Theorem cncfuni 45998

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 3931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐵)
4		eluni2 4864	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑏 ∈ 𝐵 𝑥 ∈ 𝑏)
5	3, 4	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑏 ∈ 𝐵 𝑥 ∈ 𝑏)
6		simp1l 1198	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝜑)
7		simp2 1137	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝑏 ∈ 𝐵)
8		elin 3915	. . . . . . . . . 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 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 6669	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝐹 = 𝐴)
14	13	ineq2d 4171	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑏 ∩ dom 𝐹) = (𝑏 ∩ 𝐴))
15		incom 4160	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∩ 𝐴) = (𝐴 ∩ 𝑏)
16	14, 15	eqtr2di 2785	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∩ 𝑏) = (𝑏 ∩ dom 𝐹))
17	16	reseq2d 5935	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑏)) = (𝐹 ↾ (𝑏 ∩ dom 𝐹)))
18		frel 6664	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:𝐴⟶ℂ → Rel 𝐹)
19	1, 18	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Rel 𝐹)
20		resindm 5986	. . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑏)) = (𝐹 ↾ 𝑏))
23		inss1 4188	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∩ 𝑏) ⊆ 𝐴
24	23	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 ∩ 𝑏) ⊆ 𝐴)
25		cncfuni.acn	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
26	24, 25	sstrd 3942	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∩ 𝑏) ⊆ ℂ)
27		ssidd 3955	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ℂ ⊆ ℂ)
28		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
29		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏))
30	28	cnfldtop 24708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (TopOpen‘ℂfld) ∈ Top
31		unicntop 24710	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
32	31	restid 17347	. . . . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . . 19 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
35	28, 29, 34	cncfcn 24840	. . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) Cn (TopOpen‘ℂfld)) = ((𝐴 ∩ 𝑏)–cn→ℂ))
38	22, 37	eleq12d 2827	. . . . . . . . . . . . . . 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 24707	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
44		resttopon 23086	. . . . . . . . . . . . . . 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 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 23205	. . . . . . . . . . . . 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 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → 𝑥 ∈ (𝐴 ∩ 𝑏))
53		rspa 3223	. . . . . . . . . 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 11097	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
57	56	ssex 5263	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ℂ → 𝐴 ∈ V)
58	25, 57	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ V)
59		restabs 23090	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)))
61	60	eqcomd 2739	. . . . . . . . . . . 12 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) = (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)))
62	61	oveq1d 7370	. . . . . . . . . . 11 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld)) = ((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld)))
63	62	fveq1d 6833	. . . . . . . . . 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 2835	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐹 ↾ (𝐴 ∩ 𝑏)) ∈ (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴 ∩ 𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
66		resttop 23085	. . . . . . . . . . 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 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
69	31	restuni 23087	. . . . . . . . . . . 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 3968	. . . . . . . . . 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 2733	. . . . . . . . . . . . . 14 ⊢ ∪ ((TopOpen‘ℂfld) ↾t 𝐴) = ∪ ((TopOpen‘ℂfld) ↾t 𝐴)
76	75	isopn3 22991	. . . . . . . . . . . . 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 2739	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴 ∩ 𝑏)))
80	52, 79	eleqtrd 2835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∩ 𝑏)) → 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴 ∩ 𝑏)))
81	70	feq2d 6643	. . . . . . . . . . 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 23214	. . . . . . . . 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 1373	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵 ∧ 𝑥 ∈ 𝑏) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
88	87	rexlimdv3a 3139	. . . . 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 3126	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
91		resttopon 23086	. . . . 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 23205	. . . 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 2733	. . . . 5 ⊢ ((TopOpen‘ℂfld) ↾t 𝐴) = ((TopOpen‘ℂfld) ↾t 𝐴)
97	28, 96, 34	cncfcn 24840	. . . 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 2739	. 2 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) = (𝐴–cn→ℂ))
100	95, 99	eleqtrd 2835	1 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ∩ cin 3898   ⊆ wss 3899  ∪ cuni 4860  dom cdm 5621   ↾ cres 5623  Rel wrel 5626  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355  ℂcc 11014   ↾_t crest 17334  TopOpenctopn 17335  ℂ_fldccnfld 21301  Topctop 22818  TopOnctopon 22835  intcnt 22942   Cn ccn 23149   CnP ccnp 23150  –cn→ccncf 24806

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11072  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092  ax-pre-mulgt0 11093  ax-pre-sup 11094

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-map 8761  df-en 8879  df-dom 8880  df-sdom 8881  df-fin 8882  df-fi 9305  df-sup 9336  df-inf 9337  df-pnf 11158  df-mnf 11159  df-xr 11160  df-ltxr 11161  df-le 11162  df-sub 11356  df-neg 11357  df-div 11785  df-nn 12136  df-2 12198  df-3 12199  df-4 12200  df-5 12201  df-6 12202  df-7 12203  df-8 12204  df-9 12205  df-n0 12392  df-z 12479  df-dec 12599  df-uz 12743  df-q 12857  df-rp 12901  df-xneg 13021  df-xadd 13022  df-xmul 13023  df-fz 13418  df-seq 13919  df-exp 13979  df-cj 15016  df-re 15017  df-im 15018  df-sqrt 15152  df-abs 15153  df-struct 17068  df-slot 17103  df-ndx 17115  df-base 17131  df-plusg 17184  df-mulr 17185  df-starv 17186  df-tset 17190  df-ple 17191  df-ds 17193  df-unif 17194  df-rest 17336  df-topn 17337  df-topgen 17357  df-psmet 21293  df-xmet 21294  df-met 21295  df-bl 21296  df-mopn 21297  df-cnfld 21302  df-top 22819  df-topon 22836  df-topsp 22858  df-bases 22871  df-ntr 22945  df-cn 23152  df-cnp 23153  df-xms 24245  df-ms 24246  df-cncf 24808

This theorem is referenced by:  fouriersw  46343