fourierdlem101 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem101 46198

Description: Integral by substitution for a piecewise continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem101.d	⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))
fourierdlem101.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem101.g	⊢ 𝐺 = (𝑡 ∈ (-π[,]π) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
fourierdlem101.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem101.6	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem101.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
fourierdlem101.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem101.f	⊢ (𝜑 → 𝐹:(-π[,]π)⟶ℂ)
fourierdlem101.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem101.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
fourierdlem101.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))

**Assertion**
Ref	Expression
fourierdlem101	⊢ (𝜑 → ∫(-π[,]π)((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)) d𝑠)

Distinct variable groups: 𝐷,𝑠,𝑡 𝑡,𝐹 𝑖,𝐺,𝑠,𝑡 𝑡,𝐿 𝑖,𝑀,𝑠,𝑡 𝑚,𝑀,𝑝,𝑖 𝑛,𝑁,𝑠 𝑡,𝑁 𝑄,𝑖,𝑠,𝑡 𝑄,𝑝 𝑡,𝑅 𝑖,𝑋,𝑠,𝑡 𝜑,𝑖,𝑠,𝑡 𝜑,𝑛 Allowed substitution hints: 𝜑(𝑚,𝑝) 𝐷(𝑖,𝑚,𝑛,𝑝) 𝑃(𝑡,𝑖,𝑚,𝑛,𝑠,𝑝) 𝑄(𝑚,𝑛) 𝑅(𝑖,𝑚,𝑛,𝑠,𝑝) 𝐹(𝑖,𝑚,𝑛,𝑠,𝑝) 𝐺(𝑚,𝑛,𝑝) 𝐿(𝑖,𝑚,𝑛,𝑠,𝑝) 𝑀(𝑛) 𝑁(𝑖,𝑚,𝑝) 𝑋(𝑚,𝑛,𝑝)

Proof of Theorem fourierdlem101 Dummy variables 𝑟 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → 𝑡 ∈ (-π[,]π))
2		fourierdlem101.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:(-π[,]π)⟶ℂ)
3	2	ffvelcdmda 7058	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → (𝐹‘𝑡) ∈ ℂ)
4		fourierdlem101.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → 𝑁 ∈ ℕ)
6		pire 26372	. . . . . . . . . . . 12 ⊢ π ∈ ℝ
7	6	renegcli 11489	. . . . . . . . . . 11 ⊢ -π ∈ ℝ
8		eliccre 45496	. . . . . . . . . . 11 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ ∧ 𝑡 ∈ (-π[,]π)) → 𝑡 ∈ ℝ)
9	7, 6, 8	mp3an12 1453	. . . . . . . . . 10 ⊢ (𝑡 ∈ (-π[,]π) → 𝑡 ∈ ℝ)
10	9	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → 𝑡 ∈ ℝ)
11		fourierdlem101.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ)
12	11	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → 𝑋 ∈ ℝ)
13	10, 12	resubcld 11612	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → (𝑡 − 𝑋) ∈ ℝ)
14		fourierdlem101.d	. . . . . . . . 9 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))
15	14	dirkerre 46086	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑡 − 𝑋) ∈ ℝ) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
16	5, 13, 15	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
17	16	recnd 11208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℂ)
18	3, 17	mulcld 11200	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ℂ)
19		fourierdlem101.g	. . . . . 6 ⊢ 𝐺 = (𝑡 ∈ (-π[,]π) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
20	19	fvmpt2 6981	. . . . 5 ⊢ ((𝑡 ∈ (-π[,]π) ∧ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ℂ) → (𝐺‘𝑡) = ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
21	1, 18, 20	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → (𝐺‘𝑡) = ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
22	21	eqcomd 2736	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = (𝐺‘𝑡))
23	22	itgeq2dv 25689	. 2 ⊢ (𝜑 → ∫(-π[,]π)((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) d𝑡 = ∫(-π[,]π)(𝐺‘𝑡) d𝑡)
24		fourierdlem101.p	. . 3 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
25		fveq2 6860	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝑄‘𝑗) = (𝑄‘𝑖))
26	25	oveq1d 7404	. . . 4 ⊢ (𝑗 = 𝑖 → ((𝑄‘𝑗) − 𝑋) = ((𝑄‘𝑖) − 𝑋))
27	26	cbvmptv 5213	. . 3 ⊢ (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) − 𝑋)) = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) − 𝑋))
28		fourierdlem101.6	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
29		fourierdlem101.q	. . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
30	18, 19	fmptd 7088	. . 3 ⊢ (𝜑 → 𝐺:(-π[,]π)⟶ℂ)
31	19	reseq1i 5948	. . . . 5 ⊢ (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑡 ∈ (-π[,]π) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
32		ioossicc 13400	. . . . . . 7 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
33	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → -π ∈ ℝ)
34	33	rexrd 11230	. . . . . . . . 9 ⊢ (𝜑 → -π ∈ ℝ^*)
35	34	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ^*)
36	6	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → π ∈ ℝ)
37	36	rexrd 11230	. . . . . . . . 9 ⊢ (𝜑 → π ∈ ℝ^*)
38	37	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈ ℝ^*)
39	24, 28, 29	fourierdlem15 46113	. . . . . . . . 9 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
40	39	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
41		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
42	35, 38, 40, 41	fourierdlem8 46106	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
43	32, 42	sstrid 3960	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
44	43	resmptd 6013	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ (-π[,]π) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))))
45	31, 44	eqtrid 2777	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))))
46	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:(-π[,]π)⟶ℂ)
47	46, 43	feqresmpt 6932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑡)))
48		fourierdlem101.fcn	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
49	47, 48	eqeltrrd 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑡)) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
50		eqidd 2731	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)))
51		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟)) → 𝑠 = ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟))
52		eqidd 2731	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋)) = (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋)))
53		oveq1 7396	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑟 → (𝑡 − 𝑋) = (𝑟 − 𝑋))
54	53	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑡 = 𝑟) → (𝑡 − 𝑋) = (𝑟 − 𝑋))
55		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
56		elioore 13342	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑟 ∈ ℝ)
57	56	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑟 ∈ ℝ)
58	11	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
59	57, 58	resubcld 11612	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑟 − 𝑋) ∈ ℝ)
60	59	adantlr 715	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑟 − 𝑋) ∈ ℝ)
61	52, 54, 55, 60	fvmptd 6977	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟) = (𝑟 − 𝑋))
62	61	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟)) → ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟) = (𝑟 − 𝑋))
63	51, 62	eqtrd 2765	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟)) → 𝑠 = (𝑟 − 𝑋))
64	63	fveq2d 6864	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟)) → ((𝐷‘𝑁)‘𝑠) = ((𝐷‘𝑁)‘(𝑟 − 𝑋)))
65		elioore 13342	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
66	65	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑡 ∈ ℝ)
67	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
68	66, 67	resubcld 11612	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑡 − 𝑋) ∈ ℝ)
69	68	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑡 − 𝑋) ∈ ℝ)
70		eqid 2730	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋)) = (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))
71	69, 70	fmptd 7088	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋)):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℝ)
72	71	ffvelcdmda 7058	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟) ∈ ℝ)
73	4	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑁 ∈ ℕ)
74	14	dirkerre 46086	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑟 − 𝑋) ∈ ℝ) → ((𝐷‘𝑁)‘(𝑟 − 𝑋)) ∈ ℝ)
75	73, 60, 74	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐷‘𝑁)‘(𝑟 − 𝑋)) ∈ ℝ)
76	50, 64, 72, 75	fvmptd 6977	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠))‘((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟)) = ((𝐷‘𝑁)‘(𝑟 − 𝑋)))
77	76	eqcomd 2736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐷‘𝑁)‘(𝑟 − 𝑋)) = ((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠))‘((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟)))
78	77	mpteq2dva 5202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑟 − 𝑋))) = (𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠))‘((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟))))
79	53	fveq2d 6864	. . . . . . . . 9 ⊢ (𝑡 = 𝑟 → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) = ((𝐷‘𝑁)‘(𝑟 − 𝑋)))
80	79	cbvmptv 5213	. . . . . . . 8 ⊢ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = (𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑟 − 𝑋)))
81	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = (𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑟 − 𝑋))))
82	14	dirkerre 46086	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑁)‘𝑠) ∈ ℝ)
83	4, 82	sylan 580	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑁)‘𝑠) ∈ ℝ)
84		eqid 2730	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠))
85	83, 84	fmptd 7088	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)):ℝ⟶ℝ)
86	85	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)):ℝ⟶ℝ)
87		fcompt 7107	. . . . . . . 8 ⊢ (((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)):ℝ⟶ℝ ∧ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋)):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℝ) → ((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∘ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))) = (𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠))‘((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟))))
88	86, 71, 87	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∘ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))) = (𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠))‘((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟))))
89	78, 81, 88	3eqtr4d 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = ((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∘ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))))
90		eqid 2730	. . . . . . . 8 ⊢ (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) = (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋))
91		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → 𝑡 ∈ ℂ)
92	11	recnd 11208	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℂ)
93	92	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → 𝑋 ∈ ℂ)
94	91, 93	negsubd 11545	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → (𝑡 + -𝑋) = (𝑡 − 𝑋))
95	94	eqcomd 2736	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → (𝑡 − 𝑋) = (𝑡 + -𝑋))
96	95	mpteq2dva 5202	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) = (𝑡 ∈ ℂ ↦ (𝑡 + -𝑋)))
97	92	negcld 11526	. . . . . . . . . . 11 ⊢ (𝜑 → -𝑋 ∈ ℂ)
98		eqid 2730	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℂ ↦ (𝑡 + -𝑋)) = (𝑡 ∈ ℂ ↦ (𝑡 + -𝑋))
99	98	addccncf 24816	. . . . . . . . . . 11 ⊢ (-𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ (𝑡 + -𝑋)) ∈ (ℂ–cn→ℂ))
100	97, 99	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (𝑡 + -𝑋)) ∈ (ℂ–cn→ℂ))
101	96, 100	eqeltrd 2829	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ∈ (ℂ–cn→ℂ))
102	101	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ∈ (ℂ–cn→ℂ))
103		ioossre 13374	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
104		ax-resscn 11131	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
105	103, 104	sstri 3958	. . . . . . . . 9 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
106	105	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
107	104	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ℝ ⊆ ℂ)
108	90, 102, 106, 107, 69	cncfmptssg 45862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋)) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℝ))
109	83	recnd 11208	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑁)‘𝑠) ∈ ℂ)
110	109, 84	fmptd 7088	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)):ℝ⟶ℂ)
111		ssid 3971	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
112	14	dirkerf 46088	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁):ℝ⟶ℝ)
113	4, 112	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷‘𝑁):ℝ⟶ℝ)
114	113	feqmptd 6931	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷‘𝑁) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)))
115	14	dirkercncf 46098	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁) ∈ (ℝ–cn→ℝ))
116	4, 115	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷‘𝑁) ∈ (ℝ–cn→ℝ))
117	114, 116	eqeltrrd 2830	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∈ (ℝ–cn→ℝ))
118		cncfcdm 24797	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∈ (ℝ–cn→ℝ)) → ((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∈ (ℝ–cn→ℂ) ↔ (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)):ℝ⟶ℂ))
119	111, 117, 118	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → ((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∈ (ℝ–cn→ℂ) ↔ (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)):ℝ⟶ℂ))
120	110, 119	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∈ (ℝ–cn→ℂ))
121	120	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∈ (ℝ–cn→ℂ))
122	108, 121	cncfco 24806	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∘ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
123	89, 122	eqeltrd 2829	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
124	49, 123	mulcncf 25352	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
125	45, 124	eqeltrd 2829	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
126		cncff 24792	. . . . . . . 8 ⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
127	48, 126	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
128	113	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑁):ℝ⟶ℝ)
129		elioore 13342	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℝ)
130	129	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
131	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
132	130, 131	resubcld 11612	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑠 − 𝑋) ∈ ℝ)
133	128, 132	ffvelcdmd 7059	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐷‘𝑁)‘(𝑠 − 𝑋)) ∈ ℝ)
134	133	recnd 11208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐷‘𝑁)‘(𝑠 − 𝑋)) ∈ ℂ)
135		eqid 2730	. . . . . . . . 9 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))
136	134, 135	fmptd 7088	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
137	136	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
138		eqid 2730	. . . . . . 7 ⊢ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) = (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)))
139		fourierdlem101.r	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
140		oveq1 7396	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑄‘𝑖) → (𝑡 − 𝑋) = ((𝑄‘𝑖) − 𝑋))
141	140	fveq2d 6864	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑄‘𝑖) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) = ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)))
142	141	eqcomd 2736	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑄‘𝑖) → ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
143	142	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ 𝑡 = (𝑄‘𝑖)) → ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
144		eqidd 2731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ ¬ 𝑡 = (𝑄‘𝑖)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))))
145		oveq1 7396	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (𝑠 − 𝑋) = (𝑡 − 𝑋))
146	145	fveq2d 6864	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → ((𝐷‘𝑁)‘(𝑠 − 𝑋)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
147	146	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ ¬ 𝑡 = (𝑄‘𝑖)) ∧ 𝑠 = 𝑡) → ((𝐷‘𝑁)‘(𝑠 − 𝑋)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
148		velsn 4607	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ {(𝑄‘𝑖)} ↔ 𝑡 = (𝑄‘𝑖))
149	148	notbii 320	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑡 ∈ {(𝑄‘𝑖)} ↔ ¬ 𝑡 = (𝑄‘𝑖))
150		elunnel2 4120	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ∧ ¬ 𝑡 ∈ {(𝑄‘𝑖)}) → 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
151	149, 150	sylan2br 595	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ∧ ¬ 𝑡 = (𝑄‘𝑖)) → 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
152	151	adantll 714	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ ¬ 𝑡 = (𝑄‘𝑖)) → 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
153	113	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝐷‘𝑁):ℝ⟶ℝ)
154		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 = (𝑄‘𝑖)) → 𝑡 = (𝑄‘𝑖))
155	9	ssriv 3952	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-π[,]π) ⊆ ℝ
156		fzossfz 13645	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0..^𝑀) ⊆ (0...𝑀)
157	156, 41	sselid 3946	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
158	40, 157	ffvelcdmd 7059	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (-π[,]π))
159	155, 158	sselid 3946	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
160	159	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 = (𝑄‘𝑖)) → (𝑄‘𝑖) ∈ ℝ)
161	154, 160	eqeltrd 2829	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 = (𝑄‘𝑖)) → 𝑡 ∈ ℝ)
162	161	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ 𝑡 = (𝑄‘𝑖)) → 𝑡 ∈ ℝ)
163	152, 65	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ ¬ 𝑡 = (𝑄‘𝑖)) → 𝑡 ∈ ℝ)
164	162, 163	pm2.61dan 812	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → 𝑡 ∈ ℝ)
165	11	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → 𝑋 ∈ ℝ)
166	164, 165	resubcld 11612	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝑡 − 𝑋) ∈ ℝ)
167	153, 166	ffvelcdmd 7059	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
168	167	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ ¬ 𝑡 = (𝑄‘𝑖)) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
169	144, 147, 152, 168	fvmptd 6977	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ ¬ 𝑡 = (𝑄‘𝑖)) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
170	143, 169	ifeqda 4527	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → if(𝑡 = (𝑄‘𝑖), ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
171	170	mpteq2dva 5202	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ if(𝑡 = (𝑄‘𝑖), ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) = (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
172	113	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘𝑁):ℝ⟶ℝ)
173		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}))
174		elun 4118	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↔ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝑄‘𝑖)}))
175	173, 174	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝑄‘𝑖)}))
176	175	adantlr 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝑄‘𝑖)}))
177		elsni 4608	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 ∈ {(𝑄‘𝑖)} → 𝑠 = (𝑄‘𝑖))
178	177	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝑄‘𝑖)}) → 𝑠 = (𝑄‘𝑖))
179	159	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝑄‘𝑖)}) → (𝑄‘𝑖) ∈ ℝ)
180	178, 179	eqeltrd 2829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝑄‘𝑖)}) → 𝑠 ∈ ℝ)
181	180	ex 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ {(𝑄‘𝑖)} → 𝑠 ∈ ℝ))
182	181	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝑠 ∈ {(𝑄‘𝑖)} → 𝑠 ∈ ℝ))
183		pm3.44 961	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℝ) ∧ (𝑠 ∈ {(𝑄‘𝑖)} → 𝑠 ∈ ℝ)) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝑄‘𝑖)}) → 𝑠 ∈ ℝ))
184	129, 182, 183	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝑄‘𝑖)}) → 𝑠 ∈ ℝ))
185	176, 184	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → 𝑠 ∈ ℝ)
186	11	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → 𝑋 ∈ ℝ)
187	185, 186	resubcld 11612	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝑠 − 𝑋) ∈ ℝ)
188		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋)) = (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋))
189	187, 188	fmptd 7088	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋)):(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})⟶ℝ)
190		fcompt 7107	. . . . . . . . . . . . . . 15 ⊢ (((𝐷‘𝑁):ℝ⟶ℝ ∧ (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋)):(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})⟶ℝ) → ((𝐷‘𝑁) ∘ (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋))) = (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘((𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋))‘𝑡))))
191	172, 189, 190	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑁) ∘ (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋))) = (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘((𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋))‘𝑡))))
192		eqidd 2731	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋)) = (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋)))
193	145	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ 𝑠 = 𝑡) → (𝑠 − 𝑋) = (𝑡 − 𝑋))
194		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}))
195	192, 193, 194, 166	fvmptd 6977	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → ((𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋))‘𝑡) = (𝑡 − 𝑋))
196	195	fveq2d 6864	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → ((𝐷‘𝑁)‘((𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋))‘𝑡)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
197	196	mpteq2dva 5202	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘((𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋))‘𝑡))) = (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
198	191, 197	eqtr2d 2766	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = ((𝐷‘𝑁) ∘ (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋))))
199		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℂ ↦ (𝑠 − 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 − 𝑋))
200		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
201	92	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → 𝑋 ∈ ℂ)
202	200, 201	negsubd 11545	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → (𝑠 + -𝑋) = (𝑠 − 𝑋))
203	202	eqcomd 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → (𝑠 − 𝑋) = (𝑠 + -𝑋))
204	203	mpteq2dva 5202	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 − 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + -𝑋)))
205		eqid 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℂ ↦ (𝑠 + -𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + -𝑋))
206	205	addccncf 24816	. . . . . . . . . . . . . . . . . . 19 ⊢ (-𝑋 ∈ ℂ → (𝑠 ∈ ℂ ↦ (𝑠 + -𝑋)) ∈ (ℂ–cn→ℂ))
207	97, 206	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 + -𝑋)) ∈ (ℂ–cn→ℂ))
208	204, 207	eqeltrd 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 − 𝑋)) ∈ (ℂ–cn→ℂ))
209	208	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℂ ↦ (𝑠 − 𝑋)) ∈ (ℂ–cn→ℂ))
210	159	recnd 11208	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
211	210	snssd 4775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {(𝑄‘𝑖)} ⊆ ℂ)
212	106, 211	unssd 4157	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ⊆ ℂ)
213	199, 209, 212, 107, 187	cncfmptssg 45862	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋)) ∈ ((((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})–cn→ℝ))
214	114, 120	eqeltrd 2829	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐷‘𝑁) ∈ (ℝ–cn→ℂ))
215	214	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘𝑁) ∈ (ℝ–cn→ℂ))
216	213, 215	cncfco 24806	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑁) ∘ (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋))) ∈ ((((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})–cn→ℂ))
217		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
218		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) = ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}))
219	217	cnfldtop 24677	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘ℂ_fld) ∈ Top
220		unicntop 24679	. . . . . . . . . . . . . . . . . . 19 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
221	220	restid 17402	. . . . . . . . . . . . . . . . . 18 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
222	219, 221	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
223	222	eqcomi 2739	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
224	217, 218, 223	cncfcn 24809	. . . . . . . . . . . . . . 15 ⊢ (((((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) Cn (TopOpen‘ℂ_fld)))
225	212, 111, 224	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) Cn (TopOpen‘ℂ_fld)))
226	216, 225	eleqtrd 2831	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑁) ∘ (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋))) ∈ (((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) Cn (TopOpen‘ℂ_fld)))
227	198, 226	eqeltrd 2829	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ (((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) Cn (TopOpen‘ℂ_fld)))
228	217	cnfldtopon 24676	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
229		resttopon 23054	. . . . . . . . . . . . . 14 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∈ (TopOn‘(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})))
230	228, 212, 229	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∈ (TopOn‘(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})))
231		cncnp 23173	. . . . . . . . . . . . 13 ⊢ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∈ (TopOn‘(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)) → ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ (((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) Cn (TopOpen‘ℂ_fld)) ↔ ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))):(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})⟶ℂ ∧ ∀𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})(𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘𝑠))))
232	230, 228, 231	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ (((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) Cn (TopOpen‘ℂ_fld)) ↔ ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))):(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})⟶ℂ ∧ ∀𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})(𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘𝑠))))
233	227, 232	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))):(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})⟶ℂ ∧ ∀𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})(𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘𝑠)))
234	233	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})(𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘𝑠))
235		eqidd 2731	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = (𝑄‘𝑖))
236		elsng 4605	. . . . . . . . . . . . . 14 ⊢ ((𝑄‘𝑖) ∈ ℝ → ((𝑄‘𝑖) ∈ {(𝑄‘𝑖)} ↔ (𝑄‘𝑖) = (𝑄‘𝑖)))
237	159, 236	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) ∈ {(𝑄‘𝑖)} ↔ (𝑄‘𝑖) = (𝑄‘𝑖)))
238	235, 237	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ {(𝑄‘𝑖)})
239	238	olcd 874	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ (𝑄‘𝑖) ∈ {(𝑄‘𝑖)}))
240		elun 4118	. . . . . . . . . . 11 ⊢ ((𝑄‘𝑖) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↔ ((𝑄‘𝑖) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ (𝑄‘𝑖) ∈ {(𝑄‘𝑖)}))
241	239, 240	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}))
242		fveq2 6860	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑄‘𝑖) → ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘𝑠) = ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘(𝑄‘𝑖)))
243	242	eleq2d 2815	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑄‘𝑖) → ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘𝑠) ↔ (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘(𝑄‘𝑖))))
244	243	rspccva 3590	. . . . . . . . . 10 ⊢ ((∀𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})(𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘𝑠) ∧ (𝑄‘𝑖) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘(𝑄‘𝑖)))
245	234, 241, 244	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘(𝑄‘𝑖)))
246	171, 245	eqeltrd 2829	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ if(𝑡 = (𝑄‘𝑖), ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘(𝑄‘𝑖)))
247		eqid 2730	. . . . . . . . 9 ⊢ (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ if(𝑡 = (𝑄‘𝑖), ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) = (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ if(𝑡 = (𝑄‘𝑖), ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)))
248	218, 217, 247, 137, 106, 210	ellimc 25780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))) lim_ℂ (𝑄‘𝑖)) ↔ (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ if(𝑡 = (𝑄‘𝑖), ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) CnP (TopOpen‘ℂ_fld))‘(𝑄‘𝑖))))
249	246, 248	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))) lim_ℂ (𝑄‘𝑖)))
250	127, 137, 138, 139, 249	mullimcf 45614	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋))) ∈ ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) lim_ℂ (𝑄‘𝑖)))
251		fvres 6879	. . . . . . . . . 10 ⊢ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) = (𝐹‘𝑡))
252	251	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) = (𝐹‘𝑡))
253	252	oveq1d 7404	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)) = ((𝐹‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)))
254	253	mpteq2dva 5202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) = (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))))
255	254	oveq1d 7404	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) lim_ℂ (𝑄‘𝑖)) = ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) lim_ℂ (𝑄‘𝑖)))
256	250, 255	eleqtrd 2831	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋))) ∈ ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) lim_ℂ (𝑄‘𝑖)))
257		eqidd 2731	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))))
258		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 𝑡) → 𝑠 = 𝑡)
259	258	oveq1d 7404	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 𝑡) → (𝑠 − 𝑋) = (𝑡 − 𝑋))
260	259	fveq2d 6864	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 𝑡) → ((𝐷‘𝑁)‘(𝑠 − 𝑋)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
261		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
262	113	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑁):ℝ⟶ℝ)
263	262, 69	ffvelcdmd 7059	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
264	257, 260, 261, 263	fvmptd 6977	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
265	264	oveq2d 7405	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)) = ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
266	265	mpteq2dva 5202	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) = (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))))
267	266	oveq1d 7404	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) lim_ℂ (𝑄‘𝑖)) = ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))) lim_ℂ (𝑄‘𝑖)))
268	256, 267	eleqtrd 2831	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋))) ∈ ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))) lim_ℂ (𝑄‘𝑖)))
269	45	eqcomd 2736	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
270	269	oveq1d 7404	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))) lim_ℂ (𝑄‘𝑖)) = ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
271	268, 270	eleqtrd 2831	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋))) ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
272		fourierdlem101.l	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
273		iftrue 4496	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑄‘(𝑖 + 1)) → if(𝑡 = (𝑄‘(𝑖 + 1)), ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)) = ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)))
274		oveq1 7396	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑄‘(𝑖 + 1)) → (𝑡 − 𝑋) = ((𝑄‘(𝑖 + 1)) − 𝑋))
275	274	eqcomd 2736	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑄‘(𝑖 + 1)) → ((𝑄‘(𝑖 + 1)) − 𝑋) = (𝑡 − 𝑋))
276	275	fveq2d 6864	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑄‘(𝑖 + 1)) → ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
277	273, 276	eqtrd 2765	. . . . . . . . . 10 ⊢ (𝑡 = (𝑄‘(𝑖 + 1)) → if(𝑡 = (𝑄‘(𝑖 + 1)), ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
278	277	adantl 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ 𝑡 = (𝑄‘(𝑖 + 1))) → if(𝑡 = (𝑄‘(𝑖 + 1)), ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
279		iffalse 4499	. . . . . . . . . . 11 ⊢ (¬ 𝑡 = (𝑄‘(𝑖 + 1)) → if(𝑡 = (𝑄‘(𝑖 + 1)), ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))
280	279	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → if(𝑡 = (𝑄‘(𝑖 + 1)), ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))
281		eqidd 2731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))))
282	146	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) ∧ 𝑠 = 𝑡) → ((𝐷‘𝑁)‘(𝑠 − 𝑋)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
283		elun 4118	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↔ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ 𝑡 ∈ {(𝑄‘(𝑖 + 1))}))
284	283	biimpi 216	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ 𝑡 ∈ {(𝑄‘(𝑖 + 1))}))
285	284	orcomd 871	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) → (𝑡 ∈ {(𝑄‘(𝑖 + 1))} ∨ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
286	285	ad2antlr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → (𝑡 ∈ {(𝑄‘(𝑖 + 1))} ∨ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
287		velsn 4607	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ {(𝑄‘(𝑖 + 1))} ↔ 𝑡 = (𝑄‘(𝑖 + 1)))
288	287	notbii 320	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑡 ∈ {(𝑄‘(𝑖 + 1))} ↔ ¬ 𝑡 = (𝑄‘(𝑖 + 1)))
289	288	biimpri 228	. . . . . . . . . . . . 13 ⊢ (¬ 𝑡 = (𝑄‘(𝑖 + 1)) → ¬ 𝑡 ∈ {(𝑄‘(𝑖 + 1))})
290	289	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → ¬ 𝑡 ∈ {(𝑄‘(𝑖 + 1))})
291		pm2.53 851	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ {(𝑄‘(𝑖 + 1))} ∨ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (¬ 𝑡 ∈ {(𝑄‘(𝑖 + 1))} → 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
292	286, 290, 291	sylc 65	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
293	172	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → (𝐷‘𝑁):ℝ⟶ℝ)
294	292, 65	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
295	11	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → 𝑋 ∈ ℝ)
296	294, 295	resubcld 11612	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → (𝑡 − 𝑋) ∈ ℝ)
297	293, 296	ffvelcdmd 7059	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
298	281, 282, 292, 297	fvmptd 6977	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
299	280, 298	eqtrd 2765	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → if(𝑡 = (𝑄‘(𝑖 + 1)), ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
300	278, 299	pm2.61dan 812	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → if(𝑡 = (𝑄‘(𝑖 + 1)), ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
301	300	mpteq2dva 5202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ if(𝑡 = (𝑄‘(𝑖 + 1)), ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) = (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
302		eqid 2730	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = (𝑡 ∈ ℝ ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
303	104	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℝ ⊆ ℂ)
304		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
305	11	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑋 ∈ ℝ)
306	304, 305	resubcld 11612	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (𝑡 − 𝑋) ∈ ℝ)
307	90, 101, 303, 303, 306	cncfmptssg 45862	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑡 ∈ ℝ ↦ (𝑡 − 𝑋)) ∈ (ℝ–cn→ℝ))
308	307, 214	cncfcompt 45874	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑡 ∈ ℝ ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ (ℝ–cn→ℂ))
309	308	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ℝ ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ (ℝ–cn→ℂ))
310	103	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
311		fzofzp1 13731	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
312	311	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
313	40, 312	ffvelcdmd 7059	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ (-π[,]π))
314	155, 313	sselid 3946	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
315	314	snssd 4775	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {(𝑄‘(𝑖 + 1))} ⊆ ℝ)
316	310, 315	unssd 4157	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ⊆ ℝ)
317	111	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ℂ ⊆ ℂ)
318	172	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → (𝐷‘𝑁):ℝ⟶ℝ)
319	316	sselda 3948	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → 𝑡 ∈ ℝ)
320	11	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → 𝑋 ∈ ℝ)
321	319, 320	resubcld 11612	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → (𝑡 − 𝑋) ∈ ℝ)
322	318, 321	ffvelcdmd 7059	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
323	322	recnd 11208	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℂ)
324	302, 309, 316, 317, 323	cncfmptssg 45862	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})–cn→ℂ))
325	155, 104	sstri 3958	. . . . . . . . . . . . . . 15 ⊢ (-π[,]π) ⊆ ℂ
326	325, 313	sselid 3946	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
327	326	snssd 4775	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {(𝑄‘(𝑖 + 1))} ⊆ ℂ)
328	106, 327	unssd 4157	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ⊆ ℂ)
329		eqid 2730	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) = ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}))
330	217, 329, 223	cncfcn 24809	. . . . . . . . . . . 12 ⊢ (((((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) Cn (TopOpen‘ℂ_fld)))
331	328, 111, 330	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) Cn (TopOpen‘ℂ_fld)))
332	324, 331	eleqtrd 2831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ (((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) Cn (TopOpen‘ℂ_fld)))
333		resttopon 23054	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∈ (TopOn‘(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})))
334	228, 328, 333	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∈ (TopOn‘(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})))
335		cncnp 23173	. . . . . . . . . . 11 ⊢ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∈ (TopOn‘(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)) → ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ (((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) Cn (TopOpen‘ℂ_fld)) ↔ ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))):(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})⟶ℂ ∧ ∀𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})(𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘𝑠))))
336	334, 228, 335	sylancl 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ (((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) Cn (TopOpen‘ℂ_fld)) ↔ ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))):(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})⟶ℂ ∧ ∀𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})(𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘𝑠))))
337	332, 336	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))):(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})⟶ℂ ∧ ∀𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})(𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘𝑠)))
338	337	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})(𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘𝑠))
339		eqidd 2731	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
340		elsng 4605	. . . . . . . . . . . 12 ⊢ ((𝑄‘(𝑖 + 1)) ∈ ℝ → ((𝑄‘(𝑖 + 1)) ∈ {(𝑄‘(𝑖 + 1))} ↔ (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1))))
341	314, 340	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) ∈ {(𝑄‘(𝑖 + 1))} ↔ (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1))))
342	339, 341	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ {(𝑄‘(𝑖 + 1))})
343	342	olcd 874	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ (𝑄‘(𝑖 + 1)) ∈ {(𝑄‘(𝑖 + 1))}))
344		elun 4118	. . . . . . . . 9 ⊢ ((𝑄‘(𝑖 + 1)) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↔ ((𝑄‘(𝑖 + 1)) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ (𝑄‘(𝑖 + 1)) ∈ {(𝑄‘(𝑖 + 1))}))
345	343, 344	sylibr 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}))
346		fveq2 6860	. . . . . . . . . 10 ⊢ (𝑠 = (𝑄‘(𝑖 + 1)) → ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘𝑠) = ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘(𝑄‘(𝑖 + 1))))
347	346	eleq2d 2815	. . . . . . . . 9 ⊢ (𝑠 = (𝑄‘(𝑖 + 1)) → ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘𝑠) ↔ (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘(𝑄‘(𝑖 + 1)))))
348	347	rspccva 3590	. . . . . . . 8 ⊢ ((∀𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})(𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘𝑠) ∧ (𝑄‘(𝑖 + 1)) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘(𝑄‘(𝑖 + 1))))
349	338, 345, 348	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘(𝑄‘(𝑖 + 1))))
350	301, 349	eqeltrd 2829	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ if(𝑡 = (𝑄‘(𝑖 + 1)), ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘(𝑄‘(𝑖 + 1))))
351		eqid 2730	. . . . . . 7 ⊢ (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ if(𝑡 = (𝑄‘(𝑖 + 1)), ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) = (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ if(𝑡 = (𝑄‘(𝑖 + 1)), ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡)))
352	329, 217, 351, 137, 106, 326	ellimc 25780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))) lim_ℂ (𝑄‘(𝑖 + 1))) ↔ (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↦ if(𝑡 = (𝑄‘(𝑖 + 1)), ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)), ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) ∈ ((((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) CnP (TopOpen‘ℂ_fld))‘(𝑄‘(𝑖 + 1)))))
353	350, 352	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))) lim_ℂ (𝑄‘(𝑖 + 1))))
354	127, 137, 138, 272, 353	mullimcf 45614	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐿 · ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋))) ∈ ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) lim_ℂ (𝑄‘(𝑖 + 1))))
355	266, 254, 45	3eqtr4d 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
356	355	oveq1d 7404	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) · ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋)))‘𝑡))) lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
357	354, 356	eleqtrd 2831	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐿 · ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋))) ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
358	24, 27, 28, 29, 11, 30, 125, 271, 357	fourierdlem93 46190	. 2 ⊢ (𝜑 → ∫(-π[,]π)(𝐺‘𝑡) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐺‘(𝑋 + 𝑠)) d𝑠)
359	19	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝐺 = (𝑡 ∈ (-π[,]π) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))))
360		fveq2 6860	. . . . . . 7 ⊢ (𝑡 = (𝑋 + 𝑠) → (𝐹‘𝑡) = (𝐹‘(𝑋 + 𝑠)))
361	360	oveq1d 7404	. . . . . 6 ⊢ (𝑡 = (𝑋 + 𝑠) → ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
362	361	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) ∧ 𝑡 = (𝑋 + 𝑠)) → ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
363		oveq1 7396	. . . . . . . 8 ⊢ (𝑡 = (𝑋 + 𝑠) → (𝑡 − 𝑋) = ((𝑋 + 𝑠) − 𝑋))
364	92	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑋 ∈ ℂ)
365	33, 11	resubcld 11612	. . . . . . . . . . . 12 ⊢ (𝜑 → (-π − 𝑋) ∈ ℝ)
366	365	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (-π − 𝑋) ∈ ℝ)
367	36, 11	resubcld 11612	. . . . . . . . . . . 12 ⊢ (𝜑 → (π − 𝑋) ∈ ℝ)
368	367	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (π − 𝑋) ∈ ℝ)
369		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋)))
370		eliccre 45496	. . . . . . . . . . 11 ⊢ (((-π − 𝑋) ∈ ℝ ∧ (π − 𝑋) ∈ ℝ ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ∈ ℝ)
371	366, 368, 369, 370	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ∈ ℝ)
372	371	recnd 11208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ∈ ℂ)
373	364, 372	pncan2d 11541	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → ((𝑋 + 𝑠) − 𝑋) = 𝑠)
374	363, 373	sylan9eqr 2787	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) ∧ 𝑡 = (𝑋 + 𝑠)) → (𝑡 − 𝑋) = 𝑠)
375	374	fveq2d 6864	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) ∧ 𝑡 = (𝑋 + 𝑠)) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) = ((𝐷‘𝑁)‘𝑠))
376	375	oveq2d 7405	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) ∧ 𝑡 = (𝑋 + 𝑠)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)))
377	362, 376	eqtrd 2765	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) ∧ 𝑡 = (𝑋 + 𝑠)) → ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)))
378	7	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → -π ∈ ℝ)
379	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → π ∈ ℝ)
380	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑋 ∈ ℝ)
381	380, 371	readdcld 11209	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + 𝑠) ∈ ℝ)
382	33	recnd 11208	. . . . . . . . 9 ⊢ (𝜑 → -π ∈ ℂ)
383	92, 382	pncan3d 11542	. . . . . . . 8 ⊢ (𝜑 → (𝑋 + (-π − 𝑋)) = -π)
384	383	eqcomd 2736	. . . . . . 7 ⊢ (𝜑 → -π = (𝑋 + (-π − 𝑋)))
385	384	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → -π = (𝑋 + (-π − 𝑋)))
386		elicc2 13378	. . . . . . . . . 10 ⊢ (((-π − 𝑋) ∈ ℝ ∧ (π − 𝑋) ∈ ℝ) → (𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋)) ↔ (𝑠 ∈ ℝ ∧ (-π − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (π − 𝑋))))
387	366, 368, 386	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋)) ↔ (𝑠 ∈ ℝ ∧ (-π − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (π − 𝑋))))
388	369, 387	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑠 ∈ ℝ ∧ (-π − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (π − 𝑋)))
389	388	simp2d 1143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (-π − 𝑋) ≤ 𝑠)
390	366, 371, 380, 389	leadd2dd 11799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + (-π − 𝑋)) ≤ (𝑋 + 𝑠))
391	385, 390	eqbrtrd 5131	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → -π ≤ (𝑋 + 𝑠))
392	388	simp3d 1144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ≤ (π − 𝑋))
393	371, 368, 380, 392	leadd2dd 11799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + 𝑠) ≤ (𝑋 + (π − 𝑋)))
394		picn 26373	. . . . . . . 8 ⊢ π ∈ ℂ
395	394	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → π ∈ ℂ)
396	364, 395	pncan3d 11542	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + (π − 𝑋)) = π)
397	393, 396	breqtrd 5135	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + 𝑠) ≤ π)
398	378, 379, 381, 391, 397	eliccd 45495	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + 𝑠) ∈ (-π[,]π))
399	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝐹:(-π[,]π)⟶ℂ)
400	399, 398	ffvelcdmd 7059	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
401	371, 109	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → ((𝐷‘𝑁)‘𝑠) ∈ ℂ)
402	400, 401	mulcld 11200	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)) ∈ ℂ)
403	359, 377, 398, 402	fvmptd 6977	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝐺‘(𝑋 + 𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)))
404	403	itgeq2dv 25689	. 2 ⊢ (𝜑 → ∫((-π − 𝑋)[,](π − 𝑋))(𝐺‘(𝑋 + 𝑠)) d𝑠 = ∫((-π − 𝑋)[,](π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)) d𝑠)
405	23, 358, 404	3eqtrd 2769	1 ⊢ (𝜑 → ∫(-π[,]π)((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)) d𝑠)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 {crab 3408 ∪ cun 3914 ⊆ wss 3916 ifcif 4490 {csn 4591 class class class wbr 5109 ↦ cmpt 5190 ↾ cres 5642 ∘ ccom 5644 ⟶wf 6509 ‘cfv 6513 (class class class)co 7389 ↑_m cmap 8801 ℂcc 11072 ℝcr 11073 0cc0 11074 1c1 11075 + caddc 11077 · cmul 11079 ℝ^*cxr 11213 < clt 11214 ≤ cle 11215 − cmin 11411 -cneg 11412 / cdiv 11841 ℕcn 12187 2c2 12242 (,)cioo 13312 [,]cicc 13315 ...cfz 13474 ..^cfzo 13621 mod cmo 13837 sincsin 16035 πcpi 16038 ↾_t crest 17389 TopOpenctopn 17390 ℂ_fldccnfld 21270 Topctop 22786 TopOnctopon 22803 Cn ccn 23117 CnP ccnp 23118 –cn→ccncf 24775 ∫citg 25525 lim_ℂ climc 25769

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-inf2 9600 ax-cc 10394 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-addf 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-symdif 4218 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-disj 5077 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-ofr 7656 df-om 7845 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-oadd 8440 df-omul 8441 df-er 8673 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9319 df-fi 9368 df-sup 9399 df-inf 9400 df-oi 9469 df-dju 9860 df-card 9898 df-acn 9901 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-z 12536 df-dec 12656 df-uz 12800 df-q 12914 df-rp 12958 df-xneg 13078 df-xadd 13079 df-xmul 13080 df-ioo 13316 df-ioc 13317 df-ico 13318 df-icc 13319 df-fz 13475 df-fzo 13622 df-fl 13760 df-mod 13838 df-seq 13973 df-exp 14033 df-fac 14245 df-bc 14274 df-hash 14302 df-shft 15039 df-cj 15071 df-re 15072 df-im 15073 df-sqrt 15207 df-abs 15208 df-limsup 15443 df-clim 15460 df-rlim 15461 df-sum 15659 df-ef 16039 df-sin 16041 df-cos 16042 df-pi 16044 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17391 df-topn 17392 df-0g 17410 df-gsum 17411 df-topgen 17412 df-pt 17413 df-prds 17416 df-xrs 17471 df-qtop 17476 df-imas 17477 df-xps 17479 df-mre 17553 df-mrc 17554 df-acs 17556 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18717 df-mulg 19006 df-cntz 19255 df-cmn 19718 df-psmet 21262 df-xmet 21263 df-met 21264 df-bl 21265 df-mopn 21266 df-fbas 21267 df-fg 21268 df-cnfld 21271 df-top 22787 df-topon 22804 df-topsp 22826 df-bases 22839 df-cld 22912 df-ntr 22913 df-cls 22914 df-nei 22991 df-lp 23029 df-perf 23030 df-cn 23120 df-cnp 23121 df-t1 23207 df-haus 23208 df-cmp 23280 df-tx 23455 df-hmeo 23648 df-fil 23739 df-fm 23831 df-flim 23832 df-flf 23833 df-xms 24214 df-ms 24215 df-tms 24216 df-cncf 24777 df-ovol 25371 df-vol 25372 df-mbf 25526 df-itg1 25527 df-itg2 25528 df-ibl 25529 df-itg 25530 df-0p 25577 df-ditg 25754 df-limc 25773 df-dv 25774

This theorem is referenced by: fourierdlem111 46208

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