fourierdlem91 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem91 46625

Description: Given a piecewise continuous function and changing the interval and the partition, the limit at the upper bound of each interval of the moved partition is still finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem91.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem91.t	⊢ 𝑇 = (𝐵 − 𝐴)
fourierdlem91.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem91.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem91.f	⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
fourierdlem91.6	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierdlem91.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem91.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
fourierdlem91.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem91.d	⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞))
fourierdlem91.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem91.h	⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})
fourierdlem91.n	⊢ 𝑁 = ((♯‘𝐻) − 1)
fourierdlem91.s	⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
fourierdlem91.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
fourierdlem91.J	⊢ 𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
fourierdlem91.17	⊢ (𝜑 → 𝐽 ∈ (0..^𝑁))
fourierdlem91.u	⊢ 𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))
fourierdlem91.i	⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < ))
fourierdlem91.w	⊢ 𝑊 = (𝑖 ∈ (0..^𝑀) ↦ 𝐿)

**Assertion**
Ref	Expression
fourierdlem91	⊢ (𝜑 → if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆‘𝐽))), (𝐹‘(𝐸‘(𝑆‘(𝐽 + 1))))) ∈ ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) lim_ℂ (𝑆‘(𝐽 + 1))))

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

Proof of Theorem fourierdlem91 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem91.q	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
2		fourierdlem91.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		fourierdlem91.p	. . . . . . . . . . . . . 14 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
4	3	fourierdlem2 46537	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
5	2, 4	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
6	1, 5	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
7	6	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
8		elmapi 8796	. . . . . . . . . 10 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
10		fzossfz 13633	. . . . . . . . . 10 ⊢ (0..^𝑀) ⊆ (0...𝑀)
11		fourierdlem91.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝐵 − 𝐴)
12		fourierdlem91.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
13		fourierdlem91.J	. . . . . . . . . . . . 13 ⊢ 𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
14		fourierdlem91.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < ))
15	3, 2, 1, 11, 12, 13, 14	fourierdlem37 46572	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼:ℝ⟶(0..^𝑀) ∧ (𝑥 ∈ ℝ → sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < ) ∈ {𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))})))
16	15	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼:ℝ⟶(0..^𝑀))
17		fourierdlem91.c	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ ℝ)
18		fourierdlem91.d	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞))
19		elioore 13328	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ (𝐶(,)+∞) → 𝐷 ∈ ℝ)
20	18, 19	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷 ∈ ℝ)
21		elioo4g 13359	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (𝐶(,)+∞) ↔ ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐷 ∈ ℝ) ∧ (𝐶 < 𝐷 ∧ 𝐷 < +∞)))
22	18, 21	sylib 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐷 ∈ ℝ) ∧ (𝐶 < 𝐷 ∧ 𝐷 < +∞)))
23	22	simprd 495	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐶 < 𝐷 ∧ 𝐷 < +∞))
24	23	simpld 494	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 < 𝐷)
25		fourierdlem91.o	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
26		oveq1 7374	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑥 → (𝑦 + (𝑘 · 𝑇)) = (𝑥 + (𝑘 · 𝑇)))
27	26	eleq1d 2821	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑥 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
28	27	rexbidv 3161	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
29	28	cbvrabv 3399	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}
30	29	uneq2i 4105	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
31		fourierdlem91.n	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑁 = ((♯‘𝐻) − 1)
32		fourierdlem91.h	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})
33	32	fveq2i 6843	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯‘𝐻) = (♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
34	33	oveq1i 7377	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐻) − 1) = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)
35	31, 34	eqtri 2759	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)
36		fourierdlem91.s	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
37		isoeq5 7276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) → (𝑓 Isom < , < ((0...𝑁), 𝐻) ↔ 𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))))
38	32, 37	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Isom < , < ((0...𝑁), 𝐻) ↔ 𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))
39	38	iotabii 6483	. . . . . . . . . . . . . . . . . . 19 ⊢ (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻)) = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))
40	36, 39	eqtri 2759	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))
41	11, 3, 2, 1, 17, 20, 24, 25, 30, 35, 40	fourierdlem54 46588	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))))
42	41	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
43	42	simprd 495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
44	42	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ)
45	25	fourierdlem2 46537	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))))
46	44, 45	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))))
47	43, 46	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))
48	47	simpld 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ (ℝ ↑_m (0...𝑁)))
49		elmapi 8796	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) → 𝑆:(0...𝑁)⟶ℝ)
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶ℝ)
51		fourierdlem91.17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ (0..^𝑁))
52		elfzofz 13630	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ (0...𝑁))
53	51, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
54	50, 53	ffvelcdmd 7037	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℝ)
55	16, 54	ffvelcdmd 7037	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))
56	10, 55	sselid 3919	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘(𝑆‘𝐽)) ∈ (0...𝑀))
57	9, 56	ffvelcdmd 7037	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ)
58	57	rexrd 11195	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ^*)
59	58	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ^*)
60		fzofzp1 13719	. . . . . . . . . 10 ⊢ ((𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀) → ((𝐼‘(𝑆‘𝐽)) + 1) ∈ (0...𝑀))
61	55, 60	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝐼‘(𝑆‘𝐽)) + 1) ∈ (0...𝑀))
62	9, 61	ffvelcdmd 7037	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈ ℝ)
63	62	rexrd 11195	. . . . . . 7 ⊢ (𝜑 → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈ ℝ^*)
64	63	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈ ℝ^*)
65	3, 2, 1	fourierdlem11 46546	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
66	65	simp1d 1143	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
67	66	rexrd 11195	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
68	65	simp2d 1144	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
69		iocssre 13380	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
70	67, 68, 69	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
71	65	simp3d 1145	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 𝐵)
72	66, 68, 71, 11, 12	fourierdlem4 46539	. . . . . . . . 9 ⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵))
73		fzofzp1 13719	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (0..^𝑁) → (𝐽 + 1) ∈ (0...𝑁))
74	51, 73	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 + 1) ∈ (0...𝑁))
75	50, 74	ffvelcdmd 7037	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ)
76	72, 75	ffvelcdmd 7037	. . . . . . . 8 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ (𝐴(,]𝐵))
77	70, 76	sseldd 3922	. . . . . . 7 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
78	77	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
79	66, 68	iccssred 13387	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
80	66, 68, 71, 13	fourierdlem17 46552	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍:(𝐴(,]𝐵)⟶(𝐴[,]𝐵))
81	72, 54	ffvelcdmd 7037	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸‘(𝑆‘𝐽)) ∈ (𝐴(,]𝐵))
82	80, 81	ffvelcdmd 7037	. . . . . . . . 9 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ (𝐴[,]𝐵))
83	79, 82	sseldd 3922	. . . . . . . 8 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ)
84	47	simprrd 774	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
85		fveq2 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐽 → (𝑆‘𝑖) = (𝑆‘𝐽))
86		oveq1 7374	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝐽 → (𝑖 + 1) = (𝐽 + 1))
87	86	fveq2d 6844	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐽 → (𝑆‘(𝑖 + 1)) = (𝑆‘(𝐽 + 1)))
88	85, 87	breq12d 5098	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐽 → ((𝑆‘𝑖) < (𝑆‘(𝑖 + 1)) ↔ (𝑆‘𝐽) < (𝑆‘(𝐽 + 1))))
89	88	rspccva 3563	. . . . . . . . . . . . . 14 ⊢ ((∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)) ∧ 𝐽 ∈ (0..^𝑁)) → (𝑆‘𝐽) < (𝑆‘(𝐽 + 1)))
90	84, 51, 89	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝐽) < (𝑆‘(𝐽 + 1)))
91	54, 75	posdifd 11737	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆‘𝐽) < (𝑆‘(𝐽 + 1)) ↔ 0 < ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
92	90, 91	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
93		eleq1 2824	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ (0..^𝑁) ↔ 𝐽 ∈ (0..^𝑁)))
94	93	anbi2d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ↔ (𝜑 ∧ 𝐽 ∈ (0..^𝑁))))
95		oveq1 7374	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝐽 → (𝑗 + 1) = (𝐽 + 1))
96	95	fveq2d 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝐽 → (𝑆‘(𝑗 + 1)) = (𝑆‘(𝐽 + 1)))
97	96	fveq2d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘(𝑗 + 1))) = (𝐸‘(𝑆‘(𝐽 + 1))))
98		fveq2 6840	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝐽 → (𝑆‘𝑗) = (𝑆‘𝐽))
99	98	fveq2d 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘𝑗)) = (𝐸‘(𝑆‘𝐽)))
100	99	fveq2d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝐽 → (𝑍‘(𝐸‘(𝑆‘𝑗))) = (𝑍‘(𝐸‘(𝑆‘𝐽))))
101	97, 100	oveq12d 7385	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))))
102	96, 98	oveq12d 7385	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
103	101, 102	eqeq12d 2752	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) ↔ ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
104	94, 103	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) ↔ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))))
105	11	oveq2i 7378	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 · 𝑇) = (𝑘 · (𝐵 − 𝐴))
106	105	oveq2i 7378	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑘 · (𝐵 − 𝐴)))
107	106	eleq1i 2827	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
108	107	rexbii 3084	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
109	108	rgenw 3055	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∀𝑦 ∈ (𝐶[,]𝐷)(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
110		rabbi 3419	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀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})
111	109, 110	mpbi 230	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}
112	111	uneq2i 4105	. . . . . . . . . . . . . . . . . . 19 ⊢ ({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})
113	112	fveq2i 6843	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘({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}))
114	113	oveq1i 7377	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1) = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
115	35, 114	eqtri 2759	. . . . . . . . . . . . . . . 16 ⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
116		isoeq5 7276	. . . . . . . . . . . . . . . . . . 19 ⊢ (({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}) → (𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))))
117	112, 116	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
118	117	iotabii 6483	. . . . . . . . . . . . . . . . 17 ⊢ (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
119	40, 118	eqtri 2759	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
120		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) = ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))
121	3, 11, 2, 1, 17, 18, 25, 115, 119, 12, 13, 120	fourierdlem65 46599	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))
122	104, 121	vtoclg 3499	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (0..^𝑁) → ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
123	122	anabsi7 672	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
124	51, 123	mpdan 688	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
125	92, 124	breqtrrd 5113	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))))
126	83, 77	posdifd 11737	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝐸‘(𝑆‘(𝐽 + 1))) ↔ 0 < ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽))))))
127	125, 126	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝐸‘(𝑆‘(𝐽 + 1))))
128	100, 97	oveq12d 7385	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))
129	98	fveq2d 6844	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → (𝐼‘(𝑆‘𝑗)) = (𝐼‘(𝑆‘𝐽)))
130	129	fveq2d 6844	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑄‘(𝐼‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝐽))))
131	129	oveq1d 7382	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → ((𝐼‘(𝑆‘𝑗)) + 1) = ((𝐼‘(𝑆‘𝐽)) + 1))
132	131	fveq2d 6844	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
133	130, 132	oveq12d 7385	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1))) = ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
134	128, 133	sseq12d 3955	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → (((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1))) ↔ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))))
135	94, 134	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)))) ↔ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))))
136	32, 30	eqtri 2759	. . . . . . . . . . . . . 14 ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
137		eqid 2736	. . . . . . . . . . . . . 14 ⊢ ((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)))
138	11, 3, 2, 1, 17, 20, 24, 25, 136, 31, 36, 12, 13, 137, 14	fourierdlem79 46613	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1))))
139	135, 138	vtoclg 3499	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (0..^𝑁) → ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))))
140	139	anabsi7 672	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
141	51, 140	mpdan 688	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
142	57, 62, 83, 77, 127, 141	fourierdlem10 46545	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄‘(𝐼‘(𝑆‘𝐽))) ≤ (𝑍‘(𝐸‘(𝑆‘𝐽))) ∧ (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
143	142	simpld 494	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ≤ (𝑍‘(𝐸‘(𝑆‘𝐽))))
144	57, 83, 77, 143, 127	lelttrd 11304	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝐸‘(𝑆‘(𝐽 + 1))))
145	144	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝐸‘(𝑆‘(𝐽 + 1))))
146	62	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈ ℝ)
147	142	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
148	147	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
149		neqne 2940	. . . . . . . . 9 ⊢ (¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) → (𝐸‘(𝑆‘(𝐽 + 1))) ≠ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
150	149	necomd 2987	. . . . . . . 8 ⊢ (¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ≠ (𝐸‘(𝑆‘(𝐽 + 1))))
151	150	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ≠ (𝐸‘(𝑆‘(𝐽 + 1))))
152	78, 146, 148, 151	leneltd 11300	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → (𝐸‘(𝑆‘(𝐽 + 1))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
153	59, 64, 78, 145, 152	eliood 45928	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
154		fvres 6859	. . . . 5 ⊢ ((𝐸‘(𝑆‘(𝐽 + 1))) ∈ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝐸‘(𝑆‘(𝐽 + 1)))) = (𝐹‘(𝐸‘(𝑆‘(𝐽 + 1)))))
155	153, 154	syl 17	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝐸‘(𝑆‘(𝐽 + 1)))) = (𝐹‘(𝐸‘(𝑆‘(𝐽 + 1)))))
156	155	eqcomd 2742	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → (𝐹‘(𝐸‘(𝑆‘(𝐽 + 1)))) = ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝐸‘(𝑆‘(𝐽 + 1)))))
157	156	ifeq2da 4499	. 2 ⊢ (𝜑 → if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆‘𝐽))), (𝐹‘(𝐸‘(𝑆‘(𝐽 + 1))))) = if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝐸‘(𝑆‘(𝐽 + 1))))))
158		fourierdlem91.f	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
159		fdm 6677	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℂ → dom 𝐹 = ℝ)
160	158, 159	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = ℝ)
161	160	feq2d 6652	. . . . . 6 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ↔ 𝐹:ℝ⟶ℂ))
162	158, 161	mpbird 257	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
163		ioosscn 13361	. . . . . 6 ⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ℂ
164	163	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ℂ)
165		ioossre 13360	. . . . . 6 ⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ℝ
166	165, 160	sseqtrrid 3965	. . . . 5 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ dom 𝐹)
167		fourierdlem91.u	. . . . . . 7 ⊢ 𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))
168	75, 77	resubcld 11578	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℝ)
169	167, 168	eqeltrid 2840	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ)
170	169	recnd 11173	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ℂ)
171		eqid 2736	. . . . 5 ⊢ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}
172	83, 77, 169	iooshift 45952	. . . . . 6 ⊢ (𝜑 → (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)})
173		ioossre 13360	. . . . . . 7 ⊢ (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ⊆ ℝ
174	173, 160	sseqtrrid 3965	. . . . . 6 ⊢ (𝜑 → (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ⊆ dom 𝐹)
175	172, 174	eqsstrrd 3957	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} ⊆ dom 𝐹)
176		elioore 13328	. . . . . 6 ⊢ (𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) → 𝑦 ∈ ℝ)
177	68, 66	resubcld 11578	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
178	11, 177	eqeltrid 2840	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ ℝ)
179	178	recnd 11173	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℂ)
180	66, 68	posdifd 11737	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
181	71, 180	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
182	181, 11	breqtrrdi 5127	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 𝑇)
183	182	gt0ne0d 11714	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ≠ 0)
184	170, 179, 183	divcan1d 11932	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑈 / 𝑇) · 𝑇) = 𝑈)
185	184	eqcomd 2742	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 = ((𝑈 / 𝑇) · 𝑇))
186	185	oveq2d 7383	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 + 𝑈) = (𝑦 + ((𝑈 / 𝑇) · 𝑇)))
187	186	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + 𝑈) = (𝑦 + ((𝑈 / 𝑇) · 𝑇)))
188	187	fveq2d 6844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑈)) = (𝐹‘(𝑦 + ((𝑈 / 𝑇) · 𝑇))))
189	158	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐹:ℝ⟶ℂ)
190	178	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑇 ∈ ℝ)
191	77	recnd 11173	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
192	75	recnd 11173	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℂ)
193	191, 192	negsubdi2d 11521	. . . . . . . . . . . . 13 ⊢ (𝜑 → -((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
194	193	eqcomd 2742	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) = -((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))))
195	194	oveq1d 7382	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) / 𝑇) = (-((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇))
196	167	oveq1i 7377	. . . . . . . . . . . 12 ⊢ (𝑈 / 𝑇) = (((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) / 𝑇)
197	196	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑈 / 𝑇) = (((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) / 𝑇))
198	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
199		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → 𝑥 = (𝑆‘(𝐽 + 1)))
200		oveq2 7375	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘(𝐽 + 1))))
201	200	oveq1d 7382	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇))
202	201	fveq2d 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
203	202	oveq1d 7382	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
204	199, 203	oveq12d 7385	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
205	204	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 = (𝑆‘(𝐽 + 1))) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
206	68, 75	resubcld 11578	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐵 − (𝑆‘(𝐽 + 1))) ∈ ℝ)
207	206, 178, 183	redivcld 11983	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℝ)
208	207	flcld 13757	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℤ)
209	208	zred 12633	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℝ)
210	209, 178	remulcld 11175	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℝ)
211	75, 210	readdcld 11174	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) ∈ ℝ)
212	198, 205, 75, 211	fvmptd 6955	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
213	212	oveq1d 7382	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))))
214	208	zcnd 12634	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℂ)
215	214, 179	mulcld 11165	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℂ)
216	192, 215	pncan2d 11507	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
217	213, 216	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
218	217, 215	eqeltrd 2836	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℂ)
219	218, 179, 183	divnegd 11944	. . . . . . . . . . 11 ⊢ (𝜑 → -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (-((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇))
220	195, 197, 219	3eqtr4d 2781	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈 / 𝑇) = -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇))
221	217	oveq1d 7382	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇))
222	214, 179, 183	divcan4d 11937	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
223	221, 222	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
224	223, 208	eqeltrd 2836	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ)
225	224	znegcld 12635	. . . . . . . . . 10 ⊢ (𝜑 → -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ)
226	220, 225	eqeltrd 2836	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 / 𝑇) ∈ ℤ)
227	226	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑈 / 𝑇) ∈ ℤ)
228		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
229		fourierdlem91.6	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
230	229	adantlr 716	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
231	189, 190, 227, 228, 230	fperiodmul 45737	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + ((𝑈 / 𝑇) · 𝑇))) = (𝐹‘𝑦))
232	188, 231	eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑈)) = (𝐹‘𝑦))
233	176, 232	sylan2 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) → (𝐹‘(𝑦 + 𝑈)) = (𝐹‘𝑦))
234	6	simprrd 774	. . . . . . . 8 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
235		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘𝑖) = (𝑄‘(𝐼‘(𝑆‘𝐽))))
236		oveq1 7374	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 + 1) = ((𝐼‘(𝑆‘𝐽)) + 1))
237	236	fveq2d 6844	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘(𝑖 + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
238	235, 237	breq12d 5098	. . . . . . . . 9 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
239	238	rspccva 3563	. . . . . . . 8 ⊢ ((∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
240	234, 55, 239	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
241	55	ancli 548	. . . . . . . 8 ⊢ (𝜑 → (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)))
242		eleq1 2824	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 ∈ (0..^𝑀) ↔ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)))
243	242	anbi2d 631	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))))
244	235, 237	oveq12d 7385	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
245	244	reseq2d 5944	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))))
246	244	oveq1d 7382	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) = (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ))
247	245, 246	eleq12d 2830	. . . . . . . . . 10 ⊢ (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)))
248	243, 247	imbi12d 344	. . . . . . . . 9 ⊢ (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))))
249		fourierdlem91.fcn	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
250	248, 249	vtoclg 3499	. . . . . . . 8 ⊢ ((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)))
251	55, 241, 250	sylc 65	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ∈ (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ))
252		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))
253		fourierdlem91.w	. . . . . . . . . . . . 13 ⊢ 𝑊 = (𝑖 ∈ (0..^𝑀) ↦ 𝐿)
254		nfmpt1 5184	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑖 ∈ (0..^𝑀) ↦ 𝐿)
255	253, 254	nfcxfr 2896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝑊
256		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐼‘(𝑆‘𝐽))
257	255, 256	nffv 6850	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝑊‘(𝐼‘(𝑆‘𝐽)))
258	257	nfel1 2915	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝑊‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
259	252, 258	nfim 1898	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑊‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
260	243	biimpar 477	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝜑 ∧ 𝑖 ∈ (0..^𝑀)))
261	260	3adant2 1132	. . . . . . . . . . . . 13 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝜑 ∧ 𝑖 ∈ (0..^𝑀)))
262		fourierdlem91.l	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
263	261, 262	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
264		fveq2 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑊‘𝑖) = (𝑊‘(𝐼‘(𝑆‘𝐽))))
265	264	eqcomd 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑊‘(𝐼‘(𝑆‘𝐽))) = (𝑊‘𝑖))
266	265	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑊‘(𝐼‘(𝑆‘𝐽))) = (𝑊‘𝑖))
267	260	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → 𝑖 ∈ (0..^𝑀))
268		elex 3450	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) → 𝐿 ∈ V)
269	260, 262, 268	3syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → 𝐿 ∈ V)
270	253	fvmpt2 6959	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝐿 ∈ V) → (𝑊‘𝑖) = 𝐿)
271	267, 269, 270	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑊‘𝑖) = 𝐿)
272	266, 271	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑊‘(𝐼‘(𝑆‘𝐽))) = 𝐿)
273	272	3adant2 1132	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑊‘(𝐼‘(𝑆‘𝐽))) = 𝐿)
274	245, 237	oveq12d 7385	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
275	274	eqcomd 2742	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
276	275	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
277	263, 273, 276	3eltr4d 2851	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑊‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
278	277	3exp 1120	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) → ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑊‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))))
279	262	2a1i 12	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑊‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))))
280	278, 279	impbid 212	. . . . . . . . 9 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ↔ ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑊‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))))
281	259, 280, 262	vtoclg1f 3514	. . . . . . . 8 ⊢ ((𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀) → ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑊‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))))
282	55, 241, 281	sylc 65	. . . . . . 7 ⊢ (𝜑 → (𝑊‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
283		eqid 2736	. . . . . . 7 ⊢ if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝐸‘(𝑆‘(𝐽 + 1))))) = if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝐸‘(𝑆‘(𝐽 + 1)))))
284		eqid 2736	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) ∪ {(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))})) = ((TopOpen‘ℂ_fld) ↾_t (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) ∪ {(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))}))
285	57, 62, 240, 251, 282, 83, 77, 127, 141, 283, 284	fourierdlem33 46568	. . . . . 6 ⊢ (𝜑 → if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝐸‘(𝑆‘(𝐽 + 1))))) ∈ (((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) lim_ℂ (𝐸‘(𝑆‘(𝐽 + 1)))))
286	141	resabs1d 5973	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) = (𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))))
287	286	oveq1d 7382	. . . . . 6 ⊢ (𝜑 → (((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) lim_ℂ (𝐸‘(𝑆‘(𝐽 + 1)))) = ((𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) lim_ℂ (𝐸‘(𝑆‘(𝐽 + 1)))))
288	285, 287	eleqtrd 2838	. . . . 5 ⊢ (𝜑 → if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝐸‘(𝑆‘(𝐽 + 1))))) ∈ ((𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) lim_ℂ (𝐸‘(𝑆‘(𝐽 + 1)))))
289	162, 164, 166, 170, 171, 175, 233, 288	limcperiod 46058	. . . 4 ⊢ (𝜑 → if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝐸‘(𝑆‘(𝐽 + 1))))) ∈ ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) lim_ℂ ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)))
290	167	oveq2i 7378	. . . . . 6 ⊢ ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
291	191, 192	pncan3d 11508	. . . . . 6 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (𝑆‘(𝐽 + 1)))
292	290, 291	eqtrid 2783	. . . . 5 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = (𝑆‘(𝐽 + 1)))
293	292	oveq2d 7383	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) lim_ℂ ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) = ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) lim_ℂ (𝑆‘(𝐽 + 1))))
294	289, 293	eleqtrd 2838	. . 3 ⊢ (𝜑 → if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝐸‘(𝑆‘(𝐽 + 1))))) ∈ ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) lim_ℂ (𝑆‘(𝐽 + 1))))
295	167	oveq2i 7378	. . . . . . . . 9 ⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = ((𝑍‘(𝐸‘(𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
296	295	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = ((𝑍‘(𝐸‘(𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))))
297	17, 20	iccssred 13387	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
298		ax-resscn 11095	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℂ
299	297, 298	sstrdi 3934	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ)
300	25, 44, 43	fourierdlem15 46550	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
301	300, 53	ffvelcdmd 7037	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆‘𝐽) ∈ (𝐶[,]𝐷))
302	299, 301	sseldd 3922	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℂ)
303	192, 302	subcld 11505	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) ∈ ℂ)
304	83	recnd 11173	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ℂ)
305	191, 303, 304	subsub23d 45720	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (𝑍‘(𝐸‘(𝑆‘𝐽))) ↔ ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
306	124, 305	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (𝑍‘(𝐸‘(𝑆‘𝐽))))
307	306	eqcomd 2742	. . . . . . . . 9 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
308	307	oveq1d 7382	. . . . . . . 8 ⊢ (𝜑 → ((𝑍‘(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))))))
309	191, 303	subcld 11505	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) ∈ ℂ)
310	309, 192, 191	addsub12d 11528	. . . . . . . . 9 ⊢ (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))))))
311	191, 303, 191	sub32d 11537	. . . . . . . . . . 11 ⊢ (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))))
312	191	subidd 11493	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = 0)
313	312	oveq1d 7382	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
314		df-neg 11380	. . . . . . . . . . . 12 ⊢ -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
315	192, 302	negsubdi2d 11521	. . . . . . . . . . . 12 ⊢ (𝜑 → -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
316	314, 315	eqtr3id 2785	. . . . . . . . . . 11 ⊢ (𝜑 → (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
317	311, 313, 316	3eqtrd 2775	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
318	317	oveq2d 7383	. . . . . . . . 9 ⊢ (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)))))
319	192, 302	pncan3d 11508	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1)))) = (𝑆‘𝐽))
320	310, 318, 319	3eqtrd 2775	. . . . . . . 8 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (𝑆‘𝐽))
321	296, 308, 320	3eqtrd 2775	. . . . . . 7 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = (𝑆‘𝐽))
322	321, 292	oveq12d 7385	. . . . . 6 ⊢ (𝜑 → (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) = ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))
323	172, 322	eqtr3d 2773	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} = ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))
324	323	reseq2d 5944	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) = (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))))
325	324	oveq1d 7382	. . 3 ⊢ (𝜑 → ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) lim_ℂ (𝑆‘(𝐽 + 1))) = ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) lim_ℂ (𝑆‘(𝐽 + 1))))
326	294, 325	eleqtrd 2838	. 2 ⊢ (𝜑 → if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝐸‘(𝑆‘(𝐽 + 1))))) ∈ ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) lim_ℂ (𝑆‘(𝐽 + 1))))
327	157, 326	eqeltrd 2836	1 ⊢ (𝜑 → if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆‘𝐽))), (𝐹‘(𝐸‘(𝑆‘(𝐽 + 1))))) ∈ ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) lim_ℂ (𝑆‘(𝐽 + 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 {crab 3389 Vcvv 3429 ∪ cun 3887 ⊆ wss 3889 ifcif 4466 {csn 4567 {cpr 4569 class class class wbr 5085 ↦ cmpt 5166 dom cdm 5631 ran crn 5632 ↾ cres 5633 ℩cio 6452 ⟶wf 6494 ‘cfv 6498 Isom wiso 6499 (class class class)co 7367 ↑_m cmap 8773 supcsup 9353 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 +∞cpnf 11176 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 − cmin 11377 -cneg 11378 / cdiv 11807 ℕcn 12174 2c2 12236 ℤcz 12524 (,)cioo 13298 (,]cioc 13299 [,]cicc 13301 ...cfz 13461 ..^cfzo 13608 ⌊cfl 13749 ♯chash 14292 ↾_t crest 17383 TopOpenctopn 17384 ℂ_fldccnfld 21352 –cn→ccncf 24843 lim_ℂ climc 25829

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-mulr 17234 df-starv 17235 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-rest 17385 df-topn 17386 df-topgen 17406 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-cn 23192 df-cnp 23193 df-cmp 23352 df-xms 24285 df-ms 24286 df-cncf 24845 df-limc 25833

This theorem is referenced by: fourierdlem99 46633 fourierdlem100 46634 fourierdlem107 46641 fourierdlem109 46643

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