fourierdlem101 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem101 46779

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 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → 𝑡 ∈ (-π[,]π))
2		fourierdlem101.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:(-π[,]π)⟶ℂ)
3	2	ffvelcdmda 7069	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → (𝐹‘𝑡) ∈ ℂ)
4		fourierdlem101.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5	4	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → 𝑁 ∈ ℕ)
6		pire 26577	. . . . . . . . . . . 12 ⊢ π ∈ ℝ
7	6	renegcli 11507	. . . . . . . . . . 11 ⊢ -π ∈ ℝ
8		eliccre 46079	. . . . . . . . . . 11 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ ∧ 𝑡 ∈ (-π[,]π)) → 𝑡 ∈ ℝ)
9	7, 6, 8	mp3an12 1475	. . . . . . . . . 10 ⊢ (𝑡 ∈ (-π[,]π) → 𝑡 ∈ ℝ)
10	9	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → 𝑡 ∈ ℝ)
11		fourierdlem101.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ)
12	11	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → 𝑋 ∈ ℝ)
13	10, 12	resubcld 11630	. . . . . . . 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 46667	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑡 − 𝑋) ∈ ℝ) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
16	5, 13, 15	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
17	16	recnd 11225	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℂ)
18	3, 17	mulcld 11217	. . . . 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 6991	. . . . 5 ⊢ ((𝑡 ∈ (-π[,]π) ∧ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ ℂ) → (𝐺‘𝑡) = ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
21	1, 18, 20	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → (𝐺‘𝑡) = ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
22	21	eqcomd 2771	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (-π[,]π)) → ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = (𝐺‘𝑡))
23	22	itgeq2dv 25902	. 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 6871	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝑄‘𝑗) = (𝑄‘𝑖))
26	25	oveq1d 7415	. . . 4 ⊢ (𝑗 = 𝑖 → ((𝑄‘𝑗) − 𝑋) = ((𝑄‘𝑖) − 𝑋))
27	26	cbvmptv 5209	. . 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 7099	. . 3 ⊢ (𝜑 → 𝐺:(-π[,]π)⟶ℂ)
31	19	reseq1i 5965	. . . . 5 ⊢ (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑡 ∈ (-π[,]π) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
32		ioossicc 13451	. . . . . . 7 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
33	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → -π ∈ ℝ)
34	33	rexrd 11247	. . . . . . . . 9 ⊢ (𝜑 → -π ∈ ℝ^*)
35	34	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ^*)
36	6	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → π ∈ ℝ)
37	36	rexrd 11247	. . . . . . . . 9 ⊢ (𝜑 → π ∈ ℝ^*)
38	37	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈ ℝ^*)
39	24, 28, 29	fourierdlem15 46694	. . . . . . . . 9 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
40	39	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
41		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
42	35, 38, 40, 41	fourierdlem8 46687	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
43	32, 42	sstrid 3950	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
44	43	resmptd 6033	. . . . 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 2812	. . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:(-π[,]π)⟶ℂ)
47	46, 43	feqresmpt 6940	. . . . . 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 2866	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘𝑡)) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
50		eqidd 2766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)))
51		simpr 489	. . . . . . . . . . . 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 2766	. . . . . . . . . . . . . 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 7407	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑟 → (𝑡 − 𝑋) = (𝑟 − 𝑋))
54	53	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑡 = 𝑟) → (𝑡 − 𝑋) = (𝑟 − 𝑋))
55		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
56		elioore 13393	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑟 ∈ ℝ)
57	56	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑟 ∈ ℝ)
58	11	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
59	57, 58	resubcld 11630	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑟 − 𝑋) ∈ ℝ)
60	59	adantlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑟 − 𝑋) ∈ ℝ)
61	52, 54, 55, 60	fvmptd 6987	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 2800	. . . . . . . . . . 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 6875	. . . . . . . . . 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 13393	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
66	65	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑡 ∈ ℝ)
67	11	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
68	66, 67	resubcld 11630	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑡 − 𝑋) ∈ ℝ)
69	68	adantlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑡 − 𝑋) ∈ ℝ)
70		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋)) = (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))
71	69, 70	fmptd 7099	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋)):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℝ)
72	71	ffvelcdmda 7069	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋))‘𝑟) ∈ ℝ)
73	4	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑁 ∈ ℕ)
74	14	dirkerre 46667	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑟 − 𝑋) ∈ ℝ) → ((𝐷‘𝑁)‘(𝑟 − 𝑋)) ∈ ℝ)
75	73, 60, 74	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑟 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐷‘𝑁)‘(𝑟 − 𝑋)) ∈ ℝ)
76	50, 64, 72, 75	fvmptd 6987	. . . . . . . . 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 2771	. . . . . . . 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 5198	. . . . . . 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 6875	. . . . . . . . 9 ⊢ (𝑡 = 𝑟 → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) = ((𝐷‘𝑁)‘(𝑟 − 𝑋)))
80	79	cbvmptv 5209	. . . . . . . 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 46667	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑁)‘𝑠) ∈ ℝ)
83	4, 82	sylan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑁)‘𝑠) ∈ ℝ)
84		eqid 2765	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠))
85	83, 84	fmptd 7099	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)):ℝ⟶ℝ)
86	85	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)):ℝ⟶ℝ)
87		fcompt 7119	. . . . . . . 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 595	. . . . . . 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 2810	. . . . . 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 2765	. . . . . . . 8 ⊢ (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) = (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋))
91		simpr 489	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → 𝑡 ∈ ℂ)
92	11	recnd 11225	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℂ)
93	92	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → 𝑋 ∈ ℂ)
94	91, 93	negsubd 11563	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → (𝑡 + -𝑋) = (𝑡 − 𝑋))
95	94	eqcomd 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → (𝑡 − 𝑋) = (𝑡 + -𝑋))
96	95	mpteq2dva 5198	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) = (𝑡 ∈ ℂ ↦ (𝑡 + -𝑋)))
97	92	negcld 11544	. . . . . . . . . . 11 ⊢ (𝜑 → -𝑋 ∈ ℂ)
98		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℂ ↦ (𝑡 + -𝑋)) = (𝑡 ∈ ℂ ↦ (𝑡 + -𝑋))
99	98	addccncf 25037	. . . . . . . . . . 11 ⊢ (-𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ (𝑡 + -𝑋)) ∈ (ℂ–cn→ℂ))
100	97, 99	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (𝑡 + -𝑋)) ∈ (ℂ–cn→ℂ))
101	96, 100	eqeltrd 2865	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ∈ (ℂ–cn→ℂ))
102	101	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ∈ (ℂ–cn→ℂ))
103		ioossre 13425	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
104		ax-resscn 11145	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
105	103, 104	sstri 3948	. . . . . . . . 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 46443	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑡 − 𝑋)) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℝ))
109	83	recnd 11225	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑁)‘𝑠) ∈ ℂ)
110	109, 84	fmptd 7099	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)):ℝ⟶ℂ)
111		ssid 3961	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
112	14	dirkerf 46669	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁):ℝ⟶ℝ)
113	4, 112	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷‘𝑁):ℝ⟶ℝ)
114	113	feqmptd 6939	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷‘𝑁) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)))
115	14	dirkercncf 46679	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁) ∈ (ℝ–cn→ℝ))
116	4, 115	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷‘𝑁) ∈ (ℝ–cn→ℝ))
117	114, 116	eqeltrrd 2866	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∈ (ℝ–cn→ℝ))
118		cncfcdm 25018	. . . . . . . . . 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 598	. . . . . . . . 9 ⊢ (𝜑 → ((𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∈ (ℝ–cn→ℂ) ↔ (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)):ℝ⟶ℂ))
120	110, 119	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∈ (ℝ–cn→ℂ))
121	120	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑁)‘𝑠)) ∈ (ℝ–cn→ℂ))
122	108, 121	cncfco 25027	. . . . . 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 2865	. . . . 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 25566	. . . 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 2865	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
126		cncff 25013	. . . . . . . 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 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
128	113	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑁):ℝ⟶ℝ)
129		elioore 13393	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℝ)
130	129	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
131	11	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
132	130, 131	resubcld 11630	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑠 − 𝑋) ∈ ℝ)
133	128, 132	ffvelcdmd 7070	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐷‘𝑁)‘(𝑠 − 𝑋)) ∈ ℝ)
134	133	recnd 11225	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐷‘𝑁)‘(𝑠 − 𝑋)) ∈ ℂ)
135		eqid 2765	. . . . . . . . 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 7099	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
137	136	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
138		eqid 2765	. . . . . . 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 7407	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑄‘𝑖) → (𝑡 − 𝑋) = ((𝑄‘𝑖) − 𝑋))
141	140	fveq2d 6875	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑄‘𝑖) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) = ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)))
142	141	eqcomd 2771	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑄‘𝑖) → ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
143	142	adantl 486	. . . . . . . . . . 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 2766	. . . . . . . . . . . 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 7407	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (𝑠 − 𝑋) = (𝑡 − 𝑋))
146	145	fveq2d 6875	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → ((𝐷‘𝑁)‘(𝑠 − 𝑋)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
147	146	adantl 486	. . . . . . . . . . . 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 4601	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ {(𝑄‘𝑖)} ↔ 𝑡 = (𝑄‘𝑖))
149	148	notbii 323	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑡 ∈ {(𝑄‘𝑖)} ↔ ¬ 𝑡 = (𝑄‘𝑖))
150		elunnel2 4111	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ∧ ¬ 𝑡 ∈ {(𝑄‘𝑖)}) → 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
151	149, 150	sylan2br 606	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ∧ ¬ 𝑡 = (𝑄‘𝑖)) → 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
152	151	adantll 726	. . . . . . . . . . . 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 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝐷‘𝑁):ℝ⟶ℝ)
154		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 = (𝑄‘𝑖)) → 𝑡 = (𝑄‘𝑖))
155	9	ssriv 3943	. . . . . . . . . . . . . . . . . . . 20 ⊢ (-π[,]π) ⊆ ℝ
156		fzossfz 13698	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0..^𝑀) ⊆ (0...𝑀)
157	156, 41	sselid 3937	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
158	40, 157	ffvelcdmd 7070	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (-π[,]π))
159	155, 158	sselid 3937	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
160	159	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 = (𝑄‘𝑖)) → (𝑄‘𝑖) ∈ ℝ)
161	154, 160	eqeltrd 2865	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 = (𝑄‘𝑖)) → 𝑡 ∈ ℝ)
162	161	adantlr 727	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ 𝑡 = (𝑄‘𝑖)) → 𝑡 ∈ ℝ)
163	152, 65	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ ¬ 𝑡 = (𝑄‘𝑖)) → 𝑡 ∈ ℝ)
164	162, 163	pm2.61dan 824	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → 𝑡 ∈ ℝ)
165	11	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → 𝑋 ∈ ℝ)
166	164, 165	resubcld 11630	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝑡 − 𝑋) ∈ ℝ)
167	153, 166	ffvelcdmd 7070	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
168	167	adantr 485	. . . . . . . . . . . 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 6987	. . . . . . . . . . 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 4520	. . . . . . . . . 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 5198	. . . . . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘𝑁):ℝ⟶ℝ)
173		elun 4109	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↔ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝑄‘𝑖)}))
174	173	bilani 509	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝑄‘𝑖)}))
175		elsni 4602	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 ∈ {(𝑄‘𝑖)} → 𝑠 = (𝑄‘𝑖))
176	175	adantl 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝑄‘𝑖)}) → 𝑠 = (𝑄‘𝑖))
177	159	adantr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝑄‘𝑖)}) → (𝑄‘𝑖) ∈ ℝ)
178	176, 177	eqeltrd 2865	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ {(𝑄‘𝑖)}) → 𝑠 ∈ ℝ)
179	178	ex 417	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ {(𝑄‘𝑖)} → 𝑠 ∈ ℝ))
180	179	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝑠 ∈ {(𝑄‘𝑖)} → 𝑠 ∈ ℝ))
181		pm3.44 974	. . . . . . . . . . . . . . . . . . 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))
182	129, 180, 181	sylancr 598	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ 𝑠 ∈ {(𝑄‘𝑖)}) → 𝑠 ∈ ℝ))
183	174, 182	mpd 16	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → 𝑠 ∈ ℝ)
184	11	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → 𝑋 ∈ ℝ)
185	183, 184	resubcld 11630	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → (𝑠 − 𝑋) ∈ ℝ)
186		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋)) = (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋))
187	185, 186	fmptd 7099	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋)):(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})⟶ℝ)
188		fcompt 7119	. . . . . . . . . . . . . . 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))))
189	172, 187, 188	syl2anc 595	. . . . . . . . . . . . . 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))))
190		eqidd 2766	. . . . . . . . . . . . . . . . 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)))
191	145	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∧ 𝑠 = 𝑡) → (𝑠 − 𝑋) = (𝑡 − 𝑋))
192		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) → 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}))
193	190, 191, 192, 166	fvmptd 6987	. . . . . . . . . . . . . . . 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))
194	193	fveq2d 6875	. . . . . . . . . . . . . . 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)))
195	194	mpteq2dva 5198	. . . . . . . . . . . . . 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))))
196	189, 195	eqtr2d 2801	. . . . . . . . . . . . 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))))
197		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℂ ↦ (𝑠 − 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 − 𝑋))
198		simpr 489	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
199	92	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → 𝑋 ∈ ℂ)
200	198, 199	negsubd 11563	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → (𝑠 + -𝑋) = (𝑠 − 𝑋))
201	200	eqcomd 2771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → (𝑠 − 𝑋) = (𝑠 + -𝑋))
202	201	mpteq2dva 5198	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 − 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + -𝑋)))
203		eqid 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℂ ↦ (𝑠 + -𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + -𝑋))
204	203	addccncf 25037	. . . . . . . . . . . . . . . . . . 19 ⊢ (-𝑋 ∈ ℂ → (𝑠 ∈ ℂ ↦ (𝑠 + -𝑋)) ∈ (ℂ–cn→ℂ))
205	97, 204	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 + -𝑋)) ∈ (ℂ–cn→ℂ))
206	202, 205	eqeltrd 2865	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 − 𝑋)) ∈ (ℂ–cn→ℂ))
207	206	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℂ ↦ (𝑠 − 𝑋)) ∈ (ℂ–cn→ℂ))
208	159	recnd 11225	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
209	208	snssd 4748	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {(𝑄‘𝑖)} ⊆ ℂ)
210	106, 209	unssd 4147	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ⊆ ℂ)
211	197, 207, 210, 107, 185	cncfmptssg 46443	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↦ (𝑠 − 𝑋)) ∈ ((((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})–cn→ℝ))
212	114, 120	eqeltrd 2865	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐷‘𝑁) ∈ (ℝ–cn→ℂ))
213	212	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘𝑁) ∈ (ℝ–cn→ℂ))
214	211, 213	cncfco 25027	. . . . . . . . . . . . . 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))
215		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
216		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) = ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}))
217	215	cnfldtop 24901	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘ℂ_fld) ∈ Top
218		unicntop 24903	. . . . . . . . . . . . . . . . . . 19 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
219	218	restid 17476	. . . . . . . . . . . . . . . . . 18 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
220	217, 219	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
221	220	eqcomi 2774	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
222	215, 216, 221	cncfcn 25030	. . . . . . . . . . . . . . 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)))
223	210, 111, 222	sylancl 597	. . . . . . . . . . . . . 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)))
224	214, 223	eleqtrd 2867	. . . . . . . . . . . . 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)))
225	196, 224	eqeltrd 2865	. . . . . . . . . . . 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)))
226	215	cnfldtopon 24900	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
227		resttopon 23279	. . . . . . . . . . . . . 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)})))
228	226, 210, 227	sylancr 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})) ∈ (TopOn‘(((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)})))
229		cncnp 23398	. . . . . . . . . . . . 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))))
230	228, 226, 229	sylancl 597	. . . . . . . . . . . 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))))
231	225, 230	mpbid 235	. . . . . . . . . . 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)))
232	231	simprd 500	. . . . . . . . . 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))
233		eqidd 2766	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = (𝑄‘𝑖))
234		elsng 4599	. . . . . . . . . . . . . 14 ⊢ ((𝑄‘𝑖) ∈ ℝ → ((𝑄‘𝑖) ∈ {(𝑄‘𝑖)} ↔ (𝑄‘𝑖) = (𝑄‘𝑖)))
235	159, 234	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) ∈ {(𝑄‘𝑖)} ↔ (𝑄‘𝑖) = (𝑄‘𝑖)))
236	233, 235	mpbird 260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ {(𝑄‘𝑖)})
237	236	olcd 887	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ (𝑄‘𝑖) ∈ {(𝑄‘𝑖)}))
238		elun 4109	. . . . . . . . . . 11 ⊢ ((𝑄‘𝑖) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}) ↔ ((𝑄‘𝑖) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ (𝑄‘𝑖) ∈ {(𝑄‘𝑖)}))
239	237, 238	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘𝑖)}))
240		fveq2 6871	. . . . . . . . . . . 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)))
241	240	eleq2d 2851	. . . . . . . . . . 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))))
242	241	rspccva 3583	. . . . . . . . . 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)))
243	232, 239, 242	syl2anc 595	. . . . . . . . 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)))
244	171, 243	eqeltrd 2865	. . . . . . . 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)))
245		eqid 2765	. . . . . . . . 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)))
246	216, 215, 245, 137, 106, 208	ellimc 25993	. . . . . . . 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))))
247	244, 246	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐷‘𝑁)‘(𝑠 − 𝑋))) lim_ℂ (𝑄‘𝑖)))
248	127, 137, 138, 139, 247	mullimcf 46197	. . . . . 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)))
249		fvres 6890	. . . . . . . . . 10 ⊢ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) = (𝐹‘𝑡))
250	249	adantl 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑡) = (𝐹‘𝑡))
251	250	oveq1d 7415	. . . . . . . 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)))
252	251	mpteq2dva 5198	. . . . . . 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))))
253	252	oveq1d 7415	. . . . . 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)))
254	248, 253	eleqtrd 2867	. . . . 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)))
255		eqidd 2766	. . . . . . . . 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))))
256		simpr 489	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 𝑡) → 𝑠 = 𝑡)
257	256	oveq1d 7415	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 𝑡) → (𝑠 − 𝑋) = (𝑡 − 𝑋))
258	257	fveq2d 6875	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 𝑡) → ((𝐷‘𝑁)‘(𝑠 − 𝑋)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
259		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
260	113	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑁):ℝ⟶ℝ)
261	260, 69	ffvelcdmd 7070	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
262	255, 258, 259, 261	fvmptd 6987	. . . . . . . 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)))
263	262	oveq2d 7416	. . . . . . 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))))
264	263	mpteq2dva 5198	. . . . . 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)))))
265	264	oveq1d 7415	. . . . 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)))
266	254, 265	eleqtrd 2867	. . . 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)))
267	45	eqcomd 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
268	267	oveq1d 7415	. . . 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)))
269	266, 268	eleqtrd 2867	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · ((𝐷‘𝑁)‘((𝑄‘𝑖) − 𝑋))) ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
270		fourierdlem101.l	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
271		iftrue 4489	. . . . . . . . . . 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)))
272		oveq1 7407	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑄‘(𝑖 + 1)) → (𝑡 − 𝑋) = ((𝑄‘(𝑖 + 1)) − 𝑋))
273	272	eqcomd 2771	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑄‘(𝑖 + 1)) → ((𝑄‘(𝑖 + 1)) − 𝑋) = (𝑡 − 𝑋))
274	273	fveq2d 6875	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑄‘(𝑖 + 1)) → ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋)) = ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
275	271, 274	eqtrd 2800	. . . . . . . . . 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)))
276	275	adantl 486	. . . . . . . . 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)))
277		iffalse 4492	. . . . . . . . . . 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))
278	277	adantl 486	. . . . . . . . . 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))
279		eqidd 2766	. . . . . . . . . . 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))))
280	146	adantl 486	. . . . . . . . . . 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)))
281		elun 4109	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ↔ (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ 𝑡 ∈ {(𝑄‘(𝑖 + 1))}))
282	281	biimpi 219	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) → (𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ 𝑡 ∈ {(𝑄‘(𝑖 + 1))}))
283	282	orcomd 884	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) → (𝑡 ∈ {(𝑄‘(𝑖 + 1))} ∨ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
284	283	ad2antlr 739	. . . . . . . . . . . 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)))))
285		velsn 4601	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ {(𝑄‘(𝑖 + 1))} ↔ 𝑡 = (𝑄‘(𝑖 + 1)))
286	285	notbii 323	. . . . . . . . . . . . 13 ⊢ (¬ 𝑡 ∈ {(𝑄‘(𝑖 + 1))} ↔ ¬ 𝑡 = (𝑄‘(𝑖 + 1)))
287	286	bilanri 511	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → ¬ 𝑡 ∈ {(𝑄‘(𝑖 + 1))})
288		pm2.53 864	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ {(𝑄‘(𝑖 + 1))} ∨ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (¬ 𝑡 ∈ {(𝑄‘(𝑖 + 1))} → 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
289	284, 287, 288	sylc 66	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
290	172	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → (𝐷‘𝑁):ℝ⟶ℝ)
291	289, 65	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → 𝑡 ∈ ℝ)
292	11	ad3antrrr 742	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → 𝑋 ∈ ℝ)
293	291, 292	resubcld 11630	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → (𝑡 − 𝑋) ∈ ℝ)
294	290, 293	ffvelcdmd 7070	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) ∧ ¬ 𝑡 = (𝑄‘(𝑖 + 1))) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
295	279, 280, 289, 294	fvmptd 6987	. . . . . . . . . 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)))
296	278, 295	eqtrd 2800	. . . . . . . . 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)))
297	276, 296	pm2.61dan 824	. . . . . . . 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)))
298	297	mpteq2dva 5198	. . . . . . 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))))
299		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = (𝑡 ∈ ℝ ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋)))
300	104	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℝ ⊆ ℂ)
301		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
302	11	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑋 ∈ ℝ)
303	301, 302	resubcld 11630	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (𝑡 − 𝑋) ∈ ℝ)
304	90, 101, 300, 300, 303	cncfmptssg 46443	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑡 ∈ ℝ ↦ (𝑡 − 𝑋)) ∈ (ℝ–cn→ℝ))
305	304, 212	cncfcompt 46455	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑡 ∈ ℝ ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ (ℝ–cn→ℂ))
306	305	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑡 ∈ ℝ ↦ ((𝐷‘𝑁)‘(𝑡 − 𝑋))) ∈ (ℝ–cn→ℂ))
307	103	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
308		fzofzp1 13784	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
309	308	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
310	40, 309	ffvelcdmd 7070	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ (-π[,]π))
311	155, 310	sselid 3937	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
312	311	snssd 4748	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {(𝑄‘(𝑖 + 1))} ⊆ ℝ)
313	307, 312	unssd 4147	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ⊆ ℝ)
314	111	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ℂ ⊆ ℂ)
315	172	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → (𝐷‘𝑁):ℝ⟶ℝ)
316	313	sselda 3939	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → 𝑡 ∈ ℝ)
317	11	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → 𝑋 ∈ ℝ)
318	316, 317	resubcld 11630	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → (𝑡 − 𝑋) ∈ ℝ)
319	315, 318	ffvelcdmd 7070	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℝ)
320	319	recnd 11225	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑡 ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) ∈ ℂ)
321	299, 306, 313, 314, 320	cncfmptssg 46443	. . . . . . . . . . 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))
322	155, 104	sstri 3948	. . . . . . . . . . . . . . 15 ⊢ (-π[,]π) ⊆ ℂ
323	322, 310	sselid 3937	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
324	323	snssd 4748	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {(𝑄‘(𝑖 + 1))} ⊆ ℂ)
325	106, 324	unssd 4147	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}) ⊆ ℂ)
326		eqid 2765	. . . . . . . . . . . . 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))}))
327	215, 326, 221	cncfcn 25030	. . . . . . . . . . . 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)))
328	325, 111, 327	sylancl 597	. . . . . . . . . . 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)))
329	321, 328	eleqtrd 2867	. . . . . . . . . 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)))
330		resttopon 23279	. . . . . . . . . . . 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))})))
331	226, 325, 330	sylancr 598	. . . . . . . . . . 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))})))
332		cncnp 23398	. . . . . . . . . . 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))))
333	331, 226, 332	sylancl 597	. . . . . . . . . 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))))
334	329, 333	mpbid 235	. . . . . . . . 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)))
335	334	simprd 500	. . . . . . . 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))
336		eqidd 2766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
337		elsng 4599	. . . . . . . . . . . 12 ⊢ ((𝑄‘(𝑖 + 1)) ∈ ℝ → ((𝑄‘(𝑖 + 1)) ∈ {(𝑄‘(𝑖 + 1))} ↔ (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1))))
338	311, 337	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) ∈ {(𝑄‘(𝑖 + 1))} ↔ (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1))))
339	336, 338	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ {(𝑄‘(𝑖 + 1))})
340	339	olcd 887	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∨ (𝑄‘(𝑖 + 1)) ∈ {(𝑄‘(𝑖 + 1))}))
341		elun 4109	. . . . . . . . 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))}))
342	340, 341	sylibr 237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))}))
343		fveq2 6871	. . . . . . . . . 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))))
344	343	eleq2d 2851	. . . . . . . . 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)))))
345	344	rspccva 3583	. . . . . . . 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))))
346	335, 342, 345	syl2anc 595	. . . . . . 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))))
347	298, 346	eqeltrd 2865	. . . . . 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))))
348		eqid 2765	. . . . . . 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)))
349	326, 215, 348, 137, 106, 323	ellimc 25993	. . . . . 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)))))
350	347, 349	mpbird 260	. . . . 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))))
351	127, 137, 138, 270, 350	mullimcf 46197	. . . 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))))
352	264, 252, 45	3eqtr4d 2810	. . . . 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)))))
353	352	oveq1d 7415	. . . 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))))
354	351, 353	eleqtrd 2867	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐿 · ((𝐷‘𝑁)‘((𝑄‘(𝑖 + 1)) − 𝑋))) ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
355	24, 27, 28, 29, 11, 30, 125, 269, 354	fourierdlem93 46771	. 2 ⊢ (𝜑 → ∫(-π[,]π)(𝐺‘𝑡) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐺‘(𝑋 + 𝑠)) d𝑠)
356	19	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝐺 = (𝑡 ∈ (-π[,]π) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))))
357		fveq2 6871	. . . . . . 7 ⊢ (𝑡 = (𝑋 + 𝑠) → (𝐹‘𝑡) = (𝐹‘(𝑋 + 𝑠)))
358	357	oveq1d 7415	. . . . . 6 ⊢ (𝑡 = (𝑋 + 𝑠) → ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
359	358	adantl 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) ∧ 𝑡 = (𝑋 + 𝑠)) → ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))))
360		oveq1 7407	. . . . . . . 8 ⊢ (𝑡 = (𝑋 + 𝑠) → (𝑡 − 𝑋) = ((𝑋 + 𝑠) − 𝑋))
361	92	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑋 ∈ ℂ)
362	33, 11	resubcld 11630	. . . . . . . . . . . 12 ⊢ (𝜑 → (-π − 𝑋) ∈ ℝ)
363	362	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (-π − 𝑋) ∈ ℝ)
364	36, 11	resubcld 11630	. . . . . . . . . . . 12 ⊢ (𝜑 → (π − 𝑋) ∈ ℝ)
365	364	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (π − 𝑋) ∈ ℝ)
366		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋)))
367		eliccre 46079	. . . . . . . . . . 11 ⊢ (((-π − 𝑋) ∈ ℝ ∧ (π − 𝑋) ∈ ℝ ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ∈ ℝ)
368	363, 365, 366, 367	syl3anc 1394	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ∈ ℝ)
369	368	recnd 11225	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ∈ ℂ)
370	361, 369	pncan2d 11559	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → ((𝑋 + 𝑠) − 𝑋) = 𝑠)
371	360, 370	sylan9eqr 2822	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) ∧ 𝑡 = (𝑋 + 𝑠)) → (𝑡 − 𝑋) = 𝑠)
372	371	fveq2d 6875	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) ∧ 𝑡 = (𝑋 + 𝑠)) → ((𝐷‘𝑁)‘(𝑡 − 𝑋)) = ((𝐷‘𝑁)‘𝑠))
373	372	oveq2d 7416	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) ∧ 𝑡 = (𝑋 + 𝑠)) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)))
374	359, 373	eqtrd 2800	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) ∧ 𝑡 = (𝑋 + 𝑠)) → ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)))
375	7	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → -π ∈ ℝ)
376	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → π ∈ ℝ)
377	11	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑋 ∈ ℝ)
378	377, 368	readdcld 11226	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + 𝑠) ∈ ℝ)
379	33	recnd 11225	. . . . . . . . 9 ⊢ (𝜑 → -π ∈ ℂ)
380	92, 379	pncan3d 11560	. . . . . . . 8 ⊢ (𝜑 → (𝑋 + (-π − 𝑋)) = -π)
381	380	eqcomd 2771	. . . . . . 7 ⊢ (𝜑 → -π = (𝑋 + (-π − 𝑋)))
382	381	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → -π = (𝑋 + (-π − 𝑋)))
383		elicc2 13429	. . . . . . . . . 10 ⊢ (((-π − 𝑋) ∈ ℝ ∧ (π − 𝑋) ∈ ℝ) → (𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋)) ↔ (𝑠 ∈ ℝ ∧ (-π − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (π − 𝑋))))
384	363, 365, 383	syl2anc 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋)) ↔ (𝑠 ∈ ℝ ∧ (-π − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (π − 𝑋))))
385	366, 384	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑠 ∈ ℝ ∧ (-π − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (π − 𝑋)))
386	385	simp2d 1159	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (-π − 𝑋) ≤ 𝑠)
387	363, 368, 377, 386	leadd2dd 11817	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + (-π − 𝑋)) ≤ (𝑋 + 𝑠))
388	382, 387	eqbrtrd 5127	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → -π ≤ (𝑋 + 𝑠))
389	385	simp3d 1160	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝑠 ≤ (π − 𝑋))
390	368, 365, 377, 389	leadd2dd 11817	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + 𝑠) ≤ (𝑋 + (π − 𝑋)))
391		picn 26579	. . . . . . . 8 ⊢ π ∈ ℂ
392	391	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → π ∈ ℂ)
393	361, 392	pncan3d 11560	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + (π − 𝑋)) = π)
394	390, 393	breqtrd 5131	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + 𝑠) ≤ π)
395	375, 376, 378, 388, 394	eliccd 46078	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝑋 + 𝑠) ∈ (-π[,]π))
396	2	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → 𝐹:(-π[,]π)⟶ℂ)
397	396, 395	ffvelcdmd 7070	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
398	368, 109	syldan 602	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → ((𝐷‘𝑁)‘𝑠) ∈ ℂ)
399	397, 398	mulcld 11217	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)) ∈ ℂ)
400	356, 374, 395, 399	fvmptd 6987	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ((-π − 𝑋)[,](π − 𝑋))) → (𝐺‘(𝑋 + 𝑠)) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)))
401	400	itgeq2dv 25902	. 2 ⊢ (𝜑 → ∫((-π − 𝑋)[,](π − 𝑋))(𝐺‘(𝑋 + 𝑠)) d𝑠 = ∫((-π − 𝑋)[,](π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)) d𝑠)
402	23, 355, 401	3eqtrd 2804	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 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ∀wral 3079 {crab 3417 ∪ cun 3905 ⊆ wss 3907 ifcif 4483 {csn 4585 class class class wbr 5105 ↦ cmpt 5186 ↾ cres 5654 ∘ ccom 5656 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 ↑_m cmap 8812 ℂcc 11086 ℝcr 11087 0cc0 11088 1c1 11089 + caddc 11091 · cmul 11093 ℝ^*cxr 11230 < clt 11231 ≤ cle 11232 − cmin 11429 -cneg 11430 / cdiv 11859 ℕcn 12224 2c2 12286 (,)cioo 13363 [,]cicc 13366 ...cfz 13526 ..^cfzo 13673 mod cmo 13893 sincsin 16107 πcpi 16110 ↾_t crest 17463 TopOpenctopn 17464 ℂ_fldccnfld 21482 Topctop 23011 TopOnctopon 23028 Cn ccn 23342 CnP ccnp 23343 –cn→ccncf 24996 ∫citg 25738 lim_ℂ climc 25982

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cc 10407 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-symdif 4208 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-disj 5073 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-ofr 7665 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-oadd 8445 df-omul 8446 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-dju 9875 df-card 9913 df-acn 9916 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ioc 13368 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-fac 14301 df-bc 14330 df-hash 14358 df-shft 15094 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-limsup 15512 df-clim 15529 df-rlim 15530 df-sum 15728 df-ef 16111 df-sin 16113 df-cos 16114 df-pi 16116 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-rest 17465 df-topn 17466 df-0g 17484 df-gsum 17485 df-topgen 17486 df-pt 17487 df-prds 17490 df-xrs 17546 df-qtop 17551 df-imas 17552 df-xps 17554 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-mulg 19125 df-cntz 19378 df-cmn 19843 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-fbas 21479 df-fg 21480 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cld 23137 df-ntr 23138 df-cls 23139 df-nei 23216 df-lp 23254 df-perf 23255 df-cn 23345 df-cnp 23346 df-t1 23432 df-haus 23433 df-cmp 23505 df-tx 23680 df-hmeo 23873 df-fil 23964 df-fm 24056 df-flim 24057 df-flf 24058 df-xms 24438 df-ms 24439 df-tms 24440 df-cncf 24998 df-ovol 25584 df-vol 25585 df-mbf 25739 df-itg1 25740 df-itg2 25741 df-ibl 25742 df-itg 25743 df-0p 25790 df-ditg 25967 df-limc 25986 df-dv 25987

This theorem is referenced by: fourierdlem111 46789

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