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 46200

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 46114	. . . . . . . . . . . . 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 8825	. . . . . . . . . 10 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
10		fzossfz 13646	. . . . . . . . . 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 46149	. . . . . . . . . . . 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 13343	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ (𝐶(,)+∞) → 𝐷 ∈ ℝ)
20	18, 19	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷 ∈ ℝ)
21		elioo4g 13374	. . . . . . . . . . . . . . . . . . . . 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 7397	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑥 → (𝑦 + (𝑘 · 𝑇)) = (𝑥 + (𝑘 · 𝑇)))
27	26	eleq1d 2814	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑥 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
28	27	rexbidv 3158	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄))
29	28	cbvrabv 3419	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}
30	29	uneq2i 4131	. . . . . . . . . . . . . . . . . 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 6864	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯‘𝐻) = (♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
34	33	oveq1i 7400	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐻) − 1) = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)
35	31, 34	eqtri 2753	. . . . . . . . . . . . . . . . . 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 7299	. . . . . . . . . . . . . . . . . . . . 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 6499	. . . . . . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . . . . 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 46165	. . . . . . . . . . . . . . . . 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 46114	. . . . . . . . . . . . . . . 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 8825	. . . . . . . . . . . . 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 13643	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ (0...𝑁))
53	51, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
54	50, 53	ffvelcdmd 7060	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℝ)
55	16, 54	ffvelcdmd 7060	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))
56	10, 55	sselid 3947	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘(𝑆‘𝐽)) ∈ (0...𝑀))
57	9, 56	ffvelcdmd 7060	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ)
58	57	rexrd 11231	. . . . . . 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 13732	. . . . . . . . . 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 7060	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈ ℝ)
63	62	rexrd 11231	. . . . . . 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 46123	. . . . . . . . . 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 13402	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
69	65	simp3d 1144	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 < 𝐵)
70	66, 67, 69, 13	fourierdlem17 46129	. . . . . . . . 9 ⊢ (𝜑 → 𝑍:(𝐴(,]𝐵)⟶(𝐴[,]𝐵))
71	66, 67, 69, 11, 12	fourierdlem4 46116	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵))
72	71, 54	ffvelcdmd 7060	. . . . . . . . 9 ⊢ (𝜑 → (𝐸‘(𝑆‘𝐽)) ∈ (𝐴(,]𝐵))
73	70, 72	ffvelcdmd 7060	. . . . . . . 8 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ (𝐴[,]𝐵))
74	68, 73	sseldd 3950	. . . . . . 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 11231	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
78		iocssre 13395	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
79	77, 67, 78	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
80		fzofzp1 13732	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (0..^𝑁) → (𝐽 + 1) ∈ (0...𝑁))
81	51, 80	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐽 + 1) ∈ (0...𝑁))
82	50, 81	ffvelcdmd 7060	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ)
83	71, 82	ffvelcdmd 7060	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ (𝐴(,]𝐵))
84	79, 83	sseldd 3950	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
85	47	simprrd 773	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
86		fveq2 6861	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐽 → (𝑆‘𝑖) = (𝑆‘𝐽))
87		oveq1 7397	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝐽 → (𝑖 + 1) = (𝐽 + 1))
88	87	fveq2d 6865	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐽 → (𝑆‘(𝑖 + 1)) = (𝑆‘(𝐽 + 1)))
89	86, 88	breq12d 5123	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐽 → ((𝑆‘𝑖) < (𝑆‘(𝑖 + 1)) ↔ (𝑆‘𝐽) < (𝑆‘(𝐽 + 1))))
90	89	rspccva 3590	. . . . . . . . . . . . . 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 11772	. . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . 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 7397	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝐽 → (𝑗 + 1) = (𝐽 + 1))
98	97	fveq2d 6865	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝐽 → (𝑆‘(𝑗 + 1)) = (𝑆‘(𝐽 + 1)))
99	98	fveq2d 6865	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘(𝑗 + 1))) = (𝐸‘(𝑆‘(𝐽 + 1))))
100		fveq2 6861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝐽 → (𝑆‘𝑗) = (𝑆‘𝐽))
101	100	fveq2d 6865	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘𝑗)) = (𝐸‘(𝑆‘𝐽)))
102	101	fveq2d 6865	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → (𝑍‘(𝐸‘(𝑆‘𝑗))) = (𝑍‘(𝐸‘(𝑆‘𝐽))))
103	99, 102	oveq12d 7408	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))))
104	98, 100	oveq12d 7408	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
105	103, 104	eqeq12d 2746	. . . . . . . . . . . . . . 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 7401	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 · 𝑇) = (𝑘 · (𝐵 − 𝐴))
108	107	oveq2i 7401	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑘 · (𝐵 − 𝐴)))
109	108	eleq1i 2820	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
110	109	rexbii 3077	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
111	110	rgenw 3049	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∀𝑦 ∈ (𝐶[,]𝐷)(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)
112		rabbi 3439	. . . . . . . . . . . . . . . . . . . 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 4131	. . . . . . . . . . . . . . . . . 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 6864	. . . . . . . . . . . . . . . . 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 7400	. . . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1)
118		isoeq5 7299	. . . . . . . . . . . . . . . . . 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 6499	. . . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))
122		eqid 2730	. . . . . . . . . . . . . . 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 46176	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))
124	106, 123	vtoclg 3523	. . . . . . . . . . . . 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 5138	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))))
127	74, 84	posdifd 11772	. . . . . . . . . . 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 7408	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))
131	100	fveq2d 6865	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → (𝐼‘(𝑆‘𝑗)) = (𝐼‘(𝑆‘𝐽)))
132	131	fveq2d 6865	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑄‘(𝐼‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝐽))))
133	131	oveq1d 7405	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐽 → ((𝐼‘(𝑆‘𝑗)) + 1) = ((𝐼‘(𝑆‘𝐽)) + 1))
134	133	fveq2d 6865	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
135	132, 134	oveq12d 7408	. . . . . . . . . . . . . . 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 3983	. . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . 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 46190	. . . . . . . . . . . . 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 3523	. . . . . . . . . . . 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 46122	. . . . . . . . 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 2934	. . . . . . . 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 11335	. . . . . 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 11341	. . . . . . 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 45503	. . . . 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 6880	. . . . 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 2736	. . 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 4524	. 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 6700	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℂ → dom 𝐹 = ℝ)
159	157, 158	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = ℝ)
160	159	feq2d 6675	. . . . . 6 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ↔ 𝐹:ℝ⟶ℂ))
161	157, 160	mpbird 257	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
162		ioosscn 13376	. . . . . 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 13375	. . . . . 6 ⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ℝ
165	164, 159	sseqtrrid 3993	. . . . 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 11613	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℝ)
168	166, 167	eqeltrid 2833	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ)
169	168	recnd 11209	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ℂ)
170		eqid 2730	. . . . 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 45527	. . . . . 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 13375	. . . . . . 7 ⊢ (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ⊆ ℝ
173	172, 159	sseqtrrid 3993	. . . . . 6 ⊢ (𝜑 → (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ⊆ dom 𝐹)
174	171, 173	eqsstrrd 3985	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} ⊆ dom 𝐹)
175		elioore 13343	. . . . . 6 ⊢ (𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) → 𝑦 ∈ ℝ)
176	67, 66	resubcld 11613	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
177	11, 176	eqeltrid 2833	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ ℝ)
178	177	recnd 11209	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℂ)
179	66, 67	posdifd 11772	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
180	69, 179	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
181	180, 11	breqtrrdi 5152	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 𝑇)
182	181	gt0ne0d 11749	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ≠ 0)
183	169, 178, 182	divcan1d 11966	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑈 / 𝑇) · 𝑇) = 𝑈)
184	183	eqcomd 2736	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 = ((𝑈 / 𝑇) · 𝑇))
185	184	oveq2d 7406	. . . . . . . . 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 6865	. . . . . . 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 11209	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
191	82	recnd 11209	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℂ)
192	190, 191	negsubdi2d 11556	. . . . . . . . . . . . 13 ⊢ (𝜑 → -((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
193	192	eqcomd 2736	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) = -((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))))
194	193	oveq1d 7405	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) / 𝑇) = (-((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇))
195	166	oveq1i 7400	. . . . . . . . . . . 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 7398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘(𝐽 + 1))))
200	199	oveq1d 7405	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇))
201	200	fveq2d 6865	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
202	201	oveq1d 7405	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
203	198, 202	oveq12d 7408	. . . . . . . . . . . . . . . . 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 11613	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐵 − (𝑆‘(𝐽 + 1))) ∈ ℝ)
206	205, 177, 182	redivcld 12017	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℝ)
207	206	flcld 13767	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℤ)
208	207	zred 12645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℝ)
209	208, 177	remulcld 11211	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℝ)
210	82, 209	readdcld 11210	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) ∈ ℝ)
211	197, 204, 82, 210	fvmptd 6978	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
212	211	oveq1d 7405	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))))
213	208	recnd 11209	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℂ)
214	213, 178	mulcld 11201	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℂ)
215	191, 214	pncan2d 11542	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
216	212, 215	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
217	216, 214	eqeltrd 2829	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℂ)
218	217, 178, 182	divnegd 11978	. . . . . . . . . . 11 ⊢ (𝜑 → -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (-((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇))
219	194, 196, 218	3eqtr4d 2775	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈 / 𝑇) = -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇))
220	216	oveq1d 7405	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇))
221	213, 178, 182	divcan4d 11971	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
222	220, 221	eqtrd 2765	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
223	222, 207	eqeltrd 2829	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ)
224	223	znegcld 12647	. . . . . . . . . 10 ⊢ (𝜑 → -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ)
225	219, 224	eqeltrd 2829	. . . . . . . . 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 45309	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + ((𝑈 / 𝑇) · 𝑇))) = (𝐹‘𝑦))
231	187, 230	eqtrd 2765	. . . . . 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 6861	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘𝑖) = (𝑄‘(𝐼‘(𝑆‘𝐽))))
235		oveq1 7397	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 + 1) = ((𝐼‘(𝑆‘𝐽)) + 1))
236	235	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘(𝑖 + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))
237	234, 236	breq12d 5123	. . . . . . . . 9 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
238	237	rspccva 3590	. . . . . . . 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 2817	. . . . . . . . . . 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 7408	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))
244	243	reseq2d 5953	. . . . . . . . . . 11 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))))
245	243	oveq1d 7405	. . . . . . . . . . 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 2823	. . . . . . . . . 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 3523	. . . . . . . 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 5209	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑖 ∈ (0..^𝑀) ↦ 𝑅)
254	252, 253	nfcxfr 2890	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝑉
255		nfcv 2892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝐼‘(𝑆‘𝐽))
256	254, 255	nffv 6871	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝑉‘(𝐼‘(𝑆‘𝐽)))
257	256	nfel1 2909	. . . . . . . . . 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 6861	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑉‘𝑖) = (𝑉‘(𝐼‘(𝑆‘𝐽))))
264	263	eqcomd 2736	. . . . . . . . . . . . . . 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 3471	. . . . . . . . . . . . . . . 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 6982	. . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . 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 7408	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 2844	. . . . . . . . . . 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 3539	. . . . . . . 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 2730	. . . . . . 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 2730	. . . . . . 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 46144	. . . . . 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 5982	. . . . . . 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 7405	. . . . . 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 2831	. . . . 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 45633	. . . 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 7401	. . . . . . 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 13402	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
292		ax-resscn 11132	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
293	291, 292	sstrdi 3962	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ)
294	25, 44, 43	fourierdlem15 46127	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
295	294, 53	ffvelcdmd 7060	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐽) ∈ (𝐶[,]𝐷))
296	293, 295	sseldd 3950	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℂ)
297	191, 296	subcld 11540	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) ∈ ℂ)
298	74	recnd 11209	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ℂ)
299	190, 297, 298	subsub23d 45292	. . . . . . . . 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 2736	. . . . . . 7 ⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
302	301	oveq1d 7405	. . . . . 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 11540	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) ∈ ℂ)
304	303, 191, 190	addsub12d 11563	. . . . . . 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 11572	. . . . . . . . 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 11528	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = 0)
307	306	oveq1d 7405	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))
308		df-neg 11415	. . . . . . . . . 10 ⊢ -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))
309	191, 296	negsubdi2d 11556	. . . . . . . . . 10 ⊢ (𝜑 → -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
310	308, 309	eqtr3id 2779	. . . . . . . . 9 ⊢ (𝜑 → (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
311	305, 307, 310	3eqtrd 2769	. . . . . . . 8 ⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))
312	311	oveq2d 7406	. . . . . . 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 11543	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1)))) = (𝑆‘𝐽))
314	304, 312, 313	3eqtrd 2769	. . . . . 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 2769	. . . . 5 ⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = (𝑆‘𝐽))
316	315	oveq2d 7406	. . . 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 2831	. . 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 7401	. . . . . . . 8 ⊢ ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))
319	190, 191	pncan3d 11543	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (𝑆‘(𝐽 + 1)))
320	318, 319	eqtrid 2777	. . . . . . 7 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = (𝑆‘(𝐽 + 1)))
321	315, 320	oveq12d 7408	. . . . . 6 ⊢ (𝜑 → (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) = ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))
322	171, 321	eqtr3d 2767	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} = ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))
323	322	reseq2d 5953	. . . 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 7405	. . 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 2831	. 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 2829	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 2926 ∀wral 3045 ∃wrex 3054 {crab 3408 Vcvv 3450 ∪ cun 3915 ⊆ wss 3917 ifcif 4491 {cpr 4594 class class class wbr 5110 ↦ cmpt 5191 dom cdm 5641 ran crn 5642 ↾ cres 5643 ℩cio 6465 ⟶wf 6510 ‘cfv 6514 Isom wiso 6515 (class class class)co 7390 ↑_m cmap 8802 supcsup 9398 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 +∞cpnf 11212 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 − cmin 11412 -cneg 11413 / cdiv 11842 ℕcn 12193 2c2 12248 ℤcz 12536 (,)cioo 13313 (,]cioc 13314 [,)cico 13315 [,]cicc 13316 ...cfz 13475 ..^cfzo 13622 ⌊cfl 13759 ♯chash 14302 ↾_t crest 17390 TopOpenctopn 17391 ℂ_fldccnfld 21271 –cn→ccncf 24776 lim_ℂ climc 25770

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-xnn0 12523 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17124 df-slot 17159 df-ndx 17171 df-base 17187 df-plusg 17240 df-mulr 17241 df-starv 17242 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-rest 17392 df-topn 17393 df-topgen 17413 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-cn 23121 df-cnp 23122 df-cmp 23281 df-xms 24215 df-ms 24216 df-cncf 24778 df-limc 25774

This theorem is referenced by: fourierdlem96 46207 fourierdlem100 46211 fourierdlem107 46218 fourierdlem109 46220

W3C validator