fourierdlem89 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem89 46193

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

**Hypotheses**
Ref	Expression
fourierdlem89.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem89.t	⊢ 𝑇 = (𝐵 − 𝐴)
fourierdlem89.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem89.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem89.f	⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
fourierdlem89.per	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierdlem89.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem89.limc	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
fourierdlem89.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem89.d	⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞))
fourierdlem89.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem89.12	⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})
fourierdlem89.n	⊢ 𝑁 = ((♯‘𝐻) − 1)
fourierdlem89.s	⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
fourierdlem89.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
fourierdlem89.z	⊢ 𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
fourierdlem89.j	⊢ (𝜑 → 𝐽 ∈ (0..^𝑁))
fourierdlem89.u	⊢ 𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))
fourierdlem89.20	⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < ))
fourierdlem89.21	⊢ 𝑉 = (𝑖 ∈ (0..^𝑀) ↦ 𝑅)

**Assertion**
Ref	Expression
fourierdlem89	⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) lim_ℂ (𝑆‘𝐽)))

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

Proof of Theorem fourierdlem89 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem89.q	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
2		fourierdlem89.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		fourierdlem89.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 46107	. . . . . . . . . . . . 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 8822	. . . . . . . . . 10 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
10		fzossfz 13639	. . . . . . . . . 10 ⊢ (0..^𝑀) ⊆ (0...𝑀)
11		fourierdlem89.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝐵 − 𝐴)
12		fourierdlem89.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
13		fourierdlem89.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
14		fourierdlem89.20	. . . . . . . . . . . . 13 ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < ))
15	3, 2, 1, 11, 12, 13, 14	fourierdlem37 46142	. . . . . . . . . . . 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		fourierdlem89.c	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ ℝ)
18		fourierdlem89.d	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞))
19		elioore 13336	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ (𝐶(,)+∞) → 𝐷 ∈ ℝ)
20	18, 19	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷 ∈ ℝ)
21		elioo4g 13367	. . . . . . . . . . . . . . . . . . . . 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		fourierdlem89.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 7394	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑥 → (𝑦 + (𝑘 · 𝑇)) = (𝑥 + (𝑘 · 𝑇)))
27	26	eleq1d 2813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑥 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
28	27	rexbidv 3157	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
29	28	cbvrabv 3416	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}
30	29	uneq2i 4128	. . . . . . . . . . . . . . . . . 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		fourierdlem89.n	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑁 = ((♯‘𝐻) − 1)
32		fourierdlem89.12	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})
33	32	fveq2i 6861	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯‘𝐻) = (♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
34	33	oveq1i 7397	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐻) − 1) = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)
35	31, 34	eqtri 2752	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)
36		fourierdlem89.s	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
37		isoeq5 7296	. . . . . . . . . . . . . . . . . . . . 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 6496	. . . . . . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . . . . 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 46158	. . . . . . . . . . . . . . . . 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 46107	. . . . . . . . . . . . . . . 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 8822	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) → 𝑆:(0...𝑁)⟶ℝ)
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶ℝ)
51		fourierdlem89.j	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ (0..^𝑁))
52		elfzofz 13636	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ (0...𝑁))
53	51, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
54	50, 53	ffvelcdmd 7057	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℝ)
55	16, 54	ffvelcdmd 7057	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))
56	10, 55	sselid 3944	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘(𝑆‘𝐽)) ∈ (0...𝑀))
57	9, 56	ffvelcdmd 7057	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ)
58	57	rexrd 11224	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ^*)
59	58	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ^*)
60		fzofzp1 13725	. . . . . . . . . 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 7057	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈ ℝ)
63	62	rexrd 11224	. . . . . . 7 ⊢ (𝜑 → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈ ℝ^*)
64	63	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈ ℝ^*)
65	3, 2, 1	fourierdlem11 46116	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
66	65	simp1d 1142	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
67	65	simp2d 1143	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
68	66, 67	iccssred 13395	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
69	65	simp3d 1144	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 𝐵)
70	66, 67, 69, 13	fourierdlem17 46122	. . . . . . . . 9 ⊢ (𝜑 → 𝑍:(𝐴(,]𝐵)⟶(𝐴[,]𝐵))
71	66, 67, 69, 11, 12	fourierdlem4 46109	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵))
72	71, 54	ffvelcdmd 7057	. . . . . . . . 9 ⊢ (𝜑 → (𝐸‘(𝑆‘𝐽)) ∈ (𝐴(,]𝐵))
73	70, 72	ffvelcdmd 7057	. . . . . . . 8 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ (𝐴[,]𝐵))
74	68, 73	sseldd 3947	. . . . . . 7 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ)
75	74	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ)
76	57	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ)
77	66	rexrd 11224	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
78		iocssre 13388	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
79	77, 67, 78	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
80		fzofzp1 13725	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (0..^𝑁) → (𝐽 + 1) ∈ (0...𝑁))
81	51, 80	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐽 + 1) ∈ (0...𝑁))
82	50, 81	ffvelcdmd 7057	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ)
83	71, 82	ffvelcdmd 7057	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ (𝐴(,]𝐵))
84	79, 83	sseldd 3947	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
85	47	simprrd 773	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
86		fveq2 6858	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐽 → (𝑆‘𝑖) = (𝑆‘𝐽))
87		oveq1 7394	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝐽 → (𝑖 + 1) = (𝐽 + 1))
88	87	fveq2d 6862	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐽 → (𝑆‘(𝑖 + 1)) = (𝑆‘(𝐽 + 1)))
89	86, 88	breq12d 5120	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐽 → ((𝑆‘𝑖) < (𝑆‘(𝑖 + 1)) ↔ (𝑆‘𝐽) < (𝑆‘(𝐽 + 1))))
90	89	rspccva 3587	. . . . . . . . . . . . . 14 ⊢ ((∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)) ∧ 𝐽 ∈ (0..^𝑁)) → (𝑆‘𝐽) < (𝑆‘(𝐽 + 1)))
91	85, 51, 90	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝐽) < (𝑆‘(𝐽 + 1)))
92	54, 82	posdifd 11765	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆‘𝐽) < (𝑆‘(𝐽 + 1)) ↔ 0 < ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
93	91, 92	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
94	51	ancli 548	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝜑 ∧ 𝐽 ∈ (0..^𝑁)))
95		eleq1 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ (0..^𝑁) ↔ 𝐽 ∈ (0..^𝑁)))
96	95	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ↔ (𝜑 ∧ 𝐽 ∈ (0..^𝑁))))
97		oveq1 7394	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝐽 → (𝑗 + 1) = (𝐽 + 1))
98	97	fveq2d 6862	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝐽 → (𝑆‘(𝑗 + 1)) = (𝑆‘(𝐽 + 1)))
99	98	fveq2d 6862	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘(𝑗 + 1))) = (𝐸‘(𝑆‘(𝐽 + 1))))
100		fveq2 6858	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝐽 → (𝑆‘𝑗) = (𝑆‘𝐽))
101	100	fveq2d 6862	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘𝑗)) = (𝐸‘(𝑆‘𝐽)))
102	101	fveq2d 6862	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → (𝑍‘(𝐸‘(𝑆‘𝑗))) = (𝑍‘(𝐸‘(𝑆‘𝐽))))
103	99, 102	oveq12d 7405	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))))
104	98, 100	oveq12d 7405	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
105	103, 104	eqeq12d 2745	. . . . . . . . . . . . . . 15 ⊢ (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))))
106	96, 105	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (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)))))
107	11	oveq2i 7398	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 · 𝑇) = (𝑘 · (𝐵 − 𝐴))
108	107	oveq2i 7398	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑘 · (𝐵 − 𝐴)))
109	108	eleq1i 2819	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
110	109	rexbii 3076	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
111	110	rgenw 3048	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∀𝑦 ∈ (𝐶[,]𝐷)(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
112		rabbi 3436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀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})
113	111, 112	mpbi 230	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}
114	113	uneq2i 4128	. . . . . . . . . . . . . . . . . 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})
115	114	fveq2i 6861	. . . . . . . . . . . . . . . . 17 ⊢ (♯‘({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}))
116	115	oveq1i 7397	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘({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)
117	35, 116	eqtri 2752	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
118		isoeq5 7296	. . . . . . . . . . . . . . . . . 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}) → (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}))))
119	114, 118	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (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})))
120	119	iotabii 6496	. . . . . . . . . . . . . . . 16 ⊢ (℩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})))
121	40, 120	eqtri 2752	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
122		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) = ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))
123	3, 11, 2, 1, 17, 18, 25, 117, 121, 12, 13, 122	fourierdlem65 46169	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))
124	106, 123	vtoclg 3520	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (0..^𝑁) → ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
125	51, 94, 124	sylc 65	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
126	93, 125	breqtrrd 5135	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))))
127	74, 84	posdifd 11765	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝐸‘(𝑆‘(𝐽 + 1))) ↔ 0 < ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽))))))
128	126, 127	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝐸‘(𝑆‘(𝐽 + 1))))
129		id 22	. . . . . . . . . . 11 ⊢ (𝜑 → 𝜑)
130	102, 99	oveq12d 7405	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))
131	100	fveq2d 6862	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → (𝐼‘(𝑆‘𝑗)) = (𝐼‘(𝑆‘𝐽)))
132	131	fveq2d 6862	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑄‘(𝐼‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝐽))))
133	131	oveq1d 7402	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → ((𝐼‘(𝑆‘𝑗)) + 1) = ((𝐼‘(𝑆‘𝐽)) + 1))
134	133	fveq2d 6862	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
135	132, 134	oveq12d 7405	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1))) = ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
136	130, 135	sseq12d 3980	. . . . . . . . . . . . . 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)))))
137	96, 136	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))))))
138		eqid 2729	. . . . . . . . . . . . . 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)))
139	11, 3, 2, 1, 17, 20, 24, 25, 30, 35, 40, 12, 13, 138, 14	fourierdlem79 46183	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1))))
140	137, 139	vtoclg 3520	. . . . . . . . . . . 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)))))
141	140	anabsi7 671	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
142	129, 51, 141	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
143	57, 62, 74, 84, 128, 142	fourierdlem10 46115	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄‘(𝐼‘(𝑆‘𝐽))) ≤ (𝑍‘(𝐸‘(𝑆‘𝐽))) ∧ (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
144	143	simpld 494	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ≤ (𝑍‘(𝐸‘(𝑆‘𝐽))))
145	144	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) ≤ (𝑍‘(𝐸‘(𝑆‘𝐽))))
146		neqne 2933	. . . . . . . 8 ⊢ (¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) ≠ (𝑄‘(𝐼‘(𝑆‘𝐽))))
147	146	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) ≠ (𝑄‘(𝐼‘(𝑆‘𝐽))))
148	76, 75, 145, 147	leneltd 11328	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑍‘(𝐸‘(𝑆‘𝐽))))
149	143	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
150	74, 84, 62, 128, 149	ltletrd 11334	. . . . . . 7 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
151	150	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
152	59, 64, 75, 148, 151	eliood 45496	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
153		fvres 6877	. . . . 5 ⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽)))) = (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝐽)))))
154	152, 153	syl 17	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽)))) = (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝐽)))))
155	154	eqcomd 2735	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽)))))
156	155	ifeq2da 4521	. 2 ⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) = if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))))
157		fourierdlem89.f	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
158		fdm 6697	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℂ → dom 𝐹 = ℝ)
159	157, 158	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = ℝ)
160	159	feq2d 6672	. . . . . 6 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ↔ 𝐹:ℝ⟶ℂ))
161	157, 160	mpbird 257	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
162		ioosscn 13369	. . . . . 6 ⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ℂ
163	162	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ℂ)
164		ioossre 13368	. . . . . 6 ⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ℝ
165	164, 159	sseqtrrid 3990	. . . . 5 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ dom 𝐹)
166		fourierdlem89.u	. . . . . . 7 ⊢ 𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))
167	82, 84	resubcld 11606	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℝ)
168	166, 167	eqeltrid 2832	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ)
169	168	recnd 11202	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ℂ)
170		eqid 2729	. . . . 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)}
171	74, 84, 168	iooshift 45520	. . . . . 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)})
172		ioossre 13368	. . . . . . 7 ⊢ (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ⊆ ℝ
173	172, 159	sseqtrrid 3990	. . . . . 6 ⊢ (𝜑 → (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ⊆ dom 𝐹)
174	171, 173	eqsstrrd 3982	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} ⊆ dom 𝐹)
175		elioore 13336	. . . . . 6 ⊢ (𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) → 𝑦 ∈ ℝ)
176	67, 66	resubcld 11606	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
177	11, 176	eqeltrid 2832	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ ℝ)
178	177	recnd 11202	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℂ)
179	66, 67	posdifd 11765	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
180	69, 179	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
181	180, 11	breqtrrdi 5149	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 𝑇)
182	181	gt0ne0d 11742	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ≠ 0)
183	169, 178, 182	divcan1d 11959	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑈 / 𝑇) · 𝑇) = 𝑈)
184	183	eqcomd 2735	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 = ((𝑈 / 𝑇) · 𝑇))
185	184	oveq2d 7403	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 + 𝑈) = (𝑦 + ((𝑈 / 𝑇) · 𝑇)))
186	185	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + 𝑈) = (𝑦 + ((𝑈 / 𝑇) · 𝑇)))
187	186	fveq2d 6862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑈)) = (𝐹‘(𝑦 + ((𝑈 / 𝑇) · 𝑇))))
188	157	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐹:ℝ⟶ℂ)
189	177	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑇 ∈ ℝ)
190	84	recnd 11202	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
191	82	recnd 11202	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℂ)
192	190, 191	negsubdi2d 11549	. . . . . . . . . . . . 13 ⊢ (𝜑 → -((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
193	192	eqcomd 2735	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) = -((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))))
194	193	oveq1d 7402	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) / 𝑇) = (-((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇))
195	166	oveq1i 7397	. . . . . . . . . . . 12 ⊢ (𝑈 / 𝑇) = (((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) / 𝑇)
196	195	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑈 / 𝑇) = (((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) / 𝑇))
197	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
198		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → 𝑥 = (𝑆‘(𝐽 + 1)))
199		oveq2 7395	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘(𝐽 + 1))))
200	199	oveq1d 7402	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇))
201	200	fveq2d 6862	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
202	201	oveq1d 7402	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
203	198, 202	oveq12d 7405	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
204	203	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 = (𝑆‘(𝐽 + 1))) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
205	67, 82	resubcld 11606	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐵 − (𝑆‘(𝐽 + 1))) ∈ ℝ)
206	205, 177, 182	redivcld 12010	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℝ)
207	206	flcld 13760	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℤ)
208	207	zred 12638	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℝ)
209	208, 177	remulcld 11204	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℝ)
210	82, 209	readdcld 11203	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) ∈ ℝ)
211	197, 204, 82, 210	fvmptd 6975	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
212	211	oveq1d 7402	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))))
213	208	recnd 11202	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℂ)
214	213, 178	mulcld 11194	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℂ)
215	191, 214	pncan2d 11535	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
216	212, 215	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
217	216, 214	eqeltrd 2828	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℂ)
218	217, 178, 182	divnegd 11971	. . . . . . . . . . 11 ⊢ (𝜑 → -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (-((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇))
219	194, 196, 218	3eqtr4d 2774	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈 / 𝑇) = -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇))
220	216	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇))
221	213, 178, 182	divcan4d 11964	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
222	220, 221	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
223	222, 207	eqeltrd 2828	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ)
224	223	znegcld 12640	. . . . . . . . . 10 ⊢ (𝜑 → -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ)
225	219, 224	eqeltrd 2828	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 / 𝑇) ∈ ℤ)
226	225	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑈 / 𝑇) ∈ ℤ)
227		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
228		fourierdlem89.per	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
229	228	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
230	188, 189, 226, 227, 229	fperiodmul 45302	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + ((𝑈 / 𝑇) · 𝑇))) = (𝐹‘𝑦))
231	187, 230	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑈)) = (𝐹‘𝑦))
232	175, 231	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) → (𝐹‘(𝑦 + 𝑈)) = (𝐹‘𝑦))
233	6	simprrd 773	. . . . . . . 8 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
234		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘𝑖) = (𝑄‘(𝐼‘(𝑆‘𝐽))))
235		oveq1 7394	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 + 1) = ((𝐼‘(𝑆‘𝐽)) + 1))
236	235	fveq2d 6862	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘(𝑖 + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
237	234, 236	breq12d 5120	. . . . . . . . 9 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
238	237	rspccva 3587	. . . . . . . 8 ⊢ ((∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
239	233, 55, 238	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
240	55	ancli 548	. . . . . . . 8 ⊢ (𝜑 → (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)))
241		eleq1 2816	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 ∈ (0..^𝑀) ↔ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)))
242	241	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))))
243	234, 236	oveq12d 7405	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
244	243	reseq2d 5950	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))))
245	243	oveq1d 7402	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) = (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ))
246	244, 245	eleq12d 2822	. . . . . . . . . 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)))
247	242, 246	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))))
248		fourierdlem89.fcn	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
249	247, 248	vtoclg 3520	. . . . . . . 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)))
250	55, 240, 249	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))
251		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))
252		fourierdlem89.21	. . . . . . . . . . . . 13 ⊢ 𝑉 = (𝑖 ∈ (0..^𝑀) ↦ 𝑅)
253		nfmpt1 5206	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑖 ∈ (0..^𝑀) ↦ 𝑅)
254	252, 253	nfcxfr 2889	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝑉
255		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐼‘(𝑆‘𝐽))
256	254, 255	nffv 6868	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝑉‘(𝐼‘(𝑆‘𝐽)))
257	256	nfel1 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘(𝐼‘(𝑆‘𝐽))))
258	251, 257	nfim 1896	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘(𝐼‘(𝑆‘𝐽)))))
259	242	biimpar 477	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝜑 ∧ 𝑖 ∈ (0..^𝑀)))
260	259	3adant2 1131	. . . . . . . . . . . . 13 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝜑 ∧ 𝑖 ∈ (0..^𝑀)))
261		fourierdlem89.limc	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
262	260, 261	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
263		fveq2 6858	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑉‘𝑖) = (𝑉‘(𝐼‘(𝑆‘𝐽))))
264	263	eqcomd 2735	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) = (𝑉‘𝑖))
265	264	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘(𝐼‘(𝑆‘𝐽))) = (𝑉‘𝑖))
266	259	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → 𝑖 ∈ (0..^𝑀))
267		elex 3468	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) → 𝑅 ∈ V)
268	259, 261, 267	3syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → 𝑅 ∈ V)
269	252	fvmpt2 6979	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑅 ∈ V) → (𝑉‘𝑖) = 𝑅)
270	266, 268, 269	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘𝑖) = 𝑅)
271	265, 270	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘(𝐼‘(𝑆‘𝐽))) = 𝑅)
272	271	3adant2 1131	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘(𝐼‘(𝑆‘𝐽))) = 𝑅)
273	244, 234	oveq12d 7405	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) = ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘(𝐼‘(𝑆‘𝐽)))))
274	273	eqcomd 2735	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘(𝐼‘(𝑆‘𝐽)))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
275	274	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘(𝐼‘(𝑆‘𝐽)))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
276	262, 272, 275	3eltr4d 2843	. . . . . . . . . . 11 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘(𝐼‘(𝑆‘𝐽)))))
277	276	3exp 1119	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) → ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘(𝐼‘(𝑆‘𝐽)))))))
278	261	2a1i 12	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘(𝐼‘(𝑆‘𝐽))))) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))))
279	277, 278	impbid 212	. . . . . . . . 9 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) ↔ ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘(𝐼‘(𝑆‘𝐽)))))))
280	258, 279, 261	vtoclg1f 3536	. . . . . . . 8 ⊢ ((𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀) → ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘(𝐼‘(𝑆‘𝐽))))))
281	55, 240, 280	sylc 65	. . . . . . 7 ⊢ (𝜑 → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) lim_ℂ (𝑄‘(𝐼‘(𝑆‘𝐽)))))
282		eqid 2729	. . . . . . 7 ⊢ if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) = if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽)))))
283		eqid 2729	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ↾_t ((𝑄‘(𝐼‘(𝑆‘𝐽)))[,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) = ((TopOpen‘ℂ_fld) ↾_t ((𝑄‘(𝐼‘(𝑆‘𝐽)))[,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
284	57, 62, 239, 250, 281, 74, 84, 128, 142, 282, 283	fourierdlem32 46137	. . . . . 6 ⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ (((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) lim_ℂ (𝑍‘(𝐸‘(𝑆‘𝐽)))))
285	142	resabs1d 5979	. . . . . . 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))))))
286	285	oveq1d 7402	. . . . . 6 ⊢ (𝜑 → (((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) lim_ℂ (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) lim_ℂ (𝑍‘(𝐸‘(𝑆‘𝐽)))))
287	284, 286	eleqtrd 2830	. . . . 5 ⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ ((𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) lim_ℂ (𝑍‘(𝐸‘(𝑆‘𝐽)))))
288	161, 163, 165, 169, 170, 174, 232, 287	limcperiod 45626	. . . 4 ⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) lim_ℂ ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)))
289	166	oveq2i 7398	. . . . . . 7 ⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = ((𝑍‘(𝐸‘(𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
290	289	a1i 11	. . . . . 6 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = ((𝑍‘(𝐸‘(𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))))
291	17, 20	iccssred 13395	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
292		ax-resscn 11125	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
293	291, 292	sstrdi 3959	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ)
294	25, 44, 43	fourierdlem15 46120	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
295	294, 53	ffvelcdmd 7057	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐽) ∈ (𝐶[,]𝐷))
296	293, 295	sseldd 3947	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℂ)
297	191, 296	subcld 11533	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) ∈ ℂ)
298	74	recnd 11202	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ℂ)
299	190, 297, 298	subsub23d 45285	. . . . . . . . 9 ⊢ (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))))
300	125, 299	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (𝑍‘(𝐸‘(𝑆‘𝐽))))
301	300	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
302	301	oveq1d 7402	. . . . . 6 ⊢ (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))))))
303	190, 297	subcld 11533	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) ∈ ℂ)
304	303, 191, 190	addsub12d 11556	. . . . . . 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))))))
305	190, 297, 190	sub32d 11565	. . . . . . . . 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))))
306	190	subidd 11521	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = 0)
307	306	oveq1d 7402	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
308		df-neg 11408	. . . . . . . . . 10 ⊢ -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
309	191, 296	negsubdi2d 11549	. . . . . . . . . 10 ⊢ (𝜑 → -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
310	308, 309	eqtr3id 2778	. . . . . . . . 9 ⊢ (𝜑 → (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
311	305, 307, 310	3eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
312	311	oveq2d 7403	. . . . . . 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)))))
313	191, 296	pncan3d 11536	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1)))) = (𝑆‘𝐽))
314	304, 312, 313	3eqtrd 2768	. . . . . 6 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (𝑆‘𝐽))
315	290, 302, 314	3eqtrd 2768	. . . . 5 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = (𝑆‘𝐽))
316	315	oveq2d 7403	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) lim_ℂ ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)) = ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) lim_ℂ (𝑆‘𝐽)))
317	288, 316	eleqtrd 2830	. . 3 ⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) lim_ℂ (𝑆‘𝐽)))
318	166	oveq2i 7398	. . . . . . . 8 ⊢ ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
319	190, 191	pncan3d 11536	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (𝑆‘(𝐽 + 1)))
320	318, 319	eqtrid 2776	. . . . . . 7 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = (𝑆‘(𝐽 + 1)))
321	315, 320	oveq12d 7405	. . . . . 6 ⊢ (𝜑 → (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) = ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))
322	171, 321	eqtr3d 2766	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} = ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))
323	322	reseq2d 5950	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) = (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))))
324	323	oveq1d 7402	. . 3 ⊢ (𝜑 → ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) lim_ℂ (𝑆‘𝐽)) = ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) lim_ℂ (𝑆‘𝐽)))
325	317, 324	eleqtrd 2830	. 2 ⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) lim_ℂ (𝑆‘𝐽)))
326	156, 325	eqeltrd 2828	1 ⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) lim_ℂ (𝑆‘𝐽)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3405 Vcvv 3447 ∪ cun 3912 ⊆ wss 3914 ifcif 4488 {cpr 4591 class class class wbr 5107 ↦ cmpt 5188 dom cdm 5638 ran crn 5639 ↾ cres 5640 ℩cio 6462 ⟶wf 6507 ‘cfv 6511 Isom wiso 6512 (class class class)co 7387 ↑_m cmap 8799 supcsup 9391 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 -cneg 11406 / cdiv 11835 ℕcn 12186 2c2 12241 ℤcz 12529 (,)cioo 13306 (,]cioc 13307 [,)cico 13308 [,]cicc 13309 ...cfz 13468 ..^cfzo 13615 ⌊cfl 13752 ♯chash 14295 ↾_t crest 17383 TopOpenctopn 17384 ℂ_fldccnfld 21264 –cn→ccncf 24769 lim_ℂ climc 25763

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 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 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-cn 23114 df-cnp 23115 df-cmp 23274 df-xms 24208 df-ms 24209 df-cncf 24771 df-limc 25767

This theorem is referenced by: fourierdlem96 46200 fourierdlem100 46204 fourierdlem107 46211 fourierdlem109 46213

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