Theorem cncfuni 46314

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 3921	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐵)
4		eluni2 4854	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑏 ∈ 𝐵 𝑥 ∈ 𝑏)
5	3, 4	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑏 ∈ 𝐵 𝑥 ∈ 𝑏)
6		simp1l 1199	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝜑)
7		simp2 1138	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝑏 ∈ 𝐵)
8		elin 3905	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∩ 𝑏) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏))
9	8	biimpri 228	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏) → 𝑥 ∈ (𝐴 ∩ 𝑏))
10	9	adantll 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝑏) → 𝑥 ∈ (𝐴 ∩ 𝑏))
11	10	3adant2 1132	. . . . . . 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 6678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = 𝐴)
14	13	ineq2d 4160	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑏 ∩ dom 𝐹) = (𝑏 ∩ 𝐴))
15		incom 4149	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∩ 𝐴) = (𝐴 ∩ 𝑏)
16	14, 15	eqtr2di 2788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∩ 𝑏) = (𝑏 ∩ dom 𝐹))
17	16	reseq2d 5944	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑏)) = (𝐹 ↾ (𝑏 ∩ dom 𝐹)))
18		frel 6673	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:𝐴⟶ℂ → Rel 𝐹)
19	1, 18	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Rel 𝐹)
20		resindm 5995	. . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑏)) = (𝐹 ↾ 𝑏))
23		inss1 4177	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∩ 𝑏) ⊆ 𝐴
24	23	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 ∩ 𝑏) ⊆ 𝐴)
25		cncfuni.acn	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
26	24, 25	sstrd 3932	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∩ 𝑏) ⊆ ℂ)
27		ssidd 3945	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ℂ ⊆ ℂ)
28		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
29		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏))
30	28	cnfldtop 24748	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (TopOpen‘ℂfld) ∈ Top
31		unicntop 24750	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
32	31	restid 17396	. . . . . . . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . . . . . 19 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
35	28, 29, 34	cncfcn 24877	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∩ 𝑏) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴 ∩ 𝑏)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)))
36	26, 27, 35	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐴 ∩ 𝑏)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)))
37	36	eqcomd 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)) = ((𝐴 ∩ 𝑏)–cn→ℂ))
38	22, 37	eleq12d 2830	. . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)))
42	28	cnfldtopon 24747	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
44		resttopon 23126	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐴 ∩ 𝑏) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) ∈ (TopOn‘(𝐴 ∩ 𝑏)))
45	43, 26, 44	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) ∈ (TopOn‘(𝐴 ∩ 𝑏)))
46	45	3ad2ant1 1134	. . . . . . . . . . . . 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 23245	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 1139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → 𝑥 ∈ (𝐴 ∩ 𝑏))
53		rspa 3226	. . . . . . . . . 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 585	. . . . . . . . 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 11119	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
57	56	ssex 5262	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ℂ → 𝐴 ∈ V)
58	25, 57	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ V)
59		restabs 23130	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)))
61	60	eqcomd 2742	. . . . . . . . . . . 12 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) = (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)))
62	61	oveq1d 7382	. . . . . . . . . . 11 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld)) = ((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld)))
63	62	fveq1d 6842	. . . . . . . . . 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 1134	. . . . . . . . 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 2838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
66		resttop 23125	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
67	55, 58, 66	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
68	67	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
69	31	restuni 23127	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ⊆ ℂ) → 𝐴 = ∪ ((TopOpen‘ℂfld) ↾t 𝐴))
70	55, 25, 69	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 = ∪ ((TopOpen‘ℂfld) ↾t 𝐴))
71	24, 70	sseqtrd 3958	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ 𝑏) ⊆ ∪ ((TopOpen‘ℂfld) ↾t 𝐴))
72	71	3ad2ant1 1134	. . . . . . . . 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 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐴 ∩ 𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))
75		eqid 2736	. . . . . . . . . . . . . 14 ⊢ ∪ ((TopOpen‘ℂfld) ↾t 𝐴) = ∪ ((TopOpen‘ℂfld) ↾t 𝐴)
76	75	isopn3 23031	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 2742	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴 ∩ 𝑏)))
80	52, 79	eleqtrd 2838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴 ∩ 𝑏)))
81	70	feq2d 6652	. . . . . . . . . . 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 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → 𝐹:∪ ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ)
84	75, 31	cnprest 23254	. . . . . . . . 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 839	. . . . . . . 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 1374	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
88	87	rexlimdv3a 3142	. . . . 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 3129	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
91		resttopon 23126	. . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝐴 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴))
92	43, 25, 91	syl2anc 585	. . . 4 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴))
93		cncnp 23245	. . . 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 585	. . 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 714	. 2 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
96		eqid 2736	. . . . 5 ⊢ ((TopOpen‘ℂfld) ↾t 𝐴) = ((TopOpen‘ℂfld) ↾t 𝐴)
97	28, 96, 34	cncfcn 24877	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐴–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
98	25, 27, 97	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐴–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
99	98	eqcomd 2742	. 2 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) = (𝐴–cn→ℂ))
100	95, 99	eleqtrd 2838	1 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  ∪ cuni 4850  dom cdm 5631   ↾ cres 5633  Rel wrel 5636  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ℂcc 11036   ↾_t crest 17383  TopOpenctopn 17384  ℂ_fldccnfld 21352  Topctop 22858  TopOnctopon 22875  intcnt 22982   Cn ccn 23189   CnP ccnp 23190  –cn→ccncf 24843

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

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-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-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-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fi 9324  df-sup 9355  df-inf 9356  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-fz 13462  df-seq 13964  df-exp 14024  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-mulr 17234  df-starv 17235  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-rest 17385  df-topn 17386  df-topgen 17406  df-psmet 21344  df-xmet 21345  df-met 21346  df-bl 21347  df-mopn 21348  df-cnfld 21353  df-top 22859  df-topon 22876  df-topsp 22898  df-bases 22911  df-ntr 22985  df-cn 23192  df-cnp 23193  df-xms 24285  df-ms 24286  df-cncf 24845

This theorem is referenced by:  fouriersw  46659