fourierdlem111 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem111 43648

Description: The fourier partial sum for 𝐹 is the sum of two integrals, with the same integrand involving 𝐹 and the Dirichlet Kernel 𝐷, but on two opposite intervals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem111.a	⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑡) · (cos‘(𝑛 · 𝑡))) d𝑡 / π))
fourierdlem111.b	⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑡) · (sin‘(𝑛 · 𝑡))) d𝑡 / π))
fourierdlem111.s	⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))))
fourierdlem111.d	⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))
fourierdlem111.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem111.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem111.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem111.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem111.6	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem111.fper	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierdlem111.g	⊢ 𝐺 = (𝑥 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥)))
fourierdlem111.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem111.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
fourierdlem111.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
fourierdlem111.t	⊢ 𝑇 = (2 · π)
fourierdlem111.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π − 𝑋) ∧ (𝑝‘𝑚) = (π − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem111.14	⊢ 𝑊 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) − 𝑋))

**Assertion**
Ref	Expression
fourierdlem111	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠))

Distinct variable groups: 𝐴,𝑚,𝑛 𝐵,𝑚 𝐷,𝑖,𝑚,𝑦 𝐷,𝑠,𝑡,𝑖,𝑦 𝑥,𝐷,𝑖,𝑠,𝑦 𝑖,𝐹,𝑛,𝑠,𝑡,𝑦 𝑥,𝐹,𝑛 𝑖,𝐺,𝑠,𝑥 𝐿,𝑠,𝑡 𝑥,𝐿 𝑖,𝑀,𝑚,𝑝 𝑀,𝑠,𝑡,𝑦 𝑥,𝑀 𝑄,𝑖,𝑝 𝑄,𝑠,𝑡,𝑦 𝑥,𝑄 𝑅,𝑠,𝑡 𝑥,𝑅 𝑇,𝑠,𝑥 𝑖,𝑊,𝑚,𝑝 𝑊,𝑠,𝑥,𝑦 𝑖,𝑋,𝑚,𝑛,𝑦 𝑋,𝑝 𝑋,𝑠,𝑡 𝑥,𝑋 𝜑,𝑖,𝑚,𝑛,𝑦 𝜑,𝑠,𝑡 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑝) 𝐴(𝑥,𝑦,𝑡,𝑖,𝑠,𝑝) 𝐵(𝑥,𝑦,𝑡,𝑖,𝑛,𝑠,𝑝) 𝐷(𝑛,𝑝) 𝑃(𝑥,𝑦,𝑡,𝑖,𝑚,𝑛,𝑠,𝑝) 𝑄(𝑚,𝑛) 𝑅(𝑦,𝑖,𝑚,𝑛,𝑝) 𝑆(𝑥,𝑦,𝑡,𝑖,𝑚,𝑛,𝑠,𝑝) 𝑇(𝑦,𝑡,𝑖,𝑚,𝑛,𝑝) 𝐹(𝑚,𝑝) 𝐺(𝑦,𝑡,𝑚,𝑛,𝑝) 𝐿(𝑦,𝑖,𝑚,𝑛,𝑝) 𝑀(𝑛) 𝑂(𝑥,𝑦,𝑡,𝑖,𝑚,𝑛,𝑠,𝑝) 𝑊(𝑡,𝑛)

Proof of Theorem fourierdlem111 Dummy variables 𝑘 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2826	. . . . . 6 ⊢ (𝑘 = 𝑛 → (𝑘 ∈ ℕ ↔ 𝑛 ∈ ℕ))
2	1	anbi2d 628	. . . . 5 ⊢ (𝑘 = 𝑛 → ((𝜑 ∧ 𝑘 ∈ ℕ) ↔ (𝜑 ∧ 𝑛 ∈ ℕ)))
3		fveq2 6756	. . . . . 6 ⊢ (𝑘 = 𝑛 → (𝑆‘𝑘) = (𝑆‘𝑛))
4		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝐷‘𝑘) = (𝐷‘𝑛))
5	4	fveq1d 6758	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((𝐷‘𝑘)‘(𝑡 − 𝑋)) = ((𝐷‘𝑛)‘(𝑡 − 𝑋)))
6	5	oveq2d 7271	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑡) · ((𝐷‘𝑘)‘(𝑡 − 𝑋))) = ((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))))
7	6	adantr 480	. . . . . . 7 ⊢ ((𝑘 = 𝑛 ∧ 𝑡 ∈ (-π(,)π)) → ((𝐹‘𝑡) · ((𝐷‘𝑘)‘(𝑡 − 𝑋))) = ((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))))
8	7	itgeq2dv 24851	. . . . . 6 ⊢ (𝑘 = 𝑛 → ∫(-π(,)π)((𝐹‘𝑡) · ((𝐷‘𝑘)‘(𝑡 − 𝑋))) d𝑡 = ∫(-π(,)π)((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) d𝑡)
9	3, 8	eqeq12d 2754	. . . . 5 ⊢ (𝑘 = 𝑛 → ((𝑆‘𝑘) = ∫(-π(,)π)((𝐹‘𝑡) · ((𝐷‘𝑘)‘(𝑡 − 𝑋))) d𝑡 ↔ (𝑆‘𝑛) = ∫(-π(,)π)((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) d𝑡))
10	2, 9	imbi12d 344	. . . 4 ⊢ (𝑘 = 𝑛 → (((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) = ∫(-π(,)π)((𝐹‘𝑡) · ((𝐷‘𝑘)‘(𝑡 − 𝑋))) d𝑡) ↔ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = ∫(-π(,)π)((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) d𝑡)))
11		fourierdlem111.6	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
13		eqid 2738	. . . . 5 ⊢ (-π(,)π) = (-π(,)π)
14		ioossre 13069	. . . . . . . . 9 ⊢ (-π(,)π) ⊆ ℝ
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (-π(,)π) ⊆ ℝ)
16	11, 15	feqresmpt 6820	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ (-π(,)π)) = (𝑥 ∈ (-π(,)π) ↦ (𝐹‘𝑥)))
17		ioossicc 13094	. . . . . . . . 9 ⊢ (-π(,)π) ⊆ (-π[,]π)
18	17	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (-π(,)π) ⊆ (-π[,]π))
19		ioombl 24634	. . . . . . . . 9 ⊢ (-π(,)π) ∈ dom vol
20	19	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (-π(,)π) ∈ dom vol)
21	11	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℝ)
22		pire 25520	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℝ
23	22	renegcli 11212	. . . . . . . . . . . . . 14 ⊢ -π ∈ ℝ
24	23, 22	elicc2i 13074	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (-π[,]π) ↔ (𝑡 ∈ ℝ ∧ -π ≤ 𝑡 ∧ 𝑡 ≤ π))
25	24	simp1bi 1143	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (-π[,]π) → 𝑡 ∈ ℝ)
26	25	ssriv 3921	. . . . . . . . . . 11 ⊢ (-π[,]π) ⊆ ℝ
27	26	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (-π[,]π) ⊆ ℝ)
28	27	sselda 3917	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π)) → 𝑥 ∈ ℝ)
29	21, 28	ffvelrnd 6944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (-π[,]π)) → (𝐹‘𝑥) ∈ ℝ)
30	11, 27	feqresmpt 6820	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ (-π[,]π)) = (𝑥 ∈ (-π[,]π) ↦ (𝐹‘𝑥)))
31		fourierdlem111.p	. . . . . . . . . 10 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
32		fourierdlem111.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
33		fourierdlem111.q	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
34		ax-resscn 10859	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
35	34	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ⊆ ℂ)
36	11, 35	fssd 6602	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
37	36, 27	fssresd 6625	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ↾ (-π[,]π)):(-π[,]π)⟶ℂ)
38		ioossicc 13094	. . . . . . . . . . . . 13 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
39	23	rexri 10964	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℝ^*
40	39	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ^*)
41	22	rexri 10964	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℝ^*
42	41	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈ ℝ^*)
43	31, 32, 33	fourierdlem15 43553	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
44	43	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
45		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
46	40, 42, 44, 45	fourierdlem8 43546	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
47	38, 46	sstrid 3928	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
48	47	resabs1d 5911	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
49		fourierdlem111.fcn	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
50	48, 49	eqeltrd 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
51		fourierdlem111.r	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
52	48	oveq1d 7270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
53	51, 52	eleqtrrd 2842	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
54		fourierdlem111.l	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
55	48	oveq1d 7270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
56	54, 55	eleqtrrd 2842	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
57	31, 32, 33, 37, 50, 53, 56	fourierdlem69 43606	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ (-π[,]π)) ∈ 𝐿¹)
58	30, 57	eqeltrrd 2840	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (-π[,]π) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
59	18, 20, 29, 58	iblss 24874	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (-π(,)π) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
60	16, 59	eqeltrd 2839	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (-π(,)π)) ∈ 𝐿¹)
61	60	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹 ↾ (-π(,)π)) ∈ 𝐿¹)
62		fourierdlem111.a	. . . . 5 ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑡) · (cos‘(𝑛 · 𝑡))) d𝑡 / π))
63		fourierdlem111.b	. . . . 5 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑡) · (sin‘(𝑛 · 𝑡))) d𝑡 / π))
64		fourierdlem111.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ)
65	64	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑋 ∈ ℝ)
66		fourierdlem111.s	. . . . 5 ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))))
67		fourierdlem111.d	. . . . 5 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))
68		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
69	12, 13, 61, 62, 63, 65, 66, 67, 68	fourierdlem83 43620	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) = ∫(-π(,)π)((𝐹‘𝑡) · ((𝐷‘𝑘)‘(𝑡 − 𝑋))) d𝑡)
70	10, 69	chvarvv 2003	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = ∫(-π(,)π)((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) d𝑡)
71	23	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -π ∈ ℝ)
72	22	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈ ℝ)
73	36	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → 𝐹:ℝ⟶ℂ)
74	25	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → 𝑡 ∈ ℝ)
75	73, 74	ffvelrnd 6944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → (𝐹‘𝑡) ∈ ℂ)
76	75	adantlr 711	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ (-π[,]π)) → (𝐹‘𝑡) ∈ ℂ)
77	67	dirkerf 43528	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛):ℝ⟶ℝ)
78	77	ad2antlr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ (-π[,]π)) → (𝐷‘𝑛):ℝ⟶ℝ)
79	64	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → 𝑋 ∈ ℝ)
80	74, 79	resubcld 11333	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → (𝑡 − 𝑋) ∈ ℝ)
81	80	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ (-π[,]π)) → (𝑡 − 𝑋) ∈ ℝ)
82	78, 81	ffvelrnd 6944	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ (-π[,]π)) → ((𝐷‘𝑛)‘(𝑡 − 𝑋)) ∈ ℝ)
83	82	recnd 10934	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ (-π[,]π)) → ((𝐷‘𝑛)‘(𝑡 − 𝑋)) ∈ ℂ)
84	76, 83	mulcld 10926	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ (-π[,]π)) → ((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) ∈ ℂ)
85	71, 72, 84	itgioo 24885	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫(-π(,)π)((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) d𝑡 = ∫(-π[,]π)((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) d𝑡)
86		fvres 6775	. . . . . . . 8 ⊢ (𝑡 ∈ (-π[,]π) → ((𝐹 ↾ (-π[,]π))‘𝑡) = (𝐹‘𝑡))
87	86	eqcomd 2744	. . . . . . 7 ⊢ (𝑡 ∈ (-π[,]π) → (𝐹‘𝑡) = ((𝐹 ↾ (-π[,]π))‘𝑡))
88	87	oveq1d 7270	. . . . . 6 ⊢ (𝑡 ∈ (-π[,]π) → ((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) = (((𝐹 ↾ (-π[,]π))‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))))
89	88	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ (-π[,]π)) → ((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) = (((𝐹 ↾ (-π[,]π))‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))))
90	89	itgeq2dv 24851	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫(-π[,]π)((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) d𝑡 = ∫(-π[,]π)(((𝐹 ↾ (-π[,]π))‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) d𝑡)
91		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑛 = 𝑚 ∧ 𝑦 ∈ ℝ) → 𝑛 = 𝑚)
92	91	oveq2d 7271	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑚 ∧ 𝑦 ∈ ℝ) → (2 · 𝑛) = (2 · 𝑚))
93	92	oveq1d 7270	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑚 ∧ 𝑦 ∈ ℝ) → ((2 · 𝑛) + 1) = ((2 · 𝑚) + 1))
94	93	oveq1d 7270	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑚 ∧ 𝑦 ∈ ℝ) → (((2 · 𝑛) + 1) / (2 · π)) = (((2 · 𝑚) + 1) / (2 · π)))
95	91	oveq1d 7270	. . . . . . . . . . . . 13 ⊢ ((𝑛 = 𝑚 ∧ 𝑦 ∈ ℝ) → (𝑛 + (1 / 2)) = (𝑚 + (1 / 2)))
96	95	oveq1d 7270	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑚 ∧ 𝑦 ∈ ℝ) → ((𝑛 + (1 / 2)) · 𝑦) = ((𝑚 + (1 / 2)) · 𝑦))
97	96	fveq2d 6760	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑚 ∧ 𝑦 ∈ ℝ) → (sin‘((𝑛 + (1 / 2)) · 𝑦)) = (sin‘((𝑚 + (1 / 2)) · 𝑦)))
98	97	oveq1d 7270	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑚 ∧ 𝑦 ∈ ℝ) → ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))) = ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))
99	94, 98	ifeq12d 4477	. . . . . . . . 9 ⊢ ((𝑛 = 𝑚 ∧ 𝑦 ∈ ℝ) → if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (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))))))
100	99	mpteq2dva 5170	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (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)))))))
101	100	cbvmptv 5183	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 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)))))))
102	67, 101	eqtri 2766	. . . . . 6 ⊢ 𝐷 = (𝑚 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))
103		fveq2 6756	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((𝐹 ↾ (-π[,]π))‘𝑠) = ((𝐹 ↾ (-π[,]π))‘𝑡))
104		oveq1 7262	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (𝑠 − 𝑋) = (𝑡 − 𝑋))
105	104	fveq2d 6760	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((𝐷‘𝑛)‘(𝑠 − 𝑋)) = ((𝐷‘𝑛)‘(𝑡 − 𝑋)))
106	103, 105	oveq12d 7273	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (((𝐹 ↾ (-π[,]π))‘𝑠) · ((𝐷‘𝑛)‘(𝑠 − 𝑋))) = (((𝐹 ↾ (-π[,]π))‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))))
107	106	cbvmptv 5183	. . . . . 6 ⊢ (𝑠 ∈ (-π[,]π) ↦ (((𝐹 ↾ (-π[,]π))‘𝑠) · ((𝐷‘𝑛)‘(𝑠 − 𝑋)))) = (𝑡 ∈ (-π[,]π) ↦ (((𝐹 ↾ (-π[,]π))‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))))
108	33	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ (𝑃‘𝑀))
109	32	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ ℕ)
110		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
111	64	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ ℝ)
112	37	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹 ↾ (-π[,]π)):(-π[,]π)⟶ℂ)
113	50	adantlr 711	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
114	53	adantlr 711	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
115	56	adantlr 711	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ (((𝐹 ↾ (-π[,]π)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
116	102, 31, 107, 108, 109, 110, 111, 112, 113, 114, 115	fourierdlem101 43638	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫(-π[,]π)(((𝐹 ↾ (-π[,]π))‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))(((𝐹 ↾ (-π[,]π))‘(𝑋 + 𝑦)) · ((𝐷‘𝑛)‘𝑦)) d𝑦)
117		oveq2 7263	. . . . . . . . . 10 ⊢ (𝑠 = 𝑦 → (𝑋 + 𝑠) = (𝑋 + 𝑦))
118	117	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑠 = 𝑦 → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑦)))
119		fveq2 6756	. . . . . . . . 9 ⊢ (𝑠 = 𝑦 → ((𝐷‘𝑛)‘𝑠) = ((𝐷‘𝑛)‘𝑦))
120	118, 119	oveq12d 7273	. . . . . . . 8 ⊢ (𝑠 = 𝑦 → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) = ((𝐹‘(𝑋 + 𝑦)) · ((𝐷‘𝑛)‘𝑦)))
121	120	cbvitgv 24846	. . . . . . 7 ⊢ ∫((-π − 𝑋)(,)(π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 = ∫((-π − 𝑋)(,)(π − 𝑋))((𝐹‘(𝑋 + 𝑦)) · ((𝐷‘𝑛)‘𝑦)) d𝑦
122	121	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫((-π − 𝑋)(,)(π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 = ∫((-π − 𝑋)(,)(π − 𝑋))((𝐹‘(𝑋 + 𝑦)) · ((𝐷‘𝑛)‘𝑦)) d𝑦)
123	23	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → -π ∈ ℝ)
124	123, 64	resubcld 11333	. . . . . . . 8 ⊢ (𝜑 → (-π − 𝑋) ∈ ℝ)
125	124	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (-π − 𝑋) ∈ ℝ)
126	22	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → π ∈ ℝ)
127	126, 64	resubcld 11333	. . . . . . . 8 ⊢ (𝜑 → (π − 𝑋) ∈ ℝ)
128	127	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (π − 𝑋) ∈ ℝ)
129	36	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝐹:ℝ⟶ℂ)
130	64	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑋 ∈ ℝ)
131		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋)))
132	124	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (-π − 𝑋) ∈ ℝ)
133	127	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (π − 𝑋) ∈ ℝ)
134		elicc2 13073	. . . . . . . . . . . . . 14 ⊢ (((-π − 𝑋) ∈ ℝ ∧ (π − 𝑋) ∈ ℝ) → (𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋)) ↔ (𝑦 ∈ ℝ ∧ (-π − 𝑋) ≤ 𝑦 ∧ 𝑦 ≤ (π − 𝑋))))
135	132, 133, 134	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋)) ↔ (𝑦 ∈ ℝ ∧ (-π − 𝑋) ≤ 𝑦 ∧ 𝑦 ≤ (π − 𝑋))))
136	131, 135	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑦 ∈ ℝ ∧ (-π − 𝑋) ≤ 𝑦 ∧ 𝑦 ≤ (π − 𝑋)))
137	136	simp1d 1140	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑦 ∈ ℝ)
138	130, 137	readdcld 10935	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + 𝑦) ∈ ℝ)
139	129, 138	ffvelrnd 6944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝐹‘(𝑋 + 𝑦)) ∈ ℂ)
140	139	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝐹‘(𝑋 + 𝑦)) ∈ ℂ)
141	77	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝐷‘𝑛):ℝ⟶ℝ)
142	137	adantlr 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑦 ∈ ℝ)
143	141, 142	ffvelrnd 6944	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → ((𝐷‘𝑛)‘𝑦) ∈ ℝ)
144	143	recnd 10934	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → ((𝐷‘𝑛)‘𝑦) ∈ ℂ)
145	140, 144	mulcld 10926	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → ((𝐹‘(𝑋 + 𝑦)) · ((𝐷‘𝑛)‘𝑦)) ∈ ℂ)
146	125, 128, 145	itgioo 24885	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫((-π − 𝑋)(,)(π − 𝑋))((𝐹‘(𝑋 + 𝑦)) · ((𝐷‘𝑛)‘𝑦)) d𝑦 = ∫((-π − 𝑋)[,](π − 𝑋))((𝐹‘(𝑋 + 𝑦)) · ((𝐷‘𝑛)‘𝑦)) d𝑦)
147	23	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → -π ∈ ℝ)
148	22	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → π ∈ ℝ)
149	64	recnd 10934	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ ℂ)
150	126	recnd 10934	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → π ∈ ℂ)
151	150	negcld 11249	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → -π ∈ ℂ)
152	149, 151	pncan3d 11265	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 + (-π − 𝑋)) = -π)
153	152	eqcomd 2744	. . . . . . . . . . . . . 14 ⊢ (𝜑 → -π = (𝑋 + (-π − 𝑋)))
154	153	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → -π = (𝑋 + (-π − 𝑋)))
155	136	simp2d 1141	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (-π − 𝑋) ≤ 𝑦)
156	132, 137, 130, 155	leadd2dd 11520	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + (-π − 𝑋)) ≤ (𝑋 + 𝑦))
157	154, 156	eqbrtrd 5092	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → -π ≤ (𝑋 + 𝑦))
158	136	simp3d 1142	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑦 ≤ (π − 𝑋))
159	137, 133, 130, 158	leadd2dd 11520	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + 𝑦) ≤ (𝑋 + (π − 𝑋)))
160	149	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑋 ∈ ℂ)
161	150	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → π ∈ ℂ)
162	160, 161	pncan3d 11265	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + (π − 𝑋)) = π)
163	159, 162	breqtrd 5096	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + 𝑦) ≤ π)
164	147, 148, 138, 157, 163	eliccd 42932	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + 𝑦) ∈ (-π[,]π))
165		fvres 6775	. . . . . . . . . . 11 ⊢ ((𝑋 + 𝑦) ∈ (-π[,]π) → ((𝐹 ↾ (-π[,]π))‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
166	164, 165	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → ((𝐹 ↾ (-π[,]π))‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
167	166	eqcomd 2744	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝐹‘(𝑋 + 𝑦)) = ((𝐹 ↾ (-π[,]π))‘(𝑋 + 𝑦)))
168	167	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝐹‘(𝑋 + 𝑦)) = ((𝐹 ↾ (-π[,]π))‘(𝑋 + 𝑦)))
169	168	oveq1d 7270	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((-π − 𝑋)[,](π − 𝑋))) → ((𝐹‘(𝑋 + 𝑦)) · ((𝐷‘𝑛)‘𝑦)) = (((𝐹 ↾ (-π[,]π))‘(𝑋 + 𝑦)) · ((𝐷‘𝑛)‘𝑦)))
170	169	itgeq2dv 24851	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫((-π − 𝑋)[,](π − 𝑋))((𝐹‘(𝑋 + 𝑦)) · ((𝐷‘𝑛)‘𝑦)) d𝑦 = ∫((-π − 𝑋)[,](π − 𝑋))(((𝐹 ↾ (-π[,]π))‘(𝑋 + 𝑦)) · ((𝐷‘𝑛)‘𝑦)) d𝑦)
171	122, 146, 170	3eqtrrd 2783	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫((-π − 𝑋)[,](π − 𝑋))(((𝐹 ↾ (-π[,]π))‘(𝑋 + 𝑦)) · ((𝐷‘𝑛)‘𝑦)) d𝑦 = ∫((-π − 𝑋)(,)(π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)
172	116, 171	eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫(-π[,]π)(((𝐹 ↾ (-π[,]π))‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) d𝑡 = ∫((-π − 𝑋)(,)(π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)
173	85, 90, 172	3eqtrd 2782	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫(-π(,)π)((𝐹‘𝑡) · ((𝐷‘𝑛)‘(𝑡 − 𝑋))) d𝑡 = ∫((-π − 𝑋)(,)(π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)
174		elioore 13038	. . . . . . . . 9 ⊢ (𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋)) → 𝑠 ∈ ℝ)
175	174	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → 𝑠 ∈ ℝ)
176	36	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → 𝐹:ℝ⟶ℂ)
177	64	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → 𝑋 ∈ ℝ)
178	174	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → 𝑠 ∈ ℝ)
179	177, 178	readdcld 10935	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → (𝑋 + 𝑠) ∈ ℝ)
180	176, 179	ffvelrnd 6944	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
181	180	adantlr 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
182	77	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → (𝐷‘𝑛):ℝ⟶ℝ)
183	182, 175	ffvelrnd 6944	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
184	183	recnd 10934	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
185	181, 184	mulcld 10926	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ)
186		fourierdlem111.g	. . . . . . . . . 10 ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥)))
187		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (𝑋 + 𝑥) = (𝑋 + 𝑠))
188	187	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (𝐹‘(𝑋 + 𝑥)) = (𝐹‘(𝑋 + 𝑠)))
189		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → ((𝐷‘𝑛)‘𝑥) = ((𝐷‘𝑛)‘𝑠))
190	188, 189	oveq12d 7273	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
191	190	cbvmptv 5183	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥))) = (𝑠 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
192	186, 191	eqtri 2766	. . . . . . . . 9 ⊢ 𝐺 = (𝑠 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
193	192	fvmpt2 6868	. . . . . . . 8 ⊢ ((𝑠 ∈ ℝ ∧ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ) → (𝐺‘𝑠) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
194	175, 185, 193	syl2anc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → (𝐺‘𝑠) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
195	194	eqcomd 2744	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ((-π − 𝑋)(,)(π − 𝑋))) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) = (𝐺‘𝑠))
196	195	itgeq2dv 24851	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫((-π − 𝑋)(,)(π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 = ∫((-π − 𝑋)(,)(π − 𝑋))(𝐺‘𝑠) d𝑠)
197	36	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐹:ℝ⟶ℂ)
198	64	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑋 ∈ ℝ)
199		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
200	198, 199	readdcld 10935	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑋 + 𝑥) ∈ ℝ)
201	197, 200	ffvelrnd 6944	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑋 + 𝑥)) ∈ ℂ)
202	201	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑋 + 𝑥)) ∈ ℂ)
203	77	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛):ℝ⟶ℝ)
204	203	ffvelrnda 6943	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐷‘𝑛)‘𝑥) ∈ ℝ)
205	204	recnd 10934	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐷‘𝑛)‘𝑥) ∈ ℂ)
206	202, 205	mulcld 10926	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥)) ∈ ℂ)
207	206, 186	fmptd 6970	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:ℝ⟶ℂ)
208	207	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝐺:ℝ⟶ℂ)
209	124	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (-π − 𝑋) ∈ ℝ)
210	127	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (π − 𝑋) ∈ ℝ)
211		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋)))
212		eliccre 42933	. . . . . . . . . 10 ⊢ (((-π − 𝑋) ∈ ℝ ∧ (π − 𝑋) ∈ ℝ ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ∈ ℝ)
213	209, 210, 211, 212	syl3anc 1369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ∈ ℝ)
214	213	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ∈ ℝ)
215	208, 214	ffvelrnd 6944	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝐺‘𝑠) ∈ ℂ)
216	125, 128, 215	itgioo 24885	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫((-π − 𝑋)(,)(π − 𝑋))(𝐺‘𝑠) d𝑠 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐺‘𝑠) d𝑠)
217		fveq2 6756	. . . . . . 7 ⊢ (𝑠 = 𝑥 → (𝐺‘𝑠) = (𝐺‘𝑥))
218	217	cbvitgv 24846	. . . . . 6 ⊢ ∫((-π − 𝑋)[,](π − 𝑋))(𝐺‘𝑠) d𝑠 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐺‘𝑥) d𝑥
219	216, 218	eqtrdi 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫((-π − 𝑋)(,)(π − 𝑋))(𝐺‘𝑠) d𝑠 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐺‘𝑥) d𝑥)
220	196, 219	eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫((-π − 𝑋)(,)(π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐺‘𝑥) d𝑥)
221		eqid 2738	. . . . . . 7 ⊢ ((π − 𝑋) − (-π − 𝑋)) = ((π − 𝑋) − (-π − 𝑋))
222	111	renegcld 11332	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑋 ∈ ℝ)
223		fourierdlem111.o	. . . . . . 7 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π − 𝑋) ∧ (𝑝‘𝑚) = (π − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
224	31	fourierdlem2 43540	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
225	32, 224	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
226	33, 225	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
227	226	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
228		elmapi 8595	. . . . . . . . . . . . . . 15 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
229	227, 228	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
230	229	ffvelrnda 6943	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
231	64	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
232	230, 231	resubcld 11333	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑄‘𝑖) − 𝑋) ∈ ℝ)
233		fourierdlem111.14	. . . . . . . . . . . 12 ⊢ 𝑊 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) − 𝑋))
234	232, 233	fmptd 6970	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊:(0...𝑀)⟶ℝ)
235		reex 10893	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
236		ovex 7288	. . . . . . . . . . . . 13 ⊢ (0...𝑀) ∈ V
237	235, 236	pm3.2i 470	. . . . . . . . . . . 12 ⊢ (ℝ ∈ V ∧ (0...𝑀) ∈ V)
238		elmapg 8586	. . . . . . . . . . . 12 ⊢ ((ℝ ∈ V ∧ (0...𝑀) ∈ V) → (𝑊 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑊:(0...𝑀)⟶ℝ))
239	237, 238	mp1i 13	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑊:(0...𝑀)⟶ℝ))
240	234, 239	mpbird 256	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ (ℝ ↑_m (0...𝑀)))
241	233	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) − 𝑋)))
242		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → (𝑄‘𝑖) = (𝑄‘0))
243	226	simprd 495	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
244	243	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π))
245	244	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘0) = -π)
246	242, 245	sylan9eqr 2801	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 = 0) → (𝑄‘𝑖) = -π)
247	246	oveq1d 7270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑄‘𝑖) − 𝑋) = (-π − 𝑋))
248		0zd 12261	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℤ)
249	32	nnzd 12354	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℤ)
250		0red 10909	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 0 ∈ ℝ)
251		nnre 11910	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
252		nngt0 11934	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀)
253	250, 251, 252	ltled 11053	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 ≤ 𝑀)
254	32, 253	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 𝑀)
255		eluz2 12517	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (ℤ_≥‘0) ↔ (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ≤ 𝑀))
256	248, 249, 254, 255	syl3anbrc 1341	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
257		eluzfz1 13192	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
258	256, 257	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ (0...𝑀))
259	241, 247, 258, 124	fvmptd 6864	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊‘0) = (-π − 𝑋))
260		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑀 → (𝑄‘𝑖) = (𝑄‘𝑀))
261	244	simprd 495	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘𝑀) = π)
262	260, 261	sylan9eqr 2801	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 = 𝑀) → (𝑄‘𝑖) = π)
263	262	oveq1d 7270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 = 𝑀) → ((𝑄‘𝑖) − 𝑋) = (π − 𝑋))
264		eluzfz2 13193	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
265	256, 264	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
266	241, 263, 265, 127	fvmptd 6864	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊‘𝑀) = (π − 𝑋))
267	259, 266	jca 511	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑊‘0) = (-π − 𝑋) ∧ (𝑊‘𝑀) = (π − 𝑋)))
268	229	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
269		elfzofz 13331	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
270	269	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
271	268, 270	ffvelrnd 6944	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
272		fzofzp1 13412	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
273	272	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
274	268, 273	ffvelrnd 6944	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
275	64	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
276	243	simprd 495	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
277	276	r19.21bi 3132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
278	271, 274, 275, 277	ltsub1dd 11517	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − 𝑋) < ((𝑄‘(𝑖 + 1)) − 𝑋))
279	270, 232	syldan 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − 𝑋) ∈ ℝ)
280	233	fvmpt2 6868	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑄‘𝑖) − 𝑋) ∈ ℝ) → (𝑊‘𝑖) = ((𝑄‘𝑖) − 𝑋))
281	270, 279, 280	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘𝑖) = ((𝑄‘𝑖) − 𝑋))
282		fveq2 6756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑗 → (𝑄‘𝑖) = (𝑄‘𝑗))
283	282	oveq1d 7270	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → ((𝑄‘𝑖) − 𝑋) = ((𝑄‘𝑗) − 𝑋))
284	283	cbvmptv 5183	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) − 𝑋))
285	233, 284	eqtri 2766	. . . . . . . . . . . . . 14 ⊢ 𝑊 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) − 𝑋))
286	285	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑊 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) − 𝑋)))
287		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑖 + 1) → (𝑄‘𝑗) = (𝑄‘(𝑖 + 1)))
288	287	oveq1d 7270	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑖 + 1) → ((𝑄‘𝑗) − 𝑋) = ((𝑄‘(𝑖 + 1)) − 𝑋))
289	288	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑄‘𝑗) − 𝑋) = ((𝑄‘(𝑖 + 1)) − 𝑋))
290	274, 275	resubcld 11333	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
291	286, 289, 273, 290	fvmptd 6864	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘(𝑖 + 1)) = ((𝑄‘(𝑖 + 1)) − 𝑋))
292	278, 281, 291	3brtr4d 5102	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘𝑖) < (𝑊‘(𝑖 + 1)))
293	292	ralrimiva 3107	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) < (𝑊‘(𝑖 + 1)))
294	240, 267, 293	jca32 515	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑊‘0) = (-π − 𝑋) ∧ (𝑊‘𝑀) = (π − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) < (𝑊‘(𝑖 + 1)))))
295	223	fourierdlem2 43540	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (𝑊 ∈ (𝑂‘𝑀) ↔ (𝑊 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑊‘0) = (-π − 𝑋) ∧ (𝑊‘𝑀) = (π − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) < (𝑊‘(𝑖 + 1))))))
296	32, 295	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ∈ (𝑂‘𝑀) ↔ (𝑊 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑊‘0) = (-π − 𝑋) ∧ (𝑊‘𝑀) = (π − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) < (𝑊‘(𝑖 + 1))))))
297	294, 296	mpbird 256	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ (𝑂‘𝑀))
298	297	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ (𝑂‘𝑀))
299	150, 151, 149	nnncan2d 11297	. . . . . . . . . . . 12 ⊢ (𝜑 → ((π − 𝑋) − (-π − 𝑋)) = (π − -π))
300		picn 25521	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
301	300	2timesi 12041	. . . . . . . . . . . . 13 ⊢ (2 · π) = (π + π)
302		fourierdlem111.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = (2 · π)
303	300, 300	subnegi 11230	. . . . . . . . . . . . 13 ⊢ (π − -π) = (π + π)
304	301, 302, 303	3eqtr4i 2776	. . . . . . . . . . . 12 ⊢ 𝑇 = (π − -π)
305	299, 304	eqtr4di 2797	. . . . . . . . . . 11 ⊢ (𝜑 → ((π − 𝑋) − (-π − 𝑋)) = 𝑇)
306	305	oveq2d 7271	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 + ((π − 𝑋) − (-π − 𝑋))) = (𝑥 + 𝑇))
307	306	fveq2d 6760	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘(𝑥 + ((π − 𝑋) − (-π − 𝑋)))) = (𝐺‘(𝑥 + 𝑇)))
308	307	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺‘(𝑥 + ((π − 𝑋) − (-π − 𝑋)))) = (𝐺‘(𝑥 + 𝑇)))
309		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
310	186	fvmpt2 6868	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥)) ∈ ℂ) → (𝐺‘𝑥) = ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥)))
311	309, 206, 310	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥)))
312	149	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑋 ∈ ℂ)
313	199	recnd 10934	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
314		2re 11977	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
315	314, 22	remulcli 10922	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 · π) ∈ ℝ
316	302, 315	eqeltri 2835	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑇 ∈ ℝ
317	316	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ ℝ)
318	317	recnd 10934	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ∈ ℂ)
319	318	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℂ)
320	312, 313, 319	addassd 10928	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑋 + 𝑥) + 𝑇) = (𝑋 + (𝑥 + 𝑇)))
321	320	eqcomd 2744	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑋 + (𝑥 + 𝑇)) = ((𝑋 + 𝑥) + 𝑇))
322	321	fveq2d 6760	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑋 + (𝑥 + 𝑇))) = (𝐹‘((𝑋 + 𝑥) + 𝑇)))
323		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝜑)
324	323, 200	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝜑 ∧ (𝑋 + 𝑥) ∈ ℝ))
325		eleq1 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (𝑋 + 𝑥) → (𝑠 ∈ ℝ ↔ (𝑋 + 𝑥) ∈ ℝ))
326	325	anbi2d 628	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (𝑋 + 𝑥) → ((𝜑 ∧ 𝑠 ∈ ℝ) ↔ (𝜑 ∧ (𝑋 + 𝑥) ∈ ℝ)))
327		oveq1 7262	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑋 + 𝑥) → (𝑠 + 𝑇) = ((𝑋 + 𝑥) + 𝑇))
328	327	fveq2d 6760	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (𝑋 + 𝑥) → (𝐹‘(𝑠 + 𝑇)) = (𝐹‘((𝑋 + 𝑥) + 𝑇)))
329		fveq2 6756	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (𝑋 + 𝑥) → (𝐹‘𝑠) = (𝐹‘(𝑋 + 𝑥)))
330	328, 329	eqeq12d 2754	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (𝑋 + 𝑥) → ((𝐹‘(𝑠 + 𝑇)) = (𝐹‘𝑠) ↔ (𝐹‘((𝑋 + 𝑥) + 𝑇)) = (𝐹‘(𝑋 + 𝑥))))
331	326, 330	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑋 + 𝑥) → (((𝜑 ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑠 + 𝑇)) = (𝐹‘𝑠)) ↔ ((𝜑 ∧ (𝑋 + 𝑥) ∈ ℝ) → (𝐹‘((𝑋 + 𝑥) + 𝑇)) = (𝐹‘(𝑋 + 𝑥)))))
332		eleq1 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑠 → (𝑥 ∈ ℝ ↔ 𝑠 ∈ ℝ))
333	332	anbi2d 628	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑠 → ((𝜑 ∧ 𝑥 ∈ ℝ) ↔ (𝜑 ∧ 𝑠 ∈ ℝ)))
334		oveq1 7262	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑠 → (𝑥 + 𝑇) = (𝑠 + 𝑇))
335	334	fveq2d 6760	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑠 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑠 + 𝑇)))
336		fveq2 6756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑠 → (𝐹‘𝑥) = (𝐹‘𝑠))
337	335, 336	eqeq12d 2754	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑠 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑠 + 𝑇)) = (𝐹‘𝑠)))
338	333, 337	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑠 → (((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑠 + 𝑇)) = (𝐹‘𝑠))))
339		fourierdlem111.fper	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
340	338, 339	chvarvv 2003	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑠 + 𝑇)) = (𝐹‘𝑠))
341	331, 340	vtoclg 3495	. . . . . . . . . . . . 13 ⊢ ((𝑋 + 𝑥) ∈ ℝ → ((𝜑 ∧ (𝑋 + 𝑥) ∈ ℝ) → (𝐹‘((𝑋 + 𝑥) + 𝑇)) = (𝐹‘(𝑋 + 𝑥))))
342	200, 324, 341	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘((𝑋 + 𝑥) + 𝑇)) = (𝐹‘(𝑋 + 𝑥)))
343	322, 342	eqtr2d 2779	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑋 + 𝑥)) = (𝐹‘(𝑋 + (𝑥 + 𝑇))))
344	343	adantlr 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑋 + 𝑥)) = (𝐹‘(𝑋 + (𝑥 + 𝑇))))
345	67, 302	dirkerper 43527	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑥 ∈ ℝ) → ((𝐷‘𝑛)‘(𝑥 + 𝑇)) = ((𝐷‘𝑛)‘𝑥))
346	345	eqcomd 2744	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑥 ∈ ℝ) → ((𝐷‘𝑛)‘𝑥) = ((𝐷‘𝑛)‘(𝑥 + 𝑇)))
347	346	adantll 710	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐷‘𝑛)‘𝑥) = ((𝐷‘𝑛)‘(𝑥 + 𝑇)))
348	344, 347	oveq12d 7273	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥)) = ((𝐹‘(𝑋 + (𝑥 + 𝑇))) · ((𝐷‘𝑛)‘(𝑥 + 𝑇))))
349	192	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐺 = (𝑠 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))))
350		oveq2 7263	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑥 + 𝑇) → (𝑋 + 𝑠) = (𝑋 + (𝑥 + 𝑇)))
351	350	fveq2d 6760	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑥 + 𝑇) → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + (𝑥 + 𝑇))))
352		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑥 + 𝑇) → ((𝐷‘𝑛)‘𝑠) = ((𝐷‘𝑛)‘(𝑥 + 𝑇)))
353	351, 352	oveq12d 7273	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑥 + 𝑇) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) = ((𝐹‘(𝑋 + (𝑥 + 𝑇))) · ((𝐷‘𝑛)‘(𝑥 + 𝑇))))
354	353	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑠 = (𝑥 + 𝑇)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) = ((𝐹‘(𝑋 + (𝑥 + 𝑇))) · ((𝐷‘𝑛)‘(𝑥 + 𝑇))))
355	316	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
356	309, 355	readdcld 10935	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 + 𝑇) ∈ ℝ)
357	316	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ)
358	199, 357	readdcld 10935	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 + 𝑇) ∈ ℝ)
359	198, 358	readdcld 10935	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑋 + (𝑥 + 𝑇)) ∈ ℝ)
360	197, 359	ffvelrnd 6944	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑋 + (𝑥 + 𝑇))) ∈ ℂ)
361	360	adantlr 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑋 + (𝑥 + 𝑇))) ∈ ℂ)
362	77	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐷‘𝑛):ℝ⟶ℝ)
363	362, 356	ffvelrnd 6944	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐷‘𝑛)‘(𝑥 + 𝑇)) ∈ ℝ)
364	363	recnd 10934	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐷‘𝑛)‘(𝑥 + 𝑇)) ∈ ℂ)
365	361, 364	mulcld 10926	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘(𝑋 + (𝑥 + 𝑇))) · ((𝐷‘𝑛)‘(𝑥 + 𝑇))) ∈ ℂ)
366	349, 354, 356, 365	fvmptd 6864	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺‘(𝑥 + 𝑇)) = ((𝐹‘(𝑋 + (𝑥 + 𝑇))) · ((𝐷‘𝑛)‘(𝑥 + 𝑇))))
367	366	eqcomd 2744	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘(𝑋 + (𝑥 + 𝑇))) · ((𝐷‘𝑛)‘(𝑥 + 𝑇))) = (𝐺‘(𝑥 + 𝑇)))
368	311, 348, 367	3eqtrrd 2783	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺‘(𝑥 + 𝑇)) = (𝐺‘𝑥))
369	308, 368	eqtrd 2778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺‘(𝑥 + ((π − 𝑋) − (-π − 𝑋)))) = (𝐺‘𝑥))
370	192	reseq1i 5876	. . . . . . . . . 10 ⊢ (𝐺 ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) = ((𝑠 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))
371	370	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) = ((𝑠 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))))
372		ioossre 13069	. . . . . . . . . 10 ⊢ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ⊆ ℝ
373		resmpt 5934	. . . . . . . . . 10 ⊢ (((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ⊆ ℝ → ((𝑠 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))))
374	372, 373	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
375	371, 374	eqtrdi 2795	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))))
376	271	rexrd 10956	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
377	376	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
378	274	rexrd 10956	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
379	378	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
380	64	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
381		elioore 13038	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) → 𝑠 ∈ ℝ)
382	381	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
383	380, 382	readdcld 10935	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
384	383	adantlr 711	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
385		eleq1 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑠 → (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↔ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))))
386	385	anbi2d 628	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑠 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))))
387	187	breq2d 5082	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑠 → ((𝑄‘𝑖) < (𝑋 + 𝑥) ↔ (𝑄‘𝑖) < (𝑋 + 𝑠)))
388	386, 387	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑋 + 𝑥)) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑋 + 𝑠))))
389	149	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
390	281, 279	eqeltrd 2839	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘𝑖) ∈ ℝ)
391	390	recnd 10934	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘𝑖) ∈ ℂ)
392	389, 391	addcomd 11107	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑊‘𝑖)) = ((𝑊‘𝑖) + 𝑋))
393	281	oveq1d 7270	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊‘𝑖) + 𝑋) = (((𝑄‘𝑖) − 𝑋) + 𝑋))
394	271	recnd 10934	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
395	394, 389	npcand 11266	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) − 𝑋) + 𝑋) = (𝑄‘𝑖))
396	392, 393, 395	3eqtrrd 2783	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = (𝑋 + (𝑊‘𝑖)))
397	396	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘𝑖) = (𝑋 + (𝑊‘𝑖)))
398	390	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑊‘𝑖) ∈ ℝ)
399		elioore 13038	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) → 𝑥 ∈ ℝ)
400	399	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
401	64	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
402	390	rexrd 10956	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘𝑖) ∈ ℝ^*)
403	402	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑊‘𝑖) ∈ ℝ^*)
404	291, 290	eqeltrd 2839	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘(𝑖 + 1)) ∈ ℝ)
405	404	rexrd 10956	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘(𝑖 + 1)) ∈ ℝ^*)
406	405	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑊‘(𝑖 + 1)) ∈ ℝ^*)
407		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))
408		ioogtlb 42923	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊‘𝑖) ∈ ℝ^* ∧ (𝑊‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑊‘𝑖) < 𝑥)
409	403, 406, 407, 408	syl3anc 1369	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑊‘𝑖) < 𝑥)
410	398, 400, 401, 409	ltadd2dd 11064	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + (𝑊‘𝑖)) < (𝑋 + 𝑥))
411	397, 410	eqbrtrd 5092	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑋 + 𝑥))
412	388, 411	chvarvv 2003	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑋 + 𝑠))
413	187	breq1d 5080	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑠 → ((𝑋 + 𝑥) < (𝑄‘(𝑖 + 1)) ↔ (𝑋 + 𝑠) < (𝑄‘(𝑖 + 1))))
414	386, 413	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑥) < (𝑄‘(𝑖 + 1))) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑄‘(𝑖 + 1)))))
415	404	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑊‘(𝑖 + 1)) ∈ ℝ)
416		iooltub 42938	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊‘𝑖) ∈ ℝ^* ∧ (𝑊‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑥 < (𝑊‘(𝑖 + 1)))
417	403, 406, 407, 416	syl3anc 1369	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑥 < (𝑊‘(𝑖 + 1)))
418	400, 415, 401, 417	ltadd2dd 11064	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑥) < (𝑋 + (𝑊‘(𝑖 + 1))))
419	404	recnd 10934	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘(𝑖 + 1)) ∈ ℂ)
420	389, 419	addcomd 11107	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑊‘(𝑖 + 1))) = ((𝑊‘(𝑖 + 1)) + 𝑋))
421	291	oveq1d 7270	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊‘(𝑖 + 1)) + 𝑋) = (((𝑄‘(𝑖 + 1)) − 𝑋) + 𝑋))
422	274	recnd 10934	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
423	422, 389	npcand 11266	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘(𝑖 + 1)) − 𝑋) + 𝑋) = (𝑄‘(𝑖 + 1)))
424	420, 421, 423	3eqtrd 2782	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑊‘(𝑖 + 1))) = (𝑄‘(𝑖 + 1)))
425	424	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + (𝑊‘(𝑖 + 1))) = (𝑄‘(𝑖 + 1)))
426	418, 425	breqtrd 5096	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑥) < (𝑄‘(𝑖 + 1)))
427	414, 426	chvarvv 2003	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑄‘(𝑖 + 1)))
428	377, 379, 384, 412, 427	eliood 42926	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
429	187	cbvmptv 5183	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) = (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑠))
430	429	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) = (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑠)))
431		ioossre 13069	. . . . . . . . . . . . . . 15 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
432	431	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
433	11, 432	feqresmpt 6820	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑥)))
434	433	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑥)))
435		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑋 + 𝑠) → (𝐹‘𝑥) = (𝐹‘(𝑋 + 𝑠)))
436	428, 430, 434, 435	fmptco 6983	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))) = (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))))
437		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ ↦ (𝑋 + 𝑥)) = (𝑥 ∈ ℂ ↦ (𝑋 + 𝑥))
438		ssid 3939	. . . . . . . . . . . . . . . . 17 ⊢ ℂ ⊆ ℂ
439	438	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℂ ⊆ ℂ)
440	439, 149, 439	constcncfg 43303	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝑋) ∈ (ℂ–cn→ℂ))
441		cncfmptid 23982	. . . . . . . . . . . . . . . . 17 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ ℂ ↦ 𝑥) ∈ (ℂ–cn→ℂ))
442	438, 438, 441	mp2an 688	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ ↦ 𝑥) ∈ (ℂ–cn→ℂ)
443	442	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝑥) ∈ (ℂ–cn→ℂ))
444	440, 443	addcncf 24513	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑋 + 𝑥)) ∈ (ℂ–cn→ℂ))
445	444	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ℂ ↦ (𝑋 + 𝑥)) ∈ (ℂ–cn→ℂ))
446		ioosscn 13070	. . . . . . . . . . . . . 14 ⊢ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ⊆ ℂ
447	446	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ⊆ ℂ)
448		ioosscn 13070	. . . . . . . . . . . . . 14 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
449	448	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
450	376	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
451	378	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
452	64	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
453	399	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
454	452, 453	readdcld 10935	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑥) ∈ ℝ)
455	454	adantlr 711	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑥) ∈ ℝ)
456	450, 451, 455, 411, 426	eliood 42926	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑥) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
457	437, 445, 447, 449, 456	cncfmptssg 43302	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ∈ (((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))–cn→((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
458	457, 49	cncfco 23976	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))) ∈ (((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))–cn→ℂ))
459	436, 458	eqeltrrd 2840	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))–cn→ℂ))
460	459	adantlr 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))–cn→ℂ))
461		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠)) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠))
462	77	feqmptd 6819	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠)))
463		cncfss 23968	. . . . . . . . . . . . . 14 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ))
464	34, 438, 463	mp2an 688	. . . . . . . . . . . . 13 ⊢ (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ)
465	67	dirkercncf 43538	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛) ∈ (ℝ–cn→ℝ))
466	464, 465	sselid 3915	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛) ∈ (ℝ–cn→ℂ))
467	462, 466	eqeltrrd 2840	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠)) ∈ (ℝ–cn→ℂ))
468	372	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ⊆ ℝ)
469	438	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ℂ ⊆ ℂ)
470		cncff 23962	. . . . . . . . . . . . . 14 ⊢ ((𝐷‘𝑛) ∈ (ℝ–cn→ℂ) → (𝐷‘𝑛):ℝ⟶ℂ)
471	466, 470	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛):ℝ⟶ℂ)
472	471	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝐷‘𝑛):ℝ⟶ℂ)
473	381	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
474	472, 473	ffvelrnd 6944	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
475	461, 467, 468, 469, 474	cncfmptssg 43302	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ (((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))–cn→ℂ))
476	475	ad2antlr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ (((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))–cn→ℂ))
477	460, 476	mulcncf 24515	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) ∈ (((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))–cn→ℂ))
478	375, 477	eqeltrd 2839	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) ∈ (((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))–cn→ℂ))
479	453, 201	syldan 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑥)) ∈ ℂ)
480	479	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑥)) ∈ ℂ)
481		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))) = (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))
482	480, 481	fmptd 6970	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))):((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))⟶ℂ)
483	482	adantlr 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))):((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))⟶ℂ)
484	77	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘𝑛):ℝ⟶ℝ)
485	372	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ⊆ ℝ)
486	484, 485	fssresd 6625	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))):((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))⟶ℝ)
487	34	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ℝ ⊆ ℂ)
488	486, 487	fssd 6602	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))):((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))⟶ℂ)
489		eqid 2738	. . . . . . . . 9 ⊢ (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))‘𝑠) · (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))‘𝑠))) = (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))‘𝑠) · (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))‘𝑠)))
490		fdm 6593	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹:ℝ⟶ℂ → dom 𝐹 = ℝ)
491	36, 490	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = ℝ)
492	431, 491	sseqtrrid 3970	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
493		ssdmres 5903	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
494	492, 493	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
495	494	eqcomd 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
496	495	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
497	456, 496	eleqtrd 2841	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑥) ∈ dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
498	271	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
499	498, 411	gtned 11040	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑥) ≠ (𝑄‘𝑖))
500		eldifsn 4717	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 + 𝑥) ∈ (dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘𝑖)}) ↔ ((𝑋 + 𝑥) ∈ dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ (𝑋 + 𝑥) ≠ (𝑄‘𝑖)))
501	497, 499, 500	sylanbrc 582	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑥) ∈ (dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘𝑖)}))
502	501	ralrimiva 3107	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))(𝑋 + 𝑥) ∈ (dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘𝑖)}))
503		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) = (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))
504	503	rnmptss 6978	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))(𝑋 + 𝑥) ∈ (dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘𝑖)}) → ran (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ⊆ (dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘𝑖)}))
505	502, 504	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ran (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ⊆ (dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘𝑖)}))
506		eqidd 2739	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) = (𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)))
507		oveq2 7263	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑊‘𝑖) → (𝑋 + 𝑥) = (𝑋 + (𝑊‘𝑖)))
508	507	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑊‘𝑖)) → (𝑋 + 𝑥) = (𝑋 + (𝑊‘𝑖)))
509	390	leidd 11471	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘𝑖) ≤ (𝑊‘𝑖))
510	390, 404, 292	ltled 11053	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘𝑖) ≤ (𝑊‘(𝑖 + 1)))
511	390, 404, 390, 509, 510	eliccd 42932	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘𝑖) ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))
512	396, 271	eqeltrrd 2840	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑊‘𝑖)) ∈ ℝ)
513	506, 508, 511, 512	fvmptd 6864	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘(𝑊‘𝑖)) = (𝑋 + (𝑊‘𝑖)))
514	396	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑊‘𝑖)) = (𝑄‘𝑖))
515	513, 514	eqtr2d 2779	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘(𝑊‘𝑖)))
516	390, 404	iccssred 13095	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ⊆ ℝ)
517	516, 34	sstrdi 3929	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ⊆ ℂ)
518	517	resmptd 5937	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑥 ∈ ℂ ↦ (𝑋 + 𝑥)) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) = (𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)))
519		rescncf 23966	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ⊆ ℂ → ((𝑥 ∈ ℂ ↦ (𝑋 + 𝑥)) ∈ (ℂ–cn→ℂ) → ((𝑥 ∈ ℂ ↦ (𝑋 + 𝑥)) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ∈ (((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))–cn→ℂ)))
520	517, 445, 519	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑥 ∈ ℂ ↦ (𝑋 + 𝑥)) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ∈ (((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))–cn→ℂ))
521	518, 520	eqeltrrd 2840	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ∈ (((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))–cn→ℂ))
522	521, 511	cnlimci 24958	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘(𝑊‘𝑖)) ∈ ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘𝑖)))
523	515, 522	eqeltrd 2839	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘𝑖)))
524		ioossicc 13094	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ⊆ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))
525		resmpt 5934	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ⊆ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) → ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) = (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)))
526	524, 525	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) = (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))
527	526	eqcomi 2747	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) = ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))
528	527	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) = ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))))
529	528	oveq1d 7270	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘𝑖)) = (((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)))
530	149	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) → 𝑋 ∈ ℂ)
531	390	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) → (𝑊‘𝑖) ∈ ℝ)
532	404	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) → (𝑊‘(𝑖 + 1)) ∈ ℝ)
533		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) → 𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))
534		eliccre 42933	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊‘𝑖) ∈ ℝ ∧ (𝑊‘(𝑖 + 1)) ∈ ℝ ∧ 𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
535	531, 532, 533, 534	syl3anc 1369	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) → 𝑥 ∈ ℝ)
536	535	recnd 10934	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) → 𝑥 ∈ ℂ)
537	530, 536	addcld 10925	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑥) ∈ ℂ)
538		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) = (𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))
539	537, 538	fmptd 6970	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)):((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))⟶ℂ)
540	390, 404, 292, 539	limciccioolb 43052	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)) = ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘𝑖)))
541	529, 540	eqtr2d 2779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘𝑖)) = ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘𝑖)))
542	523, 541	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘𝑖)))
543	505, 542, 51	limccog 43051	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))) lim_ℂ (𝑊‘𝑖)))
544	36, 432	fssresd 6625	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
545	544	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
546	456, 503	fmptd 6970	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)):((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))⟶((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
547		fcompt 6987	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ ∧ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)):((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))⟶((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))) = (𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘𝑦))))
548	545, 546, 547	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))) = (𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘𝑦))))
549		eqidd 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) = (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)))
550		oveq2 7263	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝑋 + 𝑥) = (𝑋 + 𝑦))
551	550	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) ∧ 𝑥 = 𝑦) → (𝑋 + 𝑥) = (𝑋 + 𝑦))
552		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))
553	64	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
554	372, 552	sselid 3915	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑦 ∈ ℝ)
555	553, 554	readdcld 10935	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑦) ∈ ℝ)
556	549, 551, 552, 555	fvmptd 6864	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘𝑦) = (𝑋 + 𝑦))
557	556	fveq2d 6760	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘𝑦)) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑋 + 𝑦)))
558	557	adantlr 711	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘𝑦)) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑋 + 𝑦)))
559	376	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
560	378	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
561	555	adantlr 711	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑦) ∈ ℝ)
562	396	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘𝑖) = (𝑋 + (𝑊‘𝑖)))
563	390	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑊‘𝑖) ∈ ℝ)
564	554	adantlr 711	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑦 ∈ ℝ)
565	64	ad2antrr 722	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
566	402	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑊‘𝑖) ∈ ℝ^*)
567	405	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑊‘(𝑖 + 1)) ∈ ℝ^*)
568		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))
569		ioogtlb 42923	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊‘𝑖) ∈ ℝ^* ∧ (𝑊‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑊‘𝑖) < 𝑦)
570	566, 567, 568, 569	syl3anc 1369	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑊‘𝑖) < 𝑦)
571	563, 564, 565, 570	ltadd2dd 11064	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + (𝑊‘𝑖)) < (𝑋 + 𝑦))
572	562, 571	eqbrtrd 5092	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑋 + 𝑦))
573	404	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑊‘(𝑖 + 1)) ∈ ℝ)
574		iooltub 42938	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊‘𝑖) ∈ ℝ^* ∧ (𝑊‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑦 < (𝑊‘(𝑖 + 1)))
575	566, 567, 568, 574	syl3anc 1369	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑦 < (𝑊‘(𝑖 + 1)))
576	564, 573, 565, 575	ltadd2dd 11064	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑦) < (𝑋 + (𝑊‘(𝑖 + 1))))
577	424	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + (𝑊‘(𝑖 + 1))) = (𝑄‘(𝑖 + 1)))
578	576, 577	breqtrd 5096	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑦) < (𝑄‘(𝑖 + 1)))
579	559, 560, 561, 572, 578	eliood 42926	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑦) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
580		fvres 6775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 + 𝑦) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
581	579, 580	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
582	558, 581	eqtrd 2778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘𝑦)) = (𝐹‘(𝑋 + 𝑦)))
583	582	mpteq2dva 5170	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘𝑦))) = (𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑦))))
584	550	fveq2d 6760	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑋 + 𝑥)) = (𝐹‘(𝑋 + 𝑦)))
585	584	cbvmptv 5183	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))) = (𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑦)))
586	583, 585	eqtr4di 2797	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑦 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘𝑦))) = (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))))
587	548, 586	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))) = (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))))
588	587	oveq1d 7270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))) lim_ℂ (𝑊‘𝑖)) = ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))) lim_ℂ (𝑊‘𝑖)))
589	543, 588	eleqtrd 2841	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))) lim_ℂ (𝑊‘𝑖)))
590	589	adantlr 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))) lim_ℂ (𝑊‘𝑖)))
591		fvres 6775	. . . . . . . . . . . . . 14 ⊢ ((𝑊‘𝑖) ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))‘(𝑊‘𝑖)) = ((𝐷‘𝑛)‘(𝑊‘𝑖)))
592	511, 591	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))‘(𝑊‘𝑖)) = ((𝐷‘𝑛)‘(𝑊‘𝑖)))
593	592	eqcomd 2744	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑛)‘(𝑊‘𝑖)) = (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))‘(𝑊‘𝑖)))
594	593	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑛)‘(𝑊‘𝑖)) = (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))‘(𝑊‘𝑖)))
595	516	adantlr 711	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ⊆ ℝ)
596	465	ad2antlr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘𝑛) ∈ (ℝ–cn→ℝ))
597		rescncf 23966	. . . . . . . . . . . . 13 ⊢ (((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ⊆ ℝ → ((𝐷‘𝑛) ∈ (ℝ–cn→ℝ) → ((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ∈ (((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))–cn→ℝ)))
598	595, 596, 597	sylc 65	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ∈ (((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))–cn→ℝ))
599	511	adantlr 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘𝑖) ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))
600	598, 599	cnlimci 24958	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))‘(𝑊‘𝑖)) ∈ (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)))
601	594, 600	eqeltrd 2839	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑛)‘(𝑊‘𝑖)) ∈ (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)))
602	524	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ⊆ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))
603	602	resabs1d 5911	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) = ((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))))
604	603	eqcomd 2744	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) = (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))))
605	604	oveq1d 7270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)) = ((((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)))
606	605	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)) = ((((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)))
607	390	adantlr 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘𝑖) ∈ ℝ)
608	404	adantlr 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘(𝑖 + 1)) ∈ ℝ)
609	292	adantlr 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘𝑖) < (𝑊‘(𝑖 + 1)))
610	471	ad2antlr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘𝑛):ℝ⟶ℂ)
611	610, 595	fssresd 6625	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))):((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))⟶ℂ)
612	607, 608, 609, 611	limciccioolb 43052	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)) = (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)))
613	606, 612	eqtr2d 2779	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)) = (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)))
614	601, 613	eleqtrd 2841	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑛)‘(𝑊‘𝑖)) ∈ (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)))
615	483, 488, 489, 590, 614	mullimcf 43054	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · ((𝐷‘𝑛)‘(𝑊‘𝑖))) ∈ ((𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))‘𝑠) · (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))‘𝑠))) lim_ℂ (𝑊‘𝑖)))
616		eqidd 2739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))) = (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))))
617	188	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) ∧ 𝑥 = 𝑠) → (𝐹‘(𝑋 + 𝑥)) = (𝐹‘(𝑋 + 𝑠)))
618		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))
619	36	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℂ)
620	619, 383	ffvelrnd 6944	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
621	620	adantlr 711	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
622	616, 617, 618, 621	fvmptd 6864	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))‘𝑠) = (𝐹‘(𝑋 + 𝑠)))
623	622	adantllr 715	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))‘𝑠) = (𝐹‘(𝑋 + 𝑠)))
624		fvres 6775	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))‘𝑠) = ((𝐷‘𝑛)‘𝑠))
625	624	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))‘𝑠) = ((𝐷‘𝑛)‘𝑠))
626	623, 625	oveq12d 7273	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))‘𝑠) · (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))‘𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
627	626	eqcomd 2744	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) = (((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))‘𝑠) · (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))‘𝑠)))
628	627	mpteq2dva 5170	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) = (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))‘𝑠) · (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))‘𝑠))))
629	375, 628	eqtr2d 2779	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))‘𝑠) · (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))‘𝑠))) = (𝐺 ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))))
630	629	oveq1d 7270	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))‘𝑠) · (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))‘𝑠))) lim_ℂ (𝑊‘𝑖)) = ((𝐺 ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)))
631	615, 630	eleqtrd 2841	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · ((𝐷‘𝑛)‘(𝑊‘𝑖))) ∈ ((𝐺 ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘𝑖)))
632	455, 426	ltned 11041	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑥) ≠ (𝑄‘(𝑖 + 1)))
633		eldifsn 4717	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 + 𝑥) ∈ (dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}) ↔ ((𝑋 + 𝑥) ∈ dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ (𝑋 + 𝑥) ≠ (𝑄‘(𝑖 + 1))))
634	497, 632, 633	sylanbrc 582	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) → (𝑋 + 𝑥) ∈ (dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}))
635	634	ralrimiva 3107	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))(𝑋 + 𝑥) ∈ (dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}))
636	503	rnmptss 6978	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))(𝑋 + 𝑥) ∈ (dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}) → ran (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ⊆ (dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}))
637	635, 636	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ran (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ⊆ (dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∖ {(𝑄‘(𝑖 + 1))}))
638	404	leidd 11471	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘(𝑖 + 1)) ≤ (𝑊‘(𝑖 + 1)))
639	390, 404, 404, 510, 638	eliccd 42932	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘(𝑖 + 1)) ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))
640	521, 639	cnlimci 24958	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘(𝑊‘(𝑖 + 1))) ∈ ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘(𝑖 + 1))))
641		oveq2 7263	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑊‘(𝑖 + 1)) → (𝑋 + 𝑥) = (𝑋 + (𝑊‘(𝑖 + 1))))
642	641	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 = (𝑊‘(𝑖 + 1))) → (𝑋 + 𝑥) = (𝑋 + (𝑊‘(𝑖 + 1))))
643	275, 404	readdcld 10935	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑊‘(𝑖 + 1))) ∈ ℝ)
644	506, 642, 639, 643	fvmptd 6864	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘(𝑊‘(𝑖 + 1))) = (𝑋 + (𝑊‘(𝑖 + 1))))
645	644, 424	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))‘(𝑊‘(𝑖 + 1))) = (𝑄‘(𝑖 + 1)))
646	528	oveq1d 7270	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘(𝑖 + 1))) = (((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))))
647	390, 404, 292, 539	limcicciooub 43068	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))) = ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘(𝑖 + 1))))
648	646, 647	eqtr2d 2779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑥 ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘(𝑖 + 1))) = ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘(𝑖 + 1))))
649	640, 645, 648	3eltr3d 2853	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥)) lim_ℂ (𝑊‘(𝑖 + 1))))
650	637, 649, 54	limccog 43051	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))) lim_ℂ (𝑊‘(𝑖 + 1))))
651	587	oveq1d 7270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∘ (𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝑋 + 𝑥))) lim_ℂ (𝑊‘(𝑖 + 1))) = ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))) lim_ℂ (𝑊‘(𝑖 + 1))))
652	650, 651	eleqtrd 2841	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))) lim_ℂ (𝑊‘(𝑖 + 1))))
653	652	adantlr 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥))) lim_ℂ (𝑊‘(𝑖 + 1))))
654	639	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑊‘(𝑖 + 1)) ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))
655	598, 654	cnlimci 24958	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))‘(𝑊‘(𝑖 + 1))) ∈ (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))))
656		fvres 6775	. . . . . . . . . . 11 ⊢ ((𝑊‘(𝑖 + 1)) ∈ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))‘(𝑊‘(𝑖 + 1))) = ((𝐷‘𝑛)‘(𝑊‘(𝑖 + 1))))
657	654, 656	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))))‘(𝑊‘(𝑖 + 1))) = ((𝐷‘𝑛)‘(𝑊‘(𝑖 + 1))))
658	607, 608, 609, 611	limcicciooub 43068	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))) = (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))))
659	658	eqcomd 2744	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))) = ((((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))))
660		resabs1 5910	. . . . . . . . . . . . 13 ⊢ (((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ⊆ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1))) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) = ((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))))
661	524, 660	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) = ((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))))
662	661	oveq1d 7270	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))) = (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))))
663	659, 662	eqtrd 2778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)[,](𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))) = (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))))
664	655, 657, 663	3eltr3d 2853	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑛)‘(𝑊‘(𝑖 + 1))) ∈ (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))))
665	483, 488, 489, 653, 664	mullimcf 43054	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐿 · ((𝐷‘𝑛)‘(𝑊‘(𝑖 + 1)))) ∈ ((𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))‘𝑠) · (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))‘𝑠))) lim_ℂ (𝑊‘(𝑖 + 1))))
666	629	oveq1d 7270	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (((𝑥 ∈ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑥)))‘𝑠) · (((𝐷‘𝑛) ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1))))‘𝑠))) lim_ℂ (𝑊‘(𝑖 + 1))) = ((𝐺 ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))))
667	665, 666	eleqtrd 2841	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐿 · ((𝐷‘𝑛)‘(𝑊‘(𝑖 + 1)))) ∈ ((𝐺 ↾ ((𝑊‘𝑖)(,)(𝑊‘(𝑖 + 1)))) lim_ℂ (𝑊‘(𝑖 + 1))))
668	125, 128, 221, 222, 223, 109, 298, 207, 369, 478, 631, 667	fourierdlem110 43647	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫(((-π − 𝑋) − -𝑋)[,]((π − 𝑋) − -𝑋))(𝐺‘𝑥) d𝑥 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐺‘𝑥) d𝑥)
669	668	eqcomd 2744	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫((-π − 𝑋)[,](π − 𝑋))(𝐺‘𝑥) d𝑥 = ∫(((-π − 𝑋) − -𝑋)[,]((π − 𝑋) − -𝑋))(𝐺‘𝑥) d𝑥)
670	124	recnd 10934	. . . . . . . . . 10 ⊢ (𝜑 → (-π − 𝑋) ∈ ℂ)
671	670, 149	subnegd 11269	. . . . . . . . 9 ⊢ (𝜑 → ((-π − 𝑋) − -𝑋) = ((-π − 𝑋) + 𝑋))
672	151, 149	npcand 11266	. . . . . . . . 9 ⊢ (𝜑 → ((-π − 𝑋) + 𝑋) = -π)
673	671, 672	eqtrd 2778	. . . . . . . 8 ⊢ (𝜑 → ((-π − 𝑋) − -𝑋) = -π)
674	127	recnd 10934	. . . . . . . . . 10 ⊢ (𝜑 → (π − 𝑋) ∈ ℂ)
675	674, 149	subnegd 11269	. . . . . . . . 9 ⊢ (𝜑 → ((π − 𝑋) − -𝑋) = ((π − 𝑋) + 𝑋))
676	150, 149	npcand 11266	. . . . . . . . 9 ⊢ (𝜑 → ((π − 𝑋) + 𝑋) = π)
677	675, 676	eqtrd 2778	. . . . . . . 8 ⊢ (𝜑 → ((π − 𝑋) − -𝑋) = π)
678	673, 677	oveq12d 7273	. . . . . . 7 ⊢ (𝜑 → (((-π − 𝑋) − -𝑋)[,]((π − 𝑋) − -𝑋)) = (-π[,]π))
679	678	itgeq1d 43388	. . . . . 6 ⊢ (𝜑 → ∫(((-π − 𝑋) − -𝑋)[,]((π − 𝑋) − -𝑋))(𝐺‘𝑥) d𝑥 = ∫(-π[,]π)(𝐺‘𝑥) d𝑥)
680	679	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫(((-π − 𝑋) − -𝑋)[,]((π − 𝑋) − -𝑋))(𝐺‘𝑥) d𝑥 = ∫(-π[,]π)(𝐺‘𝑥) d𝑥)
681	669, 680	eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫((-π − 𝑋)[,](π − 𝑋))(𝐺‘𝑥) d𝑥 = ∫(-π[,]π)(𝐺‘𝑥) d𝑥)
682		fveq2 6756	. . . . . 6 ⊢ (𝑥 = 𝑠 → (𝐺‘𝑥) = (𝐺‘𝑠))
683	682	cbvitgv 24846	. . . . 5 ⊢ ∫(-π(,)π)(𝐺‘𝑥) d𝑥 = ∫(-π(,)π)(𝐺‘𝑠) d𝑠
684	207	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (-π[,]π)) → 𝐺:ℝ⟶ℂ)
685	28	adantlr 711	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (-π[,]π)) → 𝑥 ∈ ℝ)
686	684, 685	ffvelrnd 6944	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ (-π[,]π)) → (𝐺‘𝑥) ∈ ℂ)
687	71, 72, 686	itgioo 24885	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫(-π(,)π)(𝐺‘𝑥) d𝑥 = ∫(-π[,]π)(𝐺‘𝑥) d𝑥)
688		elioore 13038	. . . . . . . 8 ⊢ (𝑠 ∈ (-π(,)π) → 𝑠 ∈ ℝ)
689	688	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)π)) → 𝑠 ∈ ℝ)
690	36	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π(,)π)) → 𝐹:ℝ⟶ℂ)
691	64	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π(,)π)) → 𝑋 ∈ ℝ)
692	688	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π(,)π)) → 𝑠 ∈ ℝ)
693	691, 692	readdcld 10935	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π(,)π)) → (𝑋 + 𝑠) ∈ ℝ)
694	690, 693	ffvelrnd 6944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
695	694	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)π)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
696	77	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)π)) → (𝐷‘𝑛):ℝ⟶ℝ)
697	696, 689	ffvelrnd 6944	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)π)) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
698	697	recnd 10934	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)π)) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
699	695, 698	mulcld 10926	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ)
700	689, 699, 193	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π(,)π)) → (𝐺‘𝑠) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
701	700	itgeq2dv 24851	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫(-π(,)π)(𝐺‘𝑠) d𝑠 = ∫(-π(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)
702	683, 687, 701	3eqtr3a 2803	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫(-π[,]π)(𝐺‘𝑥) d𝑥 = ∫(-π(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)
703	220, 681, 702	3eqtrd 2782	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫((-π − 𝑋)(,)(π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 = ∫(-π(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)
704	70, 173, 703	3eqtrd 2782	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = ∫(-π(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)
705	72	renegcld 11332	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -π ∈ ℝ)
706		0red 10909	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ)
707		0re 10908	. . . . . 6 ⊢ 0 ∈ ℝ
708		negpilt0 42708	. . . . . 6 ⊢ -π < 0
709	23, 707, 708	ltleii 11028	. . . . 5 ⊢ -π ≤ 0
710	709	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -π ≤ 0)
711		pipos 25522	. . . . . 6 ⊢ 0 < π
712	707, 22, 711	ltleii 11028	. . . . 5 ⊢ 0 ≤ π
713	712	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ π)
714	71, 72, 706, 710, 713	eliccd 42932	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ (-π[,]π))
715		ioossicc 13094	. . . . 5 ⊢ (-π(,)0) ⊆ (-π[,]0)
716	715	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (-π(,)0) ⊆ (-π[,]0))
717		ioombl 24634	. . . . 5 ⊢ (-π(,)0) ∈ dom vol
718	717	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (-π(,)0) ∈ dom vol)
719	36	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]0)) → 𝐹:ℝ⟶ℂ)
720	64	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]0)) → 𝑋 ∈ ℝ)
721	23	a1i 11	. . . . . . . . . 10 ⊢ (𝑠 ∈ (-π[,]0) → -π ∈ ℝ)
722		0red 10909	. . . . . . . . . 10 ⊢ (𝑠 ∈ (-π[,]0) → 0 ∈ ℝ)
723		id 22	. . . . . . . . . 10 ⊢ (𝑠 ∈ (-π[,]0) → 𝑠 ∈ (-π[,]0))
724		eliccre 42933	. . . . . . . . . 10 ⊢ ((-π ∈ ℝ ∧ 0 ∈ ℝ ∧ 𝑠 ∈ (-π[,]0)) → 𝑠 ∈ ℝ)
725	721, 722, 723, 724	syl3anc 1369	. . . . . . . . 9 ⊢ (𝑠 ∈ (-π[,]0) → 𝑠 ∈ ℝ)
726	725	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]0)) → 𝑠 ∈ ℝ)
727	720, 726	readdcld 10935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]0)) → (𝑋 + 𝑠) ∈ ℝ)
728	719, 727	ffvelrnd 6944	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]0)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
729	728	adantlr 711	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]0)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
730	77	ad2antlr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]0)) → (𝐷‘𝑛):ℝ⟶ℝ)
731	725	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]0)) → 𝑠 ∈ ℝ)
732	730, 731	ffvelrnd 6944	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]0)) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
733	732	recnd 10934	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]0)) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
734	729, 733	mulcld 10926	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ)
735	731, 734, 193	syl2anc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]0)) → (𝐺‘𝑠) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
736	735	eqcomd 2744	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]0)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) = (𝐺‘𝑠))
737	736	mpteq2dva 5170	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]0) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) = (𝑠 ∈ (-π[,]0) ↦ (𝐺‘𝑠)))
738	305	oveq2d 7271	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 + ((π − 𝑋) − (-π − 𝑋))) = (𝑠 + 𝑇))
739	738	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝑠 + ((π − 𝑋) − (-π − 𝑋))) = (𝑠 + 𝑇))
740	739	fveq2d 6760	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐺‘(𝑠 + ((π − 𝑋) − (-π − 𝑋)))) = (𝐺‘(𝑠 + 𝑇)))
741	186	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → 𝐺 = (𝑥 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥))))
742		oveq2 7263	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑠 + 𝑇) → (𝑋 + 𝑥) = (𝑋 + (𝑠 + 𝑇)))
743	742	fveq2d 6760	. . . . . . . . . 10 ⊢ (𝑥 = (𝑠 + 𝑇) → (𝐹‘(𝑋 + 𝑥)) = (𝐹‘(𝑋 + (𝑠 + 𝑇))))
744		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑥 = (𝑠 + 𝑇) → ((𝐷‘𝑛)‘𝑥) = ((𝐷‘𝑛)‘(𝑠 + 𝑇)))
745	743, 744	oveq12d 7273	. . . . . . . . 9 ⊢ (𝑥 = (𝑠 + 𝑇) → ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥)) = ((𝐹‘(𝑋 + (𝑠 + 𝑇))) · ((𝐷‘𝑛)‘(𝑠 + 𝑇))))
746	745	adantl 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) ∧ 𝑥 = (𝑠 + 𝑇)) → ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥)) = ((𝐹‘(𝑋 + (𝑠 + 𝑇))) · ((𝐷‘𝑛)‘(𝑠 + 𝑇))))
747		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℝ)
748	316	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ∈ ℝ)
749	747, 748	readdcld 10935	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝑠 + 𝑇) ∈ ℝ)
750	749	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝑠 + 𝑇) ∈ ℝ)
751	36	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝐹:ℝ⟶ℂ)
752	64	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑋 ∈ ℝ)
753	752, 749	readdcld 10935	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝑋 + (𝑠 + 𝑇)) ∈ ℝ)
754	751, 753	ffvelrnd 6944	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑋 + (𝑠 + 𝑇))) ∈ ℂ)
755	754	adantlr 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑋 + (𝑠 + 𝑇))) ∈ ℂ)
756	77	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐷‘𝑛):ℝ⟶ℝ)
757	756, 750	ffvelrnd 6944	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘(𝑠 + 𝑇)) ∈ ℝ)
758	757	recnd 10934	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘(𝑠 + 𝑇)) ∈ ℂ)
759	755, 758	mulcld 10926	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐹‘(𝑋 + (𝑠 + 𝑇))) · ((𝐷‘𝑛)‘(𝑠 + 𝑇))) ∈ ℂ)
760	741, 746, 750, 759	fvmptd 6864	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐺‘(𝑠 + 𝑇)) = ((𝐹‘(𝑋 + (𝑠 + 𝑇))) · ((𝐷‘𝑛)‘(𝑠 + 𝑇))))
761	149	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑋 ∈ ℂ)
762	747	recnd 10934	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
763	318	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ∈ ℂ)
764	761, 762, 763	addassd 10928	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → ((𝑋 + 𝑠) + 𝑇) = (𝑋 + (𝑠 + 𝑇)))
765	764	eqcomd 2744	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝑋 + (𝑠 + 𝑇)) = ((𝑋 + 𝑠) + 𝑇))
766	765	fveq2d 6760	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑋 + (𝑠 + 𝑇))) = (𝐹‘((𝑋 + 𝑠) + 𝑇)))
767	752, 747	readdcld 10935	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝑋 + 𝑠) ∈ ℝ)
768		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝜑)
769	768, 767	jca 511	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝜑 ∧ (𝑋 + 𝑠) ∈ ℝ))
770		eleq1 2826	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑋 + 𝑠) → (𝑥 ∈ ℝ ↔ (𝑋 + 𝑠) ∈ ℝ))
771	770	anbi2d 628	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑋 + 𝑠) → ((𝜑 ∧ 𝑥 ∈ ℝ) ↔ (𝜑 ∧ (𝑋 + 𝑠) ∈ ℝ)))
772		oveq1 7262	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑋 + 𝑠) → (𝑥 + 𝑇) = ((𝑋 + 𝑠) + 𝑇))
773	772	fveq2d 6760	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑋 + 𝑠) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘((𝑋 + 𝑠) + 𝑇)))
774	773, 435	eqeq12d 2754	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑋 + 𝑠) → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘((𝑋 + 𝑠) + 𝑇)) = (𝐹‘(𝑋 + 𝑠))))
775	771, 774	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑋 + 𝑠) → (((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ (𝑋 + 𝑠) ∈ ℝ) → (𝐹‘((𝑋 + 𝑠) + 𝑇)) = (𝐹‘(𝑋 + 𝑠)))))
776	775, 339	vtoclg 3495	. . . . . . . . . . . 12 ⊢ ((𝑋 + 𝑠) ∈ ℝ → ((𝜑 ∧ (𝑋 + 𝑠) ∈ ℝ) → (𝐹‘((𝑋 + 𝑠) + 𝑇)) = (𝐹‘(𝑋 + 𝑠))))
777	767, 769, 776	sylc 65	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝐹‘((𝑋 + 𝑠) + 𝑇)) = (𝐹‘(𝑋 + 𝑠)))
778	766, 777	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑋 + (𝑠 + 𝑇))) = (𝐹‘(𝑋 + 𝑠)))
779	778	adantlr 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑋 + (𝑠 + 𝑇))) = (𝐹‘(𝑋 + 𝑠)))
780	67, 302	dirkerper 43527	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘(𝑠 + 𝑇)) = ((𝐷‘𝑛)‘𝑠))
781	780	adantll 710	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘(𝑠 + 𝑇)) = ((𝐷‘𝑛)‘𝑠))
782	779, 781	oveq12d 7273	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐹‘(𝑋 + (𝑠 + 𝑇))) · ((𝐷‘𝑛)‘(𝑠 + 𝑇))) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
783		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℝ)
784	782, 759	eqeltrrd 2840	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ)
785	783, 784, 193	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐺‘𝑠) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
786	785	eqcomd 2744	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) = (𝐺‘𝑠))
787	782, 786	eqtrd 2778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → ((𝐹‘(𝑋 + (𝑠 + 𝑇))) · ((𝐷‘𝑛)‘(𝑠 + 𝑇))) = (𝐺‘𝑠))
788	740, 760, 787	3eqtrd 2782	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ℝ) → (𝐺‘(𝑠 + ((π − 𝑋) − (-π − 𝑋)))) = (𝐺‘𝑠))
789		0ltpnf 12787	. . . . . . . 8 ⊢ 0 < +∞
790		pnfxr 10960	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
791		elioo2 13049	. . . . . . . . 9 ⊢ ((-π ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (0 ∈ (-π(,)+∞) ↔ (0 ∈ ℝ ∧ -π < 0 ∧ 0 < +∞)))
792	39, 790, 791	mp2an 688	. . . . . . . 8 ⊢ (0 ∈ (-π(,)+∞) ↔ (0 ∈ ℝ ∧ -π < 0 ∧ 0 < +∞))
793	707, 708, 789, 792	mpbir3an 1339	. . . . . . 7 ⊢ 0 ∈ (-π(,)+∞)
794	793	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ (-π(,)+∞))
795	223, 221, 109, 298, 207, 788, 478, 631, 667, 71, 794	fourierdlem105 43642	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]0) ↦ (𝐺‘𝑠)) ∈ 𝐿¹)
796	737, 795	eqeltrd 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]0) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) ∈ 𝐿¹)
797	716, 718, 734, 796	iblss 24874	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (-π(,)0) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) ∈ 𝐿¹)
798		elioore 13038	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)π) → 𝑠 ∈ ℝ)
799	798	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → 𝑠 ∈ ℝ)
800	799, 784	syldan 590	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ)
801	799, 800, 193	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → (𝐺‘𝑠) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
802	801	eqcomd 2744	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0(,)π)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) = (𝐺‘𝑠))
803	802	mpteq2dva 5170	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (0(,)π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) = (𝑠 ∈ (0(,)π) ↦ (𝐺‘𝑠)))
804		ioossicc 13094	. . . . . 6 ⊢ (0(,)π) ⊆ (0[,]π)
805	804	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0(,)π) ⊆ (0[,]π))
806		ioombl 24634	. . . . . 6 ⊢ (0(,)π) ∈ dom vol
807	806	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0(,)π) ∈ dom vol)
808	207	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0[,]π)) → 𝐺:ℝ⟶ℂ)
809		0red 10909	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]π)) → 0 ∈ ℝ)
810	22	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]π)) → π ∈ ℝ)
811		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]π)) → 𝑠 ∈ (0[,]π))
812		eliccre 42933	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 𝑠 ∈ (0[,]π)) → 𝑠 ∈ ℝ)
813	809, 810, 811, 812	syl3anc 1369	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]π)) → 𝑠 ∈ ℝ)
814	813	adantlr 711	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0[,]π)) → 𝑠 ∈ ℝ)
815	808, 814	ffvelrnd 6944	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (0[,]π)) → (𝐺‘𝑠) ∈ ℂ)
816		0xr 10953	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
817	816	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
818	790	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ^*)
819	711	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < π)
820		ltpnf 12785	. . . . . . . 8 ⊢ (π ∈ ℝ → π < +∞)
821	22, 820	mp1i 13	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π < +∞)
822	817, 818, 72, 819, 821	eliood 42926	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈ (0(,)+∞))
823	223, 221, 109, 298, 207, 788, 478, 631, 667, 706, 822	fourierdlem105 43642	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (0[,]π) ↦ (𝐺‘𝑠)) ∈ 𝐿¹)
824	805, 807, 815, 823	iblss 24874	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (0(,)π) ↦ (𝐺‘𝑠)) ∈ 𝐿¹)
825	803, 824	eqeltrd 2839	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (0(,)π) ↦ ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠))) ∈ 𝐿¹)
826	705, 72, 714, 699, 797, 825	itgsplitioo 24907	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫(-π(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠))
827	704, 826	eqtrd 2778	1 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 {crab 3067 Vcvv 3422 ∖ cdif 3880 ⊆ wss 3883 ifcif 4456 {csn 4558 class class class wbr 5070 ↦ cmpt 5153 dom cdm 5580 ran crn 5581 ↾ cres 5582 ∘ ccom 5584 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 (,)cioo 13008 [,]cicc 13011 ...cfz 13168 ..^cfzo 13311 mod cmo 13517 Σcsu 15325 sincsin 15701 cosccos 15702 πcpi 15704 –cn→ccncf 23945 volcvol 24532 𝐿¹cibl 24686 ∫citg 24687 lim_ℂ climc 24931

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cc 10122 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-symdif 4173 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-t1 22373 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-ovol 24533 df-vol 24534 df-mbf 24688 df-itg1 24689 df-itg2 24690 df-ibl 24691 df-itg 24692 df-0p 24739 df-ditg 24916 df-limc 24935 df-dv 24936

This theorem is referenced by: fourierdlem112 43649

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