fourierdlem88 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem88 41204

Description: Given a piecewise continuous function 𝐹, a continuous function 𝐾 and a continuous function 𝑆, the function 𝐺 is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem88.1	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem88.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem88.x	⊢ (𝜑 → 𝑋 ∈ ran 𝑉)
fourierdlem88.y	⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
fourierdlem88.w	⊢ (𝜑 → 𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
fourierdlem88.h	⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
fourierdlem88.k	⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
fourierdlem88.u	⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
fourierdlem88.n	⊢ (𝜑 → 𝑁 ∈ ℝ)
fourierdlem88.s	⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠)))
fourierdlem88.g	⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
fourierdlem88.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem88.v	⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
fourierdlem88.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
fourierdlem88.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
fourierdlem88.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1))))
fourierdlem88.q	⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
fourierdlem88.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_𝑚 (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem88.i	⊢ 𝐼 = (ℝ D 𝐹)
fourierdlem88.ifn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
fourierdlem88.c	⊢ (𝜑 → 𝐶 ∈ ((𝐼 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
fourierdlem88.d	⊢ (𝜑 → 𝐷 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))

**Assertion**
Ref	Expression
fourierdlem88	⊢ (𝜑 → 𝐺 ∈ 𝐿¹)

Distinct variable groups: 𝐶,𝑠 𝐷,𝑠 𝐹,𝑠 𝑖,𝐺,𝑠 𝐻,𝑠 𝐾,𝑠 𝐿,𝑠 𝑖,𝑀,𝑚,𝑝 𝑀,𝑠 𝑁,𝑠 𝑄,𝑖,𝑝 𝑄,𝑠 𝑅,𝑠 𝑆,𝑠 𝑖,𝑉,𝑝 𝑉,𝑠 𝑊,𝑠 𝑖,𝑋,𝑚,𝑝 𝑋,𝑠 𝑌,𝑠 𝜑,𝑖,𝑠 Allowed substitution hints: 𝜑(𝑚,𝑝) 𝐶(𝑖,𝑚,𝑝) 𝐷(𝑖,𝑚,𝑝) 𝑃(𝑖,𝑚,𝑠,𝑝) 𝑄(𝑚) 𝑅(𝑖,𝑚,𝑝) 𝑆(𝑖,𝑚,𝑝) 𝑈(𝑖,𝑚,𝑠,𝑝) 𝐹(𝑖,𝑚,𝑝) 𝐺(𝑚,𝑝) 𝐻(𝑖,𝑚,𝑝) 𝐼(𝑖,𝑚,𝑠,𝑝) 𝐾(𝑖,𝑚,𝑝) 𝐿(𝑖,𝑚,𝑝) 𝑁(𝑖,𝑚,𝑝) 𝑂(𝑖,𝑚,𝑠,𝑝) 𝑉(𝑚) 𝑊(𝑖,𝑚,𝑝) 𝑌(𝑖,𝑚,𝑝)

Proof of Theorem fourierdlem88 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem88.o	. 2 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_𝑚 (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
2		fourierdlem88.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		pire 24609	. . . . 5 ⊢ π ∈ ℝ
4	3	a1i 11	. . . 4 ⊢ (𝜑 → π ∈ ℝ)
5	4	renegcld 10780	. . 3 ⊢ (𝜑 → -π ∈ ℝ)
6		fourierdlem88.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
7		fourierdlem88.1	. . . . . . . . 9 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
8	7	fourierdlem2 41119	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑉 ∈ (𝑃‘𝑀) ↔ (𝑉 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉‘𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1))))))
9	2, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑉 ∈ (𝑃‘𝑀) ↔ (𝑉 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉‘𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1))))))
10	6, 9	mpbid 224	. . . . . 6 ⊢ (𝜑 → (𝑉 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉‘𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))))
11	10	simpld 490	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑_𝑚 (0...𝑀)))
12		elmapi 8143	. . . . 5 ⊢ (𝑉 ∈ (ℝ ↑_𝑚 (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
13		frn 6283	. . . . 5 ⊢ (𝑉:(0...𝑀)⟶ℝ → ran 𝑉 ⊆ ℝ)
14	11, 12, 13	3syl 18	. . . 4 ⊢ (𝜑 → ran 𝑉 ⊆ ℝ)
15		fourierdlem88.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝑉)
16	14, 15	sseldd 3827	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
17		fourierdlem88.q	. . 3 ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
18	5, 4, 16, 7, 1, 2, 6, 17	fourierdlem14 41131	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀))
19		fourierdlem88.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
20		ioossre 12522	. . . . . . . . . 10 ⊢ (𝑋(,)+∞) ⊆ ℝ
21	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℝ)
22	19, 21	fssresd 6307	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)+∞)):(𝑋(,)+∞)⟶ℝ)
23		ax-resscn 10308	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
24	21, 23	syl6ss 3838	. . . . . . . 8 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℂ)
25		eqid 2824	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
26		pnfxr 10409	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
27	26	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → +∞ ∈ ℝ^*)
28	16	ltpnfd 12240	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 < +∞)
29	25, 27, 16, 28	lptioo1cn 40672	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝑋(,)+∞)))
30		fourierdlem88.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
31	22, 24, 29, 30	limcrecl 40655	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ℝ)
32		ioossre 12522	. . . . . . . . . 10 ⊢ (-∞(,)𝑋) ⊆ ℝ
33	32	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (-∞(,)𝑋) ⊆ ℝ)
34	19, 33	fssresd 6307	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ (-∞(,)𝑋)):(-∞(,)𝑋)⟶ℝ)
35	33, 23	syl6ss 3838	. . . . . . . 8 ⊢ (𝜑 → (-∞(,)𝑋) ⊆ ℂ)
36		mnfxr 10413	. . . . . . . . . 10 ⊢ -∞ ∈ ℝ^*
37	36	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → -∞ ∈ ℝ^*)
38	16	mnfltd 12243	. . . . . . . . 9 ⊢ (𝜑 → -∞ < 𝑋)
39	25, 37, 16, 38	lptioo2cn 40671	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(-∞(,)𝑋)))
40		fourierdlem88.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
41	34, 35, 39, 40	limcrecl 40655	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ ℝ)
42		fourierdlem88.h	. . . . . . 7 ⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
43		fourierdlem88.k	. . . . . . 7 ⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
44		fourierdlem88.u	. . . . . . 7 ⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
45	19, 16, 31, 41, 42, 43, 44	fourierdlem55 41171	. . . . . 6 ⊢ (𝜑 → 𝑈:(-π[,]π)⟶ℝ)
46	45	ffvelrnda 6607	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑈‘𝑠) ∈ ℝ)
47		fourierdlem88.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
48		fourierdlem88.s	. . . . . . . 8 ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠)))
49	48	fourierdlem5 41122	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → 𝑆:(-π[,]π)⟶ℝ)
50	47, 49	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆:(-π[,]π)⟶ℝ)
51	50	ffvelrnda 6607	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑆‘𝑠) ∈ ℝ)
52	46, 51	remulcld 10386	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ)
53	52	recnd 10384	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℂ)
54		fourierdlem88.g	. . 3 ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
55	53, 54	fmptd 6632	. 2 ⊢ (𝜑 → 𝐺:(-π[,]π)⟶ℂ)
56		ssid 3847	. . . 4 ⊢ ℂ ⊆ ℂ
57		cncfss 23071	. . . 4 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℝ) ⊆ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
58	23, 56, 57	mp2an 685	. . 3 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℝ) ⊆ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)
59	19	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℝ)
60	1, 2, 18	fourierdlem15 41132	. . . . . 6 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
61	60	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
62		elfzofz 12779	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
63	62	adantl 475	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
64	61, 63	ffvelrnd 6608	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (-π[,]π))
65		fzofzp1 12859	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
66	65	adantl 475	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
67	61, 66	ffvelrnd 6608	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ (-π[,]π))
68	16	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
69	7, 2, 6, 15	fourierdlem12 41129	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))
70	68	recnd 10384	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
71	70	addid2d 10555	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (0 + 𝑋) = 𝑋)
72	3	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈ ℝ)
73	72	renegcld 10780	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ)
74	73, 68	readdcld 10385	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (-π + 𝑋) ∈ ℝ)
75	72, 68	readdcld 10385	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (π + 𝑋) ∈ ℝ)
76	74, 75	iccssred 40525	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((-π + 𝑋)[,](π + 𝑋)) ⊆ ℝ)
77	7, 2, 6	fourierdlem15 41132	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶((-π + 𝑋)[,](π + 𝑋)))
78	77	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶((-π + 𝑋)[,](π + 𝑋)))
79	78, 63	ffvelrnd 6608	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)))
80	76, 79	sseldd 3827	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ)
81	80, 68	resubcld 10781	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
82	17	fvmpt2 6537	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ ℝ) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
83	63, 81, 82	syl2anc 581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
84	83	oveq1d 6919	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑋) = (((𝑉‘𝑖) − 𝑋) + 𝑋))
85	80	recnd 10384	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℂ)
86	85, 70	npcand 10716	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑉‘𝑖) − 𝑋) + 𝑋) = (𝑉‘𝑖))
87	84, 86	eqtrd 2860	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑋) = (𝑉‘𝑖))
88		fveq2 6432	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗))
89	88	oveq1d 6919	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋))
90	89	cbvmptv 4972	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
91	17, 90	eqtri 2848	. . . . . . . . . . . 12 ⊢ 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
92	91	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋)))
93		simpr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → 𝑗 = (𝑖 + 1))
94	93	fveq2d 6436	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1)))
95	94	oveq1d 6919	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
96	78, 66	ffvelrnd 6608	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ((-π + 𝑋)[,](π + 𝑋)))
97	76, 96	sseldd 3827	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
98	97, 68	resubcld 10781	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
99	92, 95, 66, 98	fvmptd 6534	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
100	99	oveq1d 6919	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑋) = (((𝑉‘(𝑖 + 1)) − 𝑋) + 𝑋))
101	97	recnd 10384	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℂ)
102	101, 70	npcand 10716	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑉‘(𝑖 + 1)) − 𝑋) + 𝑋) = (𝑉‘(𝑖 + 1)))
103	100, 102	eqtrd 2860	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑋) = (𝑉‘(𝑖 + 1)))
104	87, 103	oveq12d 6922	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋)) = ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))
105	71, 104	eleq12d 2899	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((0 + 𝑋) ∈ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋)) ↔ 𝑋 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))))
106	69, 105	mtbird 317	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ (0 + 𝑋) ∈ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋)))
107		0red 10359	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 0 ∈ ℝ)
108	83, 81	eqeltrd 2905	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
109	99, 98	eqeltrd 2905	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
110	107, 108, 109, 68	eliooshift 40527	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (0 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (0 + 𝑋) ∈ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋))))
111	106, 110	mtbird 317	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 0 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
112		fourierdlem88.fcn	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
113	104	reseq2d 5628	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋))) = (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))))
114	104	oveq1d 6919	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋))–cn→ℂ) = (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
115	112, 113, 114	3eltr4d 2920	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋))) ∈ ((((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋))–cn→ℂ))
116	31	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ℝ)
117	41	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ ℝ)
118	47	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑁 ∈ ℝ)
119	59, 64, 67, 68, 111, 115, 116, 117, 42, 43, 44, 118, 48, 54	fourierdlem78 41194	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℝ))
120	58, 119	sseldi 3824	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
121		eqid 2824	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠))
122		eqid 2824	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠))
123		eqid 2824	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
124	3	renegcli 10662	. . . . . . . . . . 11 ⊢ -π ∈ ℝ
125	124	rexri 10414	. . . . . . . . . 10 ⊢ -π ∈ ℝ^*
126	125	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → -π ∈ ℝ^*)
127	3	rexri 10414	. . . . . . . . . 10 ⊢ π ∈ ℝ^*
128	127	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → π ∈ ℝ^*)
129	61	adantr 474	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(-π[,]π))
130		simplr 787	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
131	126, 128, 129, 130	fourierdlem8 41125	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
132		ioossicc 12546	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
133	132	sseli 3822	. . . . . . . . 9 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
134	133	adantl 475	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
135	131, 134	sseldd 3827	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ (-π[,]π))
136	19, 16, 31, 41, 42	fourierdlem9 41126	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
137	136	ad2antrr 719	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐻:(-π[,]π)⟶ℝ)
138	137, 135	ffvelrnd 6608	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) ∈ ℝ)
139	43	fourierdlem43 41160	. . . . . . . . . 10 ⊢ 𝐾:(-π[,]π)⟶ℝ
140	139	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐾:(-π[,]π)⟶ℝ)
141	140, 135	ffvelrnd 6608	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐾‘𝑠) ∈ ℝ)
142	138, 141	remulcld 10386	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ)
143	44	fvmpt2 6537	. . . . . . 7 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
144	135, 142, 143	syl2anc 581	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
145	144, 142	eqeltrd 2905	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑈‘𝑠) ∈ ℝ)
146	145	recnd 10384	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑈‘𝑠) ∈ ℂ)
147	47, 48	fourierdlem18 41135	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ((-π[,]π)–cn→ℝ))
148		cncff 23065	. . . . . . . . 9 ⊢ (𝑆 ∈ ((-π[,]π)–cn→ℝ) → 𝑆:(-π[,]π)⟶ℝ)
149	147, 148	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆:(-π[,]π)⟶ℝ)
150	149	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑆:(-π[,]π)⟶ℝ)
151	150	adantr 474	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑆:(-π[,]π)⟶ℝ)
152	151, 135	ffvelrnd 6608	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑆‘𝑠) ∈ ℝ)
153	152	recnd 10384	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑆‘𝑠) ∈ ℂ)
154		eqid 2824	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠))
155		eqid 2824	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠))
156		eqid 2824	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
157	138	recnd 10384	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) ∈ ℂ)
158	141	recnd 10384	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐾‘𝑠) ∈ ℂ)
159		fourierdlem88.r	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
160		fourierdlem88.i	. . . . . . . 8 ⊢ 𝐼 = (ℝ D 𝐹)
161		fourierdlem88.ifn	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
162	23	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ℝ ⊆ ℂ)
163	161, 162	fssd 6291	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
164		fourierdlem88.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
165		eqid 2824	. . . . . . . 8 ⊢ if((𝑉‘𝑖) = 𝑋, 𝐷, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) = if((𝑉‘𝑖) = 𝑋, 𝐷, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖)))
166	16, 7, 19, 15, 30, 41, 42, 2, 6, 159, 17, 1, 160, 163, 164, 165	fourierdlem75 41191	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → if((𝑉‘𝑖) = 𝑋, 𝐷, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
167	136	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐻:(-π[,]π)⟶ℝ)
168	125	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ^*)
169	127	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈ ℝ^*)
170		simpr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
171	168, 169, 61, 170	fourierdlem8 41125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
172	132, 171	syl5ss 3837	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
173	167, 172	feqresmpt 6496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)))
174	173	oveq1d 6919	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) lim_ℂ (𝑄‘𝑖)))
175	166, 174	eleqtrd 2907	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → if((𝑉‘𝑖) = 𝑋, 𝐷, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) lim_ℂ (𝑄‘𝑖)))
176		limcresi 24047	. . . . . . . 8 ⊢ (𝐾 lim_ℂ (𝑄‘𝑖)) ⊆ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))
177	43	fourierdlem62 41178	. . . . . . . . . 10 ⊢ 𝐾 ∈ ((-π[,]π)–cn→ℝ)
178	177	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐾 ∈ ((-π[,]π)–cn→ℝ))
179	178, 64	cnlimci 24051	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘𝑖)) ∈ (𝐾 lim_ℂ (𝑄‘𝑖)))
180	176, 179	sseldi 3824	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘𝑖)) ∈ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
181	139	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐾:(-π[,]π)⟶ℝ)
182	181, 172	feqresmpt 6496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)))
183	182	oveq1d 6919	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)) lim_ℂ (𝑄‘𝑖)))
184	180, 183	eleqtrd 2907	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘𝑖)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)) lim_ℂ (𝑄‘𝑖)))
185	154, 155, 156, 157, 158, 175, 184	mullimc 40642	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (if((𝑉‘𝑖) = 𝑋, 𝐷, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) · (𝐾‘(𝑄‘𝑖))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) lim_ℂ (𝑄‘𝑖)))
186	144	eqcomd 2830	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐻‘𝑠) · (𝐾‘𝑠)) = (𝑈‘𝑠))
187	186	mpteq2dva 4966	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)))
188	187	oveq1d 6919	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)) lim_ℂ (𝑄‘𝑖)))
189	185, 188	eleqtrd 2907	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (if((𝑉‘𝑖) = 𝑋, 𝐷, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) · (𝐾‘(𝑄‘𝑖))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)) lim_ℂ (𝑄‘𝑖)))
190		limcresi 24047	. . . . . 6 ⊢ (𝑆 lim_ℂ (𝑄‘𝑖)) ⊆ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))
191	147	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑆 ∈ ((-π[,]π)–cn→ℝ))
192	191, 64	cnlimci 24051	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘𝑖)) ∈ (𝑆 lim_ℂ (𝑄‘𝑖)))
193	190, 192	sseldi 3824	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘𝑖)) ∈ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
194	150, 172	feqresmpt 6496	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)))
195	194	oveq1d 6919	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)) lim_ℂ (𝑄‘𝑖)))
196	193, 195	eleqtrd 2907	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘𝑖)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)) lim_ℂ (𝑄‘𝑖)))
197	121, 122, 123, 146, 153, 189, 196	mullimc 40642	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((if((𝑉‘𝑖) = 𝑋, 𝐷, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) · (𝐾‘(𝑄‘𝑖))) · (𝑆‘(𝑄‘𝑖))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) lim_ℂ (𝑄‘𝑖)))
198	52, 54	fmptd 6632	. . . . . . 7 ⊢ (𝜑 → 𝐺:(-π[,]π)⟶ℝ)
199	198	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺:(-π[,]π)⟶ℝ)
200	199, 172	feqresmpt 6496	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐺‘𝑠)))
201	145, 152	remulcld 10386	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ)
202	54	fvmpt2 6537	. . . . . . 7 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
203	135, 201, 202	syl2anc 581	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
204	203	mpteq2dva 4966	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐺‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))))
205	200, 204	eqtr2d 2861	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
206	205	oveq1d 6919	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) lim_ℂ (𝑄‘𝑖)) = ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
207	197, 206	eleqtrd 2907	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((if((𝑉‘𝑖) = 𝑋, 𝐷, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) · (𝐾‘(𝑄‘𝑖))) · (𝑆‘(𝑄‘𝑖))) ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
208		fourierdlem88.l	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1))))
209		fourierdlem88.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((𝐼 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
210		eqid 2824	. . . . . . . 8 ⊢ if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐶, ((𝐿 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1)))) = if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐶, ((𝐿 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1))))
211	16, 7, 19, 15, 31, 40, 42, 2, 6, 208, 17, 1, 160, 161, 209, 210	fourierdlem74 41190	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐶, ((𝐿 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1)))) ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
212	173	oveq1d 6919	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
213	211, 212	eleqtrd 2907	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐶, ((𝐿 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1)))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
214		limcresi 24047	. . . . . . . 8 ⊢ (𝐾 lim_ℂ (𝑄‘(𝑖 + 1))) ⊆ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))
215	178, 67	cnlimci 24051	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘(𝑖 + 1))) ∈ (𝐾 lim_ℂ (𝑄‘(𝑖 + 1))))
216	214, 215	sseldi 3824	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘(𝑖 + 1))) ∈ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
217	182	oveq1d 6919	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
218	216, 217	eleqtrd 2907	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘(𝑖 + 1))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
219	154, 155, 156, 157, 158, 213, 218	mullimc 40642	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐶, ((𝐿 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1)))) · (𝐾‘(𝑄‘(𝑖 + 1)))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) lim_ℂ (𝑄‘(𝑖 + 1))))
220	187	oveq1d 6919	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
221	219, 220	eleqtrd 2907	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐶, ((𝐿 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1)))) · (𝐾‘(𝑄‘(𝑖 + 1)))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
222		limcresi 24047	. . . . . 6 ⊢ (𝑆 lim_ℂ (𝑄‘(𝑖 + 1))) ⊆ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))
223	191, 67	cnlimci 24051	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘(𝑖 + 1))) ∈ (𝑆 lim_ℂ (𝑄‘(𝑖 + 1))))
224	222, 223	sseldi 3824	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘(𝑖 + 1))) ∈ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
225	194	oveq1d 6919	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
226	224, 225	eleqtrd 2907	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘(𝑖 + 1))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)) lim_ℂ (𝑄‘(𝑖 + 1))))
227	121, 122, 123, 146, 153, 221, 226	mullimc 40642	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐶, ((𝐿 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1)))) · (𝐾‘(𝑄‘(𝑖 + 1)))) · (𝑆‘(𝑄‘(𝑖 + 1)))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) lim_ℂ (𝑄‘(𝑖 + 1))))
228	205	oveq1d 6919	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) lim_ℂ (𝑄‘(𝑖 + 1))) = ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
229	227, 228	eleqtrd 2907	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐶, ((𝐿 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1)))) · (𝐾‘(𝑄‘(𝑖 + 1)))) · (𝑆‘(𝑄‘(𝑖 + 1)))) ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
230	1, 2, 18, 55, 120, 207, 229	fourierdlem69 41185	1 ⊢ (𝜑 → 𝐺 ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3116 {crab 3120 ⊆ wss 3797 ifcif 4305 class class class wbr 4872 ↦ cmpt 4951 ran crn 5342 ↾ cres 5343 ⟶wf 6118 ‘cfv 6122 (class class class)co 6904 ↑_𝑚 cmap 8121 ℂcc 10249 ℝcr 10250 0cc0 10251 1c1 10252 + caddc 10254 · cmul 10256 +∞cpnf 10387 -∞cmnf 10388 ℝ^*cxr 10389 < clt 10390 − cmin 10584 -cneg 10585 / cdiv 11008 ℕcn 11349 2c2 11405 (,)cioo 12462 [,]cicc 12465 ...cfz 12618 ..^cfzo 12759 sincsin 15165 πcpi 15168 TopOpenctopn 16434 ℂ_fldccnfld 20105 –cn→ccncf 23048 𝐿¹cibl 23782 lim_ℂ climc 24024 D cdv 24025

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cc 9571 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 ax-addf 10330 ax-mulf 10331

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-symdif 4069 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-iin 4742 df-disj 4841 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-ofr 7157 df-om 7326 df-1st 7427 df-2nd 7428 df-supp 7559 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-2o 7826 df-oadd 7829 df-omul 7830 df-er 8008 df-map 8123 df-pm 8124 df-ixp 8175 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-fsupp 8544 df-fi 8585 df-sup 8616 df-inf 8617 df-oi 8683 df-card 9077 df-acn 9080 df-cda 9304 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-7 11418 df-8 11419 df-9 11420 df-n0 11618 df-z 11704 df-dec 11821 df-uz 11968 df-q 12071 df-rp 12112 df-xneg 12231 df-xadd 12232 df-xmul 12233 df-ioo 12466 df-ioc 12467 df-ico 12468 df-icc 12469 df-fz 12619 df-fzo 12760 df-fl 12887 df-mod 12963 df-seq 13095 df-exp 13154 df-fac 13353 df-bc 13382 df-hash 13410 df-shft 14183 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-limsup 14578 df-clim 14595 df-rlim 14596 df-sum 14793 df-ef 15169 df-sin 15171 df-cos 15172 df-pi 15174 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-starv 16319 df-sca 16320 df-vsca 16321 df-ip 16322 df-tset 16323 df-ple 16324 df-ds 16326 df-unif 16327 df-hom 16328 df-cco 16329 df-rest 16435 df-topn 16436 df-0g 16454 df-gsum 16455 df-topgen 16456 df-pt 16457 df-prds 16460 df-xrs 16514 df-qtop 16519 df-imas 16520 df-xps 16522 df-mre 16598 df-mrc 16599 df-acs 16601 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-submnd 17688 df-mulg 17894 df-cntz 18099 df-cmn 18547 df-psmet 20097 df-xmet 20098 df-met 20099 df-bl 20100 df-mopn 20101 df-fbas 20102 df-fg 20103 df-cnfld 20106 df-top 21068 df-topon 21085 df-topsp 21107 df-bases 21120 df-cld 21193 df-ntr 21194 df-cls 21195 df-nei 21272 df-lp 21310 df-perf 21311 df-cn 21401 df-cnp 21402 df-t1 21488 df-haus 21489 df-cmp 21560 df-tx 21735 df-hmeo 21928 df-fil 22019 df-fm 22111 df-flim 22112 df-flf 22113 df-xms 22494 df-ms 22495 df-tms 22496 df-cncf 23050 df-ovol 23629 df-vol 23630 df-mbf 23784 df-itg1 23785 df-itg2 23786 df-ibl 23787 df-itg 23788 df-0p 23835 df-limc 24028 df-dv 24029

This theorem is referenced by: fourierdlem95 41211 fourierdlem103 41219 fourierdlem104 41220

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