fourierdlem85 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem85 46112

Description: Limit of the function 𝐺 at the lower bounds of the partition intervals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem85.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem85.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem85.x	⊢ (𝜑 → 𝑋 ∈ ran 𝑉)
fourierdlem85.y	⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
fourierdlem85.w	⊢ (𝜑 → 𝑊 ∈ ℝ)
fourierdlem85.h	⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
fourierdlem85.k	⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
fourierdlem85.u	⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
fourierdlem85.n	⊢ (𝜑 → 𝑁 ∈ ℝ)
fourierdlem85.s	⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠)))
fourierdlem85.g	⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
fourierdlem85.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem85.v	⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
fourierdlem85.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
fourierdlem85.q	⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
fourierdlem85.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem85.i	⊢ 𝐼 = (ℝ D 𝐹)
fourierdlem85.ifn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
fourierdlem85.e	⊢ (𝜑 → 𝐸 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
fourierdlem85.a	⊢ 𝐴 = ((if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) · (𝐾‘(𝑄‘𝑖))) · (𝑆‘(𝑄‘𝑖)))

**Assertion**
Ref	Expression
fourierdlem85	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))

Distinct variable groups: 𝐸,𝑠 𝐹,𝑠 𝐻,𝑠 𝐾,𝑠 𝑖,𝑀,𝑚,𝑝 𝑀,𝑠,𝑖 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) 𝐴(𝑖,𝑚,𝑠,𝑝) 𝑃(𝑖,𝑚,𝑠,𝑝) 𝑄(𝑚) 𝑅(𝑖,𝑚,𝑝) 𝑆(𝑖,𝑚,𝑝) 𝑈(𝑖,𝑚,𝑠,𝑝) 𝐸(𝑖,𝑚,𝑝) 𝐹(𝑖,𝑚,𝑝) 𝐺(𝑖,𝑚,𝑠,𝑝) 𝐻(𝑖,𝑚,𝑝) 𝐼(𝑖,𝑚,𝑠,𝑝) 𝐾(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 fourierdlem85**
Step	Hyp	Ref	Expression
1		fourierdlem85.a	. . 3 ⊢ 𝐴 = ((if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) · (𝐾‘(𝑄‘𝑖))) · (𝑆‘(𝑄‘𝑖)))
2		eqid 2740	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠))
3		eqid 2740	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠))
4		eqid 2740	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
5		pire 26518	. . . . . . . . . . 11 ⊢ π ∈ ℝ
6	5	renegcli 11597	. . . . . . . . . 10 ⊢ -π ∈ ℝ
7	6	rexri 11348	. . . . . . . . 9 ⊢ -π ∈ ℝ^*
8	7	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → -π ∈ ℝ^*)
9	5	rexri 11348	. . . . . . . . 9 ⊢ π ∈ ℝ^*
10	9	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → π ∈ ℝ^*)
11		fourierdlem85.o	. . . . . . . . . . 11 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
12		fourierdlem85.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ)
13	5	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → π ∈ ℝ)
14	13	renegcld 11717	. . . . . . . . . . . 12 ⊢ (𝜑 → -π ∈ ℝ)
15		fourierdlem85.v	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
16		fourierdlem85.p	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
17	16	fourierdlem2 46030	. . . . . . . . . . . . . . . . 17 ⊢ (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))))))
18	12, 17	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (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))))))
19	15, 18	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑉 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉‘𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))))
20	19	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑_m (0...𝑀)))
21		elmapi 8907	. . . . . . . . . . . . . 14 ⊢ (𝑉 ∈ (ℝ ↑_m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
22		frn 6754	. . . . . . . . . . . . . 14 ⊢ (𝑉:(0...𝑀)⟶ℝ → ran 𝑉 ⊆ ℝ)
23	20, 21, 22	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝑉 ⊆ ℝ)
24		fourierdlem85.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ ran 𝑉)
25	23, 24	sseldd 4009	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ ℝ)
26		fourierdlem85.q	. . . . . . . . . . . 12 ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
27	14, 13, 25, 16, 11, 12, 15, 26	fourierdlem14 46042	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀))
28	11, 12, 27	fourierdlem15 46043	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π))
29	28	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π))
30	29	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(-π[,]π))
31		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
32	8, 10, 30, 31	fourierdlem8 46036	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
33		ioossicc 13493	. . . . . . . . 9 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
34	33	sseli 4004	. . . . . . . 8 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
35	34	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
36	32, 35	sseldd 4009	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ (-π[,]π))
37		fourierdlem85.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
38		ioossre 13468	. . . . . . . . . . . . . 14 ⊢ (𝑋(,)+∞) ⊆ ℝ
39	38	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℝ)
40	37, 39	fssresd 6788	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)+∞)):(𝑋(,)+∞)⟶ℝ)
41		ax-resscn 11241	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
42	39, 41	sstrdi 4021	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℂ)
43		eqid 2740	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
44		pnfxr 11344	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → +∞ ∈ ℝ^*)
46	25	ltpnfd 13184	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 < +∞)
47	43, 45, 25, 46	lptioo1cn 45567	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝑋(,)+∞)))
48		fourierdlem85.y	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
49	40, 42, 47, 48	limcrecl 45550	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ℝ)
50		fourierdlem85.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ ℝ)
51		fourierdlem85.h	. . . . . . . . . . 11 ⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
52	37, 25, 49, 50, 51	fourierdlem9 46037	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
53	41	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
54	52, 53	fssd 6764	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℂ)
55	54	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐻:(-π[,]π)⟶ℂ)
56	55, 36	ffvelcdmd 7119	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) ∈ ℂ)
57		fourierdlem85.k	. . . . . . . . . . 11 ⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
58	57	fourierdlem43 46071	. . . . . . . . . 10 ⊢ 𝐾:(-π[,]π)⟶ℝ
59	58	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐾:(-π[,]π)⟶ℝ)
60	59, 36	ffvelcdmd 7119	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐾‘𝑠) ∈ ℝ)
61	60	recnd 11318	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐾‘𝑠) ∈ ℂ)
62	56, 61	mulcld 11310	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℂ)
63		fourierdlem85.u	. . . . . . 7 ⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
64	63	fvmpt2 7040	. . . . . 6 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℂ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
65	36, 62, 64	syl2anc 583	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
66	65, 62	eqeltrd 2844	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑈‘𝑠) ∈ ℂ)
67		fourierdlem85.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
68		fourierdlem85.s	. . . . . . . . . 10 ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠)))
69	67, 68	fourierdlem18 46046	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ((-π[,]π)–cn→ℝ))
70		cncff 24938	. . . . . . . . 9 ⊢ (𝑆 ∈ ((-π[,]π)–cn→ℝ) → 𝑆:(-π[,]π)⟶ℝ)
71	69, 70	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆:(-π[,]π)⟶ℝ)
72	71	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑆:(-π[,]π)⟶ℝ)
73	72	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑆:(-π[,]π)⟶ℝ)
74	73, 36	ffvelcdmd 7119	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑆‘𝑠) ∈ ℝ)
75	74	recnd 11318	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑆‘𝑠) ∈ ℂ)
76		eqid 2740	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠))
77		eqid 2740	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠))
78		eqid 2740	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
79		fourierdlem85.r	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
80		fourierdlem85.i	. . . . . . . 8 ⊢ 𝐼 = (ℝ D 𝐹)
81		fourierdlem85.ifn	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)
82		fourierdlem85.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
83		eqid 2740	. . . . . . . 8 ⊢ if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) = if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖)))
84	25, 16, 37, 24, 48, 50, 51, 12, 15, 79, 26, 11, 80, 81, 82, 83	fourierdlem75 46102	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
85	52	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐻:(-π[,]π)⟶ℝ)
86	7	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈ ℝ^*)
87	9	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈ ℝ^*)
88		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
89	86, 87, 29, 88	fourierdlem8 46036	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
90	33, 89	sstrid 4020	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
91	85, 90	feqresmpt 6991	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)))
92	91	oveq1d 7463	. . . . . . 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)))
93	84, 92	eleqtrd 2846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) lim_ℂ (𝑄‘𝑖)))
94		limcresi 25940	. . . . . . . 8 ⊢ (𝐾 lim_ℂ (𝑄‘𝑖)) ⊆ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))
95		ssid 4031	. . . . . . . . . . . 12 ⊢ ℂ ⊆ ℂ
96		cncfss 24944	. . . . . . . . . . . 12 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((-π[,]π)–cn→ℝ) ⊆ ((-π[,]π)–cn→ℂ))
97	41, 95, 96	mp2an 691	. . . . . . . . . . 11 ⊢ ((-π[,]π)–cn→ℝ) ⊆ ((-π[,]π)–cn→ℂ)
98	57	fourierdlem62 46089	. . . . . . . . . . 11 ⊢ 𝐾 ∈ ((-π[,]π)–cn→ℝ)
99	97, 98	sselii 4005	. . . . . . . . . 10 ⊢ 𝐾 ∈ ((-π[,]π)–cn→ℂ)
100	99	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐾 ∈ ((-π[,]π)–cn→ℂ))
101		elfzofz 13732	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
102	101	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
103	29, 102	ffvelcdmd 7119	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (-π[,]π))
104	100, 103	cnlimci 25944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘𝑖)) ∈ (𝐾 lim_ℂ (𝑄‘𝑖)))
105	94, 104	sselid 4006	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘𝑖)) ∈ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
106		cncff 24938	. . . . . . . . . 10 ⊢ (𝐾 ∈ ((-π[,]π)–cn→ℂ) → 𝐾:(-π[,]π)⟶ℂ)
107	99, 106	mp1i 13	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐾:(-π[,]π)⟶ℂ)
108	107, 90	feqresmpt 6991	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)))
109	108	oveq1d 7463	. . . . . . 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)))
110	105, 109	eleqtrd 2846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘𝑖)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)) lim_ℂ (𝑄‘𝑖)))
111	76, 77, 78, 56, 61, 93, 110	mullimc 45537	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (if((𝑉‘𝑖) = 𝑋, 𝐸, ((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)))
112	65	mpteq2dva 5266	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))))
113	112	oveq1d 7463	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)) lim_ℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) lim_ℂ (𝑄‘𝑖)))
114	111, 113	eleqtrrd 2847	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) · (𝐾‘(𝑄‘𝑖))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)) lim_ℂ (𝑄‘𝑖)))
115		limcresi 25940	. . . . . 6 ⊢ (𝑆 lim_ℂ (𝑄‘𝑖)) ⊆ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))
116	69	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑆 ∈ ((-π[,]π)–cn→ℝ))
117	116, 103	cnlimci 25944	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘𝑖)) ∈ (𝑆 lim_ℂ (𝑄‘𝑖)))
118	115, 117	sselid 4006	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘𝑖)) ∈ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
119	72, 90	feqresmpt 6991	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)))
120	119	oveq1d 7463	. . . . 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)))
121	118, 120	eleqtrd 2846	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘𝑖)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)) lim_ℂ (𝑄‘𝑖)))
122	2, 3, 4, 66, 75, 114, 121	mullimc 45537	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((if((𝑉‘𝑖) = 𝑋, 𝐸, ((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)))
123	1, 122	eqeltrid 2848	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) lim_ℂ (𝑄‘𝑖)))
124		fourierdlem85.g	. . . . 5 ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
125	124	reseq1i 6005	. . . 4 ⊢ (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
126	90	resmptd 6069	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))))
127	125, 126	eqtr2id 2793	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
128	127	oveq1d 7463	. 2 ⊢ ((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)))
129	123, 128	eleqtrd 2846	1 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 ⊆ wss 3976 ifcif 4548 class class class wbr 5166 ↦ cmpt 5249 ran crn 5701 ↾ cres 5702 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 +∞cpnf 11321 ℝ^*cxr 11323 < clt 11324 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 2c2 12348 (,)cioo 13407 [,]cicc 13410 ...cfz 13567 ..^cfzo 13711 sincsin 16111 πcpi 16114 TopOpenctopn 17481 ℂ_fldccnfld 21387 –cn→ccncf 24921 lim_ℂ climc 25917 D cdv 25918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-t1 23343 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: (None)

W3C validator