fourierdlem72 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem72 46875

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 7445	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ V
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ V)
4		fourierdlem72.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
5	4	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐹:ℝ⟶ℝ)
6		fourierdlem72.xre	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ ℝ)
7	6	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℝ)
8		elioore 13403	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
9	8	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
10	7, 9	readdcld 11239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
11	5, 10	ffvelcdmd 7082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
12		fourierdlem72.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
13	12	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ)
14	11, 13	resubcld 11643	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℝ)
15		ioossicc 13461	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
16	15	sseli 3934	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ (𝐴[,]𝐵))
17	16	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → 𝑠 ∈ (𝐴[,]𝐵))
18		id 23	. . . . . . . . . . . . 13 ⊢ (𝑠 ≠ 0 → 𝑠 ≠ 0)
19	18	necon1bi 2986	. . . . . . . . . . . 12 ⊢ (¬ 𝑠 ≠ 0 → 𝑠 = 0)
20	19	eleq1d 2848	. . . . . . . . . . 11 ⊢ (¬ 𝑠 ≠ 0 → (𝑠 ∈ (𝐴[,]𝐵) ↔ 0 ∈ (𝐴[,]𝐵)))
21	20	adantl 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → (𝑠 ∈ (𝐴[,]𝐵) ↔ 0 ∈ (𝐴[,]𝐵)))
22	17, 21	mpbid 235	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → 0 ∈ (𝐴[,]𝐵))
23		fourierdlem72.n0	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
24	23	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → ¬ 0 ∈ (𝐴[,]𝐵))
25	22, 24	condan 829	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)
26	14, 9, 25	redivcld 12044	. . . . . . 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 7111	. . . . . 6 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℝ)
29	28	ffvelcdmda 7081	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐻‘𝑠) ∈ ℝ)
30		2re 12316	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
32	9	rehalfcld 12492	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℝ)
33	32	resincld 16200	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℝ)
34	31, 33	remulcld 11240	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
35		2cnd 12320	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
36	9	recnd 11238	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
37	36	halfcld 12490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
38	37	sincld 16187	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
39		2ne0 12348	. . . . . . . . . 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 3938	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (-π[,]π))
43		fourierdlem44 46848	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
44	42, 25, 43	syl2anc 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
45	35, 38, 40, 44	mulne0d 11867	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
46	9, 34, 45	redivcld 12044	. . . . . . 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 7111	. . . . . 6 ⊢ (𝜑 → 𝐾:(𝐴(,)𝐵)⟶ℝ)
49	48	ffvelcdmda 7081	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐾‘𝑠) ∈ ℝ)
50	28	feqmptd 6951	. . . . 5 ⊢ (𝜑 → 𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐻‘𝑠)))
51	48	feqmptd 6951	. . . . 5 ⊢ (𝜑 → 𝐾 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠)))
52	3, 29, 49, 50, 51	offval2 7696	. . . 4 ⊢ (𝜑 → (𝐻 ∘_f · 𝐾) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))))
53	1, 52	eqtr4id 2817	. . 3 ⊢ (𝜑 → 𝑂 = (𝐻 ∘_f · 𝐾))
54	53	oveq2d 7428	. 2 ⊢ (𝜑 → (ℝ D 𝑂) = (ℝ D (𝐻 ∘_f · 𝐾)))
55		reelprrecn 11193	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57	11	recnd 11238	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
58	12	recnd 11238	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
59	58	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℂ)
60	57, 59	subcld 11570	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
61		ioossre 13435	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
62	61	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
63	62	sselda 3938	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
64	63	recnd 11238	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
65	60, 64, 25	divcld 11992	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) ∈ ℂ)
66	65, 27	fmptd 7111	. . 3 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℂ)
67	64	halfcld 12490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
68	67	sincld 16187	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
69	35, 68	mulcld 11230	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
70	64, 69, 45	divcld 11992	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
71	70, 47	fmptd 7111	. . 3 ⊢ (𝜑 → 𝐾:(𝐴(,)𝐵)⟶ℂ)
72		ax-resscn 11158	. . . . . 6 ⊢ ℝ ⊆ ℂ
73	72	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
74		ssid 3960	. . . . . 6 ⊢ ℂ ⊆ ℂ
75	74	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
76		cncfss 25039	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℝ) ⊆ ((𝐴(,)𝐵)–cn→ℂ))
77	73, 75, 76	syl2anc 595	. . . 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 3669	. . . . 5 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
81	4, 73	fssd 6725	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
82		ssid 3960	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ
83	82	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ)
84		ioossre 13435	. . . . . . . . 9 ⊢ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℝ
85	84	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℝ)
86		eqid 2763	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
87		tgioo4 24943	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
88	86, 87	dvres 26051	. . . . . . . 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 851	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))))
90		ioontr 46210	. . . . . . . 8 ⊢ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))
91	90	reseq2i 5977	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))
92	89, 91	eqtrdi 2814	. . . . . 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 46806	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . 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 499	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑_m (0...𝑀)))
100		elmapi 8847	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ (ℝ ↑_m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
101	99, 100	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶ℝ)
102		fourierdlem72.u	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ (0..^𝑀))
103		elfzofz 13706	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (0..^𝑀) → 𝑈 ∈ (0...𝑀))
104	102, 103	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ (0...𝑀))
105	101, 104	ffvelcdmd 7082	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘𝑈) ∈ ℝ)
106	105	rexrd 11260	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘𝑈) ∈ ℝ^*)
107		fzofzp1 13795	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (0..^𝑀) → (𝑈 + 1) ∈ (0...𝑀))
108	102, 107	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈 + 1) ∈ (0...𝑀))
109	101, 108	ffvelcdmd 7082	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) ∈ ℝ)
110	109	rexrd 11260	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) ∈ ℝ^*)
111		pire 26600	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℝ
112	111	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → π ∈ ℝ)
113	112	renegcld 11642	. . . . . . . . . . . . 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 46817	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝑈) = ((𝑉‘𝑈) − 𝑋) ∧ (𝑉‘𝑈) = (𝑋 + (𝑄‘𝑈))))
116	115	simprd 500	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘𝑈) = (𝑋 + (𝑄‘𝑈)))
117	115	simpld 499	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝑈) = ((𝑉‘𝑈) − 𝑋))
118	105, 6	resubcld 11643	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑉‘𝑈) − 𝑋) ∈ ℝ)
119	117, 118	eqeltrd 2863	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘𝑈) ∈ ℝ)
120	113, 112, 6, 95, 94, 93, 108, 114	fourierdlem13 46817	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑄‘(𝑈 + 1)) = ((𝑉‘(𝑈 + 1)) − 𝑋) ∧ (𝑉‘(𝑈 + 1)) = (𝑋 + (𝑄‘(𝑈 + 1)))))
121	120	simpld 499	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄‘(𝑈 + 1)) = ((𝑉‘(𝑈 + 1)) − 𝑋))
122	109, 6	resubcld 11643	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑉‘(𝑈 + 1)) − 𝑋) ∈ ℝ)
123	121, 122	eqeltrd 2863	. . . . . . . . . . . . . 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 46814	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝑈) ≤ 𝐴 ∧ 𝐵 ≤ (𝑄‘(𝑈 + 1))))
127	126	simpld 499	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘𝑈) ≤ 𝐴)
128	119, 78, 6, 127	leadd2dd 11830	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + (𝑄‘𝑈)) ≤ (𝑋 + 𝐴))
129	116, 128	eqbrtrd 5134	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘𝑈) ≤ (𝑋 + 𝐴))
130	126	simprd 500	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≤ (𝑄‘(𝑈 + 1)))
131	79, 123, 6, 130	leadd2dd 11830	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + 𝐵) ≤ (𝑋 + (𝑄‘(𝑈 + 1))))
132	120	simprd 500	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) = (𝑋 + (𝑄‘(𝑈 + 1))))
133	131, 132	breqtrrd 5140	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝐵) ≤ (𝑉‘(𝑈 + 1)))
134		ioossioo 13469	. . . . . . . . . 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 851	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1))))
136	135	resabs1d 6009	. . . . . . . 8 ⊢ (𝜑 → (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
137	136	eqcomd 2769	. . . . . . 7 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
138	102	ancli 557	. . . . . . . . 9 ⊢ (𝜑 → (𝜑 ∧ 𝑈 ∈ (0..^𝑀)))
139		eleq1 2851	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → (𝑖 ∈ (0..^𝑀) ↔ 𝑈 ∈ (0..^𝑀)))
140	139	anbi2d 641	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑈 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑈 ∈ (0..^𝑀))))
141		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑈 → (𝑉‘𝑖) = (𝑉‘𝑈))
142		oveq1 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑈 → (𝑖 + 1) = (𝑈 + 1))
143	142	fveq2d 6887	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑈 → (𝑉‘(𝑖 + 1)) = (𝑉‘(𝑈 + 1)))
144	141, 143	oveq12d 7430	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑈 → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) = ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1))))
145	144	reseq2d 5980	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))))
146	144	oveq1d 7427	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ) = (((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))–cn→ℝ))
147	145, 146	eleq12d 2857	. . . . . . . . . . 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 347	. . . . . . . . . 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 66	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ∈ (((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))–cn→ℝ))
152		rescncf 25037	. . . . . . . 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 66	. . . . . . 7 ⊢ (𝜑 → (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))
154	137, 153	eqeltrd 2863	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))
155	92, 154	eqeltrd 2863	. . . . 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 46862	. . . 4 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ))
157	77, 156	sseldd 3939	. . 3 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ))
158		iooretop 24903	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
159	158	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
160	47, 41, 80, 159	fourierdlem58 46861	. . . 4 ⊢ (𝜑 → (ℝ D 𝐾) ∈ ((𝐴(,)𝐵)–cn→ℝ))
161	77, 160	sseldd 3939	. . 3 ⊢ (𝜑 → (ℝ D 𝐾) ∈ ((𝐴(,)𝐵)–cn→ℂ))
162	56, 66, 71, 157, 161	dvmulcncf 46622	. 2 ⊢ (𝜑 → (ℝ D (𝐻 ∘_f · 𝐾)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
163	54, 162	eqeltrd 2863	1 ⊢ (𝜑 → (ℝ D 𝑂) ∈ ((𝐴(,)𝐵)–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 {crab 3416 Vcvv 3455 ⊆ wss 3906 {cpr 4592 class class class wbr 5110 ↦ cmpt 5193 ran crn 5664 ↾ cres 5665 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 ∘_f cof 7674 ↑_m cmap 8825 ℂcc 11099 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11442 -cneg 11443 / cdiv 11872 ℕcn 12234 2c2 12296 (,)cioo 13373 [,]cicc 13376 ...cfz 13536 ..^cfzo 13684 sincsin 16118 πcpi 16121 TopOpenctopn 17475 topGenctg 17491 ℂ_fldccnfld 21503 intcnt 23155 –cn→ccncf 25016 D cdv 26003

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-pm 8828 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-fi 9372 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-fac 14312 df-bc 14341 df-hash 14369 df-shft 15106 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-limsup 15524 df-clim 15541 df-rlim 15542 df-sum 15740 df-ef 16122 df-sin 16124 df-cos 16125 df-pi 16127 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 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 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-pt 17498 df-prds 17501 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-mulg 19135 df-cntz 19388 df-cmn 19853 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-fbas 21500 df-fg 21501 df-cnfld 21504 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-cld 23157 df-ntr 23158 df-cls 23159 df-nei 23236 df-lp 23274 df-perf 23275 df-cn 23365 df-cnp 23366 df-t1 23452 df-haus 23453 df-tx 23700 df-hmeo 23893 df-fil 23984 df-fm 24076 df-flim 24077 df-flf 24078 df-xms 24458 df-ms 24459 df-tms 24460 df-cncf 25018 df-limc 26006 df-dv 26007

This theorem is referenced by: fourierdlem103 46906 fourierdlem104 46907

W3C validator