Theorem cncfuni 45842

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 3995	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐵)
4		eluni2 4916	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑏 ∈ 𝐵 𝑥 ∈ 𝑏)
5	3, 4	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑏 ∈ 𝐵 𝑥 ∈ 𝑏)
6		simp1l 1196	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝜑)
7		simp2 1136	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝑏 ∈ 𝐵)
8		elin 3979	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∩ 𝑏) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏))
9	8	biimpri 228	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏) → 𝑥 ∈ (𝐴 ∩ 𝑏))
10	9	adantll 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝑏) → 𝑥 ∈ (𝐴 ∩ 𝑏))
11	10	3adant2 1130	. . . . . . 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 6747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = 𝐴)
14	13	ineq2d 4228	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑏 ∩ dom 𝐹) = (𝑏 ∩ 𝐴))
15		incom 4217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∩ 𝐴) = (𝐴 ∩ 𝑏)
16	14, 15	eqtr2di 2792	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∩ 𝑏) = (𝑏 ∩ dom 𝐹))
17	16	reseq2d 6000	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑏)) = (𝐹 ↾ (𝑏 ∩ dom 𝐹)))
18		frel 6742	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:𝐴⟶ℂ → Rel 𝐹)
19	1, 18	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Rel 𝐹)
20		resindm 6050	. . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑏)) = (𝐹 ↾ 𝑏))
23		inss1 4245	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∩ 𝑏) ⊆ 𝐴
24	23	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 ∩ 𝑏) ⊆ 𝐴)
25		cncfuni.acn	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
26	24, 25	sstrd 4006	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∩ 𝑏) ⊆ ℂ)
27		ssidd 4019	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ℂ ⊆ ℂ)
28		eqid 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
29		eqid 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏))
30	28	cnfldtop 24820	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (TopOpen‘ℂfld) ∈ Top
31		unicntop 24822	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
32	31	restid 17480	. . . . . . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
35	28, 29, 34	cncfcn 24950	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∩ 𝑏) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴 ∩ 𝑏)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)))
36	26, 27, 35	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐴 ∩ 𝑏)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)))
37	36	eqcomd 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)) = ((𝐴 ∩ 𝑏)–cn→ℂ))
38	22, 37	eleq12d 2833	. . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)))
42	28	cnfldtopon 24819	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
44		resttopon 23185	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐴 ∩ 𝑏) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) ∈ (TopOn‘(𝐴 ∩ 𝑏)))
45	43, 26, 44	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) ∈ (TopOn‘(𝐴 ∩ 𝑏)))
46	45	3ad2ant1 1132	. . . . . . . . . . . . 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 23304	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → 𝑥 ∈ (𝐴 ∩ 𝑏))
53		rspa 3246	. . . . . . . . . 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 584	. . . . . . . . 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 11234	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
57	56	ssex 5327	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ℂ → 𝐴 ∈ V)
58	25, 57	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ V)
59		restabs 23189	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)))
61	60	eqcomd 2741	. . . . . . . . . . . 12 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) = (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)))
62	61	oveq1d 7446	. . . . . . . . . . 11 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld)) = ((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld)))
63	62	fveq1d 6909	. . . . . . . . . 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 1132	. . . . . . . . 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 2841	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
66		resttop 23184	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
67	55, 58, 66	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
68	67	3ad2ant1 1132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
69	31	restuni 23186	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ⊆ ℂ) → 𝐴 = ∪ ((TopOpen‘ℂfld) ↾t 𝐴))
70	55, 25, 69	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 = ∪ ((TopOpen‘ℂfld) ↾t 𝐴))
71	24, 70	sseqtrd 4036	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ 𝑏) ⊆ ∪ ((TopOpen‘ℂfld) ↾t 𝐴))
72	71	3ad2ant1 1132	. . . . . . . . 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 1131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐴 ∩ 𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))
75		eqid 2735	. . . . . . . . . . . . . 14 ⊢ ∪ ((TopOpen‘ℂfld) ↾t 𝐴) = ∪ ((TopOpen‘ℂfld) ↾t 𝐴)
76	75	isopn3 23090	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 2741	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴 ∩ 𝑏)))
80	52, 79	eleqtrd 2841	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴 ∩ 𝑏)))
81	70	feq2d 6723	. . . . . . . . . . 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 1132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → 𝐹:∪ ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ)
84	75, 31	cnprest 23313	. . . . . . . . 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 1370	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
88	87	rexlimdv3a 3157	. . . . 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 3144	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
91		resttopon 23185	. . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝐴 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴))
92	43, 25, 91	syl2anc 584	. . . 4 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴))
93		cncnp 23304	. . . 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 584	. . 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 713	. 2 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
96		eqid 2735	. . . . 5 ⊢ ((TopOpen‘ℂfld) ↾t 𝐴) = ((TopOpen‘ℂfld) ↾t 𝐴)
97	28, 96, 34	cncfcn 24950	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐴–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
98	25, 27, 97	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐴–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
99	98	eqcomd 2741	. 2 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) = (𝐴–cn→ℂ))
100	95, 99	eleqtrd 2841	1 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ∩ cin 3962   ⊆ wss 3963  ∪ cuni 4912  dom cdm 5689   ↾ cres 5691  Rel wrel 5694  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  ℂcc 11151   ↾_t crest 17467  TopOpenctopn 17468  ℂ_fldccnfld 21382  Topctop 22915  TopOnctopon 22932  intcnt 23041   Cn ccn 23248   CnP ccnp 23249  –cn→ccncf 24916

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

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-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-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-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-sup 9480  df-inf 9481  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-fz 13545  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-starv 17313  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-rest 17469  df-topn 17470  df-topgen 17490  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-ntr 23044  df-cn 23251  df-cnp 23252  df-xms 24346  df-ms 24347  df-cncf 24918

This theorem is referenced by:  fouriersw  46187