fourierdlem59 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem59 46121

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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐹:ℝ⟶ℝ)
3		fourierdlem59.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ)
4	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℝ)
5		elioore 13414	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
6	5	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
7	4, 6	readdcld 11288	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
8	2, 7	ffvelcdmd 7105	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
9		fourierdlem59.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
10	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ)
11	8, 10	resubcld 11689	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℝ)
12		eqcom 2742	. . . . . . . . . . . 12 ⊢ (𝑠 = 0 ↔ 0 = 𝑠)
13	12	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑠 = 0 → 0 = 𝑠)
14	13	adantl 481	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (𝐴(,)𝐵) ∧ 𝑠 = 0) → 0 = 𝑠)
15		simpl 482	. . . . . . . . . 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 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ 𝑠 = 0) → 0 ∈ (𝐴(,)𝐵))
18		fourierdlem59.n0	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
19	18	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ 𝑠 = 0) → ¬ 0 ∈ (𝐴(,)𝐵))
20	17, 19	pm2.65da 817	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 𝑠 = 0)
21	20	neqned 2945	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)
22	11, 6, 21	redivcld 12093	. . . . 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 7134	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℝ)
25		ioossre 13445	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ
26	25	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
27		dvfre 26004	. . . 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 7464	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ∈ V
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ V)
31		eqidd 2736	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)))
32		eqidd 2736	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠))
33	30, 11, 6, 31, 32	offval2 7717	. . . . . . . 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 2794	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ((𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∘_f / (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)))
35	34	oveq2d 7447	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐻) = (ℝ D ((𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∘_f / (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠))))
36		reelprrecn 11245	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
37	36	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
38	11	recnd 11287	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
39		eqid 2735	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶))
40	38, 39	fmptd 7134	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)):(𝐴(,)𝐵)⟶ℂ)
41	6	recnd 11287	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
42		eldifsn 4791	. . . . . . . . 9 ⊢ (𝑠 ∈ (ℂ ∖ {0}) ↔ (𝑠 ∈ ℂ ∧ 𝑠 ≠ 0))
43	41, 21, 42	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (ℂ ∖ {0}))
44		eqid 2735	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)
45	43, 44	fmptd 7134	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠):(𝐴(,)𝐵)⟶(ℂ ∖ {0}))
46		eqidd 2736	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))))
47		eqidd 2736	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶))
48	30, 8, 10, 46, 47	offval2 7717	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∘_f − (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)))
49	48	eqcomd 2741	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) = ((𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∘_f − (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)))
50	49	oveq2d 7447	. . . . . . . 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 11287	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
52		eqid 2735	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))
53	51, 52	fmptd 7134	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))):(𝐴(,)𝐵)⟶ℂ)
54	10	recnd 11287	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℂ)
55		eqid 2735	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)
56	54, 55	fmptd 7134	. . . . . . . . 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 2735	. . . . . . . . . . 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 24933	. . . . . . . . . . . 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 46091	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))))
64		ioosscn 13446	. . . . . . . . . . . 12 ⊢ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℂ
65	64	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℂ)
66		ax-resscn 11210	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
67	66	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℂ)
68	62, 67	fssd 6754	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℂ)
69		ssid 4018	. . . . . . . . . . . . . 14 ⊢ ℂ ⊆ ℂ
70	69	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ⊆ ℂ)
71		cncfcdm 24938	. . . . . . . . . . . . 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 257	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℂ))
74		ioosscn 13446	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ ℂ
75	74	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
76	3	recnd 11287	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℂ)
77	3, 57	readdcld 11288	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + 𝐴) ∈ ℝ)
78	77	rexrd 11309	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 + 𝐴) ∈ ℝ^*)
79	78	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐴) ∈ ℝ^*)
80	3, 58	readdcld 11288	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + 𝐵) ∈ ℝ)
81	80	rexrd 11309	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 + 𝐵) ∈ ℝ^*)
82	81	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐵) ∈ ℝ^*)
83	57	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
84	83	rexrd 11309	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
85	58	rexrd 11309	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
86	85	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
87		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (𝐴(,)𝐵))
88		ioogtlb 45448	. . . . . . . . . . . . . 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 11418	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐴) < (𝑋 + 𝑠))
91	58	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
92		iooltub 45463	. . . . . . . . . . . . . 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 11418	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) < (𝑋 + 𝐵))
95	79, 82, 7, 90, 94	eliood 45451	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))
96	65, 73, 75, 76, 95	fourierdlem23 46086	. . . . . . . . . 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 24802	. . . . . . . . . . . . 13 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
99		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
100	99	tgioo2 24839	. . . . . . . . . . . . 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 11287	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℂ)
104	37, 102, 103	dvmptconst 45871	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 0))
105		0cnd 11252	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℂ)
106	75, 105, 70	constcncfg 45828	. . . . . . . . . 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 45880	. . . . . . . 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 45873	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 1))
111		1cnd 11254	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
112	75, 111, 70	constcncfg 45828	. . . . . . . 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 45883	. . . . . 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 24933	. . . . 5 ⊢ ((ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ) → (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℂ)
117		fdm 6746	. . . . 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 6723	. . 3 ⊢ (𝜑 → ((ℝ D 𝐻):dom (ℝ D 𝐻)⟶ℝ ↔ (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℝ))
120	28, 119	mpbid 232	. 2 ⊢ (𝜑 → (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℝ)
121		cncfcdm 24938	. . 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 257	1 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 {cpr 4633 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 ↾ cres 5691 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘_f cof 7695 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 ℝ^*cxr 11292 < clt 11293 − cmin 11490 / cdiv 11918 (,)cioo 13384 ↾_t crest 17467 TopOpenctopn 17468 topGenctg 17484 ℂ_fldccnfld 21382 –cn→ccncf 24916 D cdv 25913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-t1 23338 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917

This theorem is referenced by: fourierdlem72 46134

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