fourierdlem100 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem100 46244

Description: A piecewise continuous function is integrable on any closed interval. This lemma uses local definitions, so that the proof is more readable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierlemiblglemlem.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem100.t	⊢ 𝑇 = (𝐵 − 𝐴)
fourierdlem100.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem100.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem100.f	⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
fourierdlem100.per	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierdlem100.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem100.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
fourierdlem100.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
fourierdlem100.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem100.d	⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞))
fourierdlem100.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem100.n	⊢ 𝑁 = ((♯‘𝐻) − 1)
fourierdlem100.h	⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})
fourierdlem100.s	⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
fourierdlem100.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
fourierdlem100.j	⊢ 𝐽 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
fourierdlem100.i	⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < ))

**Assertion**
Ref	Expression
fourierdlem100	⊢ (𝜑 → (𝑥 ∈ (𝐶[,]𝐷) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)

Distinct variable groups: 𝐴,𝑓,𝑘,𝑦 𝐴,𝑖,𝑥,𝑘,𝑦 𝐴,𝑚,𝑝,𝑖 𝐵,𝑓,𝑘,𝑦 𝐵,𝑖,𝑥 𝐵,𝑚,𝑝 𝐶,𝑓,𝑦 𝐶,𝑖,𝑚,𝑝 𝑥,𝐶 𝐷,𝑓,𝑦 𝐷,𝑖,𝑚,𝑝 𝑥,𝐷 𝑓,𝐸,𝑘,𝑦 𝑖,𝐸,𝑥 𝑖,𝐹,𝑥,𝑦 𝑓,𝐻,𝑦 𝑥,𝐻 𝑓,𝐼,𝑘,𝑦 𝑖,𝐼,𝑥 𝑖,𝐽,𝑥,𝑦 𝑥,𝐿,𝑦 𝑖,𝑀,𝑚,𝑝 𝑥,𝑀,𝑦 𝑓,𝑁,𝑘,𝑦 𝑖,𝑁,𝑥 𝑚,𝑁,𝑝 𝑄,𝑓,𝑘,𝑦 𝑄,𝑖,𝑥 𝑄,𝑝 𝑥,𝑅,𝑦 𝑆,𝑓,𝑘,𝑦 𝑆,𝑖,𝑥 𝑆,𝑝 𝑇,𝑓,𝑘,𝑦 𝑇,𝑖,𝑥 𝜑,𝑓,𝑘,𝑦 𝜑,𝑖,𝑥 Allowed substitution hints: 𝜑(𝑚,𝑝) 𝐶(𝑘) 𝐷(𝑘) 𝑃(𝑥,𝑦,𝑓,𝑖,𝑘,𝑚,𝑝) 𝑄(𝑚) 𝑅(𝑓,𝑖,𝑘,𝑚,𝑝) 𝑆(𝑚) 𝑇(𝑚,𝑝) 𝐸(𝑚,𝑝) 𝐹(𝑓,𝑘,𝑚,𝑝) 𝐻(𝑖,𝑘,𝑚,𝑝) 𝐼(𝑚,𝑝) 𝐽(𝑓,𝑘,𝑚,𝑝) 𝐿(𝑓,𝑖,𝑘,𝑚,𝑝) 𝑀(𝑓,𝑘) 𝑂(𝑥,𝑦,𝑓,𝑖,𝑘,𝑚,𝑝)

Proof of Theorem fourierdlem100 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem100.f	. . 3 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
2		fourierdlem100.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
3		fourierdlem100.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞))
4		elioore 13270	. . . . 5 ⊢ (𝐷 ∈ (𝐶(,)+∞) → 𝐷 ∈ ℝ)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ)
6	2, 5	iccssred 13329	. . 3 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
7	1, 6	feqresmpt 6886	. 2 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ (𝐹‘𝑥)))
8		fourierdlem100.o	. . . 4 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
9		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑝‘𝑖) = (𝑝‘𝑗))
10		oveq1 7348	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
11	10	fveq2d 6821	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1)))
12	9, 11	breq12d 5099	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝‘𝑗) < (𝑝‘(𝑗 + 1))))
13	12	cbvralvw 3210	. . . . . . . 8 ⊢ (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))
14	13	anbi2i 623	. . . . . . 7 ⊢ ((((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1))))
15	14	a1i 11	. . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑_m (0...𝑚)) → ((((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))))
16	15	rabbiia 3399	. . . . 5 ⊢ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} = {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))}
17	16	mpteq2i 5182	. . . 4 ⊢ (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))})
18	8, 17	eqtri 2754	. . 3 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))})
19		fourierdlem100.t	. . . . . 6 ⊢ 𝑇 = (𝐵 − 𝐴)
20		fourierlemiblglemlem.p	. . . . . 6 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
21		fourierdlem100.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
22		fourierdlem100.q	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
23		elioo4g 13301	. . . . . . . . 9 ⊢ (𝐷 ∈ (𝐶(,)+∞) ↔ ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐷 ∈ ℝ) ∧ (𝐶 < 𝐷 ∧ 𝐷 < +∞)))
24	3, 23	sylib 218	. . . . . . . 8 ⊢ (𝜑 → ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐷 ∈ ℝ) ∧ (𝐶 < 𝐷 ∧ 𝐷 < +∞)))
25	24	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝐶 < 𝐷 ∧ 𝐷 < +∞))
26	25	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝐶 < 𝐷)
27		id 22	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
28	19	eqcomi 2740	. . . . . . . . . . . . 13 ⊢ (𝐵 − 𝐴) = 𝑇
29	28	oveq2i 7352	. . . . . . . . . . . 12 ⊢ (𝑘 · (𝐵 − 𝐴)) = (𝑘 · 𝑇)
30	29	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑘 · (𝐵 − 𝐴)) = (𝑘 · 𝑇))
31	27, 30	oveq12d 7359	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 + (𝑘 · (𝐵 − 𝐴))) = (𝑥 + (𝑘 · 𝑇)))
32	31	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
33	32	rexbidv 3156	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
34	33	cbvrabv 3405	. . . . . . 7 ⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} = {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}
35	34	uneq2i 4110	. . . . . 6 ⊢ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
36		fourierdlem100.n	. . . . . . 7 ⊢ 𝑁 = ((♯‘𝐻) − 1)
37		fourierdlem100.h	. . . . . . . . . 10 ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})
38	29	eqcomi 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 · 𝑇) = (𝑘 · (𝐵 − 𝐴))
39	38	oveq2i 7352	. . . . . . . . . . . . . . 15 ⊢ (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑘 · (𝐵 − 𝐴)))
40	39	eleq1i 2822	. . . . . . . . . . . . . 14 ⊢ ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
41	40	rexbii 3079	. . . . . . . . . . . . 13 ⊢ (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
42	41	rgenw 3051	. . . . . . . . . . . 12 ⊢ ∀𝑦 ∈ (𝐶[,]𝐷)(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
43		rabbi 3425	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ (𝐶[,]𝐷)(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄) ↔ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})
44	42, 43	mpbi 230	. . . . . . . . . . 11 ⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}
45	44	uneq2i 4110	. . . . . . . . . 10 ⊢ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})
46	37, 45	eqtri 2754	. . . . . . . . 9 ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})
47	46	fveq2i 6820	. . . . . . . 8 ⊢ (♯‘𝐻) = (♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))
48	47	oveq1i 7351	. . . . . . 7 ⊢ ((♯‘𝐻) − 1) = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
49	36, 48	eqtri 2754	. . . . . 6 ⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
50		fourierdlem100.s	. . . . . . 7 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
51		isoeq5 7250	. . . . . . . . 9 ⊢ (𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) → (𝑓 Isom < , < ((0...𝑁), 𝐻) ↔ 𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))))
52	46, 51	ax-mp 5	. . . . . . . 8 ⊢ (𝑓 Isom < , < ((0...𝑁), 𝐻) ↔ 𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
53	52	iotabii 6461	. . . . . . 7 ⊢ (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻)) = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
54	50, 53	eqtri 2754	. . . . . 6 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
55	19, 20, 21, 22, 2, 5, 26, 8, 35, 49, 54	fourierdlem54 46198	. . . . 5 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))))
56	55	simpld 494	. . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
57	56	simpld 494	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
58	56	simprd 495	. . 3 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
59	1, 6	fssresd 6685	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)):(𝐶[,]𝐷)⟶ℂ)
60		ioossicc 13328	. . . . . 6 ⊢ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1)))
61	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ∈ ℝ)
62	61	rexrd 11157	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ∈ ℝ^*)
63	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐷 ∈ (𝐶(,)+∞))
64	63, 4	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐷 ∈ ℝ)
65	64	rexrd 11157	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐷 ∈ ℝ^*)
66	8, 57, 58	fourierdlem15 46160	. . . . . . . 8 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
67	66	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
68		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0..^𝑁))
69	62, 65, 67, 68	fourierdlem8 46153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (𝐶[,]𝐷))
70	60, 69	sstrid 3941	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (𝐶[,]𝐷))
71	70	resabs1d 5952	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐹 ↾ (𝐶[,]𝐷)) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
72	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ)
73	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑄 ∈ (𝑃‘𝑀))
74	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℂ)
75		fourierdlem100.per	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
76	75	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
77		fourierdlem100.fcn	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
78	77	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
79		fourierdlem100.e	. . . . 5 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
80		fourierdlem100.j	. . . . 5 ⊢ 𝐽 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
81		eqid 2731	. . . . 5 ⊢ ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))
82		eqid 2731	. . . . 5 ⊢ (𝐹 ↾ ((𝐽‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1))))) = (𝐹 ↾ ((𝐽‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))
83		eqid 2731	. . . . 5 ⊢ (𝑦 ∈ (((𝐽‘(𝐸‘(𝑆‘𝑗))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))(,)((𝐸‘(𝑆‘(𝑗 + 1))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))) ↦ ((𝐹 ↾ ((𝐽‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))‘(𝑦 − ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))))) = (𝑦 ∈ (((𝐽‘(𝐸‘(𝑆‘𝑗))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))(,)((𝐸‘(𝑆‘(𝑗 + 1))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))) ↦ ((𝐹 ↾ ((𝐽‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))‘(𝑦 − ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))))
84		fourierdlem100.i	. . . . 5 ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < ))
85	20, 19, 72, 73, 74, 76, 78, 61, 63, 8, 37, 36, 50, 79, 80, 68, 81, 82, 83, 84	fourierdlem90 46234	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))
86	71, 85	eqeltrd 2831	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐹 ↾ (𝐶[,]𝐷)) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))
87		fourierdlem100.r	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
88	87	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
89		eqid 2731	. . . . 5 ⊢ (𝑖 ∈ (0..^𝑀) ↦ 𝑅) = (𝑖 ∈ (0..^𝑀) ↦ 𝑅)
90	20, 19, 72, 73, 74, 76, 78, 88, 61, 63, 8, 37, 36, 50, 79, 80, 68, 81, 84, 89	fourierdlem89 46233	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → if((𝐽‘(𝐸‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝑗))), ((𝑖 ∈ (0..^𝑀) ↦ 𝑅)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝐽‘(𝐸‘(𝑆‘𝑗))))) ∈ ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗)))
91	71	eqcomd 2737	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((𝐹 ↾ (𝐶[,]𝐷)) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
92	91	oveq1d 7356	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗)) = (((𝐹 ↾ (𝐶[,]𝐷)) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗)))
93	90, 92	eleqtrd 2833	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → if((𝐽‘(𝐸‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝑗))), ((𝑖 ∈ (0..^𝑀) ↦ 𝑅)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝐽‘(𝐸‘(𝑆‘𝑗))))) ∈ (((𝐹 ↾ (𝐶[,]𝐷)) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗)))
94		fourierdlem100.l	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
95	94	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
96		eqid 2731	. . . . 5 ⊢ (𝑖 ∈ (0..^𝑀) ↦ 𝐿) = (𝑖 ∈ (0..^𝑀) ↦ 𝐿)
97	20, 19, 72, 73, 74, 76, 78, 95, 61, 63, 8, 37, 36, 50, 79, 80, 68, 81, 84, 96	fourierdlem91 46235	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → if((𝐸‘(𝑆‘(𝑗 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)), ((𝑖 ∈ (0..^𝑀) ↦ 𝐿)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝐸‘(𝑆‘(𝑗 + 1))))) ∈ ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))))
98	91	oveq1d 7356	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) = (((𝐹 ↾ (𝐶[,]𝐷)) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))))
99	97, 98	eleqtrd 2833	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → if((𝐸‘(𝑆‘(𝑗 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)), ((𝑖 ∈ (0..^𝑀) ↦ 𝐿)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝐸‘(𝑆‘(𝑗 + 1))))) ∈ (((𝐹 ↾ (𝐶[,]𝐷)) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))))
100	18, 57, 58, 59, 86, 93, 99	fourierdlem69 46213	. 2 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) ∈ 𝐿¹)
101	7, 100	eqeltrrd 2832	1 ⊢ (𝜑 → (𝑥 ∈ (𝐶[,]𝐷) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 {crab 3395 ∪ cun 3895 ifcif 4470 {cpr 4573 class class class wbr 5086 ↦ cmpt 5167 ran crn 5612 ↾ cres 5613 ℩cio 6430 ⟶wf 6472 ‘cfv 6476 Isom wiso 6477 (class class class)co 7341 ↑_m cmap 8745 supcsup 9319 ℂcc 10999 ℝcr 11000 0cc0 11001 1c1 11002 + caddc 11004 · cmul 11006 +∞cpnf 11138 ℝ^*cxr 11140 < clt 11141 ≤ cle 11142 − cmin 11339 / cdiv 11769 ℕcn 12120 ℤcz 12463 (,)cioo 13240 (,]cioc 13241 [,]cicc 13243 ...cfz 13402 ..^cfzo 13549 ⌊cfl 13689 ♯chash 14232 –cn→ccncf 24791 𝐿¹cibl 25540 lim_ℂ climc 25785

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cc 10321 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-symdif 4198 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-ofr 7606 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-dju 9789 df-card 9827 df-acn 9830 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-xnn0 12450 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ioo 13244 df-ioc 13245 df-ico 13246 df-icc 13247 df-fz 13403 df-fzo 13550 df-fl 13691 df-mod 13769 df-seq 13904 df-exp 13964 df-hash 14233 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-limsup 15373 df-clim 15390 df-rlim 15391 df-sum 15589 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-hom 17180 df-cco 17181 df-rest 17321 df-topn 17322 df-0g 17340 df-gsum 17341 df-topgen 17342 df-pt 17343 df-prds 17346 df-xrs 17401 df-qtop 17406 df-imas 17407 df-xps 17409 df-mre 17483 df-mrc 17484 df-acs 17486 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19224 df-cmn 19689 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-cnfld 21287 df-top 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-cld 22929 df-ntr 22930 df-cls 22931 df-nei 23008 df-lp 23046 df-cn 23137 df-cnp 23138 df-cmp 23297 df-tx 23472 df-hmeo 23665 df-xms 24230 df-ms 24231 df-tms 24232 df-cncf 24793 df-ovol 25387 df-vol 25388 df-mbf 25542 df-itg1 25543 df-itg2 25544 df-ibl 25545 df-itg 25546 df-0p 25593 df-limc 25789

This theorem is referenced by: fourierdlem105 46249

W3C validator