fourierdlem90 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem90 47175

Description: Given a piecewise continuous function, it is still continuous with respect to an open interval of the moved partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem90.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem90.t	⊢ 𝑇 = (𝐵 − 𝐴)
fourierdlem90.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem90.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem90.f	⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
fourierdlem90.6	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierdlem90.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem90.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem90.d	⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞))
fourierdlem90.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem90.h	⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})
fourierdlem90.n	⊢ 𝑁 = ((♯‘𝐻) − 1)
fourierdlem90.s	⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
fourierdlem90.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
fourierdlem90.J	⊢ 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
fourierdlem90.17	⊢ (𝜑 → 𝐽 ∈ (0..^𝑁))
fourierdlem90.u	⊢ 𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))
fourierdlem90.g	⊢ 𝐺 = (𝐹 ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))
fourierdlem90.r	⊢ 𝑅 = (𝑦 ∈ (((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ↦ (𝐺‘(𝑦 − 𝑈)))
fourierdlem90.i	⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐿‘(𝐸‘𝑥))}, ℝ, < ))

**Assertion**
Ref	Expression
fourierdlem90	⊢ (𝜑 → (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) ∈ (((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))–cn→ℂ))

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 𝑦,𝐺 𝑓,𝐻,𝑦 𝑥,𝐻 𝑓,𝐼,𝑘 𝑖,𝐼,𝑥 𝑖,𝐽,𝑥,𝑦 𝑖,𝐿,𝑥,𝑦 𝑖,𝑀,𝑥 𝑚,𝑀,𝑝 𝑓,𝑁,𝑘,𝑦 𝑖,𝑁,𝑥 𝑚,𝑁,𝑝 𝑄,𝑓,𝑘,𝑦 𝑄,𝑖,𝑥 𝑄,𝑝 𝑆,𝑓,𝑘,𝑦 𝑆,𝑖,𝑥 𝑆,𝑝 𝑇,𝑖,𝑘,𝑥,𝑦 𝑦,𝑈 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(𝑥, 𝑦, 𝑓, 𝑖, 𝑘, 𝑚, 𝑝) 𝑆(𝑚) 𝑇(𝑓, 𝑚, 𝑝) 𝑈(𝑥, 𝑓, 𝑖, 𝑘, 𝑚, 𝑝) 𝐸(𝑚, 𝑝) 𝐹(𝑓, 𝑘, 𝑚, 𝑝) 𝐺(𝑥, 𝑓, 𝑖, 𝑘, 𝑚, 𝑝) 𝐻(𝑖, 𝑘, 𝑚, 𝑝) 𝐼(𝑦, 𝑚, 𝑝) 𝐽(𝑓, 𝑘, 𝑚, 𝑝) 𝐿(𝑓, 𝑘, 𝑚, 𝑝) 𝑀(𝑦, 𝑓, 𝑘) 𝑂(𝑥, 𝑦, 𝑓, 𝑖, 𝑘, 𝑚, 𝑝)

Proof of Theorem fourierdlem90 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem90.p	. . . . . . 7 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
2		fourierdlem90.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		fourierdlem90.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
4	1, 2, 3	fourierdlem11 47097	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
5	4	simp1d 1160	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6	4	simp2d 1161	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	5, 6	iccssred 13558	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8	4	simp3d 1162	. . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵)
9		fourierdlem90.J	. . . . . 6 ⊢ 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
10	5, 6, 8, 9	fourierdlem17 47103	. . . . 5 ⊢ (𝜑 → 𝐿:(𝐴(,]𝐵)⟶(𝐴[,]𝐵))
11		fourierdlem90.t	. . . . . . 7 ⊢ 𝑇 = (𝐵 − 𝐴)
12		fourierdlem90.e	. . . . . . 7 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
13	5, 6, 8, 11, 12	fourierdlem4 47090	. . . . . 6 ⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵))
14		fourierdlem90.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
15		fourierdlem90.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞))
16		elioore 13499	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (𝐶(,)+∞) → 𝐷 ∈ ℝ)
17	15, 16	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℝ)
18		elioo4g 13530	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (𝐶(,)+∞) ↔ ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐷 ∈ ℝ) ∧ (𝐶 < 𝐷 ∧ 𝐷 < +∞)))
19	15, 18	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐷 ∈ ℝ) ∧ (𝐶 < 𝐷 ∧ 𝐷 < +∞)))
20	19	simprd 501	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 < 𝐷 ∧ 𝐷 < +∞))
21	20	simpld 500	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 < 𝐷)
22		fourierdlem90.o	. . . . . . . . . . . . 13 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
23		fourierdlem90.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})
24		fourierdlem90.n	. . . . . . . . . . . . 13 ⊢ 𝑁 = ((♯‘𝐻) − 1)
25		fourierdlem90.s	. . . . . . . . . . . . 13 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
26	11, 1, 2, 3, 14, 17, 21, 22, 23, 24, 25	fourierdlem54 47139	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
27	26	simpld 500	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
28	27	simprd 501	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
29	27	simpld 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
30	22	fourierdlem2 47088	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))))
31	29, 30	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))))
32	28, 31	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))
33	32	simpld 500	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (ℝ ↑_m (0...𝑁)))
34		elmapi 8862	. . . . . . . 8 ⊢ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) → 𝑆:(0...𝑁)⟶ℝ)
35	33, 34	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶ℝ)
36		fourierdlem90.17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (0..^𝑁))
37		elfzofz 13803	. . . . . . . 8 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ (0...𝑁))
38	36, 37	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
39	35, 38	ffvelcdmd 7083	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℝ)
40	13, 39	ffvelcdmd 7083	. . . . 5 ⊢ (𝜑 → (𝐸‘(𝑆‘𝐽)) ∈ (𝐴(,]𝐵))
41	10, 40	ffvelcdmd 7083	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐸‘(𝑆‘𝐽))) ∈ (𝐴[,]𝐵))
42	7, 41	sseldd 3932	. . 3 ⊢ (𝜑 → (𝐿‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ)
43	5	rexrd 11352	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
44		iocssre 13551	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
45	43, 6, 44	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
46		fzofzp1 13892	. . . . . . 7 ⊢ (𝐽 ∈ (0..^𝑁) → (𝐽 + 1) ∈ (0...𝑁))
47	36, 46	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐽 + 1) ∈ (0...𝑁))
48	35, 47	ffvelcdmd 7083	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ)
49	13, 48	ffvelcdmd 7083	. . . 4 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ (𝐴(,]𝐵))
50	45, 49	sseldd 3932	. . 3 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
51		eqid 2761	. . 3 ⊢ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) = ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))
52		fourierdlem90.u	. . . 4 ⊢ 𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))
53	48, 50	resubcld 11737	. . . 4 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℝ)
54	52, 53	eqeltrid 2865	. . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ)
55		eqid 2761	. . 3 ⊢ (((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) = (((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈))
56		fourierdlem90.g	. . . 4 ⊢ 𝐺 = (𝐹 ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))
57		eleq1 2849	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ (0..^𝑁) ↔ 𝐽 ∈ (0..^𝑁)))
58	57	anbi2d 642	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ↔ (𝜑 ∧ 𝐽 ∈ (0..^𝑁))))
59		fveq2 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → (𝑆‘𝑗) = (𝑆‘𝐽))
60	59	fveq2d 6887	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘𝑗)) = (𝐸‘(𝑆‘𝐽)))
61	60	fveq2d 6887	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (𝐿‘(𝐸‘(𝑆‘𝑗))) = (𝐿‘(𝐸‘(𝑆‘𝐽))))
62		oveq1 7425	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → (𝑗 + 1) = (𝐽 + 1))
63	62	fveq2d 6887	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → (𝑆‘(𝑗 + 1)) = (𝑆‘(𝐽 + 1)))
64	63	fveq2d 6887	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘(𝑗 + 1))) = (𝐸‘(𝑆‘(𝐽 + 1))))
65	61, 64	oveq12d 7436	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → ((𝐿‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))
66	59	fveq2d 6887	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → (𝐼‘(𝑆‘𝑗)) = (𝐼‘(𝑆‘𝐽)))
67	66	fveq2d 6887	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (𝑄‘(𝐼‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝐽))))
68	66	oveq1d 7433	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → ((𝐼‘(𝑆‘𝑗)) + 1) = ((𝐼‘(𝑆‘𝐽)) + 1))
69	68	fveq2d 6887	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
70	67, 69	oveq12d 7436	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1))) = ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
71	65, 70	sseq12d 3964	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (((𝐿‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1))) ↔ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))))
72	58, 71	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐿‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)))) ↔ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))))
73	11	oveq2i 7429	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 · 𝑇) = (𝑘 · (𝐵 − 𝐴))
74	73	oveq2i 7429	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑘 · (𝐵 − 𝐴)))
75	74	eleq1i 2852	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
76	75	rexbii 3110	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
77	76	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝐶[,]𝐷) → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄))
78	77	rabbiia 3417	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}
79	78	uneq2i 4112	. . . . . . . . . . . . 13 ⊢ ({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})
80	23, 79	eqtri 2784	. . . . . . . . . . . 12 ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})
81		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
82	11	eqcomi 2770	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 − 𝐴) = 𝑇
83	82	oveq2i 7429	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 · (𝐵 − 𝐴)) = (𝑘 · 𝑇)
84	83	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑘 · (𝐵 − 𝐴)) = (𝑘 · 𝑇))
85	81, 84	oveq12d 7436	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑦 + (𝑘 · (𝐵 − 𝐴))) = (𝑥 + (𝑘 · 𝑇)))
86	85	eleq1d 2846	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ((𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
87	86	rexbidv 3187	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
88	87	cbvrabv 3423	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} = {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}
89	88	uneq2i 4112	. . . . . . . . . . . 12 ⊢ ({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})
90	80, 89	eqtri 2784	. . . . . . . . . . 11 ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
91		eqid 2761	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑗) + if(((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) < ((𝑄‘1) − 𝐴), (((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) / 2), (((𝑄‘1) − 𝐴) / 2))) = ((𝑆‘𝑗) + if(((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) < ((𝑄‘1) − 𝐴), (((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) / 2), (((𝑄‘1) − 𝐴) / 2)))
92		fourierdlem90.i	. . . . . . . . . . 11 ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐿‘(𝐸‘𝑥))}, ℝ, < ))
93	11, 1, 2, 3, 14, 17, 21, 22, 90, 24, 25, 12, 9, 91, 92	fourierdlem79 47164	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐿‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1))))
94	72, 93	vtoclg 3518	. . . . . . . . 9 ⊢ (𝐽 ∈ (0..^𝑁) → ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))))
95	94	anabsi7 684	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
96	36, 95	mpdan 700	. . . . . . 7 ⊢ (𝜑 → ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
97	96	resabs1d 5999	. . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) = (𝐹 ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))))
98	97	eqcomd 2767	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) = ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))))
99	1, 2, 3, 11, 12, 9, 92	fourierdlem37 47123	. . . . . . . . 9 ⊢ (𝜑 → (𝐼:ℝ⟶(0..^𝑀) ∧ (𝑥 ∈ ℝ → sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐿‘(𝐸‘𝑥))}, ℝ, < ) ∈ {𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐿‘(𝐸‘𝑥))})))
100	99	simpld 500	. . . . . . . 8 ⊢ (𝜑 → 𝐼:ℝ⟶(0..^𝑀))
101	100, 39	ffvelcdmd 7083	. . . . . . 7 ⊢ (𝜑 → (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))
102	101	ancli 558	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)))
103		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 ∈ (0..^𝑀) ↔ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)))
104	103	anbi2d 642	. . . . . . . . 9 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))))
105		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘𝑖) = (𝑄‘(𝐼‘(𝑆‘𝐽))))
106		oveq1 7425	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 + 1) = ((𝐼‘(𝑆‘𝐽)) + 1))
107	106	fveq2d 6887	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘(𝑖 + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
108	105, 107	oveq12d 7436	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
109	108	reseq2d 5970	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))))
110	108	oveq1d 7433	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) = (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ))
111	109, 110	eleq12d 2855	. . . . . . . . 9 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) ↔ (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ∈ (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ)))
112	104, 111	imbi12d 347	. . . . . . . 8 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) ↔ ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ∈ (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ))))
113		fourierdlem90.fcn	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
114	112, 113	vtoclg 3518	. . . . . . 7 ⊢ ((𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀) → ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ∈ (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ)))
115	101, 102, 114	sylc 66	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ∈ (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ))
116		rescncf 25211	. . . . . 6 ⊢ (((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ∈ (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) ∈ (((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))–cn→ℂ)))
117	96, 115, 116	sylc 66	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) ∈ (((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))–cn→ℂ))
118	98, 117	eqeltrd 2861	. . . 4 ⊢ (𝜑 → (𝐹 ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) ∈ (((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))–cn→ℂ))
119	56, 118	eqeltrid 2865	. . 3 ⊢ (𝜑 → 𝐺 ∈ (((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))–cn→ℂ))
120		fourierdlem90.r	. . 3 ⊢ 𝑅 = (𝑦 ∈ (((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ↦ (𝐺‘(𝑦 − 𝑈)))
121	42, 50, 51, 54, 55, 119, 120	cncfshiftioo 46871	. 2 ⊢ (𝜑 → 𝑅 ∈ ((((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈))–cn→ℂ))
122	120	a1i 11	. . 3 ⊢ (𝜑 → 𝑅 = (𝑦 ∈ (((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ↦ (𝐺‘(𝑦 − 𝑈))))
123	52	oveq2i 7429	. . . . . . 7 ⊢ ((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = ((𝐿‘(𝐸‘(𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
124	123	a1i 11	. . . . . 6 ⊢ (𝜑 → ((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = ((𝐿‘(𝐸‘(𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))))
125	64, 61	oveq12d 7436	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝐽)))))
126	63, 59	oveq12d 7436	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
127	125, 126	eqeq12d 2777	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) ↔ ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
128	58, 127	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) ↔ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))))
129	80	fveq2i 6886	. . . . . . . . . . . . . . 15 ⊢ (♯‘𝐻) = (♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))
130	129	oveq1i 7428	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐻) − 1) = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
131	24, 130	eqtri 2784	. . . . . . . . . . . . 13 ⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
132		isoeq5 7327	. . . . . . . . . . . . . . . 16 ⊢ (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}))))
133	80, 132	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Isom < , < ((0...𝑁), 𝐻) ↔ 𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
134	133	iotabii 6522	. . . . . . . . . . . . . 14 ⊢ (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻)) = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
135	25, 134	eqtri 2784	. . . . . . . . . . . . 13 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
136		eqid 2761	. . . . . . . . . . . . 13 ⊢ ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) = ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))
137	1, 11, 2, 3, 14, 15, 22, 131, 135, 12, 9, 136	fourierdlem65 47150	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))
138	128, 137	vtoclg 3518	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (0..^𝑁) → ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
139	138	anabsi7 684	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
140	36, 139	mpdan 700	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
141	50	recnd 11330	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
142	48	recnd 11330	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℂ)
143	14, 17	iccssred 13558	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
144		ax-resscn 11250	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
145	143, 144	sstrdi 3943	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ)
146	22, 29, 28	fourierdlem15 47101	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
147	146, 38	ffvelcdmd 7083	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐽) ∈ (𝐶[,]𝐷))
148	145, 147	sseldd 3932	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℂ)
149	142, 148	subcld 11662	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) ∈ ℂ)
150	42	recnd 11330	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘(𝐸‘(𝑆‘𝐽))) ∈ ℂ)
151	141, 149, 150	subsub23d 46272	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (𝐿‘(𝐸‘(𝑆‘𝐽))) ↔ ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
152	140, 151	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (𝐿‘(𝐸‘(𝑆‘𝐽))))
153	152	eqcomd 2767	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(𝐸‘(𝑆‘𝐽))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
154	153	oveq1d 7433	. . . . . 6 ⊢ (𝜑 → ((𝐿‘(𝐸‘(𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))))
155	141, 149	subcld 11662	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) ∈ ℂ)
156	155, 142, 141	addsub12d 11685	. . . . . . 7 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = ((𝑆‘(𝐽 + 1)) + (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1))))))
157	141, 149, 141	sub32d 11694	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
158	141	subidd 11650	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = 0)
159	158	oveq1d 7433	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
160		df-neg 11537	. . . . . . . . . 10 ⊢ -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
161	142, 148	negsubdi2d 11678	. . . . . . . . . 10 ⊢ (𝜑 → -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
162	160, 161	eqtr3id 2810	. . . . . . . . 9 ⊢ (𝜑 → (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
163	157, 159, 162	3eqtrd 2800	. . . . . . . 8 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
164	163	oveq2d 7434	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1))))) = ((𝑆‘(𝐽 + 1)) + ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1)))))
165	142, 148	pncan3d 11665	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1)))) = (𝑆‘𝐽))
166	156, 164, 165	3eqtrd 2800	. . . . . 6 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (𝑆‘𝐽))
167	124, 154, 166	3eqtrd 2800	. . . . 5 ⊢ (𝜑 → ((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = (𝑆‘𝐽))
168	52	oveq2i 7429	. . . . . 6 ⊢ ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
169	141, 142	pncan3d 11665	. . . . . 6 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (𝑆‘(𝐽 + 1)))
170	168, 169	eqtrid 2808	. . . . 5 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = (𝑆‘(𝐽 + 1)))
171	167, 170	oveq12d 7436	. . . 4 ⊢ (𝜑 → (((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) = ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))
172	171	mpteq1d 5195	. . 3 ⊢ (𝜑 → (𝑦 ∈ (((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ↦ (𝐺‘(𝑦 − 𝑈))) = (𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ↦ (𝐺‘(𝑦 − 𝑈))))
173		fourierdlem90.f	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
174	173	feqmptd 6951	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
175	174	reseq1d 5969	. . . 4 ⊢ (𝜑 → (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) = ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))))
176		ioossre 13531	. . . . . 6 ⊢ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ℝ
177	176	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ℝ)
178	177	resmptd 6032	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) = (𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ↦ (𝐹‘𝑦)))
179	56	fveq1i 6884	. . . . . . 7 ⊢ (𝐺‘(𝑦 − 𝑈)) = ((𝐹 ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))‘(𝑦 − 𝑈))
180	179	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐺‘(𝑦 − 𝑈)) = ((𝐹 ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))‘(𝑦 − 𝑈)))
181	42	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐿‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ)
182	181	rexrd 11352	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐿‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ^*)
183	50	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
184	183	rexrd 11352	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ^*)
185	177	sselda 3931	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑦 ∈ ℝ)
186	54	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑈 ∈ ℝ)
187	185, 186	resubcld 11737	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑦 − 𝑈) ∈ ℝ)
188	39	rexrd 11352	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℝ^*)
189	188	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑆‘𝐽) ∈ ℝ^*)
190	48	rexrd 11352	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ^*)
191	190	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ∈ ℝ^*)
192		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))
193		ioogtlb 46476	. . . . . . . . . . 11 ⊢ (((𝑆‘𝐽) ∈ ℝ^* ∧ (𝑆‘(𝐽 + 1)) ∈ ℝ^* ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑆‘𝐽) < 𝑦)
194	189, 191, 192, 193	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑆‘𝐽) < 𝑦)
195	167	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = (𝑆‘𝐽))
196	185	recnd 11330	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑦 ∈ ℂ)
197	186	recnd 11330	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑈 ∈ ℂ)
198	196, 197	npcand 11666	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝑦 − 𝑈) + 𝑈) = 𝑦)
199	194, 195, 198	3brtr4d 5137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈) < ((𝑦 − 𝑈) + 𝑈))
200	181, 187, 186	ltadd1d 11902	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝐿‘(𝐸‘(𝑆‘𝐽))) < (𝑦 − 𝑈) ↔ ((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈) < ((𝑦 − 𝑈) + 𝑈)))
201	199, 200	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐿‘(𝐸‘(𝑆‘𝐽))) < (𝑦 − 𝑈))
202		iooltub 46491	. . . . . . . . . . 11 ⊢ (((𝑆‘𝐽) ∈ ℝ^* ∧ (𝑆‘(𝐽 + 1)) ∈ ℝ^* ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑦 < (𝑆‘(𝐽 + 1)))
203	189, 191, 192, 202	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑦 < (𝑆‘(𝐽 + 1)))
204	170	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = (𝑆‘(𝐽 + 1)))
205	203, 198, 204	3brtr4d 5137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝑦 − 𝑈) + 𝑈) < ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈))
206	187, 183, 186	ltadd1d 11902	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝑦 − 𝑈) < (𝐸‘(𝑆‘(𝐽 + 1))) ↔ ((𝑦 − 𝑈) + 𝑈) < ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)))
207	205, 206	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑦 − 𝑈) < (𝐸‘(𝑆‘(𝐽 + 1))))
208	182, 184, 187, 201, 207	eliood 46479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑦 − 𝑈) ∈ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))
209		fvres 6902	. . . . . . 7 ⊢ ((𝑦 − 𝑈) ∈ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝐹 ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))‘(𝑦 − 𝑈)) = (𝐹‘(𝑦 − 𝑈)))
210	208, 209	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝐹 ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))‘(𝑦 − 𝑈)) = (𝐹‘(𝑦 − 𝑈)))
211	52	oveq2i 7429	. . . . . . . . . 10 ⊢ (𝑦 − 𝑈) = (𝑦 − ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
212	211	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑦 − 𝑈) = (𝑦 − ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))))
213	142	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ∈ ℂ)
214	141	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
215	196, 213, 214	subsub2d 11691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑦 − ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (𝑦 + ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))))
216	214, 213	subcld 11662	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℂ)
217	6, 5	resubcld 11737	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
218	11, 217	eqeltrid 2865	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ∈ ℝ)
219	218	recnd 11330	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ ℂ)
220	219	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑇 ∈ ℂ)
221	5, 6	posdifd 11896	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
222	8, 221	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
223	222, 11	breqtrrdi 5147	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑇)
224	223	gt0ne0d 11873	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ≠ 0)
225	224	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑇 ≠ 0)
226	216, 220, 225	divcan1d 12087	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) · 𝑇) = ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))))
227	226	eqcomd 2767	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) · 𝑇))
228	227	oveq2d 7434	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑦 + ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))) = (𝑦 + ((((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) · 𝑇)))
229	212, 215, 228	3eqtrd 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑦 − 𝑈) = (𝑦 + ((((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) · 𝑇)))
230	229	fveq2d 6887	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐹‘(𝑦 − 𝑈)) = (𝐹‘(𝑦 + ((((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) · 𝑇))))
231	173	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝐹:ℝ⟶ℂ)
232	218	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑇 ∈ ℝ)
233	12	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
234		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → 𝑥 = (𝑆‘(𝐽 + 1)))
235		oveq2 7426	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘(𝐽 + 1))))
236	235	oveq1d 7433	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇))
237	236	fveq2d 6887	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
238	237	oveq1d 7433	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
239	234, 238	oveq12d 7436	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
240	239	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 = (𝑆‘(𝐽 + 1))) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
241	6, 48	resubcld 11737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐵 − (𝑆‘(𝐽 + 1))) ∈ ℝ)
242	241, 218, 224	redivcld 12138	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℝ)
243	242	flcld 13931	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℤ)
244	243	zred 12796	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℝ)
245	244, 218	remulcld 11332	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℝ)
246	48, 245	readdcld 11331	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) ∈ ℝ)
247	233, 240, 48, 246	fvmptd 6999	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
248	247	oveq1d 7433	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))))
249	244	recnd 11330	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℂ)
250	249, 219	mulcld 11322	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℂ)
251	142, 250	pncan2d 11664	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
252	248, 251	eqtrd 2796	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
253	252	oveq1d 7433	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇))
254	249, 219, 224	divcan4d 12092	. . . . . . . . . . 11 ⊢ (𝜑 → (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
255	253, 254	eqtrd 2796	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
256	255, 243	eqeltrd 2861	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ)
257	256	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ)
258		fourierdlem90.6	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
259	258	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
260	231, 232, 257, 185, 259	fperiodmul 46289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐹‘(𝑦 + ((((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) · 𝑇))) = (𝐹‘𝑦))
261	230, 260	eqtrd 2796	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐹‘(𝑦 − 𝑈)) = (𝐹‘𝑦))
262	180, 210, 261	3eqtrrd 2801	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐹‘𝑦) = (𝐺‘(𝑦 − 𝑈)))
263	262	mpteq2dva 5198	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ↦ (𝐹‘𝑦)) = (𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ↦ (𝐺‘(𝑦 − 𝑈))))
264	175, 178, 263	3eqtrrd 2801	. . 3 ⊢ (𝜑 → (𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ↦ (𝐺‘(𝑦 − 𝑈))) = (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))))
265	122, 172, 264	3eqtrd 2800	. 2 ⊢ (𝜑 → 𝑅 = (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))))
266	171	oveq1d 7433	. 2 ⊢ (𝜑 → ((((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈))–cn→ℂ) = (((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))–cn→ℂ))
267	121, 265, 266	3eltr3d 2875	1 ⊢ (𝜑 → (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) ∈ (((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))–cn→ℂ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 {crab 3413 ∪ cun 3897 ⊆ wss 3899 ifcif 4482 {cpr 4586 class class class wbr 5103 ↦ cmpt 5186 ran crn 5652 ↾ cres 5653 ℩cio 6491 ⟶wf 6533 ‘cfv 6537 Isom wiso 6538 (class class class)co 7418 ↑_m cmap 8840 supcsup 9425 ℂcc 11191 ℝcr 11192 0cc0 11193 1c1 11194 + caddc 11196 · cmul 11198 +∞cpnf 11333 ℝ^*cxr 11335 < clt 11336 ≤ cle 11337 − cmin 11534 -cneg 11535 / cdiv 11966 ℕcn 12328 2c2 12390 ℤcz 12686 (,)cioo 13469 (,]cioc 13470 [,]cicc 13472 ...cfz 13632 ..^cfzo 13781 ⌊cfl 13923 ♯chash 14467 –cn→ccncf 25190

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 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-oadd 8473 df-er 8710 df-map 8842 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-dju 9975 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-n0 12600 df-xnn0 12673 df-z 12687 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-ioo 13473 df-ioc 13474 df-ico 13475 df-icc 13476 df-fz 13633 df-fzo 13782 df-fl 13925 df-seq 14138 df-exp 14198 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-rest 17586 df-topgen 17607 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-top 23205 df-topon 23222 df-bases 23257 df-cld 23330 df-ntr 23331 df-cls 23332 df-nei 23409 df-lp 23447 df-cmp 23698 df-cncf 25192

This theorem is used by: fourierdlem97 47182 fourierdlem98 47183 fourierdlem100 47185 fourierdlem107 47192 fourierdlem109 47194

W3C validator