Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fourierdlem113 Structured version   Visualization version   GIF version

Theorem fourierdlem113 41073
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 6849 . . . . . . 7 (𝑤 = 𝑦 → (𝑤 mod (2 · π)) = (𝑦 mod (2 · π)))
32eqeq1d 2767 . . . . . 6 (𝑤 = 𝑦 → ((𝑤 mod (2 · π)) = 0 ↔ (𝑦 mod (2 · π)) = 0))
4 oveq2 6850 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑘 + (1 / 2)) · 𝑤) = ((𝑘 + (1 / 2)) · 𝑦))
54fveq2d 6379 . . . . . . 7 (𝑤 = 𝑦 → (sin‘((𝑘 + (1 / 2)) · 𝑤)) = (sin‘((𝑘 + (1 / 2)) · 𝑦)))
6 oveq1 6849 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤 / 2) = (𝑦 / 2))
76fveq2d 6379 . . . . . . . 8 (𝑤 = 𝑦 → (sin‘(𝑤 / 2)) = (sin‘(𝑦 / 2)))
87oveq2d 6858 . . . . . . 7 (𝑤 = 𝑦 → ((2 · π) · (sin‘(𝑤 / 2))) = ((2 · π) · (sin‘(𝑦 / 2))))
95, 8oveq12d 6860 . . . . . 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 4268 . . . . 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 4909 . . . 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 6850 . . . . . . . 8 (𝑘 = 𝑚 → (2 · 𝑘) = (2 · 𝑚))
1312oveq1d 6857 . . . . . . 7 (𝑘 = 𝑚 → ((2 · 𝑘) + 1) = ((2 · 𝑚) + 1))
1413oveq1d 6857 . . . . . 6 (𝑘 = 𝑚 → (((2 · 𝑘) + 1) / (2 · π)) = (((2 · 𝑚) + 1) / (2 · π)))
15 oveq1 6849 . . . . . . . . 9 (𝑘 = 𝑚 → (𝑘 + (1 / 2)) = (𝑚 + (1 / 2)))
1615oveq1d 6857 . . . . . . . 8 (𝑘 = 𝑚 → ((𝑘 + (1 / 2)) · 𝑦) = ((𝑚 + (1 / 2)) · 𝑦))
1716fveq2d 6379 . . . . . . 7 (𝑘 = 𝑚 → (sin‘((𝑘 + (1 / 2)) · 𝑦)) = (sin‘((𝑚 + (1 / 2)) · 𝑦)))
1817oveq1d 6857 . . . . . 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 4263 . . . . 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 4904 . . . 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 2811 . . 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 4909 . 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 6849 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤 + (𝑗 · 𝑇)) = (𝑦 + (𝑗 · 𝑇)))
2726eleq1d 2829 . . . . . . 7 (𝑤 = 𝑦 → ((𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
2827rexbidv 3199 . . . . . 6 (𝑤 = 𝑦 → (∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
2928cbvrabv 3348 . . . . 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 3926 . . . 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 6378 . . 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 6852 . 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 6849 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑘 · 𝑇) = (𝑗 · 𝑇))
3433oveq2d 6858 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑗 · 𝑇)))
3534eleq1d 2829 . . . . . . . . . 10 (𝑘 = 𝑗 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
3635cbvrexv 3320 . . . . . . . . 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 3333 . . . . . . 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 3926 . . . . . 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 6763 . . . . . 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 6858 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (𝑤 + (𝑘 · 𝑇)) = (𝑤 + (𝑗 · 𝑇)))
4443eleq1d 2829 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → ((𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄))
4544cbvrexv 3320 . . . . . . . . . . . 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 3333 . . . . . . . . . 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 3926 . . . . . . . . 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 6378 . . . . . . . 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 6852 . . . . . . 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 6853 . . . . . 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 6762 . . . . . 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 6759 . . . 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 296 . . 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 6037 . 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 24502 . . . . 5 π ∈ ℝ
6059renegcli 10596 . . . 4 -π ∈ ℝ
6160a1i 11 . . 3 (𝜑 → -π ∈ ℝ)
6259a1i 11 . . 3 (𝜑 → π ∈ ℝ)
63 negpilt0 40132 . . . 4 -π < 0
6463a1i 11 . . 3 (𝜑 → -π < 0)
65 pipos 24504 . . . 4 0 < π
6665a1i 11 . . 3 (𝜑 → 0 < π)
67 picn 24503 . . . . 5 π ∈ ℂ
68672timesi 11417 . . . 4 (2 · π) = (π + π)
69 fourierdlem113.t . . . 4 𝑇 = (2 · π)
7067, 67subnegi 10614 . . . 4 (π − -π) = (π + π)
7168, 69, 703eqtr4i 2797 . . 3 𝑇 = (π − -π)
7223fourierdlem2 40963 . . . . . . . 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 223 . . . . . 6 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
7574simpld 488 . . . . 5 (𝜑𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)))
76 elmapi 8082 . . . . 5 (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
7775, 76syl 17 . . . 4 (𝜑𝑄:(0...𝑀)⟶ℝ)
78 fzfid 12980 . . . 4 (𝜑 → (0...𝑀) ∈ Fin)
79 rnffi 40003 . . . 4 ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
8077, 78, 79syl2anc 579 . . 3 (𝜑 → ran 𝑄 ∈ Fin)
8123, 24, 25fourierdlem15 40976 . . . 4 (𝜑𝑄:(0...𝑀)⟶(-π[,]π))
82 frn 6229 . . . 4 (𝑄:(0...𝑀)⟶(-π[,]π) → ran 𝑄 ⊆ (-π[,]π))
8381, 82syl 17 . . 3 (𝜑 → ran 𝑄 ⊆ (-π[,]π))
8474simprd 489 . . . . 5 (𝜑 → (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
8584simplrd 786 . . . 4 (𝜑 → (𝑄𝑀) = π)
86 ffun 6226 . . . . . 6 (𝑄:(0...𝑀)⟶(-π[,]π) → Fun 𝑄)
8781, 86syl 17 . . . . 5 (𝜑 → Fun 𝑄)
8824nnnn0d 11598 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
89 nn0uz 11922 . . . . . . . 8 0 = (ℤ‘0)
9088, 89syl6eleq 2854 . . . . . . 7 (𝜑𝑀 ∈ (ℤ‘0))
91 eluzfz2 12556 . . . . . . 7 (𝑀 ∈ (ℤ‘0) → 𝑀 ∈ (0...𝑀))
9290, 91syl 17 . . . . . 6 (𝜑𝑀 ∈ (0...𝑀))
93 fdm 6231 . . . . . . . 8 (𝑄:(0...𝑀)⟶(-π[,]π) → dom 𝑄 = (0...𝑀))
9481, 93syl 17 . . . . . . 7 (𝜑 → dom 𝑄 = (0...𝑀))
9594eqcomd 2771 . . . . . 6 (𝜑 → (0...𝑀) = dom 𝑄)
9692, 95eleqtrd 2846 . . . . 5 (𝜑𝑀 ∈ dom 𝑄)
97 fvelrn 6542 . . . . 5 ((Fun 𝑄𝑀 ∈ dom 𝑄) → (𝑄𝑀) ∈ ran 𝑄)
9887, 96, 97syl2anc 579 . . . 4 (𝜑 → (𝑄𝑀) ∈ ran 𝑄)
9985, 98eqeltrrd 2845 . . 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 2765 . . 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 6759 . . . . 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 2798 . . . . . 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 6763 . . . . . 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 278 . . . 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 6037 . . 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 2765 . . 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 41011 . 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 10246 . . . 4 ℝ ⊆ ℂ
113112a1i 11 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → ℝ ⊆ ℂ)
114 ioossre 12437 . . . . . . 7 ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
115114a1i 11 . . . . . 6 (𝜑 → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
1161, 115fssresd 6253 . . . . 5 (𝜑 → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℝ)
117112a1i 11 . . . . 5 (𝜑 → ℝ ⊆ ℂ)
118116, 117fssd 6237 . . . 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 6237 . . . . . . 7 (𝜑𝐹:ℝ⟶ℂ)
122121adantr 472 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℂ)
123 ssid 3783 . . . . . . 7 ℝ ⊆ ℝ
124123a1i 11 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ℝ ⊆ ℝ)
125 eqid 2765 . . . . . . 7 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
126125tgioo2 22885 . . . . . . 7 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
127125, 126dvres 23966 . . . . . 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 867 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))))
129128dmeqd 5494 . . . 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 40376 . . . . . . 7 ((int‘(topGen‘ran (,)))‘((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))
131130reseq2i 5562 . . . . . 6 ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
132131dmeqi 5493 . . . . 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 22975 . . . . 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 6231 . . . . 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 2803 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → dom (ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
139 dvcn 23975 . . 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 1492 . 2 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
141120, 113sstrd 3771 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
14277adantr 472 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
143 fzofzp1 12773 . . . . . . 7 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
144143adantl 473 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
145142, 144ffvelrnd 6550 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
146145rexrd 10343 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ*)
147 elfzofz 12693 . . . . . 6 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
148147adantl 473 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
149142, 148ffvelrnd 6550 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ)
15074simprrd 790 . . . . 5 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
151150r19.21bi 3079 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
152125, 146, 149, 151lptioo1cn 40516 . . 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 23956 . . . . . . 7 (𝜑 → dom (ℝ D 𝐹) ⊆ ℝ)
156 dvfre 24005 . . . . . . . 8 ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
1571, 154, 156syl2anc 579 . . . . . . 7 (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
158 0re 10295 . . . . . . . . . 10 0 ∈ ℝ
15960, 158, 59lttri 10417 . . . . . . . . 9 ((-π < 0 ∧ 0 < π) → -π < π)
16063, 65, 159mp2an 683 . . . . . . . 8 -π < π
161160a1i 11 . . . . . . 7 (𝜑 → -π < π)
16284simplld 784 . . . . . . 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 40484 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (℩𝑥𝑥 ∈ (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖))) ∈ (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
166149rexrd 10343 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) ∈ ℝ*)
167125, 166, 145, 151lptioo2cn 40515 . . . . . . . 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 40484 . . . . . . 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 11628 . . . . . . . . . . . 12 (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
172171adantl 473 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℤ) → 𝑘 ∈ ℝ)
173 2re 11346 . . . . . . . . . . . . . . 15 2 ∈ ℝ
174173, 59remulcli 10310 . . . . . . . . . . . . . 14 (2 · π) ∈ ℝ
175174a1i 11 . . . . . . . . . . . . 13 (𝜑 → (2 · π) ∈ ℝ)
17669, 175syl5eqel 2848 . . . . . . . . . . . 12 (𝜑𝑇 ∈ ℝ)
177176adantr 472 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℤ) → 𝑇 ∈ ℝ)
178172, 177remulcld 10324 . . . . . . . . . 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 785 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑘 ∈ ℤ)
182 simpr 477 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
183111ad4ant14 759 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))
184179, 180, 181, 182, 183fperiodmul 40157 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → (𝐹‘(𝑡 + (𝑘 · 𝑇))) = (𝐹𝑡))
185 eqid 2765 . . . . . . . . . 10 (ℝ D 𝐹) = (ℝ D 𝐹)
186170, 178, 184, 185fperdvper 40771 . . . . . . . . 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 642 . . . . . . . 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 488 . . . . . . 7 (((𝜑𝑡 ∈ dom (ℝ D 𝐹)) ∧ 𝑘 ∈ ℤ) → (𝑡 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹))
189187simprd 489 . . . . . . 7 (((𝜑𝑡 ∈ dom (ℝ D 𝐹)) ∧ 𝑘 ∈ ℤ) → ((ℝ D 𝐹)‘(𝑡 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑡))
190 fveq2 6375 . . . . . . . . 9 (𝑗 = 𝑖 → (𝑄𝑗) = (𝑄𝑖))
191 oveq1 6849 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝑗 + 1) = (𝑖 + 1))
192191fveq2d 6379 . . . . . . . . 9 (𝑗 = 𝑖 → (𝑄‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)))
193190, 192oveq12d 6860 . . . . . . . 8 (𝑗 = 𝑖 → ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
194193cbvmptv 4909 . . . . . . 7 (𝑗 ∈ (0..^𝑀) ↦ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1)))) = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
195 eqid 2765 . . . . . . 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 41031 . . . . . 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 2009 . . . . . . . . 9 𝑡(𝜑𝑖 ∈ (0..^𝑀))
199 nfra1 3088 . . . . . . . . 9 𝑡𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧
200198, 199nfan 1998 . . . . . . . 8 𝑡((𝜑𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
201128, 131syl6eq 2815 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑀)) → (ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
202201fveq1d 6377 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡) = (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡))
203 fvres 6394 . . . . . . . . . . . . 13 (𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) → (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
204202, 203sylan9eq 2819 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
205204fveq2d 6379 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (abs‘((ℝ D (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡)))
206205adantlr 706 . . . . . . . . . 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 785 . . . . . . . . . . 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 5595 . . . . . . . . . . . . . 14 (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
209137, 208sylibr 225 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹))
210209ad2antrr 717 . . . . . . . . . . . 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 477 . . . . . . . . . . . 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 3762 . . . . . . . . . . 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 3077 . . . . . . . . . . 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 579 . . . . . . . . . 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 4831 . . . . . . . . 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 401 . . . . . . . 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 3104 . . . . . . 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 401 . . . . . 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 3162 . . . . 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 40786 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) ≠ ∅)
222119, 141, 152, 221, 125ellimciota 40484 . 2 ((𝜑𝑖 ∈ (0..^𝑀)) → (℩𝑦𝑦 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖))) ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
223149, 145, 153, 138, 220ioodvbdlimc2 40788 . . 3 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) ≠ ∅)
224119, 141, 167, 223, 125ellimciota 40484 . 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 6228 . . . . . . 7 ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ → Rel (ℝ D 𝐹))
226157, 225syl 17 . . . . . 6 (𝜑 → Rel (ℝ D 𝐹))
227 resindm 5621 . . . . . 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 3993 . . . . . . 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 6253 . . . . 5 (𝜑 → ((ℝ D 𝐹) ↾ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))):((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))⟶ℝ)
232228, 231feq1dd 39994 . . . 4 (𝜑 → ((ℝ D 𝐹) ↾ (-∞(,)𝑋)):((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))⟶ℝ)
233232, 117fssd 6237 . . 3 (𝜑 → ((ℝ D 𝐹) ↾ (-∞(,)𝑋)):((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))⟶ℂ)
234 ioosscn 40358 . . . . 5 (-∞(,)𝑋) ⊆ ℂ
235 ssinss1 4001 . . . . 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 1180 . . . . . . . 8 ((𝜑𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → (𝜑𝑘 ∈ ℤ))
239 simp2 1167 . . . . . . . 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 785 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝑘 ∈ ℤ)
243 simpr 477 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
244 eleq1w 2827 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑥 ∈ ℝ ↔ 𝑦 ∈ ℝ))
245244anbi2d 622 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝜑𝑥 ∈ ℝ) ↔ (𝜑𝑦 ∈ ℝ)))
246 oveq1 6849 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 + 𝑇) = (𝑦 + 𝑇))
247246fveq2d 6379 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑦 + 𝑇)))
248 fveq2 6375 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
249247, 248eqeq12d 2780 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥) ↔ (𝐹‘(𝑦 + 𝑇)) = (𝐹𝑦)))
250245, 249imbi12d 335 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥)) ↔ ((𝜑𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑇)) = (𝐹𝑦))))
251250, 111chvarv 2369 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑇)) = (𝐹𝑦))
252251ad4ant14 759 . . . . . . . . . 10 ((((𝜑𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑇)) = (𝐹𝑦))
253240, 241, 242, 243, 252fperiodmul 40157 . . . . . . . . 9 (((𝜑𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥))
254170, 178, 253, 185fperdvper 40771 . . . . . . . 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 579 . . . . . . 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 488 . . . . . 6 ((𝜑𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹))
257 oveq2 6850 . . . . . . . . . 10 (𝑤 = 𝑥 → (π − 𝑤) = (π − 𝑥))
258257oveq1d 6857 . . . . . . . . 9 (𝑤 = 𝑥 → ((π − 𝑤) / 𝑇) = ((π − 𝑥) / 𝑇))
259258fveq2d 6379 . . . . . . . 8 (𝑤 = 𝑥 → (⌊‘((π − 𝑤) / 𝑇)) = (⌊‘((π − 𝑥) / 𝑇)))
260259oveq1d 6857 . . . . . . 7 (𝑤 = 𝑥 → ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇) = ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
261260cbvmptv 4909 . . . . . 6 (𝑤 ∈ ℝ ↦ ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇)) = (𝑥 ∈ ℝ ↦ ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
262 eqid 2765 . . . . . 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 41002 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) ∧ ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))))
264263simpld 488 . . . 4 (𝜑 → ∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)))
265125cnfldtop 22866 . . . . . . . . 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 10350 . . . . . . . . . . . 12 -∞ ∈ ℝ*
269268a1i 11 . . . . . . . . . . 11 (𝑦 ∈ ℝ → -∞ ∈ ℝ*)
270 rexr 10339 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
271 mnflt 12157 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → -∞ < 𝑦)
272269, 270, 271xrltled 12183 . . . . . . . . . . 11 (𝑦 ∈ ℝ → -∞ ≤ 𝑦)
273 iooss1 12412 . . . . . . . . . . 11 ((-∞ ∈ ℝ* ∧ -∞ ≤ 𝑦) → (𝑦(,)𝑋) ⊆ (-∞(,)𝑋))
274269, 272, 273syl2anc 579 . . . . . . . . . 10 (𝑦 ∈ ℝ → (𝑦(,)𝑋) ⊆ (-∞(,)𝑋))
2752743ad2ant2 1164 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (𝑦(,)𝑋) ⊆ (-∞(,)𝑋))
276 simp3 1168 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))
277275, 276ssind 3996 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (𝑦(,)𝑋) ⊆ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)))
278 unicntop 22868 . . . . . . . . 9 ℂ = (TopOpen‘ℂfld)
279278lpss3 21228 . . . . . . . 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 1490 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → ((limPt‘(TopOpen‘ℂfld))‘(𝑦(,)𝑋)) ⊆ ((limPt‘(TopOpen‘ℂfld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
2812803adant3l 1229 . . . . . 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 1164 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑦 ∈ ℝ*)
283583ad2ant1 1163 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ℝ)
284 simp3l 1258 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑦 < 𝑋)
285125, 282, 283, 284lptioo2cn 40515 . . . . . 6 ((𝜑𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝑦(,)𝑋)))
286281, 285sseldd 3762 . . . . 5 ((𝜑𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
287286rexlimdv3a 3180 . . . 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 489 . . . 4 ((𝜑𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → ((ℝ D 𝐹)‘(𝑥 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑥))
290 oveq2 6850 . . . . . . . 8 (𝑦 = 𝑥 → (π − 𝑦) = (π − 𝑥))
291290oveq1d 6857 . . . . . . 7 (𝑦 = 𝑥 → ((π − 𝑦) / 𝑇) = ((π − 𝑥) / 𝑇))
292291fveq2d 6379 . . . . . 6 (𝑦 = 𝑥 → (⌊‘((π − 𝑦) / 𝑇)) = (⌊‘((π − 𝑥) / 𝑇)))
293292oveq1d 6857 . . . . 5 (𝑦 = 𝑥 → ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇) = ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
294293cbvmptv 4909 . . . 4 (𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇)) = (𝑥 ∈ ℝ ↦ ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
295 id 22 . . . . . 6 (𝑧 = 𝑥𝑧 = 𝑥)
296 fveq2 6375 . . . . . 6 (𝑧 = 𝑥 → ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑧) = ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑥))
297295, 296oveq12d 6860 . . . . 5 (𝑧 = 𝑥 → (𝑧 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑧)) = (𝑥 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑥)))
298297cbvmptv 4909 . . . 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 41009 . . 3 (𝜑 → (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim 𝑋) ≠ ∅)
300233, 237, 288, 299, 125ellimciota 40484 . 2 (𝜑 → (℩𝑥𝑥 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim 𝑋)) ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim 𝑋))
301 resindm 5621 . . . . . 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 3993 . . . . . . 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 6253 . . . . 5 (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))):((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))⟶ℝ)
306302, 305feq1dd 39994 . . . 4 (𝜑 → ((ℝ D 𝐹) ↾ (𝑋(,)+∞)):((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))⟶ℝ)
307306, 117fssd 6237 . . 3 (𝜑 → ((ℝ D 𝐹) ↾ (𝑋(,)+∞)):((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))⟶ℂ)
308 ioosscn 40358 . . . . 5 (𝑋(,)+∞) ⊆ ℂ
309 ssinss1 4001 . . . . 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 489 . . . 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 10346 . . . . . . . . . . . 12 +∞ ∈ ℝ*
316315a1i 11 . . . . . . . . . . 11 (𝑦 ∈ ℝ → +∞ ∈ ℝ*)
317 ltpnf 12154 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 < +∞)
318270, 316, 317xrltled 12183 . . . . . . . . . . 11 (𝑦 ∈ ℝ → 𝑦 ≤ +∞)
319 iooss2 12413 . . . . . . . . . . 11 ((+∞ ∈ ℝ*𝑦 ≤ +∞) → (𝑋(,)𝑦) ⊆ (𝑋(,)+∞))
320316, 318, 319syl2anc 579 . . . . . . . . . 10 (𝑦 ∈ ℝ → (𝑋(,)𝑦) ⊆ (𝑋(,)+∞))
3213203ad2ant2 1164 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (𝑋(,)𝑦) ⊆ (𝑋(,)+∞))
322 simp3 1168 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))
323321, 322ssind 3996 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (𝑋(,)𝑦) ⊆ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)))
324278lpss3 21228 . . . . . . . 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 1490 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → ((limPt‘(TopOpen‘ℂfld))‘(𝑋(,)𝑦)) ⊆ ((limPt‘(TopOpen‘ℂfld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
3263253adant3l 1229 . . . . . 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 1164 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑦 ∈ ℝ*)
328583ad2ant1 1163 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ℝ)
329 simp3l 1258 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 < 𝑦)
330125, 327, 328, 329lptioo1cn 40516 . . . . . 6 ((𝜑𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘(𝑋(,)𝑦)))
331326, 330sseldd 3762 . . . . 5 ((𝜑𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂfld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
332331rexlimdv3a 3180 . . . 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 252 . . . 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 41008 . . 3 (𝜑 → (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim 𝑋) ≠ ∅)
336307, 311, 333, 335, 125ellimciota 40484 . 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 6375 . . . . . . . 8 (𝑛 = 𝑘 → (𝐴𝑛) = (𝐴𝑘))
342 oveq1 6849 . . . . . . . . 9 (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
343342fveq2d 6379 . . . . . . . 8 (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
344341, 343oveq12d 6860 . . . . . . 7 (𝑛 = 𝑘 → ((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴𝑘) · (cos‘(𝑘 · 𝑋))))
345 fveq2 6375 . . . . . . . 8 (𝑛 = 𝑘 → (𝐵𝑛) = (𝐵𝑘))
346342fveq2d 6379 . . . . . . . 8 (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
347345, 346oveq12d 6860 . . . . . . 7 (𝑛 = 𝑘 → ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵𝑘) · (sin‘(𝑘 · 𝑋))))
348344, 347oveq12d 6860 . . . . . 6 (𝑛 = 𝑘 → (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵𝑘) · (sin‘(𝑘 · 𝑋)))))
349348cbvsumv 14711 . . . . 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 6850 . . . . . . 7 (𝑗 = 𝑚 → (1...𝑗) = (1...𝑚))
351350eqcomd 2771 . . . . . 6 (𝑗 = 𝑚 → (1...𝑚) = (1...𝑗))
352351sumeq1d 14716 . . . . 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 2812 . . . 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 6858 . . 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 4909 . 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 6231 . . . . . 6 (𝐹:ℝ⟶ℝ → dom 𝐹 = ℝ)
3581, 357syl 17 . . . . 5 (𝜑 → dom 𝐹 = ℝ)
359358, 154eqsstrd 3799 . . . 4 (𝜑 → dom 𝐹 ⊆ ℝ)
360358feq2d 6209 . . . . 5 (𝜑 → (𝐹:dom 𝐹⟶ℝ ↔ 𝐹:ℝ⟶ℝ))
3611, 360mpbird 248 . . . 4 (𝜑𝐹:dom 𝐹⟶ℝ)
362359sselda 3761 . . . . . . 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 706 . . . . . . 7 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑇 ∈ ℝ)
366364, 365remulcld 10324 . . . . . 6 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑘 · 𝑇) ∈ ℝ)
367363, 366readdcld 10323 . . . . 5 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑡 + (𝑘 · 𝑇)) ∈ ℝ)
368358eqcomd 2771 . . . . . 6 (𝜑 → ℝ = dom 𝐹)
369368ad2antrr 717 . . . . 5 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → ℝ = dom 𝐹)
370367, 369eleqtrd 2846 . . . 4 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑡 + (𝑘 · 𝑇)) ∈ dom 𝐹)
371 id 22 . . . . . 6 ((𝜑𝑘 ∈ ℤ) → (𝜑𝑘 ∈ ℤ))
372371adantlr 706 . . . . 5 (((𝜑𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝜑𝑘 ∈ ℤ))
373372, 363, 184syl2anc 579 . . . 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 41031 . . 3 (𝜑 → ∃𝑢 ∈ ℝ ∀𝑡 ∈ dom 𝐹(abs‘(𝐹𝑡)) ≤ 𝑢)
375358raleqdv 3292 . . . 4 (𝜑 → (∀𝑡 ∈ dom 𝐹(abs‘(𝐹𝑡)) ≤ 𝑢 ↔ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑢))
376375rexbidv 3199 . . 3 (𝜑 → (∃𝑢 ∈ ℝ ∀𝑡 ∈ dom 𝐹(abs‘(𝐹𝑡)) ≤ 𝑢 ↔ ∃𝑢 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹𝑡)) ≤ 𝑢))
377374, 376mpbid 223 . 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 41072 1 (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  cun 3730  cin 3731  wss 3732  c0 4079  ifcif 4243  {cpr 4336   class class class wbr 4809  cmpt 4888  dom cdm 5277  ran crn 5278  cres 5279  Rel wrel 5282  cio 6029  Fun wfun 6062  wf 6064  cfv 6068   Isom wiso 6069  (class class class)co 6842  𝑚 cmap 8060  Fincfn 8160  cc 10187  cr 10188  0cc0 10189  1c1 10190   + caddc 10192   · cmul 10194  +∞cpnf 10325  -∞cmnf 10326  *cxr 10327   < clt 10328  cle 10329  cmin 10520  -cneg 10521   / cdiv 10938  cn 11274  2c2 11327  0cn0 11538  cz 11624  cuz 11886  (,)cioo 12377  (,]cioc 12378  [,)cico 12379  [,]cicc 12380  ...cfz 12533  ..^cfzo 12673  cfl 12799   mod cmo 12876  seqcseq 13008  chash 13321  abscabs 14259  cli 14500  Σcsu 14701  sincsin 15076  cosccos 15077  πcpi 15079  TopOpenctopn 16348  topGenctg 16364  fldccnfld 20019  Topctop 20977  intcnt 21101  limPtclp 21218  cnccncf 22958  citg 23676   lim climc 23917   D cdv 23918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cc 9510  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268  ax-mulf 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-symdif 4005  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-disj 4778  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ofr 7096  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-omul 7769  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-fi 8524  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-acn 9019  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-xnn0 11611  df-z 11625  df-dec 11741  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12146  df-xadd 12147  df-xmul 12148  df-ioo 12381  df-ioc 12382  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-fac 13265  df-bc 13294  df-hash 13322  df-shft 14092  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-limsup 14487  df-clim 14504  df-rlim 14505  df-sum 14702  df-ef 15080  df-sin 15082  df-cos 15083  df-pi 15085  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-starv 16229  df-sca 16230  df-vsca 16231  df-ip 16232  df-tset 16233  df-ple 16234  df-ds 16236  df-unif 16237  df-hom 16238  df-cco 16239  df-rest 16349  df-topn 16350  df-0g 16368  df-gsum 16369  df-topgen 16370  df-pt 16371  df-prds 16374  df-xrs 16428  df-qtop 16433  df-imas 16434  df-xps 16436  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-submnd 17602  df-mulg 17808  df-cntz 18013  df-cmn 18461  df-psmet 20011  df-xmet 20012  df-met 20013  df-bl 20014  df-mopn 20015  df-fbas 20016  df-fg 20017  df-cnfld 20020  df-top 20978  df-topon 20995  df-topsp 21017  df-bases 21030  df-cld 21103  df-ntr 21104  df-cls 21105  df-nei 21182  df-lp 21220  df-perf 21221  df-cn 21311  df-cnp 21312  df-t1 21398  df-haus 21399  df-cmp 21470  df-tx 21645  df-hmeo 21838  df-fil 21929  df-fm 22021  df-flim 22022  df-flf 22023  df-xms 22404  df-ms 22405  df-tms 22406  df-cncf 22960  df-ovol 23522  df-vol 23523  df-mbf 23677  df-itg1 23678  df-itg2 23679  df-ibl 23680  df-itg 23681  df-0p 23728  df-ditg 23902  df-limc 23921  df-dv 23922
This theorem is referenced by:  fourierdlem114  41074
  Copyright terms: Public domain W3C validator