fourierdlem72 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem72 43719

Description: The derivative of 𝑂 is continuous on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem72.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem72.xre	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem72.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)))})
fourierdlem72.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem72.v	⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
fourierdlem72.dvcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ))
fourierdlem72.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem72.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem72.altb	⊢ (𝜑 → 𝐴 < 𝐵)
fourierdlem72.ab	⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))
fourierdlem72.n0	⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
fourierdlem72.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem72.q	⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
fourierdlem72.u	⊢ (𝜑 → 𝑈 ∈ (0..^𝑀))
fourierdlem72.abss	⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1))))
fourierdlem72.h	⊢ 𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠))
fourierdlem72.k	⊢ 𝐾 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝑠 / (2 · (sin‘(𝑠 / 2)))))
fourierdlem72.o	⊢ 𝑂 = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))

**Assertion**
Ref	Expression
fourierdlem72	⊢ (𝜑 → (ℝ D 𝑂) ∈ ((𝐴(,)𝐵)–cn→ℂ))

Distinct variable groups: 𝐴,𝑠 𝐵,𝑠 𝐶,𝑠 𝑖,𝐹 𝐹,𝑠 𝐻,𝑠 𝐾,𝑠 𝑖,𝑀,𝑚,𝑝 𝑈,𝑖 𝑖,𝑉,𝑝 𝑖,𝑋,𝑚,𝑝 𝑋,𝑠 𝜑,𝑖 𝜑,𝑠 Allowed substitution hints: 𝜑(𝑚,𝑝) 𝐴(𝑖,𝑚,𝑝) 𝐵(𝑖,𝑚,𝑝) 𝐶(𝑖,𝑚,𝑝) 𝑃(𝑖,𝑚,𝑠,𝑝) 𝑄(𝑖,𝑚,𝑠,𝑝) 𝑈(𝑚,𝑠,𝑝) 𝐹(𝑚,𝑝) 𝐻(𝑖,𝑚,𝑝) 𝐾(𝑖,𝑚,𝑝) 𝑀(𝑠) 𝑂(𝑖,𝑚,𝑠,𝑝) 𝑉(𝑚,𝑠)

**Proof of Theorem fourierdlem72**
Step	Hyp	Ref	Expression
1		fourierdlem72.o	. . . 4 ⊢ 𝑂 = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
2		ovex 7308	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ V
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ V)
4		fourierdlem72.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
5	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐹:ℝ⟶ℝ)
6		fourierdlem72.xre	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ ℝ)
7	6	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℝ)
8		elioore 13109	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
9	8	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
10	7, 9	readdcld 11004	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
11	5, 10	ffvelrnd 6962	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
12		fourierdlem72.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
13	12	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ)
14	11, 13	resubcld 11403	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℝ)
15		ioossicc 13165	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
16	15	sseli 3917	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ (𝐴[,]𝐵))
17	16	ad2antlr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → 𝑠 ∈ (𝐴[,]𝐵))
18		id 22	. . . . . . . . . . . . 13 ⊢ (𝑠 ≠ 0 → 𝑠 ≠ 0)
19	18	necon1bi 2972	. . . . . . . . . . . 12 ⊢ (¬ 𝑠 ≠ 0 → 𝑠 = 0)
20	19	eleq1d 2823	. . . . . . . . . . 11 ⊢ (¬ 𝑠 ≠ 0 → (𝑠 ∈ (𝐴[,]𝐵) ↔ 0 ∈ (𝐴[,]𝐵)))
21	20	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → (𝑠 ∈ (𝐴[,]𝐵) ↔ 0 ∈ (𝐴[,]𝐵)))
22	17, 21	mpbid 231	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → 0 ∈ (𝐴[,]𝐵))
23		fourierdlem72.n0	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
24	23	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → ¬ 0 ∈ (𝐴[,]𝐵))
25	22, 24	condan 815	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)
26	14, 9, 25	redivcld 11803	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) ∈ ℝ)
27		fourierdlem72.h	. . . . . . 7 ⊢ 𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠))
28	26, 27	fmptd 6988	. . . . . 6 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℝ)
29	28	ffvelrnda 6961	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐻‘𝑠) ∈ ℝ)
30		2re 12047	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
32	9	rehalfcld 12220	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℝ)
33	32	resincld 15852	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℝ)
34	31, 33	remulcld 11005	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
35		2cnd 12051	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
36	9	recnd 11003	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
37	36	halfcld 12218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
38	37	sincld 15839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
39		2ne0 12077	. . . . . . . . . 10 ⊢ 2 ≠ 0
40	39	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
41		fourierdlem72.ab	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))
42	41	sselda 3921	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (-π[,]π))
43		fourierdlem44 43692	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
44	42, 25, 43	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
45	35, 38, 40, 44	mulne0d 11627	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
46	9, 34, 45	redivcld 11803	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ ℝ)
47		fourierdlem72.k	. . . . . . 7 ⊢ 𝐾 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝑠 / (2 · (sin‘(𝑠 / 2)))))
48	46, 47	fmptd 6988	. . . . . 6 ⊢ (𝜑 → 𝐾:(𝐴(,)𝐵)⟶ℝ)
49	48	ffvelrnda 6961	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐾‘𝑠) ∈ ℝ)
50	28	feqmptd 6837	. . . . 5 ⊢ (𝜑 → 𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐻‘𝑠)))
51	48	feqmptd 6837	. . . . 5 ⊢ (𝜑 → 𝐾 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠)))
52	3, 29, 49, 50, 51	offval2 7553	. . . 4 ⊢ (𝜑 → (𝐻 ∘_f · 𝐾) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))))
53	1, 52	eqtr4id 2797	. . 3 ⊢ (𝜑 → 𝑂 = (𝐻 ∘_f · 𝐾))
54	53	oveq2d 7291	. 2 ⊢ (𝜑 → (ℝ D 𝑂) = (ℝ D (𝐻 ∘_f · 𝐾)))
55		reelprrecn 10963	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57	11	recnd 11003	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
58	12	recnd 11003	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
59	58	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℂ)
60	57, 59	subcld 11332	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
61		ioossre 13140	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
62	61	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
63	62	sselda 3921	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
64	63	recnd 11003	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
65	60, 64, 25	divcld 11751	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) ∈ ℂ)
66	65, 27	fmptd 6988	. . 3 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℂ)
67	64	halfcld 12218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
68	67	sincld 15839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
69	35, 68	mulcld 10995	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
70	64, 69, 45	divcld 11751	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
71	70, 47	fmptd 6988	. . 3 ⊢ (𝜑 → 𝐾:(𝐴(,)𝐵)⟶ℂ)
72		ax-resscn 10928	. . . . . 6 ⊢ ℝ ⊆ ℂ
73	72	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
74		ssid 3943	. . . . . 6 ⊢ ℂ ⊆ ℂ
75	74	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
76		cncfss 24062	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℝ) ⊆ ((𝐴(,)𝐵)–cn→ℂ))
77	73, 75, 76	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐴(,)𝐵)–cn→ℝ) ⊆ ((𝐴(,)𝐵)–cn→ℂ))
78		fourierdlem72.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
79		fourierdlem72.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
80	25	nelrdva 3640	. . . . 5 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
81	4, 73	fssd 6618	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
82		ssid 3943	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ
83	82	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ)
84		ioossre 13140	. . . . . . . . 9 ⊢ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℝ
85	84	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℝ)
86		eqid 2738	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
87	86	tgioo2 23966	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
88	86, 87	dvres 25075	. . . . . . . 8 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:ℝ⟶ℂ) ∧ (ℝ ⊆ ℝ ∧ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℝ)) → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))))
89	73, 81, 83, 85, 88	syl22anc 836	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))))
90		ioontr 43049	. . . . . . . 8 ⊢ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))
91	90	reseq2i 5888	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))
92	89, 91	eqtrdi 2794	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
93		fourierdlem72.v	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
94		fourierdlem72.m	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℕ)
95		fourierdlem72.p	. . . . . . . . . . . . . . . . 17 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
96	95	fourierdlem2 43650	. . . . . . . . . . . . . . . 16 ⊢ (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))))))
97	94, 96	syl 17	. . . . . . . . . . . . . . 15 ⊢ (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))))))
98	93, 97	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑉 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉‘𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))))
99	98	simpld 495	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑_m (0...𝑀)))
100		elmapi 8637	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ (ℝ ↑_m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
101	99, 100	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶ℝ)
102		fourierdlem72.u	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ (0..^𝑀))
103		elfzofz 13403	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (0..^𝑀) → 𝑈 ∈ (0...𝑀))
104	102, 103	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ (0...𝑀))
105	101, 104	ffvelrnd 6962	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘𝑈) ∈ ℝ)
106	105	rexrd 11025	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘𝑈) ∈ ℝ^*)
107		fzofzp1 13484	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (0..^𝑀) → (𝑈 + 1) ∈ (0...𝑀))
108	102, 107	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈 + 1) ∈ (0...𝑀))
109	101, 108	ffvelrnd 6962	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) ∈ ℝ)
110	109	rexrd 11025	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) ∈ ℝ^*)
111		pire 25615	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℝ
112	111	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → π ∈ ℝ)
113	112	renegcld 11402	. . . . . . . . . . . . 13 ⊢ (𝜑 → -π ∈ ℝ)
114		fourierdlem72.q	. . . . . . . . . . . . 13 ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
115	113, 112, 6, 95, 94, 93, 104, 114	fourierdlem13 43661	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝑈) = ((𝑉‘𝑈) − 𝑋) ∧ (𝑉‘𝑈) = (𝑋 + (𝑄‘𝑈))))
116	115	simprd 496	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘𝑈) = (𝑋 + (𝑄‘𝑈)))
117	115	simpld 495	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝑈) = ((𝑉‘𝑈) − 𝑋))
118	105, 6	resubcld 11403	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑉‘𝑈) − 𝑋) ∈ ℝ)
119	117, 118	eqeltrd 2839	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘𝑈) ∈ ℝ)
120	113, 112, 6, 95, 94, 93, 108, 114	fourierdlem13 43661	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑄‘(𝑈 + 1)) = ((𝑉‘(𝑈 + 1)) − 𝑋) ∧ (𝑉‘(𝑈 + 1)) = (𝑋 + (𝑄‘(𝑈 + 1)))))
121	120	simpld 495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄‘(𝑈 + 1)) = ((𝑉‘(𝑈 + 1)) − 𝑋))
122	109, 6	resubcld 11403	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑉‘(𝑈 + 1)) − 𝑋) ∈ ℝ)
123	121, 122	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘(𝑈 + 1)) ∈ ℝ)
124		fourierdlem72.altb	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 < 𝐵)
125		fourierdlem72.abss	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1))))
126	119, 123, 78, 79, 124, 125	fourierdlem10 43658	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝑈) ≤ 𝐴 ∧ 𝐵 ≤ (𝑄‘(𝑈 + 1))))
127	126	simpld 495	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘𝑈) ≤ 𝐴)
128	119, 78, 6, 127	leadd2dd 11590	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + (𝑄‘𝑈)) ≤ (𝑋 + 𝐴))
129	116, 128	eqbrtrd 5096	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘𝑈) ≤ (𝑋 + 𝐴))
130	126	simprd 496	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≤ (𝑄‘(𝑈 + 1)))
131	79, 123, 6, 130	leadd2dd 11590	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + 𝐵) ≤ (𝑋 + (𝑄‘(𝑈 + 1))))
132	120	simprd 496	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) = (𝑋 + (𝑄‘(𝑈 + 1))))
133	131, 132	breqtrrd 5102	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝐵) ≤ (𝑉‘(𝑈 + 1)))
134		ioossioo 13173	. . . . . . . . . 10 ⊢ ((((𝑉‘𝑈) ∈ ℝ^* ∧ (𝑉‘(𝑈 + 1)) ∈ ℝ^*) ∧ ((𝑉‘𝑈) ≤ (𝑋 + 𝐴) ∧ (𝑋 + 𝐵) ≤ (𝑉‘(𝑈 + 1)))) → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1))))
135	106, 110, 129, 133, 134	syl22anc 836	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1))))
136	135	resabs1d 5922	. . . . . . . 8 ⊢ (𝜑 → (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
137	136	eqcomd 2744	. . . . . . 7 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
138	102	ancli 549	. . . . . . . . 9 ⊢ (𝜑 → (𝜑 ∧ 𝑈 ∈ (0..^𝑀)))
139		eleq1 2826	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → (𝑖 ∈ (0..^𝑀) ↔ 𝑈 ∈ (0..^𝑀)))
140	139	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑈 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑈 ∈ (0..^𝑀))))
141		fveq2 6774	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑈 → (𝑉‘𝑖) = (𝑉‘𝑈))
142		oveq1 7282	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑈 → (𝑖 + 1) = (𝑈 + 1))
143	142	fveq2d 6778	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑈 → (𝑉‘(𝑖 + 1)) = (𝑉‘(𝑈 + 1)))
144	141, 143	oveq12d 7293	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑈 → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) = ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1))))
145	144	reseq2d 5891	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))))
146	144	oveq1d 7290	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ) = (((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))–cn→ℝ))
147	145, 146	eleq12d 2833	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑈 → (((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ) ↔ ((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ∈ (((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))–cn→ℝ)))
148	140, 147	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑖 = 𝑈 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ)) ↔ ((𝜑 ∧ 𝑈 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ∈ (((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))–cn→ℝ))))
149		fourierdlem72.dvcn	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ))
150	148, 149	vtoclg 3505	. . . . . . . . 9 ⊢ (𝑈 ∈ (0..^𝑀) → ((𝜑 ∧ 𝑈 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ∈ (((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))–cn→ℝ)))
151	102, 138, 150	sylc 65	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ∈ (((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))–cn→ℝ))
152		rescncf 24060	. . . . . . . 8 ⊢ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1))) → (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ∈ (((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))–cn→ℝ) → (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ)))
153	135, 151, 152	sylc 65	. . . . . . 7 ⊢ (𝜑 → (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))
154	137, 153	eqeltrd 2839	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))
155	92, 154	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))
156	4, 6, 78, 79, 80, 155, 12, 27	fourierdlem59 43706	. . . 4 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ))
157	77, 156	sseldd 3922	. . 3 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ))
158		iooretop 23929	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
159	158	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
160	47, 41, 80, 159	fourierdlem58 43705	. . . 4 ⊢ (𝜑 → (ℝ D 𝐾) ∈ ((𝐴(,)𝐵)–cn→ℝ))
161	77, 160	sseldd 3922	. . 3 ⊢ (𝜑 → (ℝ D 𝐾) ∈ ((𝐴(,)𝐵)–cn→ℂ))
162	56, 66, 71, 157, 161	dvmulcncf 43466	. 2 ⊢ (𝜑 → (ℝ D (𝐻 ∘_f · 𝐾)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
163	54, 162	eqeltrd 2839	1 ⊢ (𝜑 → (ℝ D 𝑂) ∈ ((𝐴(,)𝐵)–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 {crab 3068 Vcvv 3432 ⊆ wss 3887 {cpr 4563 class class class wbr 5074 ↦ cmpt 5157 ran crn 5590 ↾ cres 5591 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∘_f cof 7531 ↑_m cmap 8615 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 ℕcn 11973 2c2 12028 (,)cioo 13079 [,]cicc 13082 ...cfz 13239 ..^cfzo 13382 sincsin 15773 πcpi 15776 TopOpenctopn 17132 topGenctg 17148 ℂ_fldccnfld 20597 intcnt 22168 –cn→ccncf 24039 D cdv 25027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-t1 22465 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031

This theorem is referenced by: fourierdlem103 43750 fourierdlem104 43751

W3C validator