Theorem cncfuni 45807

Description: A complex function on a subset of the complex numbers is continuous if its domain is the union of relatively open subsets over which the function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
cncfuni.acn	⊢ (𝜑 → 𝐴 ⊆ ℂ)
cncfuni.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
cncfuni.auni	⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵)
cncfuni.opn	⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐴 ∩ 𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))
cncfuni.fcn	⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹 ↾ 𝑏) ∈ ((𝐴 ∩ 𝑏)–cn→ℂ))

Assertion

Ref	Expression
cncfuni	⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))

Distinct variable groups:   𝐴,𝑏   𝐵,𝑏   𝐹,𝑏   𝜑,𝑏

Proof of Theorem cncfuni

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cncfuni.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
2		cncfuni.auni	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵)
3	2	sselda 4008	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐵)
4		eluni2 4935	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑏 ∈ 𝐵 𝑥 ∈ 𝑏)
5	3, 4	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑏 ∈ 𝐵 𝑥 ∈ 𝑏)
6		simp1l 1197	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝜑)
7		simp2 1137	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝑏 ∈ 𝐵)
8		elin 3992	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∩ 𝑏) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏))
9	8	biimpri 228	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏) → 𝑥 ∈ (𝐴 ∩ 𝑏))
10	9	adantll 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝑏) → 𝑥 ∈ (𝐴 ∩ 𝑏))
11	10	3adant2 1131	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝑥 ∈ (𝐴 ∩ 𝑏))
12		cncfuni.fcn	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹 ↾ 𝑏) ∈ ((𝐴 ∩ 𝑏)–cn→ℂ))
13	1	fdmd 6757	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = 𝐴)
14	13	ineq2d 4241	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑏 ∩ dom 𝐹) = (𝑏 ∩ 𝐴))
15		incom 4230	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∩ 𝐴) = (𝐴 ∩ 𝑏)
16	14, 15	eqtr2di 2797	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∩ 𝑏) = (𝑏 ∩ dom 𝐹))
17	16	reseq2d 6009	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑏)) = (𝐹 ↾ (𝑏 ∩ dom 𝐹)))
18		frel 6752	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:𝐴⟶ℂ → Rel 𝐹)
19	1, 18	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Rel 𝐹)
20		resindm 6059	. . . . . . . . . . . . . . . . . 18 ⊢ (Rel 𝐹 → (𝐹 ↾ (𝑏 ∩ dom 𝐹)) = (𝐹 ↾ 𝑏))
21	19, 20	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 ↾ (𝑏 ∩ dom 𝐹)) = (𝐹 ↾ 𝑏))
22	17, 21	eqtrd 2780	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑏)) = (𝐹 ↾ 𝑏))
23		inss1 4258	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∩ 𝑏) ⊆ 𝐴
24	23	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 ∩ 𝑏) ⊆ 𝐴)
25		cncfuni.acn	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
26	24, 25	sstrd 4019	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∩ 𝑏) ⊆ ℂ)
27		ssidd 4032	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ℂ ⊆ ℂ)
28		eqid 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
29		eqid 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏))
30	28	cnfldtop 24825	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (TopOpen‘ℂfld) ∈ Top
31		unicntop 24827	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
32	31	restid 17493	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
33	30, 32	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
34	33	eqcomi 2749	. . . . . . . . . . . . . . . . . . 19 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
35	28, 29, 34	cncfcn 24955	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∩ 𝑏) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴 ∩ 𝑏)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)))
36	26, 27, 35	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐴 ∩ 𝑏)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)))
37	36	eqcomd 2746	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)) = ((𝐴 ∩ 𝑏)–cn→ℂ))
38	22, 37	eleq12d 2838	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)) ↔ (𝐹 ↾ 𝑏) ∈ ((𝐴 ∩ 𝑏)–cn→ℂ)))
39	38	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)) ↔ (𝐹 ↾ 𝑏) ∈ ((𝐴 ∩ 𝑏)–cn→ℂ)))
40	12, 39	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)))
41	40	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)))
42	28	cnfldtopon 24824	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
44		resttopon 23190	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐴 ∩ 𝑏) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) ∈ (TopOn‘(𝐴 ∩ 𝑏)))
45	43, 26, 44	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) ∈ (TopOn‘(𝐴 ∩ 𝑏)))
46	45	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) ∈ (TopOn‘(𝐴 ∩ 𝑏)))
47	42	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
48		cncnp 23309	. . . . . . . . . . . . 13 ⊢ ((((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) ∈ (TopOn‘(𝐴 ∩ 𝑏)) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → ((𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)) ↔ ((𝐹 ↾ (𝐴 ∩ 𝑏)):(𝐴 ∩ 𝑏)⟶ℂ ∧ ∀𝑥 ∈ (𝐴 ∩ 𝑏)(𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))))
49	46, 47, 48	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → ((𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)) ↔ ((𝐹 ↾ (𝐴 ∩ 𝑏)):(𝐴 ∩ 𝑏)⟶ℂ ∧ ∀𝑥 ∈ (𝐴 ∩ 𝑏)(𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))))
50	41, 49	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → ((𝐹 ↾ (𝐴 ∩ 𝑏)):(𝐴 ∩ 𝑏)⟶ℂ ∧ ∀𝑥 ∈ (𝐴 ∩ 𝑏)(𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥)))
51	50	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → ∀𝑥 ∈ (𝐴 ∩ 𝑏)(𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
52		simp3 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → 𝑥 ∈ (𝐴 ∩ 𝑏))
53		rspa 3254	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ (𝐴 ∩ 𝑏)(𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥) ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
54	51, 52, 53	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
55	30	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ Top)
56		cnex 11265	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
57	56	ssex 5339	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ℂ → 𝐴 ∈ V)
58	25, 57	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ V)
59		restabs 23194	. . . . . . . . . . . . . 14 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴 ∩ 𝑏) ⊆ 𝐴 ∧ 𝐴 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)))
60	55, 24, 58, 59	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)))
61	60	eqcomd 2746	. . . . . . . . . . . 12 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) = (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)))
62	61	oveq1d 7463	. . . . . . . . . . 11 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld)) = ((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld)))
63	62	fveq1d 6922	. . . . . . . . . 10 ⊢ (𝜑 → ((((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥) = (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
64	63	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → ((((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥) = (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
65	54, 64	eleqtrd 2846	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
66		resttop 23189	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
67	55, 58, 66	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
68	67	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
69	31	restuni 23191	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ⊆ ℂ) → 𝐴 = ∪ ((TopOpen‘ℂfld) ↾t 𝐴))
70	55, 25, 69	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 = ∪ ((TopOpen‘ℂfld) ↾t 𝐴))
71	24, 70	sseqtrd 4049	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ 𝑏) ⊆ ∪ ((TopOpen‘ℂfld) ↾t 𝐴))
72	71	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐴 ∩ 𝑏) ⊆ ∪ ((TopOpen‘ℂfld) ↾t 𝐴))
73		cncfuni.opn	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐴 ∩ 𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))
74	73	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐴 ∩ 𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))
75		eqid 2740	. . . . . . . . . . . . . 14 ⊢ ∪ ((TopOpen‘ℂfld) ↾t 𝐴) = ∪ ((TopOpen‘ℂfld) ↾t 𝐴)
76	75	isopn3 23095	. . . . . . . . . . . . 13 ⊢ ((((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top ∧ (𝐴 ∩ 𝑏) ⊆ ∪ ((TopOpen‘ℂfld) ↾t 𝐴)) → ((𝐴 ∩ 𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴 ∩ 𝑏)) = (𝐴 ∩ 𝑏)))
77	68, 72, 76	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → ((𝐴 ∩ 𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴 ∩ 𝑏)) = (𝐴 ∩ 𝑏)))
78	74, 77	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴 ∩ 𝑏)) = (𝐴 ∩ 𝑏))
79	78	eqcomd 2746	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴 ∩ 𝑏)))
80	52, 79	eleqtrd 2846	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴 ∩ 𝑏)))
81	70	feq2d 6733	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:𝐴⟶ℂ ↔ 𝐹:∪ ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ))
82	1, 81	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:∪ ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ)
83	82	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → 𝐹:∪ ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ)
84	75, 31	cnprest 23318	. . . . . . . . 9 ⊢ (((((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top ∧ (𝐴 ∩ 𝑏) ⊆ ∪ ((TopOpen‘ℂfld) ↾t 𝐴)) ∧ (𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴 ∩ 𝑏)) ∧ 𝐹:∪ ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ)) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥) ↔ (𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥)))
85	68, 72, 80, 83, 84	syl22anc 838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥) ↔ (𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥)))
86	65, 85	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
87	6, 7, 11, 86	syl3anc 1371	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
88	87	rexlimdv3a 3165	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑏 ∈ 𝐵 𝑥 ∈ 𝑏 → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥)))
89	5, 88	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
90	89	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
91		resttopon 23190	. . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝐴 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴))
92	43, 25, 91	syl2anc 583	. . . 4 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴))
93		cncnp 23309	. . . 4 ⊢ ((((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))))
94	92, 43, 93	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))))
95	1, 90, 94	mpbir2and 712	. 2 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
96		eqid 2740	. . . . 5 ⊢ ((TopOpen‘ℂfld) ↾t 𝐴) = ((TopOpen‘ℂfld) ↾t 𝐴)
97	28, 96, 34	cncfcn 24955	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐴–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
98	25, 27, 97	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐴–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
99	98	eqcomd 2746	. 2 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) = (𝐴–cn→ℂ))
100	95, 99	eleqtrd 2846	1 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ∪ cuni 4931  dom cdm 5700   ↾ cres 5702  Rel wrel 5705  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ℂcc 11182   ↾_t crest 17480  TopOpenctopn 17481  ℂ_fldccnfld 21387  Topctop 22920  TopOnctopon 22937  intcnt 23046   Cn ccn 23253   CnP ccnp 23254  –cn→ccncf 24921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fi 9480  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-fz 13568  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-struct 17194  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-mulr 17325  df-starv 17326  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-rest 17482  df-topn 17483  df-topgen 17503  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-cnfld 21388  df-top 22921  df-topon 22938  df-topsp 22960  df-bases 22974  df-ntr 23049  df-cn 23256  df-cnp 23257  df-xms 24351  df-ms 24352  df-cncf 24923

This theorem is referenced by:  fouriersw  46152