𝑋)))
228226, 227syl 17 . . . . 5 (𝜑 → ((ℝ D 𝐹) ↾ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))) = ((ℝ D 𝐹) ↾ (-∞(,)𝑋)))
229 inss2 3977 . . . . . . 7 ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ dom (ℝ D 𝐹)
230229a1i 11 . . . . . 6 (𝜑 → ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ dom (ℝ D 𝐹))
231157, 230fssresd 6232 . . . . 5 (𝜑 → ((ℝ D 𝐹) ↾ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))):((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))⟶ℝ)
232228, 231feq1dd 39864 . . . 4 (𝜑 → ((ℝ D 𝐹) ↾ (-∞(,)𝑋)):((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))⟶ℝ)
233232, 117fssd 6218 . . 3 (𝜑 → ((ℝ D 𝐹) ↾ (-∞(,)𝑋)):((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))⟶ℂ)
234 ioosscn 40237 . . . . 5 (-∞(,)𝑋) ⊆ ℂ
235 ssinss1 3984 . . . . 5 ((-∞(,)𝑋) ⊆ ℂ → ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
236234, 235ax-mp 5 . . . 4 ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ
237236a1i 11 . . 3 (𝜑 → ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
238 3simpb 1145 . . . . . . . 8 ((𝜑𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → (𝜑𝑘 ∈ ℤ))
239 simp2 1132 . . . . . . . 8 ((𝜑𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑥 ∈ dom (ℝ D 𝐹))
240170adantr 472 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝐹:ℝ⟶ℂ)
241177adantr 472 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
242 simplr 809 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝑘 ∈ ℤ)
243 simpr 479 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
244 eleq1w 2822 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑥 ∈ ℝ ↔ 𝑦 ∈ ℝ))
245244anbi2d 742 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝜑𝑥 ∈ ℝ) ↔ (𝜑𝑦 ∈ ℝ)))
246 oveq1 6821 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 + 𝑇) = (𝑦 + 𝑇))
247246fveq2d 6357 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑦 + 𝑇)))
248 fveq2 6353 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
249247, 248eqeq12d 2775 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥) ↔ (𝐹‘(𝑦 + 𝑇)) = (𝐹𝑦)))
250245, 249imbi12d 333 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥)) ↔ ((𝜑𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑇)) = (𝐹𝑦))))
251250, 111chvarv 2408 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑇)) = (𝐹𝑦))
252251ad4ant14 1209 . . . . . . . . . 10 ((((𝜑𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑇)) = (𝐹𝑦))
253240, 241, 242, 243, 252fperiodmul 40035 . . . . . . . . 9 (((𝜑𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥))
254170, 178, 253, 185fperdvper 40654 . . . . . . . 8 (((𝜑𝑘 ∈ ℤ) ∧ 𝑥 ∈ dom (ℝ D 𝐹)) → ((𝑥 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹) ∧ ((ℝ D 𝐹)‘(𝑥 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑥)))
255238, 239, 254syl2anc 696 . . . . . . 7 ((𝜑𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → ((𝑥 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹) ∧ ((ℝ D 𝐹)‘(𝑥 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑥)))
256255simpld 477 . . . . . 6 ((𝜑𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹))
257 oveq2 6822 . . . . . . . . . 10 (𝑤 = 𝑥 → (π − 𝑤) = (π − 𝑥))
258257oveq1d 6829 . . . . . . . . 9 (𝑤 = 𝑥 → ((π − 𝑤) / 𝑇) = ((π − 𝑥) / 𝑇))
259258fveq2d 6357 . . . . . . . 8 (𝑤 = 𝑥 → (⌊‘((π − 𝑤) / 𝑇)) = (⌊‘((π − 𝑥) / 𝑇)))
260259oveq1d 6829 . . . . . . 7 (𝑤 = 𝑥 → ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇) = ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
261260cbvmptv 4902 . . . . . 6 (𝑤 ∈ ℝ ↦ ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇)) = (𝑥 ∈ ℝ ↦ ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
262 eqid 2760 . . . . . 6 (𝑥 ∈ ℝ ↦ (𝑥 + ((𝑤 ∈ ℝ ↦ ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇))‘𝑥))) = (𝑥 ∈ ℝ ↦ (𝑥 + ((𝑤 ∈ ℝ ↦ ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇))‘𝑥)))
26361, 62, 161, 71, 256, 58, 261, 262, 23, 24, 25, 209fourierdlem41 40886 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) ∧ ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))))
264263simpld 477 . . . 4 (𝜑 → ∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)))
265125cnfldtop 22808 . . . . . . . . 9 (TopOpen‘ℂfld) ∈ Top
266265a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (TopOpen‘ℂfld) ∈ Top)
267236a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
268 mnfxr 10308 . . . . . . . . . . . 12 -∞ ∈ ℝ*
269268a1i 11 . . . . . . . . . . 11 (𝑦 ∈ ℝ → -∞ ∈ ℝ*)
270 rexr 10297 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
271 mnflt 12170 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → -∞ < 𝑦)
272269, 270, 271xrltled 40003 . . . . . . . . . . 11 (𝑦 ∈ ℝ → -∞ ≤ 𝑦)
273 iooss1 12423 . . . . . . . . . . 11 ((-∞ ∈ ℝ* ∧ -∞ ≤ 𝑦) → (𝑦(,)𝑋) ⊆ (-∞(,)𝑋))
274269, 272, 273syl2anc 696 . . . . . . . . . 10 (𝑦 ∈ ℝ → (𝑦(,)𝑋) ⊆ (-∞(,)𝑋))
2752743ad2ant2 1129 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (𝑦(,)𝑋) ⊆ (-∞(,)𝑋))
276 simp3 1133 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))
277275, 276ssind 3980 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (𝑦(,)𝑋) ⊆ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)))
278 unicntop 22810 . . . . . . . . 9 ℂ = (TopOpen‘ℂfld)
279278lpss3 21170 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ ∧ (𝑦(,)𝑋) ⊆ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))) → ((limPt‘(TopOpen‘ℂfld))‘(𝑦(,)𝑋)) ⊆ ((limPt‘(TopOpen‘ℂfld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
280266, 267, 277, 279syl3anc 1477 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → ((limPt‘(TopOpen‘ℂfld))‘(𝑦(,)𝑋)) ⊆ ((limPt‘(TopOpen‘ℂfld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
2812803adant3l 1194 . . . . . 6 ((𝜑𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → ((limPt‘(TopOpen‘ℂfld))‘(𝑦(,)𝑋)) ⊆ ((limPt‘(TopOpen‘ℂfld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
2822703ad2ant2 1129 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑦 ∈ ℝ*)
283583ad2ant1 1128 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ℝ)
284 simp3l 1244 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑦 < 𝑋)
285125, 282, 283, 284lptioo2cn 40398 . . . . . 6 ((𝜑𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝑦(,)𝑋)))
286281, 285sseldd 3745 . . . . 5 ((𝜑𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
287286rexlimdv3a 3171 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)))))
288264, 287mpd 15 . . 3 (𝜑𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
289255simprd 482 . . . 4 ((𝜑𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → ((ℝ D 𝐹)‘(𝑥 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑥))
290 oveq2 6822 . . . . . . . 8 (𝑦 = 𝑥 → (π − 𝑦) = (π − 𝑥))
291290oveq1d 6829 . . . . . . 7 (𝑦 = 𝑥 → ((π − 𝑦) / 𝑇) = ((π − 𝑥) / 𝑇))
292291fveq2d 6357 . . . . . 6 (𝑦 = 𝑥 → (⌊‘((π − 𝑦) / 𝑇)) = (⌊‘((π − 𝑥) / 𝑇)))
293292oveq1d 6829 . . . . 5 (𝑦 = 𝑥 → ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇) = ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
294293cbvmptv 4902 . . . 4 (𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇)) = (𝑥 ∈ ℝ ↦ ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
295 id 22 . . . . . 6 (𝑧 = 𝑥𝑧 = 𝑥)
296 fveq2 6353 . . . . . 6 (𝑧 = 𝑥 → ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑧) = ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑥))
297295, 296oveq12d 6832 . . . . 5 (𝑧 = 𝑥 → (𝑧 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑧)) = (𝑥 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑥)))
298297cbvmptv 4902 . . . 4 (𝑧 ∈ ℝ ↦ (𝑧 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑧))) = (𝑥 ∈ ℝ ↦ (𝑥 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑥)))
29961, 62, 161, 23, 71, 24, 25, 155, 157, 256, 289, 134, 169, 58, 294, 298fourierdlem49 40893 . . 3 (𝜑 → (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim 𝑋) ≠ ∅)
300233, 237, 288, 299, 125ellimciota 40367 . 2 (𝜑 → (℩𝑥𝑥 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim 𝑋)) ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim 𝑋))
301 resindm 5602 . . . . . 6 (Rel (ℝ D 𝐹) → ((ℝ D 𝐹) ↾ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))) = ((ℝ D 𝐹) ↾ (𝑋(,)+∞)))
302226, 301syl 17 . . . . 5 (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))) = ((ℝ D 𝐹) ↾ (𝑋(,)+∞)))
303 inss2 3977 . . . . . . 7 ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ dom (ℝ D 𝐹)
304303a1i 11 . . . . . 6 (𝜑 → ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ dom (ℝ D 𝐹))
305157, 304fssresd 6232 . . . . 5 (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))):((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))⟶ℝ)
306302, 305feq1dd 39864 . . . 4 (𝜑 → ((ℝ D 𝐹) ↾ (𝑋(,)+∞)):((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))⟶ℝ)
307306, 117fssd 6218 . . 3 (𝜑 → ((ℝ D 𝐹) ↾ (𝑋(,)+∞)):((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))⟶ℂ)
308 ioosscn 40237 . . . . 5 (𝑋(,)+∞) ⊆ ℂ
309 ssinss1 3984 . . . . 5 ((𝑋(,)+∞) ⊆ ℂ → ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
310308, 309ax-mp 5 . . . 4 ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ
311310a1i 11 . . 3 (𝜑 → ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
312263simprd 482 . . . 4 (𝜑 → ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)))
313265a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (TopOpen‘ℂfld) ∈ Top)
314310a1i 11 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
315 pnfxr 10304 . . . . . . . . . . . 12 +∞ ∈ ℝ*
316315a1i 11 . . . . . . . . . . 11 (𝑦 ∈ ℝ → +∞ ∈ ℝ*)
317 ltpnf 12167 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 < +∞)
318270, 316, 317xrltled 40003 . . . . . . . . . . 11 (𝑦 ∈ ℝ → 𝑦 ≤ +∞)
319 iooss2 12424 . . . . . . . . . . 11 ((+∞ ∈ ℝ*𝑦 ≤ +∞) → (𝑋(,)𝑦) ⊆ (𝑋(,)+∞))
320316, 318, 319syl2anc 696 . . . . . . . . . 10 (𝑦 ∈ ℝ → (𝑋(,)𝑦) ⊆ (𝑋(,)+∞))
3213203ad2ant2 1129 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (𝑋(,)𝑦) ⊆ (𝑋(,)+∞))
322 simp3 1133 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))
323321, 322ssind 3980 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (𝑋(,)𝑦) ⊆ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)))
324278lpss3 21170 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ ∧ (𝑋(,)𝑦) ⊆ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))) → ((limPt‘(TopOpen‘ℂfld))‘(𝑋(,)𝑦)) ⊆ ((limPt‘(TopOpen‘ℂfld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
325313, 314, 323, 324syl3anc 1477 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → ((limPt‘(TopOpen‘ℂfld))‘(𝑋(,)𝑦)) ⊆ ((limPt‘(TopOpen‘ℂfld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
3263253adant3l 1194 . . . . . 6 ((𝜑𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → ((limPt‘(TopOpen‘ℂfld))‘(𝑋(,)𝑦)) ⊆ ((limPt‘(TopOpen‘ℂfld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
3272703ad2ant2 1129 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑦 ∈ ℝ*)
328583ad2ant1 1128 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ℝ)
329 simp3l 1244 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 < 𝑦)
330125, 327, 328, 329lptioo1cn 40399 . . . . . 6 ((𝜑𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝑋(,)𝑦)))
331326, 330sseldd 3745 . . . . 5 ((𝜑𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
332331rexlimdv3a 3171 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)))))
333312, 332mpd 15 . . 3 (𝜑𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
334 biid 251 . . . 4 (((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑤 ∈ ((𝑄𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑘 ∈ ℤ) ∧ 𝑤 = (𝑋 + (𝑘 · 𝑇))) ↔ ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑤 ∈ ((𝑄𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑘 ∈ ℤ) ∧ 𝑤 = (𝑋 + (𝑘 · 𝑇))))
33561, 62, 161, 23, 71, 24, 25, 157, 256, 289, 134, 165, 58, 294, 298, 334fourierdlem48 40892 . . 3 (𝜑 → (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim 𝑋) ≠ ∅)
336307, 311, 333, 335, 125ellimciota 40367 . 2 (𝜑 → (℩𝑥𝑥 ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim 𝑋)) ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim 𝑋))
337 fourierdlem113.l . 2 (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋))
338 fourierdlem113.r . 2 (𝜑𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))
339 fourierdlem113.a . 2 𝐴 = (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
340 fourierdlem113.b . 2 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
341 fveq2 6353 . . . . . . . 8 (𝑛 = 𝑘 → (𝐴𝑛) = (𝐴𝑘))
342 oveq1 6821 . . . . . . . . 9 (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
343342fveq2d 6357 . . . . . . . 8 (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
344341, 343oveq12d 6832 . . . . . . 7 (𝑛 = 𝑘 → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))))
345 fveq2 6353 . . . . . . . 8 (𝑛 = 𝑘 → (𝐵𝑛) = (𝐵𝑘))
346342fveq2d 6357 . . . . . . . 8 (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
347345, 346oveq12d 6832 . . . . . . 7 (𝑛 = 𝑘 → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
348344, 347oveq12d 6832 . . . . . 6 (𝑛 = 𝑘 → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
349348cbvsumv 14645 . . . . 5 Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
350 oveq2 6822 . . . . . . 7 (𝑗 = 𝑚 → (1...𝑗) = (1...𝑚))
351350eqcomd 2766 . . . . . 6 (𝑗 = 𝑚 → (1...𝑚) = (1...𝑗))
352351sumeq1d 14650 . . . . 5 (𝑗 = 𝑚 → Σ𝑘 ∈ (1...𝑚)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑗)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
353349, 352syl5req 2807 . . . 4 (𝑗 = 𝑚 → Σ𝑘 ∈ (1...𝑗)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
354353oveq2d 6830 . . 3 (𝑗 = 𝑚 → (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑗)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
355354cbvmptv 4902 . 2 (𝑗 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑗)(((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))) = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))
356 fourierdlem113.15 . 2 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))))
357 fdm 6212 . . . . . 6 (𝐹:ℝ⟶ℝ → dom 𝐹 = ℝ)
3581, 357syl 17 . . . . 5 (𝜑 → dom 𝐹 = ℝ)
359358, 154eqsstrd 3780 . . . 4 (𝜑 → dom 𝐹 ⊆ ℝ)
360358feq2d 6192 . . . . 5 (𝜑 → (𝐹:dom 𝐹⟶ℝ ↔ 𝐹:ℝ⟶ℝ))
3611, 360mpbird 247 . . . 4 (𝜑𝐹:dom 𝐹⟶ℝ)
362359sselda 3744 . . . . . . 7 ((𝜑𝑡 ∈ dom 𝐹) → 𝑡 ∈ ℝ)
363362adantr 472 . . . . . 6 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑡 ∈ ℝ)
364171adantl 473 . . . . . . 7 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℝ)
365177adantlr 753 . . . . . . 7 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑇 ∈ ℝ)
366364, 365remulcld 10282 . . . . . 6 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑘 · 𝑇) ∈ ℝ)
367363, 366readdcld 10281 . . . . 5 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑡 + (𝑘 · 𝑇)) ∈ ℝ)
368358eqcomd 2766 . . . . . 6 (𝜑 → ℝ = dom 𝐹)
369368ad2antrr 764 . . . . 5 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → ℝ = dom 𝐹)
370367, 369eleqtrd 2841 . . . 4 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑡 + (𝑘 · 𝑇)) ∈ dom 𝐹)
371 id 22 . . . . . 6 ((𝜑𝑘 ∈ ℤ) → (𝜑𝑘 ∈ ℤ))
372371adantlr 753 . . . . 5 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝜑𝑘 ∈ ℤ))
373372, 363, 184syl2anc 696 . . . 4 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑡 + (𝑘 · 𝑇))) = (𝐹𝑡))
374359, 361, 61, 62, 161, 71, 24, 77, 162, 85, 140, 222, 224, 370, 373, 194, 195fourierdlem71 40915 . . 3 (𝜑 → ∃𝑢 ∈ ℝ ∀𝑡 ∈ dom 𝐹(abs‘(𝐹𝑡)) ≤ 𝑢)
375358raleqdv 3283 . . . 4 (𝜑 → (∀𝑡 ∈ dom 𝐹(abs‘(𝐹𝑡)) ≤ 𝑢 ↔ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑢))
376375rexbidv 3190 . . 3 (𝜑 → (∃𝑢 ∈ ℝ ∀𝑡 ∈ dom 𝐹(abs‘(𝐹𝑡)) ≤ 𝑢 ↔ ∃𝑢 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑢))
377374, 376mpbid 222 . 2 (𝜑 → ∃𝑢 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑢)
3781, 22, 23, 24, 25, 32, 57, 58, 110, 69, 111, 140, 222, 224, 134, 300, 336, 337, 338, 339, 340, 355, 356, 377, 196, 58fourierdlem112 40956 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  {crab 3054  cun 3713  cin 3714  wss 3715  c0 4058  ifcif 4230  {cpr 4323   class class class wbr 4804  cmpt 4881  dom cdm 5266  ran crn 5267  cres 5268  Rel wrel 5271  cio 6010  Fun wfun 6043  wf 6045  cfv 6049   Isom wiso 6050  (class class class)co 6814  𝑚 cmap 8025  Fincfn 8123  cc 10146  cr 10147  0cc0 10148  1c1 10149   + caddc 10151   · cmul 10153  +∞cpnf 10283  -∞cmnf 10284  *cxr 10285   < clt 10286  cle 10287  cmin 10478  -cneg 10479   / cdiv 10896  cn 11232  2c2 11282  0cn0 11504  cz 11589  cuz 11899  (,)cioo 12388  (,]cioc 12389  [,)cico 12390  [,]cicc 12391  ...cfz 12539  ..^cfzo 12679  cfl 12805   mod cmo 12882  seqcseq 13015  chash 13331  abscabs 14193  cli 14434  Σcsu 14635  sincsin 15013  cosccos 15014  πcpi 15016  TopOpenctopn 16304  topGenctg 16320  fldccnfld 19968  Topctop 20920  intcnt 21043  limPtclp 21160  cnccncf 22900  citg 23606   lim climc 23845   D cdv 23846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cc 9469  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226  ax-addf 10227  ax-mulf 10228
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-disj 4773  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-ofr 7064  df-om 7232  df-1st 7334  df-2nd 7335  df-supp 7465  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-omul 7735  df-er 7913  df-map 8027  df-pm 8028  df-ixp 8077  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-fsupp 8443  df-fi 8484  df-sup 8515  df-inf 8516  df-oi 8582  df-card 8975  df-acn 8978  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-xnn0 11576  df-z 11590  df-dec 11706  df-uz 11900  df-q 12002  df-rp 12046  df-xneg 12159  df-xadd 12160  df-xmul 12161  df-ioo 12392  df-ioc 12393  df-ico 12394  df-icc 12395  df-fz 12540  df-fzo 12680  df-fl 12807  df-mod 12883  df-seq 13016  df-exp 13075  df-fac 13275  df-bc 13304  df-hash 13332  df-shft 14026  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-limsup 14421  df-clim 14438  df-rlim 14439  df-sum 14636  df-ef 15017  df-sin 15019  df-cos 15020  df-pi 15022  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-ress 16087  df-plusg 16176  df-mulr 16177  df-starv 16178  df-sca 16179  df-vsca 16180  df-ip 16181  df-tset 16182  df-ple 16183  df-ds 16186  df-unif 16187  df-hom 16188  df-cco 16189  df-rest 16305  df-topn 16306  df-0g 16324  df-gsum 16325  df-topgen 16326  df-pt 16327  df-prds 16330  df-xrs 16384  df-qtop 16389  df-imas 16390  df-xps 16392  df-mre 16468  df-mrc 16469  df-acs 16471  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-submnd 17557  df-mulg 17762  df-cntz 17970  df-cmn 18415  df-psmet 19960  df-xmet 19961  df-met 19962  df-bl 19963  df-mopn 19964  df-fbas 19965  df-fg 19966  df-cnfld 19969  df-top 20921  df-topon 20938  df-topsp 20959  df-bases 20972  df-cld 21045  df-ntr 21046  df-cls 21047  df-nei 21124  df-lp 21162  df-perf 21163  df-cn 21253  df-cnp 21254  df-t1 21340  df-haus 21341  df-cmp 21412  df-tx 21587  df-hmeo 21780  df-fil 21871  df-fm 21963  df-flim 21964  df-flf 21965  df-xms 22346  df-ms 22347  df-tms 22348  df-cncf 22902  df-ovol 23453  df-vol 23454  df-mbf 23607  df-itg1 23608  df-itg2 23609  df-ibl 23610  df-itg 23611  df-0p 23656  df-ditg 23830  df-limc 23849  df-dv 23850
This theorem is referenced by:  fourierdlem114  40958
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