fourierdlem59 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem59 46593

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 13328	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
6	5	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
7	4, 6	readdcld 11174	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
8	2, 7	ffvelcdmd 7037	. . . . . . 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 11578	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℝ)
12		eqcom 2743	. . . . . . . . . . . 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 2836	. . . . . . . . 9 ⊢ ((𝑠 ∈ (𝐴(,)𝐵) ∧ 𝑠 = 0) → 0 ∈ (𝐴(,)𝐵))
17	16	adantll 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ 𝑠 = 0) → 0 ∈ (𝐴(,)𝐵))
18		fourierdlem59.n0	. . . . . . . . 9 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
19	18	ad2antrr 727	. . . . . . . 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 2939	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)
22	11, 6, 21	redivcld 11983	. . . . 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 7066	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴(,)𝐵)⟶ℝ)
25		ioossre 13360	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ
26	25	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
27		dvfre 25918	. . . 4 ⊢ ((𝐻:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐻):dom (ℝ D 𝐻)⟶ℝ)
28	24, 26, 27	syl2anc 585	. . 3 ⊢ (𝜑 → (ℝ D 𝐻):dom (ℝ D 𝐻)⟶ℝ)
29		ovex 7400	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ∈ V
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ V)
31		eqidd 2737	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)))
32		eqidd 2737	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠))
33	30, 11, 6, 31, 32	offval2 7651	. . . . . . . 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 2790	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ((𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∘_f / (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)))
35	34	oveq2d 7383	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐻) = (ℝ D ((𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∘_f / (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠))))
36		reelprrecn 11130	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
37	36	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
38	11	recnd 11173	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
39		eqid 2736	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶))
40	38, 39	fmptd 7066	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)):(𝐴(,)𝐵)⟶ℂ)
41	6	recnd 11173	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
42		eldifsn 4731	. . . . . . . . 9 ⊢ (𝑠 ∈ (ℂ ∖ {0}) ↔ (𝑠 ∈ ℂ ∧ 𝑠 ≠ 0))
43	41, 21, 42	sylanbrc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (ℂ ∖ {0}))
44		eqid 2736	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)
45	43, 44	fmptd 7066	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠):(𝐴(,)𝐵)⟶(ℂ ∖ {0}))
46		eqidd 2737	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))))
47		eqidd 2737	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶))
48	30, 8, 10, 46, 47	offval2 7651	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∘_f − (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)))
49	48	eqcomd 2742	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) = ((𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∘_f − (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)))
50	49	oveq2d 7383	. . . . . . . 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 11173	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
52		eqid 2736	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))
53	51, 52	fmptd 7066	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))):(𝐴(,)𝐵)⟶ℂ)
54	10	recnd 11173	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℂ)
55		eqid 2736	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)
56	54, 55	fmptd 7066	. . . . . . . . 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 2736	. . . . . . . . . . 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 24860	. . . . . . . . . . . 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 46563	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))))
64		ioosscn 13361	. . . . . . . . . . . 12 ⊢ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℂ
65	64	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ⊆ ℂ)
66		ax-resscn 11095	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
67	66	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℂ)
68	62, 67	fssd 6685	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℂ)
69		ssid 3944	. . . . . . . . . . . . . 14 ⊢ ℂ ⊆ ℂ
70	69	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ⊆ ℂ)
71		cncfcdm 24865	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 13361	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ ℂ
75	74	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
76	3	recnd 11173	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℂ)
77	3, 57	readdcld 11174	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + 𝐴) ∈ ℝ)
78	77	rexrd 11195	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 + 𝐴) ∈ ℝ^*)
79	78	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐴) ∈ ℝ^*)
80	3, 58	readdcld 11174	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + 𝐵) ∈ ℝ)
81	80	rexrd 11195	. . . . . . . . . . . . 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 11195	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
85	58	rexrd 11195	. . . . . . . . . . . . . . 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 45925	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
89	84, 86, 87, 88	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
90	83, 6, 4, 89	ltadd2dd 11305	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐴) < (𝑋 + 𝑠))
91	58	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
92		iooltub 45940	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
93	84, 86, 87, 92	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
94	6, 91, 4, 93	ltadd2dd 11305	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) < (𝑋 + 𝐵))
95	79, 82, 7, 90, 94	eliood 45928	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))
96	65, 73, 75, 76, 95	fourierdlem23 46558	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
97	63, 96	eqeltrd 2836	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
98		iooretop 24730	. . . . . . . . . . . . 13 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
99		tgioo4 24770	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
100	98, 99	eleqtri 2834	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
101	100	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
102	9	recnd 11173	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℂ)
103	37, 101, 102	dvmptconst 46343	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 0))
104		0cnd 11137	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℂ)
105	75, 104, 70	constcncfg 46300	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 0) ∈ ((𝐴(,)𝐵)–cn→ℂ))
106	103, 105	eqeltrd 2836	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
107	37, 53, 56, 97, 106	dvsubcncf 46352	. . . . . . . 8 ⊢ (𝜑 → (ℝ D ((𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∘_f − (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
108	50, 107	eqeltrd 2836	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
109	37, 101	dvmptidg 46345	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 1))
110		1cnd 11139	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
111	75, 110, 70	constcncfg 46300	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
112	109, 111	eqeltrd 2836	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
113	37, 40, 45, 108, 112	dvdivcncf 46355	. . . . . 6 ⊢ (𝜑 → (ℝ D ((𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∘_f / (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
114	35, 113	eqeltrd 2836	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ))
115		cncff 24860	. . . . 5 ⊢ ((ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℂ) → (ℝ D 𝐻):(𝐴(,)𝐵)⟶ℂ)
116		fdm 6677	. . . . 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 6652	. . 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 24865	. . 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 585	. 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 1542 ∈ wcel 2114 ≠ wne 2932 Vcvv 3429 ∖ cdif 3886 ⊆ wss 3889 {csn 4567 {cpr 4569 class class class wbr 5085 ↦ cmpt 5166 dom cdm 5631 ran crn 5632 ↾ cres 5633 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∘_f cof 7629 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 ℝ^*cxr 11178 < clt 11179 − cmin 11377 / cdiv 11807 (,)cioo 13298 ↾_t crest 17383 TopOpenctopn 17384 topGenctg 17400 ℂ_fldccnfld 21352 –cn→ccncf 24843 D cdv 25830

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-icc 13305 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-t1 23279 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834

This theorem is referenced by: fourierdlem72 46606

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