fourierdlem113 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem113 46933

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 7417	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 mod (2 · π)) = (𝑦 mod (2 · π)))
3	2	eqeq1d 2765	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑤 mod (2 · π)) = 0 ↔ (𝑦 mod (2 · π)) = 0))
4		oveq2 7418	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑘 + (1 / 2)) · 𝑤) = ((𝑘 + (1 / 2)) · 𝑦))
5	4	fveq2d 6885	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (sin‘((𝑘 + (1 / 2)) · 𝑤)) = (sin‘((𝑘 + (1 / 2)) · 𝑦)))
6		fvoveq1 7433	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (sin‘(𝑤 / 2)) = (sin‘(𝑦 / 2)))
7	6	oveq2d 7426	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((2 · π) · (sin‘(𝑤 / 2))) = ((2 · π) · (sin‘(𝑦 / 2))))
8	5, 7	oveq12d 7428	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((sin‘((𝑘 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2)))) = ((sin‘((𝑘 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))
9	3, 8	ifbieq2d 4514	. . . . 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))))))
10	9	cbvmptv 5215	. . . 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))))))
11		oveq2 7418	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (2 · 𝑘) = (2 · 𝑚))
12	11	oveq1d 7425	. . . . . . 7 ⊢ (𝑘 = 𝑚 → ((2 · 𝑘) + 1) = ((2 · 𝑚) + 1))
13	12	oveq1d 7425	. . . . . 6 ⊢ (𝑘 = 𝑚 → (((2 · 𝑘) + 1) / (2 · π)) = (((2 · 𝑚) + 1) / (2 · π)))
14		oveq1 7417	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (𝑘 + (1 / 2)) = (𝑚 + (1 / 2)))
15	14	fvoveq1d 7432	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (sin‘((𝑘 + (1 / 2)) · 𝑦)) = (sin‘((𝑚 + (1 / 2)) · 𝑦)))
16	15	oveq1d 7425	. . . . . 6 ⊢ (𝑘 = 𝑚 → ((sin‘((𝑘 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))) = ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))
17	13, 16	ifeq12d 4509	. . . . 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))))))
18	17	mpteq2dv 5205	. . . 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)))))))
19	10, 18	eqtrid 2810	. . 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)))))))
20	19	cbvmptv 5215	. 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)))))))
21		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)))})
22		fourierdlem113.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
23		fourierdlem113.q	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
24		oveq1 7417	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤 + (𝑗 · 𝑇)) = (𝑦 + (𝑗 · 𝑇)))
25	24	eleq1d 2848	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
26	25	rexbidv 3189	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
27	26	cbvrabv 3426	. . . . 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}
28	27	uneq2i 4119	. . . 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})
29	28	fveq2i 6884	. . 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}))
30	29	oveq1i 7420	. 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)
31		oveq1 7417	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝑘 · 𝑇) = (𝑗 · 𝑇))
32	31	oveq2d 7426	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑗 · 𝑇)))
33	32	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
34	33	cbvrexvw 3244	. . . . . . . 8 ⊢ (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄)
35	34	rabbii 3421	. . . . . . 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}
36	35	uneq2i 4119	. . . . . 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})
37		isoeq5 7319	. . . . . 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}))))
38	36, 37	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})))
39	38	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}))))
40	31	oveq2d 7426	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑤 + (𝑘 · 𝑇)) = (𝑤 + (𝑗 · 𝑇)))
41	40	eleq1d 2848	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄))
42	41	cbvrexvw 3244	. . . . . . . . . . 11 ⊢ (∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑗 ∈ ℤ (𝑤 + (𝑗 · 𝑇)) ∈ ran 𝑄)
43	42	rabbii 3421	. . . . . . . . . 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}
44	43	uneq2i 4119	. . . . . . . . 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})
45	44	fveq2i 6884	. . . . . . . 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}))
46	45	oveq1i 7420	. . . . . . 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)
47	46	oveq2i 7421	. . . . . 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))
48		isoeq4 7318	. . . . . 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}))))
49	47, 48	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})))
50	49	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}))))
51		isoeq1 7315	. . . 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}))))
52	39, 50, 51	3bitrd 308	. . 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}))))
53	52	cbviotavw 6500	. 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})))
54		fourierdlem113.x	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ)
55		pire 26619	. . . . 5 ⊢ π ∈ ℝ
56	55	renegcli 11514	. . . 4 ⊢ -π ∈ ℝ
57	56	a1i 11	. . 3 ⊢ (𝜑 → -π ∈ ℝ)
58	55	a1i 11	. . 3 ⊢ (𝜑 → π ∈ ℝ)
59		negpilt0 46000	. . . 4 ⊢ -π < 0
60	59	a1i 11	. . 3 ⊢ (𝜑 → -π < 0)
61		pipos 26623	. . . 4 ⊢ 0 < π
62	61	a1i 11	. . 3 ⊢ (𝜑 → 0 < π)
63		picn 26621	. . . . 5 ⊢ π ∈ ℂ
64	63	2timesi 12373	. . . 4 ⊢ (2 · π) = (π + π)
65		fourierdlem113.t	. . . 4 ⊢ 𝑇 = (2 · π)
66	63, 63	subnegi 11532	. . . 4 ⊢ (π − -π) = (π + π)
67	64, 65, 66	3eqtr4i 2796	. . 3 ⊢ 𝑇 = (π − -π)
68	21	fourierdlem2 46823	. . . . . . . 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))))))
69	22, 68	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
70	23, 69	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
71	70	simpld 499	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
72		elmapi 8842	. . . . 5 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
73	71, 72	syl 18	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
74		fzfid 14005	. . . 4 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
75		rnffi 45893	. . . 4 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
76	73, 74, 75	syl2anc 595	. . 3 ⊢ (𝜑 → ran 𝑄 ∈ Fin)
77	21, 22, 23	fourierdlem15 46836	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
78	77	frnd 6714	. . 3 ⊢ (𝜑 → ran 𝑄 ⊆ (-π[,]π))
79	70	simprd 500	. . . . 5 ⊢ (𝜑 → (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
80	79	simplrd 781	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) = π)
81	77	ffund 6710	. . . . 5 ⊢ (𝜑 → Fun 𝑄)
82	22	nnnn0d 12560	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
83		nn0uz 12895	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
84	82, 83	eleqtrdi 2873	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
85		eluzfz2 13555	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
86	84, 85	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
87	77	fdmd 6716	. . . . . . 7 ⊢ (𝜑 → dom 𝑄 = (0...𝑀))
88	87	eqcomd 2769	. . . . . 6 ⊢ (𝜑 → (0...𝑀) = dom 𝑄)
89	86, 88	eleqtrd 2865	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ dom 𝑄)
90		fvelrn 7071	. . . . 5 ⊢ ((Fun 𝑄 ∧ 𝑀 ∈ dom 𝑄) → (𝑄‘𝑀) ∈ ran 𝑄)
91	81, 89, 90	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ran 𝑄)
92	80, 91	eqeltrrd 2864	. . 3 ⊢ (𝜑 → π ∈ ran 𝑄)
93		fourierdlem113.e	. . 3 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇)))
94		fourierdlem113.exq	. . 3 ⊢ (𝜑 → (𝐸‘𝑋) ∈ ran 𝑄)
95		eqid 2763	. . 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})
96		isoeq1 7315	. . . . 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}))))
97	28, 44, 36	3eqtr4ri 2797	. . . . . 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})
98		isoeq5 7319	. . . . . 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}))))
99	97, 98	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})))
100	96, 99	bitrdi 290	. . . 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}))))
101	100	cbviotavw 6500	. . 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})))
102		eqid 2763	. . 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}
103	57, 58, 60, 62, 67, 76, 78, 92, 93, 54, 94, 95, 101, 102	fourierdlem51 46871	. 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}))))
104		fourierdlem113.per	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
105		ax-resscn 11152	. . . 4 ⊢ ℝ ⊆ ℂ
106	105	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ℝ ⊆ ℂ)
107		ioossre 13429	. . . . . . 7 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
108	107	a1i 11	. . . . . 6 ⊢ (𝜑 → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
109	1, 108	fssresd 6745	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℝ)
110	105	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
111	109, 110	fssd 6723	. . . 4 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
112	111	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
113	107	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
114	1, 110	fssd 6723	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
115	114	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℂ)
116		ssidd 3960	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ℝ ⊆ ℝ)
117		eqid 2763	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
118		tgioo4 24962	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
119	117, 118	dvres 26070	. . . . . 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))))))
120	106, 115, 116, 113, 119	syl22anc 851	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))))
121	120	dmeqd 5895	. . . 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))))))
122		ioontr 46227	. . . . . . 7 ⊢ ((int‘(topGen‘ran (,)))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))
123	122	reseq2i 5975	. . . . . 6 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
124	123	dmeqi 5894	. . . . 5 ⊢ dom ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = dom ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
125	124	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)))))
126		fourierdlem113.dvcn	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
127		cncff 25052	. . . . 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)
128		fdm 6715	. . . . 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))))
129	126, 127, 128	3syl 19	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
130	121, 125, 129	3eqtrd 2802	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom (ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
131		dvcn 26080	. . 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))
132	106, 112, 113, 130, 131	syl31anc 1400	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
133	113, 106	sstrd 3947	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
134	73	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
135		fzofzp1 13789	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
136	135	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
137	134, 136	ffvelcdmd 7080	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
138	137	rexrd 11254	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
139		elfzofz 13700	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
140	139	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
141	134, 140	ffvelcdmd 7080	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
142	70	simprrd 785	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
143	142	r19.21bi 3257	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
144	117, 138, 141, 143	lptioo1cn 46360	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
145	109	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℝ)
146		ssidd 3960	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ)
147	110, 114, 146	dvbss 26060	. . . . . . 7 ⊢ (𝜑 → dom (ℝ D 𝐹) ⊆ ℝ)
148		dvfre 26110	. . . . . . . 8 ⊢ ((𝐹:ℝ⟶ℝ ∧ ℝ ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
149	1, 146, 148	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
150		0re 11205	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
151	56, 150, 55	lttri 11331	. . . . . . . . 9 ⊢ ((-π < 0 ∧ 0 < π) → -π < π)
152	59, 61, 151	mp2an 704	. . . . . . . 8 ⊢ -π < π
153	152	a1i 11	. . . . . . 7 ⊢ (𝜑 → -π < π)
154	79	simplld 779	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) = -π)
155	126, 127	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
156		fourierdlem113.dvlb	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅)
157	155, 133, 144, 156, 117	ellimciota 46330	. . . . . . 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)))
158	141	rexrd 11254	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
159	117, 158, 137, 143	lptioo2cn 46359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
160		fourierdlem113.dvub	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅)
161	155, 133, 159, 160, 117	ellimciota 46330	. . . . . . 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))))
162	114	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → 𝐹:ℝ⟶ℂ)
163		zre 12590	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
164	163	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℝ)
165		2pire 26620	. . . . . . . . . . . . . 14 ⊢ (2 · π) ∈ ℝ
166	165	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · π) ∈ ℝ)
167	65, 166	eqeltrid 2867	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℝ)
168	167	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → 𝑇 ∈ ℝ)
169	164, 168	remulcld 11234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝑘 · 𝑇) ∈ ℝ)
170	162	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝐹:ℝ⟶ℂ)
171	168	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑇 ∈ ℝ)
172		simplr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑘 ∈ ℤ)
173		simpr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
174	104	ad4ant14 764	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
175	170, 171, 172, 173, 174	fperiodmul 46023	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → (𝐹‘(𝑡 + (𝑘 · 𝑇))) = (𝐹‘𝑡))
176		eqid 2763	. . . . . . . . . 10 ⊢ (ℝ D 𝐹) = (ℝ D 𝐹)
177	162, 169, 175, 176	fperdvper 46633	. . . . . . . . 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)))
178	177	an32s 664	. . . . . . . 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)))
179	178	simpld 499	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ dom (ℝ D 𝐹)) ∧ 𝑘 ∈ ℤ) → (𝑡 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹))
180	178	simprd 500	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ dom (ℝ D 𝐹)) ∧ 𝑘 ∈ ℤ) → ((ℝ D 𝐹)‘(𝑡 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑡))
181		fveq2 6881	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑄‘𝑗) = (𝑄‘𝑖))
182		fvoveq1 7433	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑄‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)))
183	181, 182	oveq12d 7428	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
184	183	cbvmptv 5215	. . . . . . 7 ⊢ (𝑗 ∈ (0..^𝑀) ↦ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1)))) = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
185		eqid 2763	. . . . . . 7 ⊢ (𝑡 ∈ ℝ ↦ (𝑡 + ((⌊‘((π − 𝑡) / 𝑇)) · 𝑇))) = (𝑡 ∈ ℝ ↦ (𝑡 + ((⌊‘((π − 𝑡) / 𝑇)) · 𝑇)))
186	147, 149, 57, 58, 153, 67, 22, 73, 154, 80, 126, 157, 161, 179, 180, 184, 185	fourierdlem71 46891	. . . . . 6 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
187	186	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
188		nfv 1944	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑖 ∈ (0..^𝑀))
189		nfra1 3289	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧
190	188, 189	nfan 1929	. . . . . . . 8 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
191	120, 123	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
192	191	fveq1d 6883	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡) = (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡))
193		fvres 6900	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
194	192, 193	sylan9eq 2818	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
195	194	fveq2d 6885	. . . . . . . . . 10 ⊢ (((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)))
196	195	adantlr 727	. . . . . . . . 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)) = (abs‘((ℝ D 𝐹)‘𝑡)))
197		simplr 780	. . . . . . . . . 10 ⊢ ((((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)
198		ssdmres 6012	. . . . . . . . . . . . 13 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
199	129, 198	sylibr 237	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom (ℝ D 𝐹))
200	199	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((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))
201		simpr 489	. . . . . . . . . . 11 ⊢ ((((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))))
202	200, 201	sseldd 3938	. . . . . . . . . 10 ⊢ ((((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))
203		rspa 3254	. . . . . . . . . 10 ⊢ ((∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧 ∧ 𝑡 ∈ dom (ℝ D 𝐹)) → (abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)
204	197, 202, 203	syl2anc 595	. . . . . . . . 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)‘𝑡)) ≤ 𝑧)
205	196, 204	eqbrtrd 5133	. . . . . . . 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)
206	190, 205	ralrimia 3264	. . . . . . 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)
207	206	ex 417	. . . . . 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))
208	207	reximdv 3180	. . . . 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))
209	187, 208	mpd 16	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((ℝ D (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))‘𝑡)) ≤ 𝑧)
210	141, 137, 145, 130, 209	ioodvbdlimc1 46647	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅)
211	112, 133, 144, 210, 117	ellimciota 46330	. 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)))
212	141, 137, 145, 130, 209	ioodvbdlimc2 46649	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅)
213	112, 133, 159, 212, 117	ellimciota 46330	. 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))))
214		resindm 6029	. . . . . 6 ⊢ ((ℝ D 𝐹) ↾ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))) = ((ℝ D 𝐹) ↾ (-∞(,)𝑋))
215	214	a1i 11	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))) = ((ℝ D 𝐹) ↾ (-∞(,)𝑋)))
216		inss2 4190	. . . . . . 7 ⊢ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ dom (ℝ D 𝐹)
217	216	a1i 11	. . . . . 6 ⊢ (𝜑 → ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ dom (ℝ D 𝐹))
218	149, 217	fssresd 6745	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))):((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))⟶ℝ)
219	215, 218	feq1dd 6688	. . . 4 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (-∞(,)𝑋)):((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))⟶ℝ)
220	219, 110	fssd 6723	. . 3 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (-∞(,)𝑋)):((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))⟶ℂ)
221		ioosscn 13430	. . . . 5 ⊢ (-∞(,)𝑋) ⊆ ℂ
222		ssinss1 4198	. . . . 5 ⊢ ((-∞(,)𝑋) ⊆ ℂ → ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
223	221, 222	ax-mp 5	. . . 4 ⊢ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ
224	223	a1i 11	. . 3 ⊢ (𝜑 → ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
225		3simpb 1167	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → (𝜑 ∧ 𝑘 ∈ ℤ))
226		simp2 1155	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑥 ∈ dom (ℝ D 𝐹))
227	162	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝐹:ℝ⟶ℂ)
228	168	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
229		simplr 780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝑘 ∈ ℤ)
230		simpr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
231		eleq1w 2846	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ℝ ↔ 𝑦 ∈ ℝ))
232	231	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ ℝ) ↔ (𝜑 ∧ 𝑦 ∈ ℝ)))
233		fvoveq1 7433	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑦 + 𝑇)))
234		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
235	233, 234	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦)))
236	232, 235	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))))
237	236, 104	chvarvv 2019	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))
238	237	ad4ant14 764	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))
239	227, 228, 229, 230, 238	fperiodmul 46023	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥))
240	162, 169, 239, 176	fperdvper 46633	. . . . . . . 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)))
241	225, 226, 240	syl2anc 595	. . . . . . 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)))
242	241	simpld 499	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom (ℝ D 𝐹))
243		oveq2 7418	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (π − 𝑤) = (π − 𝑥))
244	243	fvoveq1d 7432	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (⌊‘((π − 𝑤) / 𝑇)) = (⌊‘((π − 𝑥) / 𝑇)))
245	244	oveq1d 7425	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇) = ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
246	245	cbvmptv 5215	. . . . . 6 ⊢ (𝑤 ∈ ℝ ↦ ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇)) = (𝑥 ∈ ℝ ↦ ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
247		eqid 2763	. . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ (𝑥 + ((𝑤 ∈ ℝ ↦ ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇))‘𝑥))) = (𝑥 ∈ ℝ ↦ (𝑥 + ((𝑤 ∈ ℝ ↦ ((⌊‘((π − 𝑤) / 𝑇)) · 𝑇))‘𝑥)))
248	57, 58, 153, 67, 242, 54, 246, 247, 21, 22, 23, 199	fourierdlem41 46862	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) ∧ ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))))
249	248	simpld 499	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)))
250	117	cnfldtop 24940	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Top
251		mnfxr 11261	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
252		rexr 11250	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
253	252	mnfled 13156	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → -∞ ≤ 𝑦)
254		iooss1 13402	. . . . . . . . . . 11 ⊢ ((-∞ ∈ ℝ^* ∧ -∞ ≤ 𝑦) → (𝑦(,)𝑋) ⊆ (-∞(,)𝑋))
255	251, 253, 254	sylancr 598	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (𝑦(,)𝑋) ⊆ (-∞(,)𝑋))
256	255	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (𝑦(,)𝑋) ⊆ (-∞(,)𝑋))
257		simp3 1156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))
258	256, 257	ssind 4193	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → (𝑦(,)𝑋) ⊆ ((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)))
259		unicntop 24942	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
260	259	lpss3 23301	. . . . . . . 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))))
261	250, 223, 258, 260	mp3an12i 1494	. . . . . . 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))))
262	261	3adant3l 1199	. . . . . 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))))
263	252	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑦 ∈ ℝ^*)
264	54	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ℝ)
265		simp3l 1220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑦 < 𝑋)
266	117, 263, 264, 265	lptioo2cn 46359	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝑦(,)𝑋)))
267	262, 266	sseldd 3938	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
268	267	rexlimdv3a 3170	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ dom (ℝ D 𝐹)) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹)))))
269	249, 268	mpd 16	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((-∞(,)𝑋) ∩ dom (ℝ D 𝐹))))
270	241	simprd 500	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (ℝ D 𝐹) ∧ 𝑘 ∈ ℤ) → ((ℝ D 𝐹)‘(𝑥 + (𝑘 · 𝑇))) = ((ℝ D 𝐹)‘𝑥))
271		oveq2 7418	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (π − 𝑦) = (π − 𝑥))
272	271	fvoveq1d 7432	. . . . . 6 ⊢ (𝑦 = 𝑥 → (⌊‘((π − 𝑦) / 𝑇)) = (⌊‘((π − 𝑥) / 𝑇)))
273	272	oveq1d 7425	. . . . 5 ⊢ (𝑦 = 𝑥 → ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇) = ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
274	273	cbvmptv 5215	. . . 4 ⊢ (𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇)) = (𝑥 ∈ ℝ ↦ ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))
275		id 23	. . . . . 6 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
276		fveq2 6881	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑧) = ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑥))
277	275, 276	oveq12d 7428	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑧)) = (𝑥 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑥)))
278	277	cbvmptv 5215	. . . 4 ⊢ (𝑧 ∈ ℝ ↦ (𝑧 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑧))) = (𝑥 ∈ ℝ ↦ (𝑥 + ((𝑦 ∈ ℝ ↦ ((⌊‘((π − 𝑦) / 𝑇)) · 𝑇))‘𝑥)))
279	57, 58, 153, 21, 67, 22, 23, 147, 149, 242, 270, 126, 161, 54, 274, 278	fourierdlem49 46869	. . 3 ⊢ (𝜑 → (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅)
280	220, 224, 269, 279, 117	ellimciota 46330	. 2 ⊢ (𝜑 → (℩𝑥𝑥 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
281		resindm 6029	. . . . . 6 ⊢ ((ℝ D 𝐹) ↾ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))) = ((ℝ D 𝐹) ↾ (𝑋(,)+∞))
282	281	a1i 11	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))) = ((ℝ D 𝐹) ↾ (𝑋(,)+∞)))
283		inss2 4190	. . . . . . 7 ⊢ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ dom (ℝ D 𝐹)
284	283	a1i 11	. . . . . 6 ⊢ (𝜑 → ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ dom (ℝ D 𝐹))
285	149, 284	fssresd 6745	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))):((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))⟶ℝ)
286	282, 285	feq1dd 6688	. . . 4 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝑋(,)+∞)):((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))⟶ℝ)
287	286, 110	fssd 6723	. . 3 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝑋(,)+∞)):((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))⟶ℂ)
288		ioosscn 13430	. . . . 5 ⊢ (𝑋(,)+∞) ⊆ ℂ
289		ssinss1 4198	. . . . 5 ⊢ ((𝑋(,)+∞) ⊆ ℂ → ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
290	288, 289	ax-mp 5	. . . 4 ⊢ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ
291	290	a1i 11	. . 3 ⊢ (𝜑 → ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)) ⊆ ℂ)
292	248	simprd 500	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)))
293		pnfxr 11258	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
294	252	pnfged 13151	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → 𝑦 ≤ +∞)
295		iooss2 13403	. . . . . . . . . . 11 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑦 ≤ +∞) → (𝑋(,)𝑦) ⊆ (𝑋(,)+∞))
296	293, 294, 295	sylancr 598	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (𝑋(,)𝑦) ⊆ (𝑋(,)+∞))
297	296	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (𝑋(,)𝑦) ⊆ (𝑋(,)+∞))
298		simp3 1156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))
299	297, 298	ssind 4193	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → (𝑋(,)𝑦) ⊆ ((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)))
300	259	lpss3 23301	. . . . . . . 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))))
301	250, 290, 299, 300	mp3an12i 1494	. . . . . . 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))))
302	301	3adant3l 1199	. . . . . 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))))
303	252	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑦 ∈ ℝ^*)
304	54	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ℝ)
305		simp3l 1220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 < 𝑦)
306	117, 303, 304, 305	lptioo1cn 46360	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝑋(,)𝑦)))
307	302, 306	sseldd 3938	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹))) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
308	307	rexlimdv3a 3170	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ dom (ℝ D 𝐹)) → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹)))))
309	292, 308	mpd 16	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘((𝑋(,)+∞) ∩ dom (ℝ D 𝐹))))
310		biid 264	. . . 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))))
311	57, 58, 153, 21, 67, 22, 23, 149, 242, 270, 126, 157, 54, 274, 278, 310	fourierdlem48 46868	. . 3 ⊢ (𝜑 → (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅)
312	287, 291, 309, 311, 117	ellimciota 46330	. 2 ⊢ (𝜑 → (℩𝑥𝑥 ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
313		fourierdlem113.l	. 2 ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
314		fourierdlem113.r	. 2 ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
315		fourierdlem113.a	. 2 ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
316		fourierdlem113.b	. 2 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
317		fveq2 6881	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐴‘𝑛) = (𝐴‘𝑘))
318		fvoveq1 7433	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (cos‘(𝑛 · 𝑋)) = (cos‘(𝑘 · 𝑋)))
319	317, 318	oveq12d 7428	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) = ((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))))
320		fveq2 6881	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐵‘𝑛) = (𝐵‘𝑘))
321		fvoveq1 7433	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (sin‘(𝑛 · 𝑋)) = (sin‘(𝑘 · 𝑋)))
322	320, 321	oveq12d 7428	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))) = ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))
323	319, 322	oveq12d 7428	. . . . . 6 ⊢ (𝑛 = 𝑘 → (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))) = (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋)))))
324	323	cbvsumv 15743	. . . . 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))))
325		oveq2 7418	. . . . . . 7 ⊢ (𝑗 = 𝑚 → (1...𝑗) = (1...𝑚))
326	325	eqcomd 2769	. . . . . 6 ⊢ (𝑗 = 𝑚 → (1...𝑚) = (1...𝑗))
327	326	sumeq1d 15747	. . . . 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)))))
328	324, 327	eqtr2id 2811	. . . 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)))))
329	328	oveq2d 7426	. . 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))))))
330	329	cbvmptv 5215	. 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))))))
331		fourierdlem113.15	. 2 ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))
332	1	fdmd 6716	. . . . 5 ⊢ (𝜑 → dom 𝐹 = ℝ)
333	332	eqimssd 3993	. . . 4 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
334	1	ffdmd 6736	. . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
335	333	sselda 3937	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ dom 𝐹) → 𝑡 ∈ ℝ)
336	335	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑡 ∈ ℝ)
337	163	adantl 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℝ)
338	168	adantlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → 𝑇 ∈ ℝ)
339	337, 338	remulcld 11234	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑘 · 𝑇) ∈ ℝ)
340	336, 339	readdcld 11233	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑡 + (𝑘 · 𝑇)) ∈ ℝ)
341	332	eqcomd 2769	. . . . . 6 ⊢ (𝜑 → ℝ = dom 𝐹)
342	341	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → ℝ = dom 𝐹)
343	340, 342	eleqtrd 2865	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑡 + (𝑘 · 𝑇)) ∈ dom 𝐹)
344		id 23	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝜑 ∧ 𝑘 ∈ ℤ))
345	344	adantlr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝜑 ∧ 𝑘 ∈ ℤ))
346	345, 336, 175	syl2anc 595	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑡 + (𝑘 · 𝑇))) = (𝐹‘𝑡))
347	333, 334, 57, 58, 153, 67, 22, 73, 154, 80, 132, 211, 213, 343, 346, 184, 185	fourierdlem71 46891	. . 3 ⊢ (𝜑 → ∃𝑢 ∈ ℝ ∀𝑡 ∈ dom 𝐹(abs‘(𝐹‘𝑡)) ≤ 𝑢)
348	332	raleqdv 3323	. . . 4 ⊢ (𝜑 → (∀𝑡 ∈ dom 𝐹(abs‘(𝐹‘𝑡)) ≤ 𝑢 ↔ ∀𝑡 ∈ ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑢))
349	348	rexbidv 3189	. . 3 ⊢ (𝜑 → (∃𝑢 ∈ ℝ ∀𝑡 ∈ dom 𝐹(abs‘(𝐹‘𝑡)) ≤ 𝑢 ↔ ∃𝑢 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑢))
350	347, 349	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑢 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑢)
351	1, 20, 21, 22, 23, 30, 53, 54, 103, 65, 104, 132, 211, 213, 126, 280, 312, 313, 314, 315, 316, 330, 331, 350, 186, 54	fourierdlem112 46932	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 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 {crab 3416 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 ifcif 4487 {cpr 4591 class class class wbr 5109 ↦ cmpt 5192 dom cdm 5661 ran crn 5662 ↾ cres 5663 ℩cio 6490 Fun wfun 6530 ⟶wf 6532 ‘cfv 6536 Isom wiso 6537 (class class class)co 7410 ↑_m cmap 8820 Fincfn 8939 ℂcc 11093 ℝcr 11094 0cc0 11095 1c1 11096 + caddc 11098 · cmul 11100 +∞cpnf 11235 -∞cmnf 11236 ℝ^*cxr 11237 < clt 11238 ≤ cle 11239 − cmin 11436 -cneg 11437 / cdiv 11866 ℕcn 12228 2c2 12290 ℕ₀cn0 12499 ℤcz 12586 ℤ_≥cuz 12857 (,)cioo 13367 (,]cioc 13368 [,)cico 13369 [,]cicc 13370 ...cfz 13530 ..^cfzo 13678 ⌊cfl 13819 mod cmo 13898 seqcseq 14033 ♯chash 14362 abscabs 15281 ⇝ cli 15531 Σcsu 15733 sincsin 16112 cosccos 16113 πcpi 16115 TopOpenctopn 17469 topGenctg 17485 ℂ_fldccnfld 21522 Topctop 23050 intcnt 23174 limPtclp 23291 –cn→ccncf 25035 ∫citg 25777 lim_ℂ climc 26021 D cdv 26022

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cc 10414 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-symdif 4206 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-disj 5077 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-ofr 7675 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-omul 8454 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-dju 9883 df-card 9921 df-acn 9924 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-shft 15100 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-limsup 15518 df-clim 15535 df-rlim 15536 df-sum 15734 df-ef 16116 df-sin 16118 df-cos 16119 df-pi 16121 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-lp 23293 df-perf 23294 df-cn 23384 df-cnp 23385 df-t1 23471 df-haus 23472 df-cmp 23544 df-tx 23719 df-hmeo 23912 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037 df-ovol 25623 df-vol 25624 df-mbf 25778 df-itg1 25779 df-itg2 25780 df-ibl 25781 df-itg 25782 df-0p 25829 df-ditg 26006 df-limc 26025 df-dv 26026

This theorem is referenced by: fourierdlem114 46934

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