fourierdlem84 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem84 46111

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 46042	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀))
10		fourierdlem84.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℝ)
12	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℝ)
13	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
14	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
15		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ (𝐴[,]𝐵))
16		eliccre 45423	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ)
17	13, 14, 15, 16	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ)
18	12, 17	readdcld 11319	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
19	11, 18	ffvelcdmd 7119	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
20		fourierdlem84.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (ℝ–cn→ℝ))
21		cncff 24938	. . . . . . . 8 ⊢ (𝐷 ∈ (ℝ–cn→ℝ) → 𝐷:ℝ⟶ℝ)
22	20, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐷:ℝ⟶ℝ)
23	22	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐷:ℝ⟶ℝ)
24	23, 17	ffvelcdmd 7119	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐷‘𝑠) ∈ ℝ)
25	19, 24	remulcld 11320	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠)) ∈ ℝ)
26	25	recnd 11318	. . 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 7148	. 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 6008	. . . 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 13493	. . . . . 6 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
32	3	rexrd 11340	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
33	32	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ^*)
34	4	rexrd 11340	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
35	34	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈ ℝ^*)
36	1, 2, 9	fourierdlem15 46043	. . . . . . . 8 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
37	36	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
38		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
39	33, 35, 37, 38	fourierdlem8 46036	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
40	31, 39	sstrid 4020	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
41	40	resmptd 6069	. . . 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 2780	. . 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 11319	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 + 𝑋) ∈ ℝ)
44	4, 5	readdcld 11319	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 + 𝑋) ∈ ℝ)
45	43, 44	iccssred 13494	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴 + 𝑋)[,](𝐵 + 𝑋)) ⊆ ℝ)
46	45	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐴 + 𝑋)[,](𝐵 + 𝑋)) ⊆ ℝ)
47	6, 2, 7	fourierdlem15 46043	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶((𝐴 + 𝑋)[,](𝐵 + 𝑋)))
48	47	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶((𝐴 + 𝑋)[,](𝐵 + 𝑋)))
49		elfzofz 13732	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
50	49	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
51	48, 50	ffvelcdmd 7119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ((𝐴 + 𝑋)[,](𝐵 + 𝑋)))
52	46, 51	sseldd 4009	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ)
53	52	rexrd 11340	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ^*)
54	53	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) ∈ ℝ^*)
55		fzofzp1 13814	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
56	55	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
57	48, 56	ffvelcdmd 7119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ((𝐴 + 𝑋)[,](𝐵 + 𝑋)))
58	46, 57	sseldd 4009	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
59	58	rexrd 11340	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ^*)
60	59	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘(𝑖 + 1)) ∈ ℝ^*)
61	5	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
62		elioore 13437	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℝ)
63	62	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
64	61, 63	readdcld 11319	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
65	5	recnd 11318	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℂ)
66	65	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
67	3, 4	iccssred 13494	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
68	67	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐴[,]𝐵) ⊆ ℝ)
69	37, 50	ffvelcdmd 7119	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (𝐴[,]𝐵))
70	68, 69	sseldd 4009	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
71	70	recnd 11318	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
72	66, 71	addcomd 11492	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘𝑖)) = ((𝑄‘𝑖) + 𝑋))
73	5	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
74	52, 73	resubcld 11718	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
75	8	fvmpt2 7040	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ ℝ) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
76	50, 74, 75	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
77	76	oveq1d 7463	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑋) = (((𝑉‘𝑖) − 𝑋) + 𝑋))
78	52	recnd 11318	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℂ)
79	78, 66	npcand 11651	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑉‘𝑖) − 𝑋) + 𝑋) = (𝑉‘𝑖))
80	72, 77, 79	3eqtrrd 2785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) = (𝑋 + (𝑄‘𝑖)))
81	80	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) = (𝑋 + (𝑄‘𝑖)))
82	70	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ)
83	70	rexrd 11340	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
84	83	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ^*)
85	37, 68	fssd 6764	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
86	85, 56	ffvelcdmd 7119	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
87	86	rexrd 11340	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
88	87	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
89		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
90		ioogtlb 45413	. . . . . . . . . . . 12 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠)
91	84, 88, 89, 90	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠)
92	82, 63, 61, 91	ltadd2dd 11449	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘𝑖)) < (𝑋 + 𝑠))
93	81, 92	eqbrtrd 5188	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) < (𝑋 + 𝑠))
94	86	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
95		iooltub 45428	. . . . . . . . . . . 12 ⊢ (((𝑄‘𝑖) ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ^* ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1)))
96	84, 88, 89, 95	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1)))
97	63, 94, 61, 96	ltadd2dd 11449	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑋 + (𝑄‘(𝑖 + 1))))
98		fveq2 6920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗))
99	98	oveq1d 7463	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋))
100	99	cbvmptv 5279	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
101	8, 100	eqtri 2768	. . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑖 + 1) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1)))
104	103	oveq1d 7463	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑖 + 1) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
105	104	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
106	58, 73	resubcld 11718	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
107	102, 105, 56, 106	fvmptd 7036	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
108	107	oveq2d 7464	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)))
109	58	recnd 11318	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℂ)
110	66, 109	pncan3d 11650	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)) = (𝑉‘(𝑖 + 1)))
111	108, 110	eqtrd 2780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1)))
112	111	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1)))
113	97, 112	breqtrd 5192	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑉‘(𝑖 + 1)))
114	54, 60, 64, 93, 113	eliood 45416	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))
115		fvres 6939	. . . . . . . 8 ⊢ ((𝑋 + 𝑠) ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠)))
116	114, 115	syl 17	. . . . . . 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 2746	. . . . . 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 5266	. . . . 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 13469	. . . . . . 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 13469	. . . . . . 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 46051	. . . . 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 2844	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
126		eqid 2740	. . . . 5 ⊢ (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) = (𝑠 ∈ ℝ ↦ (𝐷‘𝑠))
127		ax-resscn 11241	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
128		ssid 4031	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
129		cncfss 24944	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ))
130	127, 128, 129	mp2an 691	. . . . . . 7 ⊢ (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ)
131	22	feqmptd 6990	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 = (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)))
132	131	eqcomd 2746	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) = 𝐷)
133	132, 20	eqeltrd 2844	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ∈ (ℝ–cn→ℝ))
134	130, 133	sselid 4006	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ∈ (ℝ–cn→ℂ))
135	134	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ∈ (ℝ–cn→ℂ))
136	40, 68	sstrd 4019	. . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐷:ℝ⟶ℝ)
139	62	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ)
140	138, 139	ffvelcdmd 7119	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑠) ∈ ℝ)
141	140	recnd 11318	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑠) ∈ ℂ)
142	141	adantlr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑠) ∈ ℂ)
143	126, 135, 136, 137, 142	cncfmptssg 45792	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
144	125, 143	mulcncf 25499	. . 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 2844	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
146		eqid 2740	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠)))
147		eqid 2740	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠))
148		eqid 2740	. . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℝ)
150	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ)
151	150, 139	readdcld 11319	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ)
152	149, 151	ffvelcdmd 7119	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
153	152	recnd 11318	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
154	153	adantlr 714	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
155	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℝ)
156		ioossre 13468	. . . . . 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 11425	. . . . 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 7464	. . . . . 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 2846	. . . . 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 46080	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) lim_ℂ (𝑄‘𝑖)))
163		limcresi 25940	. . . . . 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 4006	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ (ℝ–cn→ℂ))
165	164	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐷 ∈ (ℝ–cn→ℂ))
166	165, 70	cnlimci 25944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ (𝐷 lim_ℂ (𝑄‘𝑖)))
167	131	oveq1d 7463	. . . . . . . 8 ⊢ (𝜑 → (𝐷 lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘𝑖)))
168	167	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷 lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘𝑖)))
169	166, 168	eleqtrd 2846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘𝑖)))
170	163, 169	sselid 4006	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
171	136	resmptd 6069	. . . . . 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 7463	. . . . 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 2846	. . . 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 45537	. . 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 6005	. . . . 5 ⊢ (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
176	175, 41	eqtr2id 2793	. . . 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 7463	. . 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 2846	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · (𝐷‘(𝑄‘𝑖))) ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
179	63, 96	ltned 11426	. . . . 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 2746	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) = (𝑋 + (𝑄‘(𝑖 + 1))))
182	181	oveq2d 7464	. . . . . 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 2846	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑋 + (𝑄‘(𝑖 + 1)))))
184	86	recnd 11318	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
185	155, 73, 136, 146, 114, 157, 179, 183, 184	fourierdlem53 46080	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) lim_ℂ (𝑄‘(𝑖 + 1))))
186		limcresi 25940	. . . . . 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 25944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘(𝑖 + 1))) ∈ (𝐷 lim_ℂ (𝑄‘(𝑖 + 1))))
188	131	oveq1d 7463	. . . . . . . 8 ⊢ (𝜑 → (𝐷 lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
189	188	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷 lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
190	187, 189	eleqtrd 2846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘(𝑖 + 1))) ∈ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
191	186, 190	sselid 4006	. . . . 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 7463	. . . . 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 2846	. . . 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 45537	. . 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 7463	. . 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 2846	. 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 46096	1 ⊢ (𝜑 → 𝐺 ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 ⊆ wss 3976 class class class wbr 5166 ↦ cmpt 5249 ↾ cres 5702 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 ℝ^*cxr 11323 < clt 11324 − cmin 11520 ℕcn 12293 (,)cioo 13407 [,]cicc 13410 ...cfz 13567 ..^cfzo 13711 –cn→ccncf 24921 𝐿¹cibl 25671 lim_ℂ climc 25917

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-symdif 4272 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-cn 23256 df-cnp 23257 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-ovol 25518 df-vol 25519 df-mbf 25673 df-itg1 25674 df-itg2 25675 df-ibl 25676 df-itg 25677 df-0p 25724 df-limc 25921

This theorem is referenced by: fourierdlem103 46130 fourierdlem104 46131 fourierdlem112 46139

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