fourierdlem88 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem88 47477

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 26840	. . . . 5 ⊢ π ∈ ℝ
4	3	a1i 11	. . . 4 ⊢ (𝜑 → π ∈ ℝ)
5	4	renegcld 11782	. . 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 47392	. . . . . . . 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 18	. . . . . . 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 235	. . . . . 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 500	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑_m (0...𝑀)))
12		elmapi 8907	. . . . 5 ⊢ (𝑉 ∈ (ℝ ↑_m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
13		frn 6747	. . . . 5 ⊢ (𝑉:(0...𝑀)⟶ℝ → ran 𝑉 ⊆ ℝ)
14	11, 12, 13	3syl 19	. . . 4 ⊢ (𝜑 → ran 𝑉 ⊆ ℝ)
15		fourierdlem88.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝑉)
16	14, 15	sseldd 3914	. . 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 47404	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀))
19		fourierdlem88.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
20		ioossre 13578	. . . . . . . . . 10 ⊢ (𝑋(,)+∞) ⊆ ℝ
21	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℝ)
22	19, 21	fssresd 6779	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)+∞)):(𝑋(,)+∞)⟶ℝ)
23		ax-resscn 11296	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
24	21, 23	sstrdi 3925	. . . . . . . 8 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℂ)
25		eqid 2763	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
26		pnfxr 11402	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
27	26	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → +∞ ∈ ℝ^*)
28	16	ltpnfd 13290	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 < +∞)
29	25, 27, 16, 28	lptioo1cn 46929	. . . . . . . 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 46914	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ℝ)
32		ioossre 13578	. . . . . . . . . 10 ⊢ (-∞(,)𝑋) ⊆ ℝ
33	32	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (-∞(,)𝑋) ⊆ ℝ)
34	19, 33	fssresd 6779	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ (-∞(,)𝑋)):(-∞(,)𝑋)⟶ℝ)
35	33, 23	sstrdi 3925	. . . . . . . 8 ⊢ (𝜑 → (-∞(,)𝑋) ⊆ ℂ)
36		mnfxr 11405	. . . . . . . . . 10 ⊢ -∞ ∈ ℝ^*
37	36	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → -∞ ∈ ℝ^*)
38	16	mnfltd 13293	. . . . . . . . 9 ⊢ (𝜑 → -∞ < 𝑋)
39	25, 37, 16, 38	lptioo2cn 46928	. . . . . . . 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 46914	. . . . . . 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 47444	. . . . . 6 ⊢ (𝜑 → 𝑈:( -π[,]π)⟶ℝ)
46	45	ffvelcdmda 7116	. . . . 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 47395	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → 𝑆:( -π[,]π)⟶ℝ)
50	47, 49	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑆:( -π[,]π)⟶ℝ)
51	50	ffvelcdmda 7116	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ( -π[,]π)) → (𝑆‘𝑠) ∈ ℝ)
52	46, 51	remulcld 11378	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ( -π[,]π)) → ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ)
53	52	recnd 11376	. . 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 7146	. 2 ⊢ (𝜑 → 𝐺:( -π[,]π)⟶ℂ)
56		ssid 3935	. . . 4 ⊢ ℂ ⊆ ℂ
57		cncfss 25275	. . . 4 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℝ) ⊆ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
58	23, 56, 57	mp2an 705	. . 3 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℝ) ⊆ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)
59	19	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℝ)
60	1, 2, 18	fourierdlem15 47405	. . . . . 6 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶( -π[,]π))
61	60	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶( -π[,]π))
62		elfzofz 13850	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
63	62	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
64	61, 63	ffvelcdmd 7117	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ( -π[,]π))
65		fzofzp1 13939	. . . . . 6 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
66	65	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
67	61, 66	ffvelcdmd 7117	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ( -π[,]π))
68	16	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
69	7, 2, 6, 15	fourierdlem12 47402	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))
70	68	recnd 11376	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ)
71	70	addlidd 11550	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (0 + 𝑋) = 𝑋)
72	3	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈ ℝ)
73	72	renegcld 11782	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ)
74	73, 68	readdcld 11377	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ( -π + 𝑋) ∈ ℝ)
75	72, 68	readdcld 11377	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (π + 𝑋) ∈ ℝ)
76	74, 75	iccssred 13605	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (( -π + 𝑋)[,](π + 𝑋)) ⊆ ℝ)
77	7, 2, 6	fourierdlem15 47405	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶(( -π + 𝑋)[,](π + 𝑋)))
78	77	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶(( -π + 𝑋)[,](π + 𝑋)))
79	78, 63	ffvelcdmd 7117	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ (( -π + 𝑋)[,](π + 𝑋)))
80	76, 79	sseldd 3914	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ)
81	80, 68	resubcld 11783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
82	17	fvmpt2 7036	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ ℝ) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
83	63, 81, 82	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
84	83	oveq1d 7466	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑋) = (((𝑉‘𝑖) − 𝑋) + 𝑋))
85	80	recnd 11376	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℂ)
86	85, 70	npcand 11712	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑉‘𝑖) − 𝑋) + 𝑋) = (𝑉‘𝑖))
87	84, 86	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑋) = (𝑉‘𝑖))
88		fveq2 6916	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗))
89	88	oveq1d 7466	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋))
90	89	cbvmptv 5215	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
91	17, 90	eqtri 2786	. . . . . . . . . . . 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 490	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → 𝑗 = (𝑖 + 1))
94	93	fveq2d 6920	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1)))
95	94	oveq1d 7466	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
96	78, 66	ffvelcdmd 7117	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ (( -π + 𝑋)[,](π + 𝑋)))
97	76, 96	sseldd 3914	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
98	97, 68	resubcld 11783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
99	92, 95, 66, 98	fvmptd 7032	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
100	99	oveq1d 7466	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑋) = (((𝑉‘(𝑖 + 1)) − 𝑋) + 𝑋))
101	97	recnd 11376	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℂ)
102	101, 70	npcand 11712	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑉‘(𝑖 + 1)) − 𝑋) + 𝑋) = (𝑉‘(𝑖 + 1)))
103	100, 102	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑋) = (𝑉‘(𝑖 + 1)))
104	87, 103	oveq12d 7469	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋)) = ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))
105	71, 104	eleq12d 2858	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((0 + 𝑋) ∈ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋)) ↔ 𝑋 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))))
106	69, 105	mtbird 328	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ (0 + 𝑋) ∈ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋)))
107		0red 11350	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 0 ∈ ℝ)
108	83, 81	eqeltrd 2864	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
109	99, 98	eqeltrd 2864	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
110	107, 108, 109, 68	eliooshift 46791	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (0 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (0 + 𝑋) ∈ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋))))
111	106, 110	mtbird 328	. . . 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 5986	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ (((𝑄‘𝑖) + 𝑋)(,)((𝑄‘(𝑖 + 1)) + 𝑋))) = (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))))
114	104	oveq1d 7466	. . . . 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 2879	. . . 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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ℝ)
117	41	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ ℝ)
118	47	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑁 ∈ ℝ)
119	59, 64, 67, 68, 111, 115, 116, 117, 42, 43, 44, 118, 48, 54	fourierdlem78 47467	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℝ))
120	58, 119	sselid 3911	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
121		eqid 2763	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠))
122		eqid 2763	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠))
123		eqid 2763	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
124	3	renegcli 11658	. . . . . . . . . . 11 ⊢ -π ∈ ℝ
125	124	rexri 11406	. . . . . . . . . 10 ⊢ -π ∈ ℝ^*
126	125	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → -π ∈ ℝ^*)
127	3	rexri 11406	. . . . . . . . . 10 ⊢ π ∈ ℝ^*
128	127	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → π ∈ ℝ^*)
129	61	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶( -π[,]π))
130		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
131	126, 128, 129, 130	fourierdlem8 47398	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ ( -π[,]π))
132		ioossicc 13604	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
133	132	sseli 3909	. . . . . . . . 9 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
134	133	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
135	131, 134	sseldd 3914	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ( -π[,]π))
136	19, 16, 31, 41, 42	fourierdlem9 47399	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻:( -π[,]π)⟶ℝ)
137	136	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐻:( -π[,]π)⟶ℝ)
138	137, 135	ffvelcdmd 7117	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) ∈ ℝ)
139	43	fourierdlem43 47433	. . . . . . . . . 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	ffvelcdmd 7117	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐾‘𝑠) ∈ ℝ)
142	138, 141	remulcld 11378	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ)
143	44	fvmpt2 7036	. . . . . . 7 ⊢ ((𝑠 ∈ ( -π[,]π) ∧ ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
144	135, 142, 143	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
145	144, 142	eqeltrd 2864	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑈‘𝑠) ∈ ℝ)
146	145	recnd 11376	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑈‘𝑠) ∈ ℂ)
147	47, 48	fourierdlem18 47408	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (( -π[,]π)–cn→ℝ))
148		cncff 25269	. . . . . . . . 9 ⊢ (𝑆 ∈ (( -π[,]π)–cn→ℝ) → 𝑆:( -π[,]π)⟶ℝ)
149	147, 148	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑆:( -π[,]π)⟶ℝ)
150	149	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑆:( -π[,]π)⟶ℝ)
151	150	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑆:( -π[,]π)⟶ℝ)
152	151, 135	ffvelcdmd 7117	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑆‘𝑠) ∈ ℝ)
153	152	recnd 11376	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑆‘𝑠) ∈ ℂ)
154		eqid 2763	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠))
155		eqid 2763	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠))
156		eqid 2763	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
157	138	recnd 11376	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) ∈ ℂ)
158	141	recnd 11376	. . . . . 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 6757	. . . . . . . 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 2763	. . . . . . . 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 47464	. . . . . . 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 486	. . . . . . . . 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 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
171	168, 169, 61, 170	fourierdlem8 47398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ ( -π[,]π))
172	132, 171	sstrid 3924	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ( -π[,]π))
173	167, 172	feqresmpt 6985	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)))
174	173	oveq1d 7466	. . . . . . 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 2866	. . . . . 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 26260	. . . . . . . 8 ⊢ (𝐾 lim_ℂ (𝑄‘𝑖)) ⊆ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))
177	43	fourierdlem62 47451	. . . . . . . . . 10 ⊢ 𝐾 ∈ (( -π[,]π)–cn→ℝ)
178	177	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐾 ∈ (( -π[,]π)–cn→ℝ))
179	178, 64	cnlimci 26264	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘𝑖)) ∈ (𝐾 lim_ℂ (𝑄‘𝑖)))
180	176, 179	sselid 3911	. . . . . . 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 6985	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)))
183	182	oveq1d 7466	. . . . . . 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 2866	. . . . . 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 46901	. . . . 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 2769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐻‘𝑠) · (𝐾‘𝑠)) = (𝑈‘𝑠))
187	186	mpteq2dva 5204	. . . . . 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 7466	. . . . 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 2866	. . . 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 26260	. . . . . 6 ⊢ (𝑆 lim_ℂ (𝑄‘𝑖)) ⊆ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))
191	147	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑆 ∈ (( -π[,]π)–cn→ℝ))
192	191, 64	cnlimci 26264	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘𝑖)) ∈ (𝑆 lim_ℂ (𝑄‘𝑖)))
193	190, 192	sselid 3911	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘𝑖)) ∈ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
194	150, 172	feqresmpt 6985	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)))
195	194	oveq1d 7466	. . . . 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 2866	. . . 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 46901	. . 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 7146	. . . . . . 7 ⊢ (𝜑 → 𝐺:( -π[,]π)⟶ℝ)
199	198	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺:( -π[,]π)⟶ℝ)
200	199, 172	feqresmpt 6985	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐺‘𝑠)))
201	145, 152	remulcld 11378	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ)
202	54	fvmpt2 7036	. . . . . . 7 ⊢ ((𝑠 ∈ ( -π[,]π) ∧ ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
203	135, 201, 202	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
204	203	mpteq2dva 5204	. . . . 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 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
206	205	oveq1d 7466	. . 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 2866	. 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 2763	. . . . . . . 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 47463	. . . . . . 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 7466	. . . . . . 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 2866	. . . . . 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 26260	. . . . . . . 8 ⊢ (𝐾 lim_ℂ (𝑄‘(𝑖 + 1))) ⊆ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))
215	178, 67	cnlimci 26264	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘(𝑖 + 1))) ∈ (𝐾 lim_ℂ (𝑄‘(𝑖 + 1))))
216	214, 215	sselid 3911	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘(𝑖 + 1))) ∈ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
217	182	oveq1d 7466	. . . . . . 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 2866	. . . . . 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 46901	. . . . 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 7466	. . . . 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 2866	. . . 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 26260	. . . . . 6 ⊢ (𝑆 lim_ℂ (𝑄‘(𝑖 + 1))) ⊆ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))
223	191, 67	cnlimci 26264	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘(𝑖 + 1))) ∈ (𝑆 lim_ℂ (𝑄‘(𝑖 + 1))))
224	222, 223	sselid 3911	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘(𝑖 + 1))) ∈ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
225	194	oveq1d 7466	. . . . 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 2866	. . . 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 46901	. . 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 7466	. . 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 2866	. 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 47458	1 ⊢ (𝜑 → 𝐺 ∈ 𝐿¹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∀wral 3080 {crab 3416 ⊆ wss 3900 ifcif 4485 class class class wbr 5108 ↦ cmpt 5192 ran crn 5659 ↾ cres 5660 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ↑_m cmap 8885 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 + caddc 11242 · cmul 11244 +∞cpnf 11379 -∞cmnf 11380 ℝ^*cxr 11381 < clt 11382 − cmin 11580 -cneg 11581 / cdiv 12013 ℕcn 12375 2c2 12437 (,)cioo 13516 [,]cicc 13519 ...cfz 13679 ..^cfzo 13828 sincsin 16273 πcpi 16276 TopOpenctopn 17642 ℂ_fldccnfld 21729 –cn→ccncf 25252 𝐿¹cibl 25993 lim_ℂ climc 26237 D cdv 26238

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cc 10552 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-symdif 4201 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-disj 5076 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-ofr 7726 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-omul 8519 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-acn 10060 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-sum 15898 df-ef 16277 df-sin 16279 df-cos 16280 df-pi 16282 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-t1 23687 df-haus 23688 df-cmp 23760 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-ovol 25840 df-vol 25841 df-mbf 25995 df-itg1 25996 df-itg2 25997 df-ibl 25998 df-itg 25999 df-0p 26046 df-limc 26241 df-dv 26242

This theorem is used by: fourierdlem95 47484 fourierdlem103 47492 fourierdlem104 47493

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