fourierdlem89 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem89		Structured version Visualization version GIF version

Theorem fourierdlem89 40412

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: 𝑥,𝑘 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 40326	. . . . . . . . . . . . 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 222	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
7	6	simpld 475	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)))
8		elmapi 7879	. . . . . . . . . 10 ⊢ (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
10		fzossfz 12488	. . . . . . . . . 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 40361	. . . . . . . . . . . 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 475	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼:ℝ⟶(0..^𝑀))
17		fourierdlem89.c	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ ℝ)
18		fourierdlem89.d	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞))
19		elioore 12205	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ (𝐶(,)+∞) → 𝐷 ∈ ℝ)
20	18, 19	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷 ∈ ℝ)
21		elioo4g 12234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (𝐶(,)+∞) ↔ ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐷 ∈ ℝ) ∧ (𝐶 < 𝐷 ∧ 𝐷 < +∞)))
22	18, 21	sylib 208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐷 ∈ ℝ) ∧ (𝐶 < 𝐷 ∧ 𝐷 < +∞)))
23	22	simprd 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐶 < 𝐷 ∧ 𝐷 < +∞))
24	23	simpld 475	. . . . . . . . . . . . . . . . . 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 6657	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑥 → (𝑦 + (𝑘 · 𝑇)) = (𝑥 + (𝑘 · 𝑇)))
27	26	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑥 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
28	27	rexbidv 3052	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
29	28	cbvrabv 3199	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}
30	29	uneq2i 3764	. . . . . . . . . . . . . . . . . 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 6194	. . . . . . . . . . . . . . . . . . . 20 ⊢ (#‘𝐻) = (#‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
34	33	oveq1i 6660	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝐻) − 1) = ((#‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)
35	31, 34	eqtri 2644	. . . . . . . . . . . . . . . . . 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 6571	. . . . . . . . . . . . . . . . . . . . 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 5873	. . . . . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . . . . 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 40377	. . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
43	42	simprd 479	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
44	42	simpld 475	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ)
45	25	fourierdlem2 40326	. . . . . . . . . . . . . . . 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 222	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆 ∈ (ℝ ↑_𝑚 (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))
48	47	simpld 475	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ (ℝ ↑_𝑚 (0...𝑁)))
49		elmapi 7879	. . . . . . . . . . . . 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 12485	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ (0...𝑁))
53	51, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
54	50, 53	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℝ)
55	16, 54	ffvelrnd 6360	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))
56	10, 55	sseldi 3601	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘(𝑆‘𝐽)) ∈ (0...𝑀))
57	9, 56	ffvelrnd 6360	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ)
58	57	rexrd 10089	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ^*)
59	58	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ^*)
60		fzofzp1 12565	. . . . . . . . . 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	ffvelrnd 6360	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈ ℝ)
63	62	rexrd 10089	. . . . . . 7 ⊢ (𝜑 → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈ ℝ^*)
64	63	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈ ℝ^*)
65	3, 2, 1	fourierdlem11 40335	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
66	65	simp1d 1073	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
67	65	simp2d 1074	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
68	66, 67	iccssred 39727	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
69	65	simp3d 1075	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 𝐵)
70	66, 67, 69, 13	fourierdlem17 40341	. . . . . . . . 9 ⊢ (𝜑 → 𝑍:(𝐴(,]𝐵)⟶(𝐴[,]𝐵))
71	66, 67, 69, 11, 12	fourierdlem4 40328	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵))
72	71, 54	ffvelrnd 6360	. . . . . . . . 9 ⊢ (𝜑 → (𝐸‘(𝑆‘𝐽)) ∈ (𝐴(,]𝐵))
73	70, 72	ffvelrnd 6360	. . . . . . . 8 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ (𝐴[,]𝐵))
74	68, 73	sseldd 3604	. . . . . . 7 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ)
75	74	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ)
76	57	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ)
77	66	rexrd 10089	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
78		iocssre 12253	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
79	77, 67, 78	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
80		fzofzp1 12565	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (0..^𝑁) → (𝐽 + 1) ∈ (0...𝑁))
81	51, 80	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐽 + 1) ∈ (0...𝑁))
82	50, 81	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ)
83	71, 82	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ (𝐴(,]𝐵))
84	79, 83	sseldd 3604	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
85	47	simprrd 797	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
86		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐽 → (𝑆‘𝑖) = (𝑆‘𝐽))
87		oveq1 6657	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝐽 → (𝑖 + 1) = (𝐽 + 1))
88	87	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐽 → (𝑆‘(𝑖 + 1)) = (𝑆‘(𝐽 + 1)))
89	86, 88	breq12d 4666	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐽 → ((𝑆‘𝑖) < (𝑆‘(𝑖 + 1)) ↔ (𝑆‘𝐽) < (𝑆‘(𝐽 + 1))))
90	89	rspccva 3308	. . . . . . . . . . . . . 14 ⊢ ((∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)) ∧ 𝐽 ∈ (0..^𝑁)) → (𝑆‘𝐽) < (𝑆‘(𝐽 + 1)))
91	85, 51, 90	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝐽) < (𝑆‘(𝐽 + 1)))
92	54, 82	posdifd 10614	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆‘𝐽) < (𝑆‘(𝐽 + 1)) ↔ 0 < ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
93	91, 92	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
94	51	ancli 574	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝜑 ∧ 𝐽 ∈ (0..^𝑁)))
95		eleq1 2689	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ (0..^𝑁) ↔ 𝐽 ∈ (0..^𝑁)))
96	95	anbi2d 740	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ↔ (𝜑 ∧ 𝐽 ∈ (0..^𝑁))))
97		oveq1 6657	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝐽 → (𝑗 + 1) = (𝐽 + 1))
98	97	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝐽 → (𝑆‘(𝑗 + 1)) = (𝑆‘(𝐽 + 1)))
99	98	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘(𝑗 + 1))) = (𝐸‘(𝑆‘(𝐽 + 1))))
100		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝐽 → (𝑆‘𝑗) = (𝑆‘𝐽))
101	100	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘𝑗)) = (𝐸‘(𝑆‘𝐽)))
102	101	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → (𝑍‘(𝐸‘(𝑆‘𝑗))) = (𝑍‘(𝐸‘(𝑆‘𝐽))))
103	99, 102	oveq12d 6668	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))))
104	98, 100	oveq12d 6668	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
105	103, 104	eqeq12d 2637	. . . . . . . . . . . . . . 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 334	. . . . . . . . . . . . . 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 6661	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 · 𝑇) = (𝑘 · (𝐵 − 𝐴))
108	107	oveq2i 6661	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑘 · (𝐵 − 𝐴)))
109	108	eleq1i 2692	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
110	109	rexbii 3041	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
111	110	rgenw 2924	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∀𝑦 ∈ (𝐶[,]𝐷)(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
112		rabbi 3120	. . . . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . . . 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 3764	. . . . . . . . . . . . . . . . . 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 6194	. . . . . . . . . . . . . . . . 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 6660	. . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = ((#‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
118		isoeq5 6571	. . . . . . . . . . . . . . . . . 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 5873	. . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
122		eqid 2622	. . . . . . . . . . . . . . 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 40388	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))
124	106, 123	vtoclg 3266	. . . . . . . . . . . . 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 4681	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))))
127	74, 84	posdifd 10614	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝐸‘(𝑆‘(𝐽 + 1))) ↔ 0 < ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽))))))
128	126, 127	mpbird 247	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝐸‘(𝑆‘(𝐽 + 1))))
129		id 22	. . . . . . . . . . 11 ⊢ (𝜑 → 𝜑)
130	102, 99	oveq12d 6668	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))
131	100	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → (𝐼‘(𝑆‘𝑗)) = (𝐼‘(𝑆‘𝐽)))
132	131	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑄‘(𝐼‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝐽))))
133	131	oveq1d 6665	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → ((𝐼‘(𝑆‘𝑗)) + 1) = ((𝐼‘(𝑆‘𝐽)) + 1))
134	133	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
135	132, 134	oveq12d 6668	. . . . . . . . . . . . . . 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 3634	. . . . . . . . . . . . . 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 334	. . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . 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 40402	. . . . . . . . . . . . 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 3266	. . . . . . . . . . . 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 860	. . . . . . . . . . 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 693	. . . . . . . . . 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 40334	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄‘(𝐼‘(𝑆‘𝐽))) ≤ (𝑍‘(𝐸‘(𝑆‘𝐽))) ∧ (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
144	143	simpld 475	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ≤ (𝑍‘(𝐸‘(𝑆‘𝐽))))
145	144	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) ≤ (𝑍‘(𝐸‘(𝑆‘𝐽))))
146		neqne 2802	. . . . . . . 8 ⊢ (¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) ≠ (𝑄‘(𝐼‘(𝑆‘𝐽))))
147	146	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) ≠ (𝑄‘(𝐼‘(𝑆‘𝐽))))
148	76, 75, 145, 147	leneltd 10191	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑍‘(𝐸‘(𝑆‘𝐽))))
149	143	simprd 479	. . . . . . . 8 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
150	74, 84, 62, 128, 149	ltletrd 10197	. . . . . . 7 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
151	150	adantr 481	. . . . . 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 39720	. . . . 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 6207	. . . . 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 2628	. . 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 4117	. 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 6051	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℂ → dom 𝐹 = ℝ)
159	157, 158	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = ℝ)
160	159	feq2d 6031	. . . . . 6 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ↔ 𝐹:ℝ⟶ℂ))
161	157, 160	mpbird 247	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
162		ioosscn 39716	. . . . . 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 12235	. . . . . 6 ⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ℝ
165	164, 159	syl5sseqr 3654	. . . . 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 10458	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℝ)
168	166, 167	syl5eqel 2705	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ)
169	168	recnd 10068	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ℂ)
170		eqid 2622	. . . . 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 39748	. . . . . 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 12235	. . . . . . 7 ⊢ (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ⊆ ℝ
173	172, 159	syl5sseqr 3654	. . . . . 6 ⊢ (𝜑 → (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ⊆ dom 𝐹)
174	171, 173	eqsstr3d 3640	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} ⊆ dom 𝐹)
175		elioore 12205	. . . . . 6 ⊢ (𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) → 𝑦 ∈ ℝ)
176	67, 66	resubcld 10458	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
177	11, 176	syl5eqel 2705	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ ℝ)
178	177	recnd 10068	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℂ)
179	66, 67	posdifd 10614	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
180	69, 179	mpbid 222	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
181	180, 11	syl6breqr 4695	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 𝑇)
182	181	gt0ne0d 10592	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ≠ 0)
183	169, 178, 182	divcan1d 10802	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑈 / 𝑇) · 𝑇) = 𝑈)
184	183	eqcomd 2628	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 = ((𝑈 / 𝑇) · 𝑇))
185	184	oveq2d 6666	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 + 𝑈) = (𝑦 + ((𝑈 / 𝑇) · 𝑇)))
186	185	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + 𝑈) = (𝑦 + ((𝑈 / 𝑇) · 𝑇)))
187	186	fveq2d 6195	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑈)) = (𝐹‘(𝑦 + ((𝑈 / 𝑇) · 𝑇))))
188	157	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐹:ℝ⟶ℂ)
189	177	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑇 ∈ ℝ)
190	84	recnd 10068	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
191	82	recnd 10068	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℂ)
192	190, 191	negsubdi2d 10408	. . . . . . . . . . . . 13 ⊢ (𝜑 → -((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
193	192	eqcomd 2628	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) = -((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))))
194	193	oveq1d 6665	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) / 𝑇) = (-((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇))
195	166	oveq1i 6660	. . . . . . . . . . . 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 6658	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘(𝐽 + 1))))
200	199	oveq1d 6665	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇))
201	200	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
202	201	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
203	198, 202	oveq12d 6668	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
204	203	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 = (𝑆‘(𝐽 + 1))) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
205	67, 82	resubcld 10458	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐵 − (𝑆‘(𝐽 + 1))) ∈ ℝ)
206	205, 177, 182	redivcld 10853	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℝ)
207	206	flcld 12599	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℤ)
208	207	zred 11482	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℝ)
209	208, 177	remulcld 10070	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℝ)
210	82, 209	readdcld 10069	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) ∈ ℝ)
211	197, 204, 82, 210	fvmptd 6288	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
212	211	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))))
213	208	recnd 10068	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℂ)
214	213, 178	mulcld 10060	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℂ)
215	191, 214	pncan2d 10394	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
216	212, 215	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
217	216, 214	eqeltrd 2701	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℂ)
218	217, 178, 182	divnegd 10814	. . . . . . . . . . 11 ⊢ (𝜑 → -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (-((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇))
219	194, 196, 218	3eqtr4d 2666	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈 / 𝑇) = -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇))
220	216	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇))
221	213, 178, 182	divcan4d 10807	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
222	220, 221	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
223	222, 207	eqeltrd 2701	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ)
224	223	znegcld 11484	. . . . . . . . . 10 ⊢ (𝜑 → -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ)
225	219, 224	eqeltrd 2701	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 / 𝑇) ∈ ℤ)
226	225	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑈 / 𝑇) ∈ ℤ)
227		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
228		fourierdlem89.per	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
229	228	adantlr 751	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
230	188, 189, 226, 227, 229	fperiodmul 39518	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + ((𝑈 / 𝑇) · 𝑇))) = (𝐹‘𝑦))
231	187, 230	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑈)) = (𝐹‘𝑦))
232	175, 231	sylan2 491	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) → (𝐹‘(𝑦 + 𝑈)) = (𝐹‘𝑦))
233	6	simprrd 797	. . . . . . . 8 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
234		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘𝑖) = (𝑄‘(𝐼‘(𝑆‘𝐽))))
235		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 + 1) = ((𝐼‘(𝑆‘𝐽)) + 1))
236	235	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘(𝑖 + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
237	234, 236	breq12d 4666	. . . . . . . . 9 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
238	237	rspccva 3308	. . . . . . . 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 693	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
240	55	ancli 574	. . . . . . . 8 ⊢ (𝜑 → (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)))
241		eleq1 2689	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 ∈ (0..^𝑀) ↔ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)))
242	241	anbi2d 740	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))))
243	234, 236	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
244	243	reseq2d 5396	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))))
245	243	oveq1d 6665	. . . . . . . . . . 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 2695	. . . . . . . . . 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 334	. . . . . . . . 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 3266	. . . . . . . 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 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))
252		fourierdlem89.21	. . . . . . . . . . . . 13 ⊢ 𝑉 = (𝑖 ∈ (0..^𝑀) ↦ 𝑅)
253		nfmpt1 4747	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑖 ∈ (0..^𝑀) ↦ 𝑅)
254	252, 253	nfcxfr 2762	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝑉
255		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐼‘(𝑆‘𝐽))
256	254, 255	nffv 6198	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝑉‘(𝐼‘(𝑆‘𝐽)))
257	256	nfel1 2779	. . . . . . . . . 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 1825	. . . . . . . . 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 502	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝜑 ∧ 𝑖 ∈ (0..^𝑀)))
260	259	3adant2 1080	. . . . . . . . . . . . 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 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑉‘𝑖) = (𝑉‘(𝐼‘(𝑆‘𝐽))))
264	263	eqcomd 2628	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) = (𝑉‘𝑖))
265	264	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘(𝐼‘(𝑆‘𝐽))) = (𝑉‘𝑖))
266	259	simprd 479	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → 𝑖 ∈ (0..^𝑀))
267		elex 3212	. . . . . . . . . . . . . . . 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 6291	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑅 ∈ V) → (𝑉‘𝑖) = 𝑅)
270	266, 268, 269	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘𝑖) = 𝑅)
271	265, 270	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘(𝐼‘(𝑆‘𝐽))) = 𝑅)
272	271	3adant2 1080	. . . . . . . . . . . 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 6668	. . . . . . . . . . . . . 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 2628	. . . . . . . . . . . . 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 1082	. . . . . . . . . . . 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 2716	. . . . . . . . . . 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 1264	. . . . . . . . . 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 202	. . . . . . . . 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 3265	. . . . . . . 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 2622	. . . . . . 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 2622	. . . . . . 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 40356	. . . . . 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 5428	. . . . . . 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 6665	. . . . . 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 2703	. . . . 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 39860	. . . 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 6661	. . . . . . 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 39727	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
292		ax-resscn 9993	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
293	291, 292	syl6ss 3615	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ)
294	25, 44, 43	fourierdlem15 40339	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
295	294, 53	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐽) ∈ (𝐶[,]𝐷))
296	293, 295	sseldd 3604	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℂ)
297	191, 296	subcld 10392	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) ∈ ℂ)
298	74	recnd 10068	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ℂ)
299	190, 297, 298	subsub23d 39499	. . . . . . . . 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 247	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (𝑍‘(𝐸‘(𝑆‘𝐽))))
301	300	eqcomd 2628	. . . . . . 7 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
302	301	oveq1d 6665	. . . . . 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 10392	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) ∈ ℂ)
304	303, 191, 190	addsub12d 10415	. . . . . . 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 10424	. . . . . . . . 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 10380	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = 0)
307	306	oveq1d 6665	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
308		df-neg 10269	. . . . . . . . . 10 ⊢ -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
309	191, 296	negsubdi2d 10408	. . . . . . . . . 10 ⊢ (𝜑 → -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
310	308, 309	syl5eqr 2670	. . . . . . . . 9 ⊢ (𝜑 → (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
311	305, 307, 310	3eqtrd 2660	. . . . . . . 8 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
312	311	oveq2d 6666	. . . . . . 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 10395	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1)))) = (𝑆‘𝐽))
314	304, 312, 313	3eqtrd 2660	. . . . . 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 2660	. . . . 5 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = (𝑆‘𝐽))
316	315	oveq2d 6666	. . . 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 2703	. . 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 6661	. . . . . . . 8 ⊢ ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
319	190, 191	pncan3d 10395	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (𝑆‘(𝐽 + 1)))
320	318, 319	syl5eq 2668	. . . . . . 7 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = (𝑆‘(𝐽 + 1)))
321	315, 320	oveq12d 6668	. . . . . 6 ⊢ (𝜑 → (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) = ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))
322	171, 321	eqtr3d 2658	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} = ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))
323	322	reseq2d 5396	. . . 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 6665	. . 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 2703	. 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 2701	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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 ifcif 4086 {cpr 4179 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 ↾ cres 5116 ℩cio 5849 ⟶wf 5884 ‘cfv 5888 Isom wiso 5889 (class class class)co 6650 ↑_𝑚 cmap 7857 supcsup 8346 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℤcz 11377 (,)cioo 12175 (,]cioc 12176 [,)cico 12177 [,]cicc 12178 ...cfz 12326 ..^cfzo 12465 ⌊cfl 12591 #chash 13117 ↾_t crest 16081 TopOpenctopn 16082 ℂ_fldccnfld 19746 –cn→ccncf 22679 lim_ℂ climc 23626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-cn 21031 df-cnp 21032 df-cmp 21190 df-xms 22125 df-ms 22126 df-cncf 22681 df-limc 23630

This theorem is referenced by: fourierdlem96 40419 fourierdlem100 40423 fourierdlem107 40430 fourierdlem109 40432

W3C validator