fourierdlem72 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem72 43609

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 7288	. . . . . 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 13038	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
9	8	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
10	7, 9	readdcld 10935	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
11	5, 10	ffvelrnd 6944	. . . . . . . . 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 11333	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℝ)
15		ioossicc 13094	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
16	15	sseli 3913	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ (𝐴[,]𝐵))
17	16	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → 𝑠 ∈ (𝐴[,]𝐵))
18		id 22	. . . . . . . . . . . . 13 ⊢ (𝑠 ≠ 0 → 𝑠 ≠ 0)
19	18	necon1bi 2971	. . . . . . . . . . . 12 ⊢ (¬ 𝑠 ≠ 0 → 𝑠 = 0)
20	19	eleq1d 2823	. . . . . . . . . . 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 231	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → 0 ∈ (𝐴[,]𝐵))
23		fourierdlem72.n0	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
24	23	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → ¬ 0 ∈ (𝐴[,]𝐵))
25	22, 24	condan 814	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)
26	14, 9, 25	redivcld 11733	. . . . . . 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 6970	. . . . . 6 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℝ)
29	28	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐻‘𝑠) ∈ ℝ)
30		2re 11977	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
32	9	rehalfcld 12150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℝ)
33	32	resincld 15780	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℝ)
34	31, 33	remulcld 10936	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
35		2cnd 11981	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
36	9	recnd 10934	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
37	36	halfcld 12148	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
38	37	sincld 15767	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
39		2ne0 12007	. . . . . . . . . 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 3917	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (-π[,]π))
43		fourierdlem44 43582	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
44	42, 25, 43	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
45	35, 38, 40, 44	mulne0d 11557	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
46	9, 34, 45	redivcld 11733	. . . . . . 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 6970	. . . . . 6 ⊢ (𝜑 → 𝐾:(𝐴(,)𝐵)⟶ℝ)
49	48	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐾‘𝑠) ∈ ℝ)
50	28	feqmptd 6819	. . . . 5 ⊢ (𝜑 → 𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐻‘𝑠)))
51	48	feqmptd 6819	. . . . 5 ⊢ (𝜑 → 𝐾 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠)))
52	3, 29, 49, 50, 51	offval2 7531	. . . 4 ⊢ (𝜑 → (𝐻 ∘_f · 𝐾) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))))
53	1, 52	eqtr4id 2798	. . 3 ⊢ (𝜑 → 𝑂 = (𝐻 ∘_f · 𝐾))
54	53	oveq2d 7271	. 2 ⊢ (𝜑 → (ℝ D 𝑂) = (ℝ D (𝐻 ∘_f · 𝐾)))
55		reelprrecn 10894	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57	11	recnd 10934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
58	12	recnd 10934	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
59	58	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℂ)
60	57, 59	subcld 11262	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
61		ioossre 13069	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
62	61	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
63	62	sselda 3917	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
64	63	recnd 10934	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
65	60, 64, 25	divcld 11681	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) ∈ ℂ)
66	65, 27	fmptd 6970	. . 3 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℂ)
67	64	halfcld 12148	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
68	67	sincld 15767	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
69	35, 68	mulcld 10926	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
70	64, 69, 45	divcld 11681	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
71	70, 47	fmptd 6970	. . 3 ⊢ (𝜑 → 𝐾:(𝐴(,)𝐵)⟶ℂ)
72		ax-resscn 10859	. . . . . 6 ⊢ ℝ ⊆ ℂ
73	72	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
74		ssid 3939	. . . . . 6 ⊢ ℂ ⊆ ℂ
75	74	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
76		cncfss 23968	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℝ) ⊆ ((𝐴(,)𝐵)–cn→ℂ))
77	73, 75, 76	syl2anc 583	. . . 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 3635	. . . . 5 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
81	4, 73	fssd 6602	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
82		ssid 3939	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ
83	82	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ)
84		ioossre 13069	. . . . . . . . 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 23872	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
88	86, 87	dvres 24980	. . . . . . . 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 835	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))))
90		ioontr 42939	. . . . . . . 8 ⊢ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))
91	90	reseq2i 5877	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))
92	89, 91	eqtrdi 2795	. . . . . 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 43540	. . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑_m (0...𝑀)))
100		elmapi 8595	. . . . . . . . . . . . 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 13331	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (0..^𝑀) → 𝑈 ∈ (0...𝑀))
104	102, 103	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ (0...𝑀))
105	101, 104	ffvelrnd 6944	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘𝑈) ∈ ℝ)
106	105	rexrd 10956	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘𝑈) ∈ ℝ^*)
107		fzofzp1 13412	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (0..^𝑀) → (𝑈 + 1) ∈ (0...𝑀))
108	102, 107	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈 + 1) ∈ (0...𝑀))
109	101, 108	ffvelrnd 6944	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) ∈ ℝ)
110	109	rexrd 10956	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) ∈ ℝ^*)
111		pire 25520	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℝ
112	111	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → π ∈ ℝ)
113	112	renegcld 11332	. . . . . . . . . . . . 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 43551	. . . . . . . . . . . 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 11333	. . . . . . . . . . . . 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 43551	. . . . . . . . . . . . . . . 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 11333	. . . . . . . . . . . . . . 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 43548	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝑈) ≤ 𝐴 ∧ 𝐵 ≤ (𝑄‘(𝑈 + 1))))
127	126	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘𝑈) ≤ 𝐴)
128	119, 78, 6, 127	leadd2dd 11520	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + (𝑄‘𝑈)) ≤ (𝑋 + 𝐴))
129	116, 128	eqbrtrd 5092	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘𝑈) ≤ (𝑋 + 𝐴))
130	126	simprd 495	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≤ (𝑄‘(𝑈 + 1)))
131	79, 123, 6, 130	leadd2dd 11520	. . . . . . . . . . 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 5098	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝐵) ≤ (𝑉‘(𝑈 + 1)))
134		ioossioo 13102	. . . . . . . . . 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 835	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1))))
136	135	resabs1d 5911	. . . . . . . 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 548	. . . . . . . . 9 ⊢ (𝜑 → (𝜑 ∧ 𝑈 ∈ (0..^𝑀)))
139		eleq1 2826	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → (𝑖 ∈ (0..^𝑀) ↔ 𝑈 ∈ (0..^𝑀)))
140	139	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑈 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑈 ∈ (0..^𝑀))))
141		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑈 → (𝑉‘𝑖) = (𝑉‘𝑈))
142		oveq1 7262	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑈 → (𝑖 + 1) = (𝑈 + 1))
143	142	fveq2d 6760	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑈 → (𝑉‘(𝑖 + 1)) = (𝑉‘(𝑈 + 1)))
144	141, 143	oveq12d 7273	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑈 → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) = ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1))))
145	144	reseq2d 5880	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))))
146	144	oveq1d 7270	. . . . . . . . . . . 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 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 3495	. . . . . . . . 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 23966	. . . . . . . 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 43596	. . . 4 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ))
157	77, 156	sseldd 3918	. . 3 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ))
158		iooretop 23835	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
159	158	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
160	47, 41, 80, 159	fourierdlem58 43595	. . . 4 ⊢ (𝜑 → (ℝ D 𝐾) ∈ ((𝐴(,)𝐵)–cn→ℝ))
161	77, 160	sseldd 3918	. . 3 ⊢ (𝜑 → (ℝ D 𝐾) ∈ ((𝐴(,)𝐵)–cn→ℂ))
162	56, 66, 71, 157, 161	dvmulcncf 43356	. 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 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 {crab 3067 Vcvv 3422 ⊆ wss 3883 {cpr 4560 class class class wbr 5070 ↦ cmpt 5153 ran crn 5581 ↾ cres 5582 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∘_f cof 7509 ↑_m cmap 8573 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 (,)cioo 13008 [,]cicc 13011 ...cfz 13168 ..^cfzo 13311 sincsin 15701 πcpi 15704 TopOpenctopn 17049 topGenctg 17065 ℂ_fldccnfld 20510 intcnt 22076 –cn→ccncf 23945 D cdv 24932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-t1 22373 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936

This theorem is referenced by: fourierdlem103 43640 fourierdlem104 43641

W3C validator