fourierdlem90 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem90 46181

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 46103	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
5	4	simp1d 1142	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6	4	simp2d 1143	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	5, 6	iccssred 13355	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8	4	simp3d 1144	. . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵)
9		fourierdlem90.J	. . . . . 6 ⊢ 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
10	5, 6, 8, 9	fourierdlem17 46109	. . . . 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 46096	. . . . . 6 ⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵))
14		fourierdlem90.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
15		fourierdlem90.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞))
16		elioore 13296	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (𝐶(,)+∞) → 𝐷 ∈ ℝ)
17	15, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℝ)
18		elioo4g 13327	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (𝐶(,)+∞) ↔ ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐷 ∈ ℝ) ∧ (𝐶 < 𝐷 ∧ 𝐷 < +∞)))
19	15, 18	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐷 ∈ ℝ) ∧ (𝐶 < 𝐷 ∧ 𝐷 < +∞)))
20	19	simprd 495	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 < 𝐷 ∧ 𝐷 < +∞))
21	20	simpld 494	. . . . . . . . . . . . 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 46145	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
27	26	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
28	27	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
29	27	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
30	22	fourierdlem2 46094	. . . . . . . . . . 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 17	. . . . . . . . . 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 232	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))
33	32	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (ℝ ↑_m (0...𝑁)))
34		elmapi 8783	. . . . . . . 8 ⊢ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) → 𝑆:(0...𝑁)⟶ℝ)
35	33, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶ℝ)
36		fourierdlem90.17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (0..^𝑁))
37		elfzofz 13596	. . . . . . . 8 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ (0...𝑁))
38	36, 37	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
39	35, 38	ffvelcdmd 7023	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℝ)
40	13, 39	ffvelcdmd 7023	. . . . 5 ⊢ (𝜑 → (𝐸‘(𝑆‘𝐽)) ∈ (𝐴(,]𝐵))
41	10, 40	ffvelcdmd 7023	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐸‘(𝑆‘𝐽))) ∈ (𝐴[,]𝐵))
42	7, 41	sseldd 3938	. . 3 ⊢ (𝜑 → (𝐿‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ)
43	5	rexrd 11184	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
44		iocssre 13348	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
45	43, 6, 44	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
46		fzofzp1 13685	. . . . . . 7 ⊢ (𝐽 ∈ (0..^𝑁) → (𝐽 + 1) ∈ (0...𝑁))
47	36, 46	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐽 + 1) ∈ (0...𝑁))
48	35, 47	ffvelcdmd 7023	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ)
49	13, 48	ffvelcdmd 7023	. . . 4 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ (𝐴(,]𝐵))
50	45, 49	sseldd 3938	. . 3 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
51		eqid 2729	. . 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 11566	. . . 4 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℝ)
54	52, 53	eqeltrid 2832	. . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ)
55		eqid 2729	. . 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 2816	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ (0..^𝑁) ↔ 𝐽 ∈ (0..^𝑁)))
58	57	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ↔ (𝜑 ∧ 𝐽 ∈ (0..^𝑁))))
59		fveq2 6826	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → (𝑆‘𝑗) = (𝑆‘𝐽))
60	59	fveq2d 6830	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘𝑗)) = (𝐸‘(𝑆‘𝐽)))
61	60	fveq2d 6830	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (𝐿‘(𝐸‘(𝑆‘𝑗))) = (𝐿‘(𝐸‘(𝑆‘𝐽))))
62		oveq1 7360	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → (𝑗 + 1) = (𝐽 + 1))
63	62	fveq2d 6830	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → (𝑆‘(𝑗 + 1)) = (𝑆‘(𝐽 + 1)))
64	63	fveq2d 6830	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘(𝑗 + 1))) = (𝐸‘(𝑆‘(𝐽 + 1))))
65	61, 64	oveq12d 7371	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → ((𝐿‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))
66	59	fveq2d 6830	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → (𝐼‘(𝑆‘𝑗)) = (𝐼‘(𝑆‘𝐽)))
67	66	fveq2d 6830	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (𝑄‘(𝐼‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝐽))))
68	66	oveq1d 7368	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → ((𝐼‘(𝑆‘𝑗)) + 1) = ((𝐼‘(𝑆‘𝐽)) + 1))
69	68	fveq2d 6830	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
70	67, 69	oveq12d 7371	. . . . . . . . . . . 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 3971	. . . . . . . . . . 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 344	. . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 · 𝑇) = (𝑘 · (𝐵 − 𝐴))
74	73	oveq2i 7364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑘 · (𝐵 − 𝐴)))
75	74	eleq1i 2819	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
76	75	rexbii 3076	. . . . . . . . . . . . . . . 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 3400	. . . . . . . . . . . . . 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 4118	. . . . . . . . . . . . 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 2752	. . . . . . . . . . . 12 ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})
81		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
82	11	eqcomi 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 − 𝐴) = 𝑇
83	82	oveq2i 7364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 · (𝐵 − 𝐴)) = (𝑘 · 𝑇)
84	83	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑘 · (𝐵 − 𝐴)) = (𝑘 · 𝑇))
85	81, 84	oveq12d 7371	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑦 + (𝑘 · (𝐵 − 𝐴))) = (𝑥 + (𝑘 · 𝑇)))
86	85	eleq1d 2813	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ((𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
87	86	rexbidv 3153	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
88	87	cbvrabv 3407	. . . . . . . . . . . . 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 4118	. . . . . . . . . . . 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 2752	. . . . . . . . . . 11 ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
91		eqid 2729	. . . . . . . . . . 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 46170	. . . . . . . . . 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 3511	. . . . . . . . 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 671	. . . . . . . 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 687	. . . . . . 7 ⊢ (𝜑 → ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
97	96	resabs1d 5963	. . . . . 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 2735	. . . . 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 46129	. . . . . . . . 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 494	. . . . . . . 8 ⊢ (𝜑 → 𝐼:ℝ⟶(0..^𝑀))
101	100, 39	ffvelcdmd 7023	. . . . . . 7 ⊢ (𝜑 → (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))
102	101	ancli 548	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)))
103		eleq1 2816	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 ∈ (0..^𝑀) ↔ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)))
104	103	anbi2d 630	. . . . . . . . 9 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))))
105		fveq2 6826	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘𝑖) = (𝑄‘(𝐼‘(𝑆‘𝐽))))
106		oveq1 7360	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 + 1) = ((𝐼‘(𝑆‘𝐽)) + 1))
107	106	fveq2d 6830	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘(𝑖 + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
108	105, 107	oveq12d 7371	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
109	108	reseq2d 5934	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))))
110	108	oveq1d 7368	. . . . . . . . . 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 2822	. . . . . . . . 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 344	. . . . . . . 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 3511	. . . . . . 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 65	. . . . . 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 24806	. . . . . 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 65	. . . . 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 2828	. . . 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 2832	. . 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 45877	. 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 7364	. . . . . . 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 7371	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝐽)))))
126	63, 59	oveq12d 7371	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
127	125, 126	eqeq12d 2745	. . . . . . . . . . . . 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 344	. . . . . . . . . . . 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 6829	. . . . . . . . . . . . . . 15 ⊢ (♯‘𝐻) = (♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))
130	129	oveq1i 7363	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐻) − 1) = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
131	24, 130	eqtri 2752	. . . . . . . . . . . . 13 ⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
132		isoeq5 7262	. . . . . . . . . . . . . . . 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 6471	. . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . 13 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
136		eqid 2729	. . . . . . . . . . . . 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 46156	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))
138	128, 137	vtoclg 3511	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (0..^𝑁) → ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
139	138	anabsi7 671	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
140	36, 139	mpdan 687	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
141	50	recnd 11162	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
142	48	recnd 11162	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℂ)
143	14, 17	iccssred 13355	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
144		ax-resscn 11085	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
145	143, 144	sstrdi 3950	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ)
146	22, 29, 28	fourierdlem15 46107	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
147	146, 38	ffvelcdmd 7023	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐽) ∈ (𝐶[,]𝐷))
148	145, 147	sseldd 3938	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℂ)
149	142, 148	subcld 11493	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) ∈ ℂ)
150	42	recnd 11162	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘(𝐸‘(𝑆‘𝐽))) ∈ ℂ)
151	141, 149, 150	subsub23d 45272	. . . . . . . . 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 257	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (𝐿‘(𝐸‘(𝑆‘𝐽))))
153	152	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(𝐸‘(𝑆‘𝐽))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
154	153	oveq1d 7368	. . . . . 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 11493	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) ∈ ℂ)
156	155, 142, 141	addsub12d 11516	. . . . . . 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 11525	. . . . . . . . 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 11481	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = 0)
159	158	oveq1d 7368	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
160		df-neg 11368	. . . . . . . . . 10 ⊢ -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
161	142, 148	negsubdi2d 11509	. . . . . . . . . 10 ⊢ (𝜑 → -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
162	160, 161	eqtr3id 2778	. . . . . . . . 9 ⊢ (𝜑 → (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
163	157, 159, 162	3eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
164	163	oveq2d 7369	. . . . . . 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 11496	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1)))) = (𝑆‘𝐽))
166	156, 164, 165	3eqtrd 2768	. . . . . 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 2768	. . . . 5 ⊢ (𝜑 → ((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = (𝑆‘𝐽))
168	52	oveq2i 7364	. . . . . 6 ⊢ ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
169	141, 142	pncan3d 11496	. . . . . 6 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (𝑆‘(𝐽 + 1)))
170	168, 169	eqtrid 2776	. . . . 5 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = (𝑆‘(𝐽 + 1)))
171	167, 170	oveq12d 7371	. . . 4 ⊢ (𝜑 → (((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) = ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))
172	171	mpteq1d 5185	. . 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 6895	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝐹‘𝑦)))
175	174	reseq1d 5933	. . . 4 ⊢ (𝜑 → (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) = ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))))
176		ioossre 13328	. . . . . 6 ⊢ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ℝ
177	176	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ℝ)
178	177	resmptd 5995	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (𝐹‘𝑦)) ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) = (𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ↦ (𝐹‘𝑦)))
179	56	fveq1i 6827	. . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐿‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ)
182	181	rexrd 11184	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐿‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ^*)
183	50	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
184	183	rexrd 11184	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ^*)
185	177	sselda 3937	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑦 ∈ ℝ)
186	54	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑈 ∈ ℝ)
187	185, 186	resubcld 11566	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑦 − 𝑈) ∈ ℝ)
188	39	rexrd 11184	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℝ^*)
189	188	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑆‘𝐽) ∈ ℝ^*)
190	48	rexrd 11184	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ^*)
191	190	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ∈ ℝ^*)
192		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))
193		ioogtlb 45480	. . . . . . . . . . 11 ⊢ (((𝑆‘𝐽) ∈ ℝ^* ∧ (𝑆‘(𝐽 + 1)) ∈ ℝ^* ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑆‘𝐽) < 𝑦)
194	189, 191, 192, 193	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑆‘𝐽) < 𝑦)
195	167	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = (𝑆‘𝐽))
196	185	recnd 11162	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑦 ∈ ℂ)
197	186	recnd 11162	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑈 ∈ ℂ)
198	196, 197	npcand 11497	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝑦 − 𝑈) + 𝑈) = 𝑦)
199	194, 195, 198	3brtr4d 5127	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈) < ((𝑦 − 𝑈) + 𝑈))
200	181, 187, 186	ltadd1d 11731	. . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐿‘(𝐸‘(𝑆‘𝐽))) < (𝑦 − 𝑈))
202		iooltub 45495	. . . . . . . . . . 11 ⊢ (((𝑆‘𝐽) ∈ ℝ^* ∧ (𝑆‘(𝐽 + 1)) ∈ ℝ^* ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑦 < (𝑆‘(𝐽 + 1)))
203	189, 191, 192, 202	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑦 < (𝑆‘(𝐽 + 1)))
204	170	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = (𝑆‘(𝐽 + 1)))
205	203, 198, 204	3brtr4d 5127	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝑦 − 𝑈) + 𝑈) < ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈))
206	187, 183, 186	ltadd1d 11731	. . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑦 − 𝑈) < (𝐸‘(𝑆‘(𝐽 + 1))))
208	182, 184, 187, 201, 207	eliood 45483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑦 − 𝑈) ∈ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))
209		fvres 6845	. . . . . . 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 17	. . . . . 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 7364	. . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ∈ ℂ)
214	141	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
215	196, 213, 214	subsub2d 11522	. . . . . . . . 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 11493	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℂ)
217	6, 5	resubcld 11566	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
218	11, 217	eqeltrid 2832	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ∈ ℝ)
219	218	recnd 11162	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ ℂ)
220	219	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑇 ∈ ℂ)
221	5, 6	posdifd 11725	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
222	8, 221	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
223	222, 11	breqtrrdi 5137	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑇)
224	223	gt0ne0d 11702	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ≠ 0)
225	224	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝑇 ≠ 0)
226	216, 220, 225	divcan1d 11919	. . . . . . . . . . 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 2735	. . . . . . . . . 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 7369	. . . . . . . . 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 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝑦 − 𝑈) = (𝑦 + ((((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) · 𝑇)))
230	229	fveq2d 6830	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → 𝐹:ℝ⟶ℂ)
232	218	adantr 480	. . . . . . . 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 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → 𝑥 = (𝑆‘(𝐽 + 1)))
235		oveq2 7361	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘(𝐽 + 1))))
236	235	oveq1d 7368	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇))
237	236	fveq2d 6830	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
238	237	oveq1d 7368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
239	234, 238	oveq12d 7371	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
240	239	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 = (𝑆‘(𝐽 + 1))) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
241	6, 48	resubcld 11566	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐵 − (𝑆‘(𝐽 + 1))) ∈ ℝ)
242	241, 218, 224	redivcld 11970	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℝ)
243	242	flcld 13720	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℤ)
244	243	zred 12598	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℝ)
245	244, 218	remulcld 11164	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℝ)
246	48, 245	readdcld 11163	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) ∈ ℝ)
247	233, 240, 48, 246	fvmptd 6941	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
248	247	oveq1d 7368	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))))
249	244	recnd 11162	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℂ)
250	249, 219	mulcld 11154	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℂ)
251	142, 250	pncan2d 11495	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
252	248, 251	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
253	252	oveq1d 7368	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇))
254	249, 219, 224	divcan4d 11924	. . . . . . . . . . 11 ⊢ (𝜑 → (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
255	253, 254	eqtrd 2764	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
256	255, 243	eqeltrd 2828	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ)
257	256	adantr 480	. . . . . . . 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 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
260	231, 232, 257, 185, 259	fperiodmul 45289	. . . . . . 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 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐹‘(𝑦 − 𝑈)) = (𝐹‘𝑦))
262	180, 210, 261	3eqtrrd 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) → (𝐹‘𝑦) = (𝐺‘(𝑦 − 𝑈)))
263	262	mpteq2dva 5188	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ↦ (𝐹‘𝑦)) = (𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ↦ (𝐺‘(𝑦 − 𝑈))))
264	175, 178, 263	3eqtrrd 2769	. . 3 ⊢ (𝜑 → (𝑦 ∈ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ↦ (𝐺‘(𝑦 − 𝑈))) = (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))))
265	122, 172, 264	3eqtrd 2768	. 2 ⊢ (𝜑 → 𝑅 = (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))))
266	171	oveq1d 7368	. 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 2842	1 ⊢ (𝜑 → (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) ∈ (((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3396 ∪ cun 3903 ⊆ wss 3905 ifcif 4478 {cpr 4581 class class class wbr 5095 ↦ cmpt 5176 ran crn 5624 ↾ cres 5625 ℩cio 6440 ⟶wf 6482 ‘cfv 6486 Isom wiso 6487 (class class class)co 7353 ↑_m cmap 8760 supcsup 9349 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 +∞cpnf 11165 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 − cmin 11365 -cneg 11366 / cdiv 11795 ℕcn 12146 2c2 12201 ℤcz 12489 (,)cioo 13266 (,]cioc 13267 [,]cicc 13269 ...cfz 13428 ..^cfzo 13575 ⌊cfl 13712 ♯chash 14255 –cn→ccncf 24785

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-dju 9816 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-xnn0 12476 df-z 12490 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-rest 17344 df-topgen 17365 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-top 22797 df-topon 22814 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-cmp 23290 df-cncf 24787

This theorem is referenced by: fourierdlem97 46188 fourierdlem98 46189 fourierdlem100 46191 fourierdlem107 46198 fourierdlem109 46200

W3C validator