fourierdlem85 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem85 46640

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 2737	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑈‘𝑠))
3		eqid 2737	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠))
4		eqid 2737	. . . 4 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
5		pire 26437	. . . . . . . . . . 11 ⊢ π ∈ ℝ
6	5	renegcli 11449	. . . . . . . . . 10 ⊢ -π ∈ ℝ
7	6	rexri 11197	. . . . . . . . 9 ⊢ -π ∈ ℝ^*
8	7	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → -π ∈ ℝ^*)
9	5	rexri 11197	. . . . . . . . 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 11571	. . . . . . . . . . . 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 46558	. . . . . . . . . . . . . . . . 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 8790	. . . . . . . . . . . . . 14 ⊢ (𝑉 ∈ (ℝ ↑_m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
22		frn 6670	. . . . . . . . . . . . . 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 3923	. . . . . . . . . . . 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 46570	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀))
28	11, 12, 27	fourierdlem15 46571	. . . . . . . . . 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 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
32	8, 10, 30, 31	fourierdlem8 46564	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
33		ioossicc 13380	. . . . . . . . 9 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
34	33	sseli 3918	. . . . . . . 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 3923	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ (-π[,]π))
37		fourierdlem85.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
38		ioossre 13354	. . . . . . . . . . . . . 14 ⊢ (𝑋(,)+∞) ⊆ ℝ
39	38	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℝ)
40	37, 39	fssresd 6702	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)+∞)):(𝑋(,)+∞)⟶ℝ)
41		ax-resscn 11089	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
42	39, 41	sstrdi 3935	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℂ)
43		eqid 2737	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
44		pnfxr 11193	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → +∞ ∈ ℝ^*)
46	25	ltpnfd 13066	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 < +∞)
47	43, 45, 25, 46	lptioo1cn 46095	. . . . . . . . . . . 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 46080	. . . . . . . . . . 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 46565	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
53	41	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
54	52, 53	fssd 6680	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℂ)
55	54	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐻:(-π[,]π)⟶ℂ)
56	55, 36	ffvelcdmd 7032	. . . . . . 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 46599	. . . . . . . . . 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 7032	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐾‘𝑠) ∈ ℝ)
61	60	recnd 11167	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐾‘𝑠) ∈ ℂ)
62	56, 61	mulcld 11159	. . . . . 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 6954	. . . . . 6 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℂ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
65	36, 62, 64	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
66	65, 62	eqeltrd 2837	. . . 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 46574	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ((-π[,]π)–cn→ℝ))
70		cncff 24873	. . . . . . . . 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 7032	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑆‘𝑠) ∈ ℝ)
75	74	recnd 11167	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑆‘𝑠) ∈ ℂ)
76		eqid 2737	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠))
77		eqid 2737	. . . . . 6 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠))
78		eqid 2737	. . . . . 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 2737	. . . . . . . 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 46630	. . . . . . 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 46564	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
90	33, 89	sstrid 3934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (-π[,]π))
91	85, 90	feqresmpt 6904	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)))
92	91	oveq1d 7376	. . . . . . 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 2839	. . . . . 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 25865	. . . . . . . 8 ⊢ (𝐾 lim_ℂ (𝑄‘𝑖)) ⊆ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))
95		ssid 3945	. . . . . . . . . . . 12 ⊢ ℂ ⊆ ℂ
96		cncfss 24879	. . . . . . . . . . . 12 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((-π[,]π)–cn→ℝ) ⊆ ((-π[,]π)–cn→ℂ))
97	41, 95, 96	mp2an 693	. . . . . . . . . . 11 ⊢ ((-π[,]π)–cn→ℝ) ⊆ ((-π[,]π)–cn→ℂ)
98	57	fourierdlem62 46617	. . . . . . . . . . 11 ⊢ 𝐾 ∈ ((-π[,]π)–cn→ℝ)
99	97, 98	sselii 3919	. . . . . . . . . 10 ⊢ 𝐾 ∈ ((-π[,]π)–cn→ℂ)
100	99	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐾 ∈ ((-π[,]π)–cn→ℂ))
101		elfzofz 13624	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
102	101	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
103	29, 102	ffvelcdmd 7032	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (-π[,]π))
104	100, 103	cnlimci 25869	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘𝑖)) ∈ (𝐾 lim_ℂ (𝑄‘𝑖)))
105	94, 104	sselid 3920	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾‘(𝑄‘𝑖)) ∈ ((𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
106		cncff 24873	. . . . . . . . . 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 6904	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐾 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐾‘𝑠)))
109	108	oveq1d 7376	. . . . . . 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 2839	. . . . . 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 46067	. . . . 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 5179	. . . . . 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 7376	. . . . 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 2840	. . . 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 25865	. . . . . 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 25869	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘𝑖)) ∈ (𝑆 lim_ℂ (𝑄‘𝑖)))
118	115, 117	sselid 3920	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑄‘𝑖)) ∈ ((𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
119	72, 90	feqresmpt 6904	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝑆‘𝑠)))
120	119	oveq1d 7376	. . . . 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 2839	. . . 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 46067	. . 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 2841	. 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 5935	. . . 4 ⊢ (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
126	90	resmptd 6000	. . . 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 2785	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
128	127	oveq1d 7376	. 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 2839	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 1542 ∈ wcel 2114 ∀wral 3052 {crab 3390 ⊆ wss 3890 ifcif 4467 class class class wbr 5086 ↦ cmpt 5167 ran crn 5626 ↾ cres 5627 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 ↑_m cmap 8767 ℂcc 11030 ℝcr 11031 0cc0 11032 1c1 11033 + caddc 11035 · cmul 11037 +∞cpnf 11170 ℝ^*cxr 11172 < clt 11173 − cmin 11371 -cneg 11372 / cdiv 11801 ℕcn 12168 2c2 12230 (,)cioo 13292 [,]cicc 13295 ...cfz 13455 ..^cfzo 13602 sincsin 16022 πcpi 16025 TopOpenctopn 17378 ℂ_fldccnfld 21347 –cn→ccncf 24856 lim_ℂ climc 25842 D cdv 25843

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13296 df-ioc 13297 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-mod 13823 df-seq 13958 df-exp 14018 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15023 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-limsup 15427 df-clim 15444 df-rlim 15445 df-sum 15643 df-ef 16026 df-sin 16028 df-cos 16029 df-pi 16031 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-rest 17379 df-topn 17380 df-0g 17398 df-gsum 17399 df-topgen 17400 df-pt 17401 df-prds 17404 df-xrs 17460 df-qtop 17465 df-imas 17466 df-xps 17468 df-mre 17542 df-mrc 17543 df-acs 17545 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-mulg 19038 df-cntz 19286 df-cmn 19751 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-cld 22997 df-ntr 22998 df-cls 22999 df-nei 23076 df-lp 23114 df-perf 23115 df-cn 23205 df-cnp 23206 df-t1 23292 df-haus 23293 df-cmp 23365 df-tx 23540 df-hmeo 23733 df-fil 23824 df-fm 23916 df-flim 23917 df-flf 23918 df-xms 24298 df-ms 24299 df-tms 24300 df-cncf 24858 df-limc 25846 df-dv 25847

This theorem is referenced by: (None)

W3C validator