fourierdlem72 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem72 39071

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		ovex 6577	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ V
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ V)
3		fourierdlem72.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
4	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐹:ℝ⟶ℝ)
5		fourierdlem72.xre	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ ℝ)
6	5	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℝ)
7		elioore 12076	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
8	7	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
9	6, 8	readdcld 9948	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
10	4, 9	ffvelrnd 6268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
11		fourierdlem72.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
12	11	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ)
13	10, 12	resubcld 10337	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℝ)
14		ioossicc 12130	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
15	14	sseli 3564	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ (𝐴[,]𝐵))
16	15	ad2antlr 759	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → 𝑠 ∈ (𝐴[,]𝐵))
17		id 22	. . . . . . . . . . . . 13 ⊢ (𝑠 ≠ 0 → 𝑠 ≠ 0)
18	17	necon1bi 2810	. . . . . . . . . . . 12 ⊢ (¬ 𝑠 ≠ 0 → 𝑠 = 0)
19	18	eleq1d 2672	. . . . . . . . . . 11 ⊢ (¬ 𝑠 ≠ 0 → (𝑠 ∈ (𝐴[,]𝐵) ↔ 0 ∈ (𝐴[,]𝐵)))
20	19	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → (𝑠 ∈ (𝐴[,]𝐵) ↔ 0 ∈ (𝐴[,]𝐵)))
21	16, 20	mpbid 221	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → 0 ∈ (𝐴[,]𝐵))
22		fourierdlem72.n0	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
23	22	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑠 ≠ 0) → ¬ 0 ∈ (𝐴[,]𝐵))
24	21, 23	condan 831	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)
25	13, 8, 24	redivcld 10732	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) ∈ ℝ)
26		fourierdlem72.h	. . . . . . 7 ⊢ 𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠))
27	25, 26	fmptd 6292	. . . . . 6 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℝ)
28	27	ffvelrnda 6267	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐻‘𝑠) ∈ ℝ)
29		2re 10967	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
31	8	rehalfcld 11156	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℝ)
32	31	resincld 14712	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℝ)
33	30, 32	remulcld 9949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
34		2cnd 10970	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
35	8	recnd 9947	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
36	35	halfcld 11154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
37	36	sincld 14699	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
38		2ne0 10990	. . . . . . . . . 10 ⊢ 2 ≠ 0
39	38	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
40		fourierdlem72.ab	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))
41	40	sselda 3568	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (-π[,]π))
42		fourierdlem44 39044	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
43	41, 24, 42	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
44	34, 37, 39, 43	mulne0d 10558	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
45	8, 33, 44	redivcld 10732	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ ℝ)
46		fourierdlem72.k	. . . . . . 7 ⊢ 𝐾 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝑠 / (2 · (sin‘(𝑠 / 2)))))
47	45, 46	fmptd 6292	. . . . . 6 ⊢ (𝜑 → 𝐾:(𝐴(,)𝐵)⟶ℝ)
48	47	ffvelrnda 6267	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐾‘𝑠) ∈ ℝ)
49	27	feqmptd 6159	. . . . 5 ⊢ (𝜑 → 𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐻‘𝑠)))
50	47	feqmptd 6159	. . . . 5 ⊢ (𝜑 → 𝐾 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠)))
51	2, 28, 48, 49, 50	offval2 6812	. . . 4 ⊢ (𝜑 → (𝐻 ∘_𝑓 · 𝐾) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))))
52		fourierdlem72.o	. . . 4 ⊢ 𝑂 = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
53	51, 52	syl6reqr 2663	. . 3 ⊢ (𝜑 → 𝑂 = (𝐻 ∘_𝑓 · 𝐾))
54	53	oveq2d 6565	. 2 ⊢ (𝜑 → (ℝ D 𝑂) = (ℝ D (𝐻 ∘_𝑓 · 𝐾)))
55		reelprrecn 9907	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57	10	recnd 9947	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
58	11	recnd 9947	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
59	58	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℂ)
60	57, 59	subcld 10271	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
61		ioossre 12106	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
62	61	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
63	62	sselda 3568	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
64	63	recnd 9947	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
65	60, 64, 24	divcld 10680	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) ∈ ℂ)
66	65, 26	fmptd 6292	. . 3 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℂ)
67	64	halfcld 11154	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
68	67	sincld 14699	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
69	34, 68	mulcld 9939	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
70	64, 69, 44	divcld 10680	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
71	70, 46	fmptd 6292	. . 3 ⊢ (𝜑 → 𝐾:(𝐴(,)𝐵)⟶ℂ)
72		ax-resscn 9872	. . . . . 6 ⊢ ℝ ⊆ ℂ
73	72	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
74		ssid 3587	. . . . . 6 ⊢ ℂ ⊆ ℂ
75	74	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
76		cncfss 22510	. . . . 5 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℝ) ⊆ ((𝐴(,)𝐵)–cn→ℂ))
77	73, 75, 76	syl2anc 691	. . . 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	24	nelrdva 3384	. . . . 5 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
81	3, 73	fssd 5970	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
82		ssid 3587	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ
83	82	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ)
84		ioossre 12106	. . . . . . . . 9 ⊢ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℝ
85	84	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℝ)
86		eqid 2610	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
87	86	tgioo2 22414	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
88	86, 87	dvres 23481	. . . . . . . 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 1319	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))))
90		ioontr 38583	. . . . . . . 8 ⊢ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))
91	90	reseq2i 5314	. . . . . . 7 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))
92	89, 91	syl6eq 2660	. . . . . 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 39002	. . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . 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 474	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑_𝑚 (0...𝑀)))
100		elmapi 7765	. . . . . . . . . . . . 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 12354	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (0..^𝑀) → 𝑈 ∈ (0...𝑀))
104	102, 103	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ (0...𝑀))
105	101, 104	ffvelrnd 6268	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘𝑈) ∈ ℝ)
106	105	rexrd 9968	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘𝑈) ∈ ℝ^*)
107		fzofzp1 12431	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (0..^𝑀) → (𝑈 + 1) ∈ (0...𝑀))
108	102, 107	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈 + 1) ∈ (0...𝑀))
109	101, 108	ffvelrnd 6268	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) ∈ ℝ)
110	109	rexrd 9968	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) ∈ ℝ^*)
111		pire 24014	. . . . . . . . . . . . . . 15 ⊢ π ∈ ℝ
112	111	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → π ∈ ℝ)
113	112	renegcld 10336	. . . . . . . . . . . . 13 ⊢ (𝜑 → -π ∈ ℝ)
114		fourierdlem72.q	. . . . . . . . . . . . 13 ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
115	113, 112, 5, 95, 94, 93, 104, 114	fourierdlem13 39013	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝑈) = ((𝑉‘𝑈) − 𝑋) ∧ (𝑉‘𝑈) = (𝑋 + (𝑄‘𝑈))))
116	115	simprd 478	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘𝑈) = (𝑋 + (𝑄‘𝑈)))
117	115	simpld 474	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝑈) = ((𝑉‘𝑈) − 𝑋))
118	105, 5	resubcld 10337	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑉‘𝑈) − 𝑋) ∈ ℝ)
119	117, 118	eqeltrd 2688	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘𝑈) ∈ ℝ)
120	113, 112, 5, 95, 94, 93, 108, 114	fourierdlem13 39013	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑄‘(𝑈 + 1)) = ((𝑉‘(𝑈 + 1)) − 𝑋) ∧ (𝑉‘(𝑈 + 1)) = (𝑋 + (𝑄‘(𝑈 + 1)))))
121	120	simpld 474	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄‘(𝑈 + 1)) = ((𝑉‘(𝑈 + 1)) − 𝑋))
122	109, 5	resubcld 10337	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑉‘(𝑈 + 1)) − 𝑋) ∈ ℝ)
123	121, 122	eqeltrd 2688	. . . . . . . . . . . . . 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 39010	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝑈) ≤ 𝐴 ∧ 𝐵 ≤ (𝑄‘(𝑈 + 1))))
127	126	simpld 474	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘𝑈) ≤ 𝐴)
128	119, 78, 5, 127	leadd2dd 10521	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + (𝑄‘𝑈)) ≤ (𝑋 + 𝐴))
129	116, 128	eqbrtrd 4605	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉‘𝑈) ≤ (𝑋 + 𝐴))
130	126	simprd 478	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≤ (𝑄‘(𝑈 + 1)))
131	79, 123, 5, 130	leadd2dd 10521	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + 𝐵) ≤ (𝑋 + (𝑄‘(𝑈 + 1))))
132	120	simprd 478	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉‘(𝑈 + 1)) = (𝑋 + (𝑄‘(𝑈 + 1))))
133	131, 132	breqtrrd 4611	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝐵) ≤ (𝑉‘(𝑈 + 1)))
134		ioossioo 12136	. . . . . . . . . 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 1319	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1))))
136	135	resabs1d 5348	. . . . . . . 8 ⊢ (𝜑 → (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
137	136	eqcomd 2616	. . . . . . 7 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) = (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
138	102	ancli 572	. . . . . . . . 9 ⊢ (𝜑 → (𝜑 ∧ 𝑈 ∈ (0..^𝑀)))
139		eleq1 2676	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → (𝑖 ∈ (0..^𝑀) ↔ 𝑈 ∈ (0..^𝑀)))
140	139	anbi2d 736	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑈 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑈 ∈ (0..^𝑀))))
141		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑈 → (𝑉‘𝑖) = (𝑉‘𝑈))
142		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑈 → (𝑖 + 1) = (𝑈 + 1))
143	142	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑈 → (𝑉‘(𝑖 + 1)) = (𝑉‘(𝑈 + 1)))
144	141, 143	oveq12d 6567	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑈 → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) = ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1))))
145	144	reseq2d 5317	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) = ((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))))
146	144	oveq1d 6564	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ) = (((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))–cn→ℝ))
147	145, 146	eleq12d 2682	. . . . . . . . . . 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 333	. . . . . . . . . 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 3239	. . . . . . . . 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 63	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ∈ (((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))–cn→ℝ))
152		rescncf 22508	. . . . . . . 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 63	. . . . . . 7 ⊢ (𝜑 → (((ℝ D 𝐹) ↾ ((𝑉‘𝑈)(,)(𝑉‘(𝑈 + 1)))) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))
154	137, 153	eqeltrd 2688	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))
155	92, 154	eqeltrd 2688	. . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))
156	3, 5, 78, 79, 80, 155, 11, 26	fourierdlem59 39058	. . . 4 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ))
157	77, 156	sseldd 3569	. . 3 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ))
158		iooretop 22379	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
159	158	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
160	46, 40, 80, 159	fourierdlem58 39057	. . . 4 ⊢ (𝜑 → (ℝ D 𝐾) ∈ ((𝐴(,)𝐵)–cn→ℝ))
161	77, 160	sseldd 3569	. . 3 ⊢ (𝜑 → (ℝ D 𝐾) ∈ ((𝐴(,)𝐵)–cn→ℂ))
162	56, 66, 71, 157, 161	dvmulcncf 38815	. 2 ⊢ (𝜑 → (ℝ D (𝐻 ∘_𝑓 · 𝐾)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
163	54, 162	eqeltrd 2688	1 ⊢ (𝜑 → (ℝ D 𝑂) ∈ ((𝐴(,)𝐵)–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 Vcvv 3173 ⊆ wss 3540 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 ↑_𝑚 cmap 7744 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 (,)cioo 12046 [,]cicc 12049 ...cfz 12197 ..^cfzo 12334 sincsin 14633 πcpi 14636 TopOpenctopn 15905 topGenctg 15921 ℂ_fldccnfld 19567 intcnt 20631 –cn→ccncf 22487 D cdv 23433

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-t1 20928 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437

This theorem is referenced by: fourierdlem103 39102 fourierdlem104 39103

W3C validator