fourierdlem84 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem84 47018

Description: If 𝐹 is piecewise continuous and 𝐷 is continuous, then 𝐺 is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem84.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem84.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem84.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem84.xre	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem84.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝‘𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem84.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem84.v	⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
fourierdlem84.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
fourierdlem84.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
fourierdlem84.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1))))
fourierdlem84.q	⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
fourierdlem84.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem84.d	⊢ (𝜑 → 𝐷 ∈ (ℝ–cn→ℝ))
fourierdlem84.g	⊢ 𝐺 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠)))

**Assertion**
Ref	Expression
fourierdlem84	⊢ (𝜑 → 𝐺 ∈ 𝐿¹)

Distinct variable groups: 𝐴,𝑖,𝑚,𝑝 𝐴,𝑠,𝑖 𝐵,𝑖,𝑚,𝑝 𝐵,𝑠 𝐷,𝑠 𝐹,𝑠 𝑖,𝐺 𝐿,𝑠 𝑖,𝑀,𝑠 𝑚,𝑀,𝑝 𝑄,𝑖,𝑠 𝑄,𝑝 𝑅,𝑠 𝑖,𝑉,𝑠 𝑉,𝑝 𝑖,𝑋,𝑠 𝑚,𝑋,𝑝 𝜑,𝑖,𝑠 Allowed substitution hints: 𝜑(𝑚, 𝑝) 𝐷(𝑖, 𝑚, 𝑝) 𝑃(𝑖, 𝑚, 𝑠, 𝑝) 𝑄(𝑚) 𝑅(𝑖, 𝑚, 𝑝) 𝐹(𝑖, 𝑚, 𝑝) 𝐺(𝑚, 𝑠, 𝑝) 𝐿(𝑖, 𝑚, 𝑝) 𝑂(𝑖, 𝑚, 𝑠, 𝑝) 𝑉(𝑚)

Proof of Theorem fourierdlem84 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem84.o	. 2 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
2		fourierdlem84.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		fourierdlem84.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		fourierdlem84.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5		fourierdlem84.xre	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
6		fourierdlem84.p	. . 3 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝‘𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
7		fourierdlem84.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
8		fourierdlem84.q	. . 3 ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
9	3, 4, 5, 6, 1, 2, 7, 8	fourierdlem14 46949	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀))
10		fourierdlem84.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
11	10	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℝ)
12	5	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℝ)
13	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
14	4	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
15		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ (𝐴[,]𝐵))
16		eliccre 46335	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ)
17	13, 14, 15, 16	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ)
18	12, 17	readdcld 11262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
19	11, 18	ffvelcdmd 7078	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
20		fourierdlem84.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (ℝ–cn→ℝ))
21		cncff 25121	. . . . . . . 8 ⊢ (𝐷 ∈ (ℝ–cn→ℝ) → 𝐷:ℝ⟶ℝ)
22	20, 21	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐷:ℝ⟶ℝ)
23	22	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐷:ℝ⟶ℝ)
24	23, 17	ffvelcdmd 7078	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐷‘𝑠) ∈ ℝ)
25	19, 24	remulcld 11263	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠)) ∈ ℝ)
26	25	recnd 11261	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠)) ∈ ℂ)
27		fourierdlem84.g	. . 3 ⊢ 𝐺 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠)))
28	26, 27	fmptd 7107	. 2 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
29	27	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))))
30	29	reseq1d 5971	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
31		ioossicc 13486	. . . . . 6 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
32	3	rexrd 11283	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
33	32	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ^*)
34	4	rexrd 11283	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
35	34	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈ ℝ^*)
36	1, 2, 9	fourierdlem15 46950	. . . . . . . 8 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
37	36	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
38		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
39	33, 35, 37, 38	fourierdlem8 46943	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
40	31, 39	sstrid 3942	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
41	40	resmptd 6036	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))))
42	30, 41	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))))
43	3, 5	readdcld 11262	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 + 𝑋) ∈ ℝ)
44	4, 5	readdcld 11262	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 + 𝑋) ∈ ℝ)
45	43, 44	iccssred 13487	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴 + 𝑋)[,](𝐵 + 𝑋)) ⊆ ℝ)
46	45	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐴 + 𝑋)[,](𝐵 + 𝑋)) ⊆ ℝ)
47	6, 2, 7	fourierdlem15 46950	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶((𝐴 + 𝑋)[,](𝐵 + 𝑋)))
48	47	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶((𝐴 + 𝑋)[,](𝐵 + 𝑋)))
49		elfzofz 13731	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
50	49	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
51	48, 50	ffvelcdmd 7078	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ((𝐴 + 𝑋)[,](𝐵 + 𝑋)))
52	46, 51	sseldd 3932	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ)
53	52	rexrd 11283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ^*)
54	53	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) ∈ ℝ^*)
55		fzofzp1 13820	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
56	55	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
57	48, 56	ffvelcdmd 7078	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ((𝐴 + 𝑋)[,](𝐵 + 𝑋)))
58	46, 57	sseldd 3932	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
59	58	rexrd 11283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ^*)
60	59	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘(𝑖 + 1)) ∈ ℝ^*)
61	5	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
62		elioore 13428	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℝ)
63	62	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
64	61, 63	readdcld 11262	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
65	5	recnd 11261	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℂ)
66	65	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
67	3, 4	iccssred 13487	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
68	67	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐴[,]𝐵) ⊆ ℝ)
69	37, 50	ffvelcdmd 7078	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (𝐴[,]𝐵))
70	68, 69	sseldd 3932	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
71	70	recnd 11261	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
72	66, 71	addcomd 11436	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘𝑖)) = ((𝑄‘𝑖) + 𝑋))
73	5	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
74	52, 73	resubcld 11666	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
75	8	fvmpt2 6998	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ ℝ) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
76	50, 74, 75	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
77	76	oveq1d 7428	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑋) = (((𝑉‘𝑖) − 𝑋) + 𝑋))
78	52	recnd 11261	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℂ)
79	78, 66	npcand 11597	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑉‘𝑖) − 𝑋) + 𝑋) = (𝑉‘𝑖))
80	72, 77, 79	3eqtrrd 2800	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) = (𝑋 + (𝑄‘𝑖)))
81	80	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) = (𝑋 + (𝑄‘𝑖)))
82	70	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
83	70	rexrd 11283	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
84	83	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
85	37, 68	fssd 6720	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
86	85, 56	ffvelcdmd 7078	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
87	86	rexrd 11283	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
88	87	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
89		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
90		ioogtlb 46325	. . . . . . . . . . . 12 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠)
91	84, 88, 89, 90	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠)
92	82, 63, 61, 91	ltadd2dd 11393	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘𝑖)) < (𝑋 + 𝑠))
93	81, 92	eqbrtrd 5127	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) < (𝑋 + 𝑠))
94	86	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
95		iooltub 46340	. . . . . . . . . . . 12 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1)))
96	84, 88, 89, 95	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1)))
97	63, 94, 61, 96	ltadd2dd 11393	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑋 + (𝑄‘(𝑖 + 1))))
98		fveq2 6878	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗))
99	98	oveq1d 7428	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋))
100	99	cbvmptv 5209	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
101	8, 100	eqtri 2783	. . . . . . . . . . . . . . 15 ⊢ 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
102	101	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋)))
103		fveq2 6878	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑖 + 1) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1)))
104	103	oveq1d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑖 + 1) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
105	104	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
106	58, 73	resubcld 11666	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
107	102, 105, 56, 106	fvmptd 6994	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
108	107	oveq2d 7429	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)))
109	58	recnd 11261	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℂ)
110	66, 109	pncan3d 11596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)) = (𝑉‘(𝑖 + 1)))
111	108, 110	eqtrd 2795	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1)))
112	111	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1)))
113	97, 112	breqtrd 5131	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑉‘(𝑖 + 1)))
114	54, 60, 64, 93, 113	eliood 46328	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))
115		fvres 6897	. . . . . . . 8 ⊢ ((𝑋 + 𝑠) ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠)))
116	114, 115	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠)))
117	116	eqcomd 2766	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) = ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)))
118	117	mpteq2dva 5198	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠))))
119		ioosscn 13461	. . . . . . 7 ⊢ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℂ
120	119	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℂ)
121		fourierdlem84.fcn	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
122		ioosscn 13461	. . . . . . 7 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
123	122	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
124	120, 121, 123, 66, 114	fourierdlem23 46958	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
125	118, 124	eqeltrd 2860	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
126		eqid 2760	. . . . 5 ⊢ (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) = (𝑠 ∈ ℝ ↦ (𝐷‘𝑠))
127		ax-resscn 11181	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
128		ssid 3953	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
129		cncfss 25127	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ))
130	127, 128, 129	mp2an 705	. . . . . . 7 ⊢ (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ)
131	22	feqmptd 6946	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 = (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)))
132	131	eqcomd 2766	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) = 𝐷)
133	132, 20	eqeltrd 2860	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ∈ (ℝ–cn→ℝ))
134	130, 133	sselid 3929	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ∈ (ℝ–cn→ℂ))
135	134	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ∈ (ℝ–cn→ℂ))
136	40, 68	sstrd 3941	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ)
137	128	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ℂ ⊆ ℂ)
138	22	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐷:ℝ⟶ℝ)
139	62	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
140	138, 139	ffvelcdmd 7078	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑠) ∈ ℝ)
141	140	recnd 11261	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑠) ∈ ℂ)
142	141	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑠) ∈ ℂ)
143	126, 135, 136, 137, 142	cncfmptssg 46699	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
144	125, 143	mulcncf 25674	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
145	42, 144	eqeltrd 2860	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
146		eqid 2760	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠)))
147		eqid 2760	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠))
148		eqid 2760	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠)))
149	10	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℝ)
150	5	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
151	150, 139	readdcld 11262	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
152	149, 151	ffvelcdmd 7078	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
153	152	recnd 11261	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
154	153	adantlr 728	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
155	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℝ)
156		ioossre 13460	. . . . . 6 ⊢ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ
157	156	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ)
158	82, 91	gtned 11369	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ (𝑄‘𝑖))
159		fourierdlem84.r	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
160	80	oveq2d 7429	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)) = ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑋 + (𝑄‘𝑖))))
161	159, 160	eleqtrd 2862	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑋 + (𝑄‘𝑖))))
162	155, 73, 136, 146, 114, 157, 158, 161, 71	fourierdlem53 46987	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) lim_ℂ (𝑄‘𝑖)))
163		limcresi 26112	. . . . . 6 ⊢ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘𝑖)) ⊆ (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))
164	130, 20	sselid 3929	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ (ℝ–cn→ℂ))
165	164	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐷 ∈ (ℝ–cn→ℂ))
166	165, 70	cnlimci 26116	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ (𝐷 lim_ℂ (𝑄‘𝑖)))
167	131	oveq1d 7428	. . . . . . . 8 ⊢ (𝜑 → (𝐷 lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘𝑖)))
168	167	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷 lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘𝑖)))
169	166, 168	eleqtrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘𝑖)))
170	163, 169	sselid 3929	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
171	136	resmptd 6036	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)))
172	171	oveq1d 7428	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘𝑖)))
173	170, 172	eleqtrd 2862	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘𝑖)))
174	146, 147, 148, 154, 142, 162, 173	mullimc 46446	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · (𝐷‘(𝑄‘𝑖))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) lim_ℂ (𝑄‘𝑖)))
175	27	reseq1i 5968	. . . . 5 ⊢ (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
176	175, 41	eqtr2id 2808	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
177	176	oveq1d 7428	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) lim_ℂ (𝑄‘𝑖)) = ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
178	174, 177	eleqtrd 2862	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · (𝐷‘(𝑄‘𝑖))) ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
179	63, 96	ltned 11370	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ (𝑄‘(𝑖 + 1)))
180		fourierdlem84.l	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1))))
181	111	eqcomd 2766	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) = (𝑋 + (𝑄‘(𝑖 + 1))))
182	181	oveq2d 7429	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑋 + (𝑄‘(𝑖 + 1)))))
183	180, 182	eleqtrd 2862	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑋 + (𝑄‘(𝑖 + 1)))))
184	86	recnd 11261	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
185	155, 73, 136, 146, 114, 157, 179, 183, 184	fourierdlem53 46987	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) lim_ℂ (𝑄‘(𝑖 + 1))))
186		limcresi 26112	. . . . . 6 ⊢ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))) ⊆ (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))
187	165, 86	cnlimci 26116	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘(𝑖 + 1))) ∈ (𝐷 lim_ℂ (𝑄‘(𝑖 + 1))))
188	131	oveq1d 7428	. . . . . . . 8 ⊢ (𝜑 → (𝐷 lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
189	188	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷 lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
190	187, 189	eleqtrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘(𝑖 + 1))) ∈ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
191	186, 190	sselid 3929	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘(𝑖 + 1))) ∈ (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
192	171	oveq1d 7428	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
193	191, 192	eleqtrd 2862	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘(𝑖 + 1))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
194	146, 147, 148, 154, 142, 185, 193	mullimc 46446	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐿 · (𝐷‘(𝑄‘(𝑖 + 1)))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) lim_ℂ (𝑄‘(𝑖 + 1))))
195	176	oveq1d 7428	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
196	194, 195	eleqtrd 2862	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐿 · (𝐷‘(𝑄‘(𝑖 + 1)))) ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
197	1, 2, 9, 28, 145, 178, 196	fourierdlem69 47003	1 ⊢ (𝜑 → 𝐺 ∈ 𝐿¹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 {crab 3412 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 ↾ cres 5657 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ↑_m cmap 8826 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 ℝ^*cxr 11266 < clt 11267 − cmin 11465 ℕcn 12257 (,)cioo 13398 [,]cicc 13401 ...cfz 13561 ..^cfzo 13709 –cn→ccncf 25104 𝐿¹cibl 25845 lim_ℂ climc 26089

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cc 10437 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-ofr 7679 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-omul 8460 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-acn 9947 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-sum 15774 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-cn 23452 df-cnp 23453 df-cmp 23612 df-tx 23788 df-hmeo 23981 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-ovol 25692 df-vol 25693 df-mbf 25847 df-itg1 25848 df-itg2 25849 df-ibl 25850 df-itg 25851 df-0p 25898 df-limc 26093

This theorem is used by: fourierdlem103 47037 fourierdlem104 47038 fourierdlem112 47046

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