fourierdlem88 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem88 43625

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 25520	. . . . 5 ⊢ π ∈ ℝ
4	3	a1i 11	. . . 4 ⊢ (𝜑 → π ∈ ℝ)
5	4	renegcld 11332	. . 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 43540	. . . . . . . 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 231	. . . . . 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 494	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑_m (0...𝑀)))
12		elmapi 8595	. . . . 5 ⊢ (𝑉 ∈ (ℝ ↑_m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
13		frn 6591	. . . . 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 3918	. . 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 43552	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀))
19		fourierdlem88.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
20		ioossre 13069	. . . . . . . . . 10 ⊢ (𝑋(,)+∞) ⊆ ℝ
21	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℝ)
22	19, 21	fssresd 6625	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)+∞)):(𝑋(,)+∞)⟶ℝ)
23		ax-resscn 10859	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
24	21, 23	sstrdi 3929	. . . . . . . 8 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℂ)
25		eqid 2738	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
26		pnfxr 10960	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
27	26	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → +∞ ∈ ℝ^*)
28	16	ltpnfd 12786	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 < +∞)
29	25, 27, 16, 28	lptioo1cn 43077	. . . . . . . 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 43060	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ℝ)
32		ioossre 13069	. . . . . . . . . 10 ⊢ (-∞(,)𝑋) ⊆ ℝ
33	32	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (-∞(,)𝑋) ⊆ ℝ)
34	19, 33	fssresd 6625	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ (-∞(,)𝑋)):(-∞(,)𝑋)⟶ℝ)
35	33, 23	sstrdi 3929	. . . . . . . 8 ⊢ (𝜑 → (-∞(,)𝑋) ⊆ ℂ)
36		mnfxr 10963	. . . . . . . . . 10 ⊢ -∞ ∈ ℝ^*
37	36	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → -∞ ∈ ℝ^*)
38	16	mnfltd 12789	. . . . . . . . 9 ⊢ (𝜑 → -∞ < 𝑋)
39	25, 37, 16, 38	lptioo2cn 43076	. . . . . . . 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 43060	. . . . . . 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 43592	. . . . . 6 ⊢ (𝜑 → 𝑈:(-π[,]π)⟶ℝ)
46	45	ffvelrnda 6943	. . . . 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 43543	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → 𝑆:(-π[,]π)⟶ℝ)
50	47, 49	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆:(-π[,]π)⟶ℝ)
51	50	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑆‘𝑠) ∈ ℝ)
52	46, 51	remulcld 10936	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ)
53	52	recnd 10934	. . 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 6970	. 2 ⊢ (𝜑 → 𝐺:(-π[,]π)⟶ℂ)
56		ssid 3939	. . . 4 ⊢ ℂ ⊆ ℂ
57		cncfss 23968	. . . 4 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℝ) ⊆ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
58	23, 56, 57	mp2an 688	. . 3 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℝ) ⊆ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)
59	19	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℝ)
60	1, 2, 18	fourierdlem15 43553	. . . . . 6 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
61	60	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
62		elfzofz 13331	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
63	62	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
64	61, 63	ffvelrnd 6944	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (-π[,]π))
65		fzofzp1 13412	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
66	65	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
67	61, 66	ffvelrnd 6944	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ (-π[,]π))
68	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
69	7, 2, 6, 15	fourierdlem12 43550	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))
70	68	recnd 10934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
71	70	addid2d 11106	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (0 + 𝑋) = 𝑋)
72	3	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈ ℝ)
73	72	renegcld 11332	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ)
74	73, 68	readdcld 10935	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (-π + 𝑋) ∈ ℝ)
75	72, 68	readdcld 10935	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (π + 𝑋) ∈ ℝ)
76	74, 75	iccssred 13095	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((-π + 𝑋)[,](π + 𝑋)) ⊆ ℝ)
77	7, 2, 6	fourierdlem15 43553	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶((-π + 𝑋)[,](π + 𝑋)))
78	77	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶((-π + 𝑋)[,](π + 𝑋)))
79	78, 63	ffvelrnd 6944	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)))
80	76, 79	sseldd 3918	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ)
81	80, 68	resubcld 11333	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
82	17	fvmpt2 6868	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ ℝ) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
83	63, 81, 82	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
84	83	oveq1d 7270	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑋) = (((𝑉‘𝑖) − 𝑋) + 𝑋))
85	80	recnd 10934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℂ)
86	85, 70	npcand 11266	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑉‘𝑖) − 𝑋) + 𝑋) = (𝑉‘𝑖))
87	84, 86	eqtrd 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑋) = (𝑉‘𝑖))
88		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗))
89	88	oveq1d 7270	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋))
90	89	cbvmptv 5183	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
91	17, 90	eqtri 2766	. . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → 𝑗 = (𝑖 + 1))
94	93	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1)))
95	94	oveq1d 7270	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
96	78, 66	ffvelrnd 6944	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ((-π + 𝑋)[,](π + 𝑋)))
97	76, 96	sseldd 3918	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
98	97, 68	resubcld 11333	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
99	92, 95, 66, 98	fvmptd 6864	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
100	99	oveq1d 7270	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑋) = (((𝑉‘(𝑖 + 1)) − 𝑋) + 𝑋))
101	97	recnd 10934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℂ)
102	101, 70	npcand 11266	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑉‘(𝑖 + 1)) − 𝑋) + 𝑋) = (𝑉‘(𝑖 + 1)))
103	100, 102	eqtrd 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑋) = (𝑉‘(𝑖 + 1)))
104	87, 103	oveq12d 7273	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋)) = ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))
105	71, 104	eleq12d 2833	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((0 + 𝑋) ∈ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋)) ↔ 𝑋 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))))
106	69, 105	mtbird 324	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ (0 + 𝑋) ∈ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋)))
107		0red 10909	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 0 ∈ ℝ)
108	83, 81	eqeltrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
109	99, 98	eqeltrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
110	107, 108, 109, 68	eliooshift 42934	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (0 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (0 + 𝑋) ∈ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋))))
111	106, 110	mtbird 324	. . . 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 5880	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋))) = (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))))
114	104	oveq1d 7270	. . . . 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 2854	. . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ℝ)
117	41	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ ℝ)
118	47	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑁 ∈ ℝ)
119	59, 64, 67, 68, 111, 115, 116, 117, 42, 43, 44, 118, 48, 54	fourierdlem78 43615	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℝ))
120	58, 119	sselid 3915	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
121		eqid 2738	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠))
122		eqid 2738	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠))
123		eqid 2738	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
124	3	renegcli 11212	. . . . . . . . . . 11 ⊢ -π ∈ ℝ
125	124	rexri 10964	. . . . . . . . . 10 ⊢ -π ∈ ℝ^*
126	125	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → -π ∈ ℝ^*)
127	3	rexri 10964	. . . . . . . . . 10 ⊢ π ∈ ℝ^*
128	127	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → π ∈ ℝ^*)
129	61	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(-π[,]π))
130		simplr 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
131	126, 128, 129, 130	fourierdlem8 43546	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
132		ioossicc 13094	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
133	132	sseli 3913	. . . . . . . . 9 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
134	133	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
135	131, 134	sseldd 3918	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ (-π[,]π))
136	19, 16, 31, 41, 42	fourierdlem9 43547	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
137	136	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐻:(-π[,]π)⟶ℝ)
138	137, 135	ffvelrnd 6944	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) ∈ ℝ)
139	43	fourierdlem43 43581	. . . . . . . . . 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 6944	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐾‘𝑠) ∈ ℝ)
142	138, 141	remulcld 10936	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ)
143	44	fvmpt2 6868	. . . . . . 7 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
144	135, 142, 143	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
145	144, 142	eqeltrd 2839	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑈‘𝑠) ∈ ℝ)
146	145	recnd 10934	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑈‘𝑠) ∈ ℂ)
147	47, 48	fourierdlem18 43556	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ((-π[,]π)–cn→ℝ))
148		cncff 23962	. . . . . . . . 9 ⊢ (𝑆 ∈ ((-π[,]π)–cn→ℝ) → 𝑆:(-π[,]π)⟶ℝ)
149	147, 148	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆:(-π[,]π)⟶ℝ)
150	149	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑆:(-π[,]π)⟶ℝ)
151	150	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑆:(-π[,]π)⟶ℝ)
152	151, 135	ffvelrnd 6944	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑆‘𝑠) ∈ ℝ)
153	152	recnd 10934	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑆‘𝑠) ∈ ℂ)
154		eqid 2738	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠))
155		eqid 2738	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠))
156		eqid 2738	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
157	138	recnd 10934	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) ∈ ℂ)
158	141	recnd 10934	. . . . . 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 6602	. . . . . . . 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 2738	. . . . . . . 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 43612	. . . . . . 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 480	. . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
171	168, 169, 61, 170	fourierdlem8 43546	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
172	132, 171	sstrid 3928	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
173	167, 172	feqresmpt 6820	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)))
174	173	oveq1d 7270	. . . . . . 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 2841	. . . . . 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 24954	. . . . . . . 8 ⊢ (𝐾 lim_ℂ (𝑄‘𝑖)) ⊆ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))
177	43	fourierdlem62 43599	. . . . . . . . . 10 ⊢ 𝐾 ∈ ((-π[,]π)–cn→ℝ)
178	177	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐾 ∈ ((-π[,]π)–cn→ℝ))
179	178, 64	cnlimci 24958	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘𝑖)) ∈ (𝐾 lim_ℂ (𝑄‘𝑖)))
180	176, 179	sselid 3915	. . . . . . 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 6820	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)))
183	182	oveq1d 7270	. . . . . . 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 2841	. . . . . 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 43047	. . . . 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 2744	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐻‘𝑠) · (𝐾‘𝑠)) = (𝑈‘𝑠))
187	186	mpteq2dva 5170	. . . . . 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 7270	. . . . 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 2841	. . . 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 24954	. . . . . 6 ⊢ (𝑆 lim_ℂ (𝑄‘𝑖)) ⊆ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))
191	147	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑆 ∈ ((-π[,]π)–cn→ℝ))
192	191, 64	cnlimci 24958	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘𝑖)) ∈ (𝑆 lim_ℂ (𝑄‘𝑖)))
193	190, 192	sselid 3915	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘𝑖)) ∈ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
194	150, 172	feqresmpt 6820	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)))
195	194	oveq1d 7270	. . . . 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 2841	. . . 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 43047	. . 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 6970	. . . . . . 7 ⊢ (𝜑 → 𝐺:(-π[,]π)⟶ℝ)
199	198	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺:(-π[,]π)⟶ℝ)
200	199, 172	feqresmpt 6820	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐺‘𝑠)))
201	145, 152	remulcld 10936	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ)
202	54	fvmpt2 6868	. . . . . . 7 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
203	135, 201, 202	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
204	203	mpteq2dva 5170	. . . . 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 2779	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
206	205	oveq1d 7270	. . 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 2841	. 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 2738	. . . . . . . 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 43611	. . . . . . 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 7270	. . . . . . 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 2841	. . . . . 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 24954	. . . . . . . 8 ⊢ (𝐾 lim_ℂ (𝑄‘(𝑖 + 1))) ⊆ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))
215	178, 67	cnlimci 24958	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘(𝑖 + 1))) ∈ (𝐾 lim_ℂ (𝑄‘(𝑖 + 1))))
216	214, 215	sselid 3915	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘(𝑖 + 1))) ∈ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
217	182	oveq1d 7270	. . . . . . 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 2841	. . . . . 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 43047	. . . . 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 7270	. . . . 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 2841	. . . 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 24954	. . . . . 6 ⊢ (𝑆 lim_ℂ (𝑄‘(𝑖 + 1))) ⊆ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))
223	191, 67	cnlimci 24958	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘(𝑖 + 1))) ∈ (𝑆 lim_ℂ (𝑄‘(𝑖 + 1))))
224	222, 223	sselid 3915	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘(𝑖 + 1))) ∈ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
225	194	oveq1d 7270	. . . . 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 2841	. . . 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 43047	. . 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 7270	. . 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 2841	. 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 43606	1 ⊢ (𝜑 → 𝐺 ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 {crab 3067 ⊆ wss 3883 ifcif 4456 class class class wbr 5070 ↦ cmpt 5153 ran crn 5581 ↾ cres 5582 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 +∞cpnf 10937 -∞cmnf 10938 ℝ^*cxr 10939 < clt 10940 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 (,)cioo 13008 [,]cicc 13011 ...cfz 13168 ..^cfzo 13311 sincsin 15701 πcpi 15704 TopOpenctopn 17049 ℂ_fldccnfld 20510 –cn→ccncf 23945 𝐿¹cibl 24686 lim_ℂ climc 24931 D cdv 24932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cc 10122 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-symdif 4173 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-t1 22373 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-ovol 24533 df-vol 24534 df-mbf 24688 df-itg1 24689 df-itg2 24690 df-ibl 24691 df-itg 24692 df-0p 24739 df-limc 24935 df-dv 24936

This theorem is referenced by: fourierdlem95 43632 fourierdlem103 43640 fourierdlem104 43641

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