fourierdlem113 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem113 46217

Description: Fourier series convergence for periodic, piecewise smooth functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem113.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem113.t	⊢ 𝑇 = (2 · π)
fourierdlem113.per	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierdlem113.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem113.l	⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
fourierdlem113.r	⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
fourierdlem113.p	⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem113.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem113.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem113.dvcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem113.dvlb	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅)
fourierdlem113.dvub	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅)
fourierdlem113.a	⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
fourierdlem113.b	⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
fourierdlem113.15	⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))
fourierdlem113.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇)))
fourierdlem113.exq	⊢ (𝜑 → (𝐸‘𝑋) ∈ ran 𝑄)

**Assertion**
Ref	Expression
fourierdlem113	⊢ (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (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.

Step	Hyp	Ref	Expression
1		fourierdlem113.f	. 2 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2		oveq1 7394	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 mod (2 · π)) = (𝑦 mod (2 · π)))
3	2	eqeq1d 2731	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑤 mod (2 · π)) = 0 ↔ (𝑦 mod (2 · π)) = 0))
4		oveq2 7395	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑘 + (1 / 2)) · 𝑤) = ((𝑘 + (1 / 2)) · 𝑦))
5	4	fveq2d 6862	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (sin‘((𝑘 + (1 / 2)) · 𝑤)) = (sin‘((𝑘 + (1 / 2)) · 𝑦)))
6		oveq1 7394	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤 / 2) = (𝑦 / 2))
7	6	fveq2d 6862	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (sin‘(𝑤 / 2)) = (sin‘(𝑦 / 2)))
8	7	oveq2d 7403	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((2 · π) · (sin‘(𝑤 / 2))) = ((2 · π) · (sin‘(𝑦 / 2))))
9	5, 8	oveq12d 7405	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((sin‘((𝑘 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2)))) = ((sin‘((𝑘 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))
10	3, 9	ifbieq2d 4515	. . . . 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))))))
11	10	cbvmptv 5211	. . . 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 7395	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (2 · 𝑘) = (2 · 𝑚))
13	12	oveq1d 7402	. . . . . . 7 ⊢ (𝑘 = 𝑚 → ((2 · 𝑘) + 1) = ((2 · 𝑚) + 1))
14	13	oveq1d 7402	. . . . . 6 ⊢ (𝑘 = 𝑚 → (((2 · 𝑘) + 1) / (2 · π)) = (((2 · 𝑚) + 1) / (2 · π)))
15		oveq1 7394	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝑘 + (1 / 2)) = (𝑚 + (1 / 2)))
16	15	oveq1d 7402	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝑘 + (1 / 2)) · 𝑦) = ((𝑚 + (1 / 2)) · 𝑦))
17	16	fveq2d 6862	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (sin‘((𝑘 + (1 / 2)) · 𝑦)) = (sin‘((𝑚 + (1 / 2)) · 𝑦)))
18	17	oveq1d 7402	. . . . . 6 ⊢ (𝑘 = 𝑚 → ((sin‘((𝑘 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))) = ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))
19	14, 18	ifeq12d 4510	. . . . 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))))))
20	19	mpteq2dv 5201	. . . 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)))))))
21	11, 20	eqtrid 2776	. . 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)))))))
22	21	cbvmptv 5211	. 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 ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
24		fourierdlem113.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
25		fourierdlem113.q	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
26		oveq1 7394	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤 + (𝑗 · 𝑇)) = (𝑦 + (𝑗 · 𝑇)))
27	26	eleq1d 2813	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
28	27	rexbidv 3157	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
29	28	cbvrabv 3416	. . . . 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}
30	29	uneq2i 4128	. . . 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})
31	30	fveq2i 6861	. . 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}))
32	31	oveq1i 7397	. 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 7394	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑘 · 𝑇) = (𝑗 · 𝑇))
34	33	oveq2d 7403	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑗 · 𝑇)))
35	34	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
36	35	cbvrexvw 3216	. . . . . . . . 9 ⊢ (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄)
37	36	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
38	37	rabbiia 3409	. . . . . . 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}
39	38	uneq2i 4128	. . . . . 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 7296	. . . . . 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}))))
41	39, 40	ax-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})))
42	41	a1i 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}))))
43	33	oveq2d 7403	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑤 + (𝑘 · 𝑇)) = (𝑤 + (𝑗 · 𝑇)))
44	43	eleq1d 2813	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → ((𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄))
45	44	cbvrexvw 3216	. . . . . . . . . . . 12 ⊢ (∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄)
46	45	a1i 11	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ((-π + 𝑋)[,](π + 𝑋)) → (∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄))
47	46	rabbiia 3409	. . . . . . . . . 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}
48	47	uneq2i 4128	. . . . . . . . 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})
49	48	fveq2i 6861	. . . . . . . 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}))
50	49	oveq1i 7397	. . . . . . 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)
51	50	oveq2i 7398	. . . . . 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 7295	. . . . . 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}))))
53	51, 52	ax-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})))
54	53	a1i 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 7292	. . . 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}))))
56	42, 54, 55	3bitrd 305	. . 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}))))
57	56	cbviotavw 6472	. 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 ⊢ (𝜑 → 𝑋 ∈ ℝ)
59		pire 26366	. . . . 5 ⊢ π ∈ ℝ
60	59	renegcli 11483	. . . 4 ⊢ -π ∈ ℝ
61	60	a1i 11	. . 3 ⊢ (𝜑 → -π ∈ ℝ)
62	59	a1i 11	. . 3 ⊢ (𝜑 → π ∈ ℝ)
63		negpilt0 45279	. . . 4 ⊢ -π < 0
64	63	a1i 11	. . 3 ⊢ (𝜑 → -π < 0)
65		pipos 26368	. . . 4 ⊢ 0 < π
66	65	a1i 11	. . 3 ⊢ (𝜑 → 0 < π)
67		picn 26367	. . . . 5 ⊢ π ∈ ℂ
68	67	2timesi 12319	. . . 4 ⊢ (2 · π) = (π + π)
69		fourierdlem113.t	. . . 4 ⊢ 𝑇 = (2 · π)
70	67, 67	subnegi 11501	. . . 4 ⊢ (π − -π) = (π + π)
71	68, 69, 70	3eqtr4i 2762	. . 3 ⊢ 𝑇 = (π − -π)
72	23	fourierdlem2 46107	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
73	24, 72	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
74	25, 73	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
75	74	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
76		elmapi 8822	. . . . 5 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
77	75, 76	syl 17	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
78		fzfid 13938	. . . 4 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
79		rnffi 45169	. . . 4 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
80	77, 78, 79	syl2anc 584	. . 3 ⊢ (𝜑 → ran 𝑄 ∈ Fin)
81	23, 24, 25	fourierdlem15 46120	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
82		frn 6695	. . . 4 ⊢ (𝑄:(0...𝑀)⟶(-π[,]π) → ran 𝑄 ⊆ (-π[,]π))
83	81, 82	syl 17	. . 3 ⊢ (𝜑 → ran 𝑄 ⊆ (-π[,]π))
84	74	simprd 495	. . . . 5 ⊢ (𝜑 → (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
85	84	simplrd 769	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) = π)
86		ffun 6691	. . . . . 6 ⊢ (𝑄:(0...𝑀)⟶(-π[,]π) → Fun 𝑄)
87	81, 86	syl 17	. . . . 5 ⊢ (𝜑 → Fun 𝑄)
88	24	nnnn0d 12503	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
89		nn0uz 12835	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
90	88, 89	eleqtrdi 2838	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
91		eluzfz2 13493	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
92	90, 91	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
93		fdm 6697	. . . . . . . 8 ⊢ (𝑄:(0...𝑀)⟶(-π[,]π) → dom 𝑄 = (0...𝑀))
94	81, 93	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝑄 = (0...𝑀))
95	94	eqcomd 2735	. . . . . 6 ⊢ (𝜑 → (0...𝑀) = dom 𝑄)
96	92, 95	eleqtrd 2830	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ dom 𝑄)
97		fvelrn 7048	. . . . 5 ⊢ ((Fun 𝑄 ∧ 𝑀 ∈ dom 𝑄) → (𝑄‘𝑀) ∈ ran 𝑄)
98	87, 96, 97	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ran 𝑄)
99	85, 98	eqeltrrd 2829	. . 3 ⊢ (𝜑 → π ∈ ran 𝑄)
100		fourierdlem113.e	. . 3 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇)))
101		fourierdlem113.exq	. . 3 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ran 𝑄)
102		eqid 2729	. . 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 7292	. . . . 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}))))
104	30, 48, 39	3eqtr4ri 2763	. . . . . 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 7296	. . . . . 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}))))
106	104, 105	ax-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})))
107	103, 106	bitrdi 287	. . . 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}))))
108	107	cbviotavw 6472	. . 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 2729	. . 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}
110	61, 62, 64, 66, 71, 80, 83, 99, 100, 58, 101, 102, 108, 109	fourierdlem51 46155	. 2 ⊢ (𝜑 → 𝑋 ∈ 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 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
112		ax-resscn 11125	. . . 4 ⊢ ℝ ⊆ ℂ
113	112	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ℝ ⊆ ℂ)
114		ioossre 13368	. . . . . . 7 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
115	114	a1i 11	. . . . . 6 ⊢ (𝜑 → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
116	1, 115	fssresd 6727	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℝ)
117	112	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
118	116, 117	fssd 6705	. . . 4 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
119	118	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
120	114	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
121	1, 117	fssd 6705	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
122	121	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℂ)
123		ssid 3969	. . . . . . 7 ⊢ ℝ ⊆ ℝ
124	123	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ℝ ⊆ ℝ)
125		eqid 2729	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
126		tgioo4 24693	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
127	125, 126	dvres 25812	. . . . . 6 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:ℝ⟶ℂ) ∧ (ℝ ⊆ ℝ ∧ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)) → (ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))))
128	113, 122, 124, 120, 127	syl22anc 838	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))))
129	128	dmeqd 5869	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom (ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = dom ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))))
130		ioontr 45509	. . . . . . 7 ⊢ ((int‘(topGen‘ran (,)))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))
131	130	reseq2i 5947	. . . . . 6 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
132	131	dmeqi 5868	. . . . 5 ⊢ dom ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = dom ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
133	132	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = dom ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
134		fourierdlem113.dvcn	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
135		cncff 24786	. . . . 5 ⊢ (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
136		fdm 6697	. . . . 5 ⊢ (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ → dom ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
137	134, 135, 136	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
138	129, 133, 137	3eqtrd 2768	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom (ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
139		dvcn 25823	. . 3 ⊢ (((ℝ ⊆ ℂ ∧ (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ ∧ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ) ∧ dom (ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
140	113, 119, 120, 138, 139	syl31anc 1375	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
141	120, 113	sstrd 3957	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
142	77	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
143		fzofzp1 13725	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
144	143	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
145	142, 144	ffvelcdmd 7057	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
146	145	rexrd 11224	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
147		elfzofz 13636	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
148	147	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
149	142, 148	ffvelcdmd 7057	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
150	74	simprrd 773	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
151	150	r19.21bi 3229	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
152	125, 146, 149, 151	lptioo1cn 45644	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
153	116	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℝ)
154	123	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ)
155	117, 121, 154	dvbss 25802	. . . . . . 7 ⊢ (𝜑 → dom (ℝ D 𝐹) ⊆ ℝ)
156		dvfre 25855	. . . . . . . 8 ⊢ ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
157	1, 154, 156	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
158		0re 11176	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
159	60, 158, 59	lttri 11300	. . . . . . . . 9 ⊢ ((-π < 0 ∧ 0 < π) → -π < π)
160	63, 65, 159	mp2an 692	. . . . . . . 8 ⊢ -π < π
161	160	a1i 11	. . . . . . 7 ⊢ (𝜑 → -π < π)
162	84	simplld 767	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) = -π)
163	134, 135	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
164		fourierdlem113.dvlb	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅)
165	163, 141, 152, 164, 125	ellimciota 45612	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (℩𝑥𝑥 ∈ (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) ∈ (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
166	149	rexrd 11224	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
167	125, 166, 145, 151	lptioo2cn 45643	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
168		fourierdlem113.dvub	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅)
169	163, 141, 167, 168, 125	ellimciota 45612	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (℩𝑥𝑥 ∈ (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ∈ (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
170	121	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → 𝐹:ℝ⟶ℂ)
171		zre 12533	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
172	171	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℝ)
173		2re 12260	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
174	173, 59	remulcli 11190	. . . . . . . . . . . . . 14 ⊢ (2 · π) ∈ ℝ
175	174	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · π) ∈ ℝ)
176	69, 175	eqeltrid 2832	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℝ)
177	176	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → 𝑇 ∈ ℝ)
178	172, 177	remulcld 11204	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝑘 · 𝑇) ∈ ℝ)
179	170	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝐹:ℝ⟶ℂ)
180	177	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑇 ∈ ℝ)
181		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑘 ∈ ℤ)
182		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
183	111	ad4ant14 752	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
184	179, 180, 181, 182, 183	fperiodmul 45302	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → (𝐹‘(𝑡 + (𝑘 · 𝑇))) = (𝐹‘𝑡))
185		eqid 2729	. . . . . . . . . 10 ⊢ (ℝ D 𝐹) = (ℝ D 𝐹)
186	170, 178, 184, 185	fperdvper 45917	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ dom (ℝ D 𝐹)) → ((𝑡 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹) ∧ ((ℝ D 𝐹)‘(𝑡 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑡)))
187	186	an32s 652	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ dom (ℝ D 𝐹)) ∧ 𝑘 ∈ ℤ) → ((𝑡 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹) ∧ ((ℝ D 𝐹)‘(𝑡 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑡)))
188	187	simpld 494	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ dom (ℝ D 𝐹)) ∧ 𝑘 ∈ ℤ) → (𝑡 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹))
189	187	simprd 495	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ dom (ℝ D 𝐹)) ∧ 𝑘 ∈ ℤ) → ((ℝ D 𝐹)‘(𝑡 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑡))
190		fveq2 6858	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑄‘𝑗) = (𝑄‘𝑖))
191		oveq1 7394	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑗 + 1) = (𝑖 + 1))
192	191	fveq2d 6862	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑄‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)))
193	190, 192	oveq12d 7405	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
194	193	cbvmptv 5211	. . . . . . 7 ⊢ (𝑗 ∈ (0..^𝑀) ↦ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
195		eqid 2729	. . . . . . 7 ⊢ (𝑡 ∈ ℝ ↦ (𝑡 + ((⌊‘((π − 𝑡) / 𝑇)) · 𝑇))) = (𝑡 ∈ ℝ ↦ (𝑡 + ((⌊‘((π − 𝑡) / 𝑇)) · 𝑇)))
196	155, 157, 61, 62, 161, 71, 24, 77, 162, 85, 134, 165, 169, 188, 189, 194, 195	fourierdlem71 46175	. . . . . 6 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
197	196	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
198		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑖 ∈ (0..^𝑀))
199		nfra1 3261	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧
200	198, 199	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
201	128, 131	eqtrdi 2780	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
202	201	fveq1d 6860	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡) = (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡))
203		fvres 6877	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
204	202, 203	sylan9eq 2784	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
205	204	fveq2d 6862	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (abs‘((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡)))
206	205	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (abs‘((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) = (abs‘((ℝ D 𝐹)‘𝑡)))
207		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
208		ssdmres 5984	. . . . . . . . . . . . . 14 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
209	137, 208	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹))
210	209	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹))
211		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
212	210, 211	sseldd 3947	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑡 ∈ dom (ℝ D 𝐹))
213		rspa 3226	. . . . . . . . . . 11 ⊢ ((∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 ∧ 𝑡 ∈ dom (ℝ D 𝐹)) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
214	207, 212, 213	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
215	206, 214	eqbrtrd 5129	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (abs‘((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧)
216	215	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧))
217	200, 216	ralrimi 3235	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) → ∀𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧)
218	217	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 → ∀𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧))
219	218	reximdv 3148	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧))
220	197, 219	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧)
221	149, 145, 153, 138, 220	ioodvbdlimc1 45931	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅)
222	119, 141, 152, 221, 125	ellimciota 45612	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (℩𝑦𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
223	149, 145, 153, 138, 220	ioodvbdlimc2 45933	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅)
224	119, 141, 167, 223, 125	ellimciota 45612	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (℩𝑦𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
225		frel 6693	. . . . . . 7 ⊢ ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ → Rel (ℝ D 𝐹))
226	157, 225	syl 17	. . . . . 6 ⊢ (𝜑 → Rel (ℝ D 𝐹))
227		resindm 6001	. . . . . 6 ⊢ (Rel (ℝ D 𝐹) → ((ℝ D 𝐹) ↾ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))) = ((ℝ D 𝐹) ↾ (-∞(,)𝑋)))
228	226, 227	syl 17	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))) = ((ℝ D 𝐹) ↾ (-∞(,)𝑋)))
229		inss2 4201	. . . . . . 7 ⊢ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ dom (ℝ D 𝐹)
230	229	a1i 11	. . . . . 6 ⊢ (𝜑 → ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ dom (ℝ D 𝐹))
231	157, 230	fssresd 6727	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))):((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))⟶ℝ)
232	228, 231	feq1dd 6671	. . . 4 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (-∞(,)𝑋)):((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))⟶ℝ)
233	232, 117	fssd 6705	. . 3 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (-∞(,)𝑋)):((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))⟶ℂ)
234		ioosscn 13369	. . . . 5 ⊢ (-∞(,)𝑋) ⊆ ℂ
235		ssinss1 4209	. . . . 5 ⊢ ((-∞(,)𝑋) ⊆ ℂ → ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
236	234, 235	ax-mp 5	. . . 4 ⊢ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ
237	236	a1i 11	. . 3 ⊢ (𝜑 → ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
238		3simpb 1149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → (𝜑 ∧ 𝑘 ∈ ℤ))
239		simp2 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑥 ∈ dom (ℝ D 𝐹))
240	170	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝐹:ℝ⟶ℂ)
241	177	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
242		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝑘 ∈ ℤ)
243		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
244		eleq1w 2811	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ℝ ↔ 𝑦 ∈ ℝ))
245	244	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ ℝ) ↔ (𝜑 ∧ 𝑦 ∈ ℝ)))
246		oveq1 7394	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑇) = (𝑦 + 𝑇))
247	246	fveq2d 6862	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑦 + 𝑇)))
248		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
249	247, 248	eqeq12d 2745	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦)))
250	245, 249	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))))
251	250, 111	chvarvv 1989	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))
252	251	ad4ant14 752	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))
253	240, 241, 242, 243, 252	fperiodmul 45302	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥))
254	170, 178, 253, 185	fperdvper 45917	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ dom (ℝ D 𝐹)) → ((𝑥 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹) ∧ ((ℝ D 𝐹)‘(𝑥 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑥)))
255	238, 239, 254	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → ((𝑥 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹) ∧ ((ℝ D 𝐹)‘(𝑥 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑥)))
256	255	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹))
257		oveq2 7395	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (π − 𝑤) = (π − 𝑥))
258	257	oveq1d 7402	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → ((π − 𝑤) / 𝑇) = ((π − 𝑥) / 𝑇))
259	258	fveq2d 6862	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (⌊‘((π − 𝑤) / 𝑇)) = (⌊‘((π − 𝑥) / 𝑇)))
260	259	oveq1d 7402	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇) = ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
261	260	cbvmptv 5211	. . . . . 6 ⊢ (𝑤 ∈ ℝ ↦ ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇)) = (𝑥 ∈ ℝ ↦ ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
262		eqid 2729	. . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ (𝑥 + ((𝑤 ∈ ℝ ↦ ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇))‘𝑥))) = (𝑥 ∈ ℝ ↦ (𝑥 + ((𝑤 ∈ ℝ ↦ ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇))‘𝑥)))
263	61, 62, 161, 71, 256, 58, 261, 262, 23, 24, 25, 209	fourierdlem41 46146	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) ∧ ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))))
264	263	simpld 494	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)))
265	125	cnfldtop 24671	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ Top
266	265	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (TopOpen‘ℂ_fld) ∈ Top)
267	236	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
268		mnfxr 11231	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
269	268	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → -∞ ∈ ℝ^*)
270		rexr 11220	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
271		mnflt 13083	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → -∞ < 𝑦)
272	269, 270, 271	xrltled 13110	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → -∞ ≤ 𝑦)
273		iooss1 13341	. . . . . . . . . . 11 ⊢ ((-∞ ∈ ℝ^* ∧ -∞ ≤ 𝑦) → (𝑦(,)𝑋) ⊆ (-∞(,)𝑋))
274	269, 272, 273	syl2anc 584	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (𝑦(,)𝑋) ⊆ (-∞(,)𝑋))
275	274	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (𝑦(,)𝑋) ⊆ (-∞(,)𝑋))
276		simp3 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))
277	275, 276	ssind 4204	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (𝑦(,)𝑋) ⊆ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)))
278		unicntop 24673	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
279	278	lpss3 23031	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ ∧ (𝑦(,)𝑋) ⊆ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))) → ((limPt‘(TopOpen‘ℂ_fld))‘(𝑦(,)𝑋)) ⊆ ((limPt‘(TopOpen‘ℂ_fld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
280	266, 267, 277, 279	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → ((limPt‘(TopOpen‘ℂ_fld))‘(𝑦(,)𝑋)) ⊆ ((limPt‘(TopOpen‘ℂ_fld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
281	280	3adant3l 1181	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → ((limPt‘(TopOpen‘ℂ_fld))‘(𝑦(,)𝑋)) ⊆ ((limPt‘(TopOpen‘ℂ_fld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
282	270	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑦 ∈ ℝ^*)
283	58	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ℝ)
284		simp3l 1202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑦 < 𝑋)
285	125, 282, 283, 284	lptioo2cn 45643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝑦(,)𝑋)))
286	281, 285	sseldd 3947	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
287	286	rexlimdv3a 3138	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)))))
288	264, 287	mpd 15	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
289	255	simprd 495	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → ((ℝ D 𝐹)‘(𝑥 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑥))
290		oveq2 7395	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (π − 𝑦) = (π − 𝑥))
291	290	oveq1d 7402	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((π − 𝑦) / 𝑇) = ((π − 𝑥) / 𝑇))
292	291	fveq2d 6862	. . . . . 6 ⊢ (𝑦 = 𝑥 → (⌊‘((π − 𝑦) / 𝑇)) = (⌊‘((π − 𝑥) / 𝑇)))
293	292	oveq1d 7402	. . . . 5 ⊢ (𝑦 = 𝑥 → ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇) = ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
294	293	cbvmptv 5211	. . . 4 ⊢ (𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇)) = (𝑥 ∈ ℝ ↦ ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
295		id 22	. . . . . 6 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
296		fveq2 6858	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑧) = ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑥))
297	295, 296	oveq12d 7405	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑧)) = (𝑥 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑥)))
298	297	cbvmptv 5211	. . . 4 ⊢ (𝑧 ∈ ℝ ↦ (𝑧 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑧))) = (𝑥 ∈ ℝ ↦ (𝑥 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑥)))
299	61, 62, 161, 23, 71, 24, 25, 155, 157, 256, 289, 134, 169, 58, 294, 298	fourierdlem49 46153	. . 3 ⊢ (𝜑 → (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅)
300	233, 237, 288, 299, 125	ellimciota 45612	. 2 ⊢ (𝜑 → (℩𝑥𝑥 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
301		resindm 6001	. . . . . 6 ⊢ (Rel (ℝ D 𝐹) → ((ℝ D 𝐹) ↾ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))) = ((ℝ D 𝐹) ↾ (𝑋(,)+∞)))
302	226, 301	syl 17	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))) = ((ℝ D 𝐹) ↾ (𝑋(,)+∞)))
303		inss2 4201	. . . . . . 7 ⊢ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ dom (ℝ D 𝐹)
304	303	a1i 11	. . . . . 6 ⊢ (𝜑 → ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ dom (ℝ D 𝐹))
305	157, 304	fssresd 6727	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))):((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))⟶ℝ)
306	302, 305	feq1dd 6671	. . . 4 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝑋(,)+∞)):((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))⟶ℝ)
307	306, 117	fssd 6705	. . 3 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝑋(,)+∞)):((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))⟶ℂ)
308		ioosscn 13369	. . . . 5 ⊢ (𝑋(,)+∞) ⊆ ℂ
309		ssinss1 4209	. . . . 5 ⊢ ((𝑋(,)+∞) ⊆ ℂ → ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
310	308, 309	ax-mp 5	. . . 4 ⊢ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ
311	310	a1i 11	. . 3 ⊢ (𝜑 → ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
312	263	simprd 495	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)))
313	265	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (TopOpen‘ℂ_fld) ∈ Top)
314	310	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
315		pnfxr 11228	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
316	315	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → +∞ ∈ ℝ^*)
317		ltpnf 13080	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 < +∞)
318	270, 316, 317	xrltled 13110	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → 𝑦 ≤ +∞)
319		iooss2 13342	. . . . . . . . . . 11 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑦 ≤ +∞) → (𝑋(,)𝑦) ⊆ (𝑋(,)+∞))
320	316, 318, 319	syl2anc 584	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (𝑋(,)𝑦) ⊆ (𝑋(,)+∞))
321	320	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (𝑋(,)𝑦) ⊆ (𝑋(,)+∞))
322		simp3 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))
323	321, 322	ssind 4204	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (𝑋(,)𝑦) ⊆ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)))
324	278	lpss3 23031	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ ∧ (𝑋(,)𝑦) ⊆ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))) → ((limPt‘(TopOpen‘ℂ_fld))‘(𝑋(,)𝑦)) ⊆ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
325	313, 314, 323, 324	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → ((limPt‘(TopOpen‘ℂ_fld))‘(𝑋(,)𝑦)) ⊆ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
326	325	3adant3l 1181	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → ((limPt‘(TopOpen‘ℂ_fld))‘(𝑋(,)𝑦)) ⊆ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
327	270	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑦 ∈ ℝ^*)
328	58	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ℝ)
329		simp3l 1202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 < 𝑦)
330	125, 327, 328, 329	lptioo1cn 45644	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝑋(,)𝑦)))
331	326, 330	sseldd 3947	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
332	331	rexlimdv3a 3138	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)))))
333	312, 332	mpd 15	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
334		biid 261	. . . 4 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑘 ∈ ℤ) ∧ 𝑤 = (𝑋 + (𝑘 · 𝑇))) ↔ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑤 ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑘 ∈ ℤ) ∧ 𝑤 = (𝑋 + (𝑘 · 𝑇))))
335	61, 62, 161, 23, 71, 24, 25, 157, 256, 289, 134, 165, 58, 294, 298, 334	fourierdlem48 46152	. . 3 ⊢ (𝜑 → (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅)
336	307, 311, 333, 335, 125	ellimciota 45612	. 2 ⊢ (𝜑 → (℩𝑥𝑥 ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
337		fourierdlem113.l	. 2 ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
338		fourierdlem113.r	. 2 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
339		fourierdlem113.a	. 2 ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
340		fourierdlem113.b	. 2 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
341		fveq2 6858	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐴‘𝑛) = (𝐴‘𝑘))
342		oveq1 7394	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑛 · 𝑋) = (𝑘 · 𝑋))
343	342	fveq2d 6862	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
344	341, 343	oveq12d 7405	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))))
345		fveq2 6858	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘))
346	342	fveq2d 6862	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
347	345, 346	oveq12d 7405	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
348	344, 347	oveq12d 7405	. . . . . 6 ⊢ (𝑛 = 𝑘 → (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))
349	348	cbvsumv 15662	. . . . 5 ⊢ Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) = Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
350		oveq2 7395	. . . . . . 7 ⊢ (𝑗 = 𝑚 → (1...𝑗) = (1...𝑚))
351	350	eqcomd 2735	. . . . . 6 ⊢ (𝑗 = 𝑚 → (1...𝑚) = (1...𝑗))
352	351	sumeq1d 15666	. . . . 5 ⊢ (𝑗 = 𝑚 → Σ𝑘 ∈ (1...𝑚)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑘 ∈ (1...𝑗)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))
353	349, 352	eqtr2id 2777	. . . 4 ⊢ (𝑗 = 𝑚 → Σ𝑘 ∈ (1...𝑗)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))) = Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))
354	353	oveq2d 7403	. . 3 ⊢ (𝑗 = 𝑚 → (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑗)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) = (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))))
355	354	cbvmptv 5211	. 2 ⊢ (𝑗 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑘 ∈ (1...𝑗)(((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))) = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))))
356		fourierdlem113.15	. 2 ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))
357		fdm 6697	. . . . . 6 ⊢ (𝐹:ℝ⟶ℝ → dom 𝐹 = ℝ)
358	1, 357	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝐹 = ℝ)
359	358, 154	eqsstrd 3981	. . . 4 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
360	358	feq2d 6672	. . . . 5 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℝ ↔ 𝐹:ℝ⟶ℝ))
361	1, 360	mpbird 257	. . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
362	359	sselda 3946	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ dom 𝐹) → 𝑡 ∈ ℝ)
363	362	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑡 ∈ ℝ)
364	171	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℝ)
365	177	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑇 ∈ ℝ)
366	364, 365	remulcld 11204	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑘 · 𝑇) ∈ ℝ)
367	363, 366	readdcld 11203	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑡 + (𝑘 · 𝑇)) ∈ ℝ)
368	358	eqcomd 2735	. . . . . 6 ⊢ (𝜑 → ℝ = dom 𝐹)
369	368	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → ℝ = dom 𝐹)
370	367, 369	eleqtrd 2830	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑡 + (𝑘 · 𝑇)) ∈ dom 𝐹)
371		id 22	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝜑 ∧ 𝑘 ∈ ℤ))
372	371	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝜑 ∧ 𝑘 ∈ ℤ))
373	372, 363, 184	syl2anc 584	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑡 + (𝑘 · 𝑇))) = (𝐹‘𝑡))
374	359, 361, 61, 62, 161, 71, 24, 77, 162, 85, 140, 222, 224, 370, 373, 194, 195	fourierdlem71 46175	. . 3 ⊢ (𝜑 → ∃𝑢 ∈ ℝ ∀𝑡 ∈ dom 𝐹(abs‘(𝐹‘𝑡)) ≤ 𝑢)
375	358	raleqdv 3299	. . . 4 ⊢ (𝜑 → (∀𝑡 ∈ dom 𝐹(abs‘(𝐹‘𝑡)) ≤ 𝑢 ↔ ∀𝑡 ∈ ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑢))
376	375	rexbidv 3157	. . 3 ⊢ (𝜑 → (∃𝑢 ∈ ℝ ∀𝑡 ∈ dom 𝐹(abs‘(𝐹‘𝑡)) ≤ 𝑢 ↔ ∃𝑢 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑢))
377	374, 376	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑢 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑢)
378	1, 22, 23, 24, 25, 32, 57, 58, 110, 69, 111, 140, 222, 224, 134, 300, 336, 337, 338, 339, 340, 355, 356, 377, 196, 58	fourierdlem112 46216	1 ⊢ (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3405 ∪ cun 3912 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 ifcif 4488 {cpr 4591 class class class wbr 5107 ↦ cmpt 5188 dom cdm 5638 ran crn 5639 ↾ cres 5640 Rel wrel 5643 ℩cio 6462 Fun wfun 6505 ⟶wf 6507 ‘cfv 6511 Isom wiso 6512 (class class class)co 7387 ↑_m cmap 8799 Fincfn 8918 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 +∞cpnf 11205 -∞cmnf 11206 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 -cneg 11406 / cdiv 11835 ℕcn 12186 2c2 12241 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 (,)cioo 13306 (,]cioc 13307 [,)cico 13308 [,]cicc 13309 ...cfz 13468 ..^cfzo 13615 ⌊cfl 13752 mod cmo 13831 seqcseq 13966 ♯chash 14295 abscabs 15200 ⇝ cli 15450 Σcsu 15652 sincsin 16029 cosccos 16030 πcpi 16032 TopOpenctopn 17384 topGenctg 17400 ℂ_fldccnfld 21264 Topctop 22780 intcnt 22904 limPtclp 23021 –cn→ccncf 24769 ∫citg 25519 lim_ℂ climc 25763 D cdv 25764

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cc 10388 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-symdif 4216 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-t1 23201 df-haus 23202 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-ovol 25365 df-vol 25366 df-mbf 25520 df-itg1 25521 df-itg2 25522 df-ibl 25523 df-itg 25524 df-0p 25571 df-ditg 25748 df-limc 25767 df-dv 25768

This theorem is referenced by: fourierdlem114 46218

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