fourierdlem59 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem59		Structured version Visualization version GIF version

Theorem fourierdlem59 46203

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 13270	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
6	5	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
7	4, 6	readdcld 11136	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
8	2, 7	ffvelcdmd 7013	. . . . . . 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 11540	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℝ)
12		eqcom 2738	. . . . . . . . . . . 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 2831	. . . . . . . . 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 816	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 𝑠 = 0)
21	20	neqned 2935	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)
22	11, 6, 21	redivcld 11944	. . . . 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 7042	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℝ)
25		ioossre 13302	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ
26	25	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
27		dvfre 25877	. . . 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 7374	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ∈ V
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ V)
31		eqidd 2732	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)))
32		eqidd 2732	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠))
33	30, 11, 6, 31, 32	offval2 7625	. . . . . . . 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 2785	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ((𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∘_f / (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)))
35	34	oveq2d 7357	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐻) = (ℝ D ((𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∘_f / (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠))))
36		reelprrecn 11093	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
37	36	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
38	11	recnd 11135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
39		eqid 2731	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶))
40	38, 39	fmptd 7042	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)):(𝐴(,)𝐵)⟶ℂ)
41	6	recnd 11135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
42		eldifsn 4733	. . . . . . . . 9 ⊢ (𝑠 ∈ (ℂ ∖ {0}) ↔ (𝑠 ∈ ℂ ∧ 𝑠 ≠ 0))
43	41, 21, 42	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (ℂ ∖ {0}))
44		eqid 2731	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)
45	43, 44	fmptd 7042	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠):(𝐴(,)𝐵)⟶(ℂ ∖ {0}))
46		eqidd 2732	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))))
47		eqidd 2732	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶))
48	30, 8, 10, 46, 47	offval2 7625	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∘_f − (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)))
49	48	eqcomd 2737	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) = ((𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∘_f − (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)))
50	49	oveq2d 7357	. . . . . . . 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 11135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
52		eqid 2731	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))
53	51, 52	fmptd 7042	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))):(𝐴(,)𝐵)⟶ℂ)
54	10	recnd 11135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℂ)
55		eqid 2731	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)
56	54, 55	fmptd 7042	. . . . . . . . 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 2731	. . . . . . . . . . 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 24808	. . . . . . . . . . . 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 46173	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))))
64		ioosscn 13303	. . . . . . . . . . . 12 ⊢ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℂ
65	64	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℂ)
66		ax-resscn 11058	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
67	66	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℂ)
68	62, 67	fssd 6663	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℂ)
69		ssid 3952	. . . . . . . . . . . . . 14 ⊢ ℂ ⊆ ℂ
70	69	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ⊆ ℂ)
71		cncfcdm 24813	. . . . . . . . . . . . 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 13303	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ ℂ
75	74	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
76	3	recnd 11135	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℂ)
77	3, 57	readdcld 11136	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + 𝐴) ∈ ℝ)
78	77	rexrd 11157	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 + 𝐴) ∈ ℝ^*)
79	78	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐴) ∈ ℝ^*)
80	3, 58	readdcld 11136	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + 𝐵) ∈ ℝ)
81	80	rexrd 11157	. . . . . . . . . . . . 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 11157	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
85	58	rexrd 11157	. . . . . . . . . . . . . . 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 45535	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
89	84, 86, 87, 88	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
90	83, 6, 4, 89	ltadd2dd 11267	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐴) < (𝑋 + 𝑠))
91	58	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
92		iooltub 45550	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
93	84, 86, 87, 92	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
94	6, 91, 4, 93	ltadd2dd 11267	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) < (𝑋 + 𝐵))
95	79, 82, 7, 90, 94	eliood 45538	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))
96	65, 73, 75, 76, 95	fourierdlem23 46168	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
97	63, 96	eqeltrd 2831	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
98		iooretop 24675	. . . . . . . . . . . . 13 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
99		tgioo4 24715	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
100	98, 99	eleqtri 2829	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
101	100	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
102	9	recnd 11135	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℂ)
103	37, 101, 102	dvmptconst 45953	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 0))
104		0cnd 11100	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℂ)
105	75, 104, 70	constcncfg 45910	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 0) ∈ ((𝐴(,)𝐵)–cn→ℂ))
106	103, 105	eqeltrd 2831	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
107	37, 53, 56, 97, 106	dvsubcncf 45962	. . . . . . . 8 ⊢ (𝜑 → (ℝ D ((𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∘_f − (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
108	50, 107	eqeltrd 2831	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
109	37, 101	dvmptidg 45955	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 1))
110		1cnd 11102	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
111	75, 110, 70	constcncfg 45910	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
112	109, 111	eqeltrd 2831	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
113	37, 40, 45, 108, 112	dvdivcncf 45965	. . . . . 6 ⊢ (𝜑 → (ℝ D ((𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∘_f / (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
114	35, 113	eqeltrd 2831	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ))
115		cncff 24808	. . . . 5 ⊢ ((ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ) → (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℂ)
116		fdm 6655	. . . . 5 ⊢ ((ℝ D 𝐻):(𝐴(,)𝐵)⟶ℂ → dom (ℝ D 𝐻) = (𝐴(,)𝐵))
117	114, 115, 116	3syl 18	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐻) = (𝐴(,)𝐵))
118	117	feq2d 6630	. . 3 ⊢ (𝜑 → ((ℝ D 𝐻):dom (ℝ D 𝐻)⟶ℝ ↔ (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℝ))
119	28, 118	mpbid 232	. 2 ⊢ (𝜑 → (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℝ)
120		cncfcdm 24813	. . 3 ⊢ ((ℝ ⊆ ℂ ∧ (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ)) → ((ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℝ))
121	67, 114, 120	syl2anc 584	. 2 ⊢ (𝜑 → ((ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℝ))
122	119, 121	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 1541 ∈ wcel 2111 ≠ wne 2928 Vcvv 3436 ∖ cdif 3894 ⊆ wss 3897 {csn 4571 {cpr 4573 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5611 ran crn 5612 ↾ cres 5613 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 ∘_f cof 7603 ℂcc 10999 ℝcr 11000 0cc0 11001 1c1 11002 + caddc 11004 ℝ^*cxr 11140 < clt 11141 − cmin 11339 / cdiv 11769 (,)cioo 13240 ↾_t crest 17319 TopOpenctopn 17320 topGenctg 17336 ℂ_fldccnfld 21286 –cn→ccncf 24791 D cdv 25786

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ioo 13244 df-icc 13247 df-fz 13403 df-fzo 13550 df-seq 13904 df-exp 13964 df-hash 14233 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-hom 17180 df-cco 17181 df-rest 17321 df-topn 17322 df-0g 17340 df-gsum 17341 df-topgen 17342 df-pt 17343 df-prds 17346 df-xrs 17401 df-qtop 17406 df-imas 17407 df-xps 17409 df-mre 17483 df-mrc 17484 df-acs 17486 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19224 df-cmn 19689 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-cld 22929 df-ntr 22930 df-cls 22931 df-nei 23008 df-lp 23046 df-perf 23047 df-cn 23137 df-cnp 23138 df-t1 23224 df-haus 23225 df-tx 23472 df-hmeo 23665 df-fil 23756 df-fm 23848 df-flim 23849 df-flf 23850 df-xms 24230 df-ms 24231 df-tms 24232 df-cncf 24793 df-limc 25789 df-dv 25790

This theorem is referenced by: fourierdlem72 46216

W3C validator