fourierdlem59 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem59 43706

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

**Hypotheses**
Ref	Expression
fourierdlem59.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem59.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem59.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem59.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem59.n0	⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
fourierdlem59.fdv	⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))
fourierdlem59.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem59.h	⊢ 𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠))

**Assertion**
Ref	Expression
fourierdlem59	⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ))

Distinct variable groups: 𝐴,𝑠 𝐵,𝑠 𝐶,𝑠 𝐹,𝑠 𝑋,𝑠 𝜑,𝑠 Allowed substitution hint: 𝐻(𝑠)

**Proof of Theorem fourierdlem59**
Step	Hyp	Ref	Expression
1		fourierdlem59.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐹:ℝ⟶ℝ)
3		fourierdlem59.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ)
4	3	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℝ)
5		elioore 13109	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
6	5	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
7	4, 6	readdcld 11004	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
8	2, 7	ffvelrnd 6962	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
9		fourierdlem59.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
10	9	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ)
11	8, 10	resubcld 11403	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℝ)
12		eqcom 2745	. . . . . . . . . . . 12 ⊢ (𝑠 = 0 ↔ 0 = 𝑠)
13	12	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑠 = 0 → 0 = 𝑠)
14	13	adantl 482	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (𝐴(,)𝐵) ∧ 𝑠 = 0) → 0 = 𝑠)
15		simpl 483	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (𝐴(,)𝐵) ∧ 𝑠 = 0) → 𝑠 ∈ (𝐴(,)𝐵))
16	14, 15	eqeltrd 2839	. . . . . . . . 9 ⊢ ((𝑠 ∈ (𝐴(,)𝐵) ∧ 𝑠 = 0) → 0 ∈ (𝐴(,)𝐵))
17	16	adantll 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ 𝑠 = 0) → 0 ∈ (𝐴(,)𝐵))
18		fourierdlem59.n0	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
19	18	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ 𝑠 = 0) → ¬ 0 ∈ (𝐴(,)𝐵))
20	17, 19	pm2.65da 814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 𝑠 = 0)
21	20	neqned 2950	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)
22	11, 6, 21	redivcld 11803	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) ∈ ℝ)
23		fourierdlem59.h	. . . . 5 ⊢ 𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠))
24	22, 23	fmptd 6988	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℝ)
25		ioossre 13140	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ
26	25	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
27		dvfre 25115	. . . 4 ⊢ ((𝐻:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐻):dom (ℝ D 𝐻)⟶ℝ)
28	24, 26, 27	syl2anc 584	. . 3 ⊢ (𝜑 → (ℝ D 𝐻):dom (ℝ D 𝐻)⟶ℝ)
29		ovex 7308	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ∈ V
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ V)
31		eqidd 2739	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)))
32		eqidd 2739	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠))
33	30, 11, 6, 31, 32	offval2 7553	. . . . . . . 8 ⊢ (𝜑 → ((𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∘_f / (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠)))
34	23, 33	eqtr4id 2797	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ((𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∘_f / (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)))
35	34	oveq2d 7291	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐻) = (ℝ D ((𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∘_f / (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠))))
36		reelprrecn 10963	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
37	36	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
38	11	recnd 11003	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
39		eqid 2738	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶))
40	38, 39	fmptd 6988	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)):(𝐴(,)𝐵)⟶ℂ)
41	6	recnd 11003	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
42		eldifsn 4720	. . . . . . . . 9 ⊢ (𝑠 ∈ (ℂ ∖ {0}) ↔ (𝑠 ∈ ℂ ∧ 𝑠 ≠ 0))
43	41, 21, 42	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (ℂ ∖ {0}))
44		eqid 2738	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)
45	43, 44	fmptd 6988	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠):(𝐴(,)𝐵)⟶(ℂ ∖ {0}))
46		eqidd 2739	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))))
47		eqidd 2739	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶))
48	30, 8, 10, 46, 47	offval2 7553	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∘_f − (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)))
49	48	eqcomd 2744	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) = ((𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∘_f − (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)))
50	49	oveq2d 7291	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) = (ℝ D ((𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∘_f − (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶))))
51	8	recnd 11003	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
52		eqid 2738	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))
53	51, 52	fmptd 6988	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))):(𝐴(,)𝐵)⟶ℂ)
54	10	recnd 11003	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℂ)
55		eqid 2738	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)
56	54, 55	fmptd 6988	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶):(𝐴(,)𝐵)⟶ℂ)
57		fourierdlem59.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
58		fourierdlem59.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
59		eqid 2738	. . . . . . . . . . 11 ⊢ (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
60		fourierdlem59.fdv	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))
61		cncff 24056	. . . . . . . . . . . 12 ⊢ ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ) → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ)
62	60, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ)
63	1, 3, 57, 58, 59, 62	fourierdlem28 43676	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))))
64		ioosscn 13141	. . . . . . . . . . . 12 ⊢ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℂ
65	64	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℂ)
66		ax-resscn 10928	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
67	66	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℂ)
68	62, 67	fssd 6618	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℂ)
69		ssid 3943	. . . . . . . . . . . . . 14 ⊢ ℂ ⊆ ℂ
70	69	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ⊆ ℂ)
71		cncffvrn 24061	. . . . . . . . . . . . 13 ⊢ ((ℂ ⊆ ℂ ∧ (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ)) → ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℂ) ↔ (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℂ))
72	70, 60, 71	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℂ) ↔ (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℂ))
73	68, 72	mpbird 256	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℂ))
74		ioosscn 13141	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ ℂ
75	74	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
76	3	recnd 11003	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℂ)
77	3, 57	readdcld 11004	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + 𝐴) ∈ ℝ)
78	77	rexrd 11025	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 + 𝐴) ∈ ℝ^*)
79	78	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐴) ∈ ℝ^*)
80	3, 58	readdcld 11004	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + 𝐵) ∈ ℝ)
81	80	rexrd 11025	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 + 𝐵) ∈ ℝ^*)
82	81	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐵) ∈ ℝ^*)
83	57	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
84	83	rexrd 11025	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
85	58	rexrd 11025	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
86	85	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
87		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (𝐴(,)𝐵))
88		ioogtlb 43033	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
89	84, 86, 87, 88	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
90	83, 6, 4, 89	ltadd2dd 11134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐴) < (𝑋 + 𝑠))
91	58	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
92		iooltub 43048	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
93	84, 86, 87, 92	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
94	6, 91, 4, 93	ltadd2dd 11134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) < (𝑋 + 𝐵))
95	79, 82, 7, 90, 94	eliood 43036	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))
96	65, 73, 75, 76, 95	fourierdlem23 43671	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
97	63, 96	eqeltrd 2839	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
98		iooretop 23929	. . . . . . . . . . . . 13 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
99		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
100	99	tgioo2 23966	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
101	98, 100	eleqtri 2837	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
102	101	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
103	9	recnd 11003	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℂ)
104	37, 102, 103	dvmptconst 43456	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 0))
105		0cnd 10968	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℂ)
106	75, 105, 70	constcncfg 43413	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 0) ∈ ((𝐴(,)𝐵)–cn→ℂ))
107	104, 106	eqeltrd 2839	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
108	37, 53, 56, 97, 107	dvsubcncf 43465	. . . . . . . 8 ⊢ (𝜑 → (ℝ D ((𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∘_f − (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
109	50, 108	eqeltrd 2839	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
110	37, 102	dvmptidg 43458	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 1))
111		1cnd 10970	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
112	75, 111, 70	constcncfg 43413	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
113	110, 112	eqeltrd 2839	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
114	37, 40, 45, 109, 113	dvdivcncf 43468	. . . . . 6 ⊢ (𝜑 → (ℝ D ((𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∘_f / (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
115	35, 114	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ))
116		cncff 24056	. . . . 5 ⊢ ((ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ) → (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℂ)
117		fdm 6609	. . . . 5 ⊢ ((ℝ D 𝐻):(𝐴(,)𝐵)⟶ℂ → dom (ℝ D 𝐻) = (𝐴(,)𝐵))
118	115, 116, 117	3syl 18	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐻) = (𝐴(,)𝐵))
119	118	feq2d 6586	. . 3 ⊢ (𝜑 → ((ℝ D 𝐻):dom (ℝ D 𝐻)⟶ℝ ↔ (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℝ))
120	28, 119	mpbid 231	. 2 ⊢ (𝜑 → (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℝ)
121		cncffvrn 24061	. . 3 ⊢ ((ℝ ⊆ ℂ ∧ (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ)) → ((ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℝ))
122	67, 115, 121	syl2anc 584	. 2 ⊢ (𝜑 → ((ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℝ))
123	120, 122	mpbird 256	1 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3432 ∖ cdif 3884 ⊆ wss 3887 {csn 4561 {cpr 4563 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 ↾ cres 5591 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∘_f cof 7531 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 ℝ^*cxr 11008 < clt 11009 − cmin 11205 / cdiv 11632 (,)cioo 13079 ↾_t crest 17131 TopOpenctopn 17132 topGenctg 17148 ℂ_fldccnfld 20597 –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-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-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 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: fourierdlem72 43719

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