fourierdlem72 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem72 46183

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 7423	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ V
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ V)
4		fourierdlem72.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
5	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐹:ℝ⟶ℝ)
6		fourierdlem72.xre	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ ℝ)
7	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℝ)
8		elioore 13343	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
9	8	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
10	7, 9	readdcld 11210	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
11	5, 10	ffvelcdmd 7060	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
12		fourierdlem72.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
13	12	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ)
14	11, 13	resubcld 11613	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℝ)
15		ioossicc 13401	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
16	15	sseli 3945	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ (𝐴[,]𝐵))
17	16	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → 𝑠 ∈ (𝐴[,]𝐵))
18		id 22	. . . . . . . . . . . . 13 ⊢ (𝑠 ≠ 0 → 𝑠 ≠ 0)
19	18	necon1bi 2954	. . . . . . . . . . . 12 ⊢ (¬ 𝑠 ≠ 0 → 𝑠 = 0)
20	19	eleq1d 2814	. . . . . . . . . . 11 ⊢ (¬ 𝑠 ≠ 0 → (𝑠 ∈ (𝐴[,]𝐵) ↔ 0 ∈ (𝐴[,]𝐵)))
21	20	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → (𝑠 ∈ (𝐴[,]𝐵) ↔ 0 ∈ (𝐴[,]𝐵)))
22	17, 21	mpbid 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → 0 ∈ (𝐴[,]𝐵))
23		fourierdlem72.n0	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
24	23	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → ¬ 0 ∈ (𝐴[,]𝐵))
25	22, 24	condan 817	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)
26	14, 9, 25	redivcld 12017	. . . . . . 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 7089	. . . . . 6 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℝ)
29	28	ffvelcdmda 7059	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐻‘𝑠) ∈ ℝ)
30		2re 12267	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
32	9	rehalfcld 12436	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℝ)
33	32	resincld 16118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℝ)
34	31, 33	remulcld 11211	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
35		2cnd 12271	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
36	9	recnd 11209	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
37	36	halfcld 12434	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
38	37	sincld 16105	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
39		2ne0 12297	. . . . . . . . . 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 3949	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (-π[,]π))
43		fourierdlem44 46156	. . . . . . . . . 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 11837	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
46	9, 34, 45	redivcld 12017	. . . . . . 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 7089	. . . . . 6 ⊢ (𝜑 → 𝐾:(𝐴(,)𝐵)⟶ℝ)
49	48	ffvelcdmda 7059	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐾‘𝑠) ∈ ℝ)
50	28	feqmptd 6932	. . . . 5 ⊢ (𝜑 → 𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐻‘𝑠)))
51	48	feqmptd 6932	. . . . 5 ⊢ (𝜑 → 𝐾 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠)))
52	3, 29, 49, 50, 51	offval2 7676	. . . 4 ⊢ (𝜑 → (𝐻 ∘_f · 𝐾) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))))
53	1, 52	eqtr4id 2784	. . 3 ⊢ (𝜑 → 𝑂 = (𝐻 ∘_f · 𝐾))
54	53	oveq2d 7406	. 2 ⊢ (𝜑 → (ℝ D 𝑂) = (ℝ D (𝐻 ∘_f · 𝐾)))
55		reelprrecn 11167	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57	11	recnd 11209	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
58	12	recnd 11209	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
59	58	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℂ)
60	57, 59	subcld 11540	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
61		ioossre 13375	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
62	61	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
63	62	sselda 3949	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
64	63	recnd 11209	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
65	60, 64, 25	divcld 11965	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) ∈ ℂ)
66	65, 27	fmptd 7089	. . 3 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℂ)
67	64	halfcld 12434	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
68	67	sincld 16105	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
69	35, 68	mulcld 11201	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
70	64, 69, 45	divcld 11965	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
71	70, 47	fmptd 7089	. . 3 ⊢ (𝜑 → 𝐾:(𝐴(,)𝐵)⟶ℂ)
72		ax-resscn 11132	. . . . . 6 ⊢ ℝ ⊆ ℂ
73	72	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
74		ssid 3972	. . . . . 6 ⊢ ℂ ⊆ ℂ
75	74	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
76		cncfss 24799	. . . . 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 3679	. . . . 5 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
81	4, 73	fssd 6708	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
82		ssid 3972	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ
83	82	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ)
84		ioossre 13375	. . . . . . . . 9 ⊢ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℝ
85	84	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℝ)
86		eqid 2730	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
87		tgioo4 24700	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
88	86, 87	dvres 25819	. . . . . . . 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 838	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))))
90		ioontr 45516	. . . . . . . 8 ⊢ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))
91	90	reseq2i 5950	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))
92	89, 91	eqtrdi 2781	. . . . . 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 46114	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑_m (0...𝑀)))
100		elmapi 8825	. . . . . . . . . . . . 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 13643	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (0..^𝑀) → 𝑈 ∈ (0...𝑀))
104	102, 103	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ (0...𝑀))
105	101, 104	ffvelcdmd 7060	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘𝑈) ∈ ℝ)
106	105	rexrd 11231	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘𝑈) ∈ ℝ^*)
107		fzofzp1 13732	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (0..^𝑀) → (𝑈 + 1) ∈ (0...𝑀))
108	102, 107	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈 + 1) ∈ (0...𝑀))
109	101, 108	ffvelcdmd 7060	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) ∈ ℝ)
110	109	rexrd 11231	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) ∈ ℝ^*)
111		pire 26373	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℝ
112	111	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → π ∈ ℝ)
113	112	renegcld 11612	. . . . . . . . . . . . 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 46125	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝑈) = ((𝑉‘𝑈) − 𝑋) ∧ (𝑉‘𝑈) = (𝑋 + (𝑄‘𝑈))))
116	115	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘𝑈) = (𝑋 + (𝑄‘𝑈)))
117	115	simpld 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝑈) = ((𝑉‘𝑈) − 𝑋))
118	105, 6	resubcld 11613	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑉‘𝑈) − 𝑋) ∈ ℝ)
119	117, 118	eqeltrd 2829	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘𝑈) ∈ ℝ)
120	113, 112, 6, 95, 94, 93, 108, 114	fourierdlem13 46125	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑄‘(𝑈 + 1)) = ((𝑉‘(𝑈 + 1)) − 𝑋) ∧ (𝑉‘(𝑈 + 1)) = (𝑋 + (𝑄‘(𝑈 + 1)))))
121	120	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄‘(𝑈 + 1)) = ((𝑉‘(𝑈 + 1)) − 𝑋))
122	109, 6	resubcld 11613	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑉‘(𝑈 + 1)) − 𝑋) ∈ ℝ)
123	121, 122	eqeltrd 2829	. . . . . . . . . . . . . 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 46122	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝑈) ≤ 𝐴 ∧ 𝐵 ≤ (𝑄‘(𝑈 + 1))))
127	126	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘𝑈) ≤ 𝐴)
128	119, 78, 6, 127	leadd2dd 11800	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + (𝑄‘𝑈)) ≤ (𝑋 + 𝐴))
129	116, 128	eqbrtrd 5132	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘𝑈) ≤ (𝑋 + 𝐴))
130	126	simprd 495	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≤ (𝑄‘(𝑈 + 1)))
131	79, 123, 6, 130	leadd2dd 11800	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + 𝐵) ≤ (𝑋 + (𝑄‘(𝑈 + 1))))
132	120	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) = (𝑋 + (𝑄‘(𝑈 + 1))))
133	131, 132	breqtrrd 5138	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝐵) ≤ (𝑉‘(𝑈 + 1)))
134		ioossioo 13409	. . . . . . . . . 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 838	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1))))
136	135	resabs1d 5982	. . . . . . . 8 ⊢ (𝜑 → (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
137	136	eqcomd 2736	. . . . . . 7 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
138	102	ancli 548	. . . . . . . . 9 ⊢ (𝜑 → (𝜑 ∧ 𝑈 ∈ (0..^𝑀)))
139		eleq1 2817	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → (𝑖 ∈ (0..^𝑀) ↔ 𝑈 ∈ (0..^𝑀)))
140	139	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑈 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑈 ∈ (0..^𝑀))))
141		fveq2 6861	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑈 → (𝑉‘𝑖) = (𝑉‘𝑈))
142		oveq1 7397	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑈 → (𝑖 + 1) = (𝑈 + 1))
143	142	fveq2d 6865	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑈 → (𝑉‘(𝑖 + 1)) = (𝑉‘(𝑈 + 1)))
144	141, 143	oveq12d 7408	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑈 → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) = ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1))))
145	144	reseq2d 5953	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))))
146	144	oveq1d 7405	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ) = (((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))–cn→ℝ))
147	145, 146	eleq12d 2823	. . . . . . . . . . 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 344	. . . . . . . . . 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 3523	. . . . . . . . 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 24797	. . . . . . . 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 2829	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))
155	92, 154	eqeltrd 2829	. . . . 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 46170	. . . 4 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ))
157	77, 156	sseldd 3950	. . 3 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ))
158		iooretop 24660	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
159	158	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
160	47, 41, 80, 159	fourierdlem58 46169	. . . 4 ⊢ (𝜑 → (ℝ D 𝐾) ∈ ((𝐴(,)𝐵)–cn→ℝ))
161	77, 160	sseldd 3950	. . 3 ⊢ (𝜑 → (ℝ D 𝐾) ∈ ((𝐴(,)𝐵)–cn→ℂ))
162	56, 66, 71, 157, 161	dvmulcncf 45930	. 2 ⊢ (𝜑 → (ℝ D (𝐻 ∘_f · 𝐾)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
163	54, 162	eqeltrd 2829	1 ⊢ (𝜑 → (ℝ D 𝑂) ∈ ((𝐴(,)𝐵)–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 {crab 3408 Vcvv 3450 ⊆ wss 3917 {cpr 4594 class class class wbr 5110 ↦ cmpt 5191 ran crn 5642 ↾ cres 5643 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ∘_f cof 7654 ↑_m cmap 8802 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 − cmin 11412 -cneg 11413 / cdiv 11842 ℕcn 12193 2c2 12248 (,)cioo 13313 [,]cicc 13316 ...cfz 13475 ..^cfzo 13622 sincsin 16036 πcpi 16039 TopOpenctopn 17391 topGenctg 17407 ℂ_fldccnfld 21271 intcnt 22911 –cn→ccncf 24776 D cdv 25771

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660 df-ef 16040 df-sin 16042 df-cos 16043 df-pi 16045 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-t1 23208 df-haus 23209 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-limc 25774 df-dv 25775

This theorem is referenced by: fourierdlem103 46214 fourierdlem104 46215

W3C validator