Theorem ucnextcn 22915

Description: Extension by continuity. Theorem 2 of [BourbakiTop1] p. II.20. Given an uniform space on a set 𝑋, a subset 𝐴 dense in 𝑋, and a function 𝐹 uniformly continuous from 𝐴 to 𝑌, that function can be extended by continuity to the whole 𝑋, and its extension is uniformly continuous. (Contributed by Thierry Arnoux, 25-Jan-2018.)

Hypotheses

Ref	Expression
ucnextcn.x	⊢ 𝑋 = (Base‘𝑉)
ucnextcn.y	⊢ 𝑌 = (Base‘𝑊)
ucnextcn.j	⊢ 𝐽 = (TopOpen‘𝑉)
ucnextcn.k	⊢ 𝐾 = (TopOpen‘𝑊)
ucnextcn.s	⊢ 𝑆 = (UnifSt‘𝑉)
ucnextcn.t	⊢ 𝑇 = (UnifSt‘(𝑉 ↾s 𝐴))
ucnextcn.u	⊢ 𝑈 = (UnifSt‘𝑊)
ucnextcn.v	⊢ (𝜑 → 𝑉 ∈ TopSp)
ucnextcn.r	⊢ (𝜑 → 𝑉 ∈ UnifSp)
ucnextcn.w	⊢ (𝜑 → 𝑊 ∈ TopSp)
ucnextcn.z	⊢ (𝜑 → 𝑊 ∈ CUnifSp)
ucnextcn.h	⊢ (𝜑 → 𝐾 ∈ Haus)
ucnextcn.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
ucnextcn.f	⊢ (𝜑 → 𝐹 ∈ (𝑇 Cnu𝑈))
ucnextcn.c	⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)

Assertion

Ref	Expression
ucnextcn	⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Proof of Theorem ucnextcn

Dummy variables 𝑎 𝑏 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ucnextcn.x	. 2 ⊢ 𝑋 = (Base‘𝑉)
2		ucnextcn.y	. 2 ⊢ 𝑌 = (Base‘𝑊)
3		ucnextcn.j	. 2 ⊢ 𝐽 = (TopOpen‘𝑉)
4		ucnextcn.k	. 2 ⊢ 𝐾 = (TopOpen‘𝑊)
5		ucnextcn.u	. 2 ⊢ 𝑈 = (UnifSt‘𝑊)
6		ucnextcn.v	. 2 ⊢ (𝜑 → 𝑉 ∈ TopSp)
7		ucnextcn.w	. 2 ⊢ (𝜑 → 𝑊 ∈ TopSp)
8		ucnextcn.z	. 2 ⊢ (𝜑 → 𝑊 ∈ CUnifSp)
9		ucnextcn.h	. 2 ⊢ (𝜑 → 𝐾 ∈ Haus)
10		ucnextcn.a	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
11		ucnextcn.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑇 Cnu𝑈))
12		ucnextcn.r	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ UnifSp)
13		ucnextcn.t	. . . . . . 7 ⊢ 𝑇 = (UnifSt‘(𝑉 ↾s 𝐴))
14	1, 13	ressust 22875	. . . . . 6 ⊢ ((𝑉 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
15	12, 10, 14	syl2anc 586	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (UnifOn‘𝐴))
16		cuspusp 22911	. . . . . . . 8 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
17	8, 16	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ UnifSp)
18	2, 5, 4	isusp 22872	. . . . . . 7 ⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑌) ∧ 𝐾 = (unifTop‘𝑈)))
19	17, 18	sylib 220	. . . . . 6 ⊢ (𝜑 → (𝑈 ∈ (UnifOn‘𝑌) ∧ 𝐾 = (unifTop‘𝑈)))
20	19	simpld 497	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑌))
21		isucn 22889	. . . . 5 ⊢ ((𝑇 ∈ (UnifOn‘𝐴) ∧ 𝑈 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑇 Cnu𝑈) ↔ (𝐹:𝐴⟶𝑌 ∧ ∀𝑤 ∈ 𝑈 ∃𝑣 ∈ 𝑇 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑣𝑧 → (𝐹‘𝑦)𝑤(𝐹‘𝑧)))))
22	15, 20, 21	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑇 Cnu𝑈) ↔ (𝐹:𝐴⟶𝑌 ∧ ∀𝑤 ∈ 𝑈 ∃𝑣 ∈ 𝑇 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑣𝑧 → (𝐹‘𝑦)𝑤(𝐹‘𝑧)))))
23	11, 22	mpbid 234	. . 3 ⊢ (𝜑 → (𝐹:𝐴⟶𝑌 ∧ ∀𝑤 ∈ 𝑈 ∃𝑣 ∈ 𝑇 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑣𝑧 → (𝐹‘𝑦)𝑤(𝐹‘𝑧))))
24	23	simpld 497	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝑌)
25		ucnextcn.c	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
26	20	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑈 ∈ (UnifOn‘𝑌))
27	26	elfvexd 6706	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ V)
28		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
29	25	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((cls‘𝐽)‘𝐴) = 𝑋)
30	28, 29	eleqtrrd 2918	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
31	1, 3	istps 21544	. . . . . . . . 9 ⊢ (𝑉 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
32	6, 31	sylib 220	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
33	32	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34	10	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ⊆ 𝑋)
35		trnei 22502	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
36	33, 34, 28, 35	syl3anc 1367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
37	30, 36	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
38		filfbas 22458	. . . . 5 ⊢ ((((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
39	37, 38	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
40	24	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹:𝐴⟶𝑌)
41		fmval 22553	. . . 4 ⊢ ((𝑌 ∈ V ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴) ∧ 𝐹:𝐴⟶𝑌) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎))))
42	27, 39, 40, 41	syl3anc 1367	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎))))
43	15	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
44	11	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹 ∈ (𝑇 Cnu𝑈))
45		ucnextcn.s	. . . . . . . . . . 11 ⊢ 𝑆 = (UnifSt‘𝑉)
46	1, 45, 3	isusp 22872	. . . . . . . . . 10 ⊢ (𝑉 ∈ UnifSp ↔ (𝑆 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑆)))
47	12, 46	sylib 220	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑆)))
48	47	simpld 497	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (UnifOn‘𝑋))
49	48	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ (UnifOn‘𝑋))
50	12	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑉 ∈ UnifSp)
51	6	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑉 ∈ TopSp)
52	1, 3, 45	neipcfilu 22907	. . . . . . . 8 ⊢ ((𝑉 ∈ UnifSp ∧ 𝑉 ∈ TopSp ∧ 𝑥 ∈ 𝑋) → ((nei‘𝐽)‘{𝑥}) ∈ (CauFilu‘𝑆))
53	50, 51, 28, 52	syl3anc 1367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((nei‘𝐽)‘{𝑥}) ∈ (CauFilu‘𝑆))
54		0nelfb 22441	. . . . . . . 8 ⊢ ((((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴) → ¬ ∅ ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))
55	39, 54	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ ∅ ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))
56		trcfilu 22905	. . . . . . 7 ⊢ ((𝑆 ∈ (UnifOn‘𝑋) ∧ (((nei‘𝐽)‘{𝑥}) ∈ (CauFilu‘𝑆) ∧ ¬ ∅ ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu‘(𝑆 ↾t (𝐴 × 𝐴))))
57	49, 53, 55, 34, 56	syl121anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu‘(𝑆 ↾t (𝐴 × 𝐴))))
58	43	elfvexd 6706	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ V)
59		ressuss 22874	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (UnifSt‘(𝑉 ↾s 𝐴)) = ((UnifSt‘𝑉) ↾t (𝐴 × 𝐴)))
60	45	oveq1i 7168	. . . . . . . . 9 ⊢ (𝑆 ↾t (𝐴 × 𝐴)) = ((UnifSt‘𝑉) ↾t (𝐴 × 𝐴))
61	59, 13, 60	3eqtr4g 2883	. . . . . . . 8 ⊢ (𝐴 ∈ V → 𝑇 = (𝑆 ↾t (𝐴 × 𝐴)))
62	61	fveq2d 6676	. . . . . . 7 ⊢ (𝐴 ∈ V → (CauFilu‘𝑇) = (CauFilu‘(𝑆 ↾t (𝐴 × 𝐴))))
63	58, 62	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (CauFilu‘𝑇) = (CauFilu‘(𝑆 ↾t (𝐴 × 𝐴))))
64	57, 63	eleqtrrd 2918	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu‘𝑇))
65		imaeq2 5927	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝐹 “ 𝑎) = (𝐹 “ 𝑏))
66	65	cbvmptv 5171	. . . . . 6 ⊢ (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎)) = (𝑏 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑏))
67	66	rneqi 5809	. . . . 5 ⊢ ran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎)) = ran (𝑏 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑏))
68	43, 26, 44, 64, 67	fmucnd 22903	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎)) ∈ (CauFilu‘𝑈))
69		cfilufg 22904	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑌) ∧ ran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎)) ∈ (CauFilu‘𝑈)) → (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎))) ∈ (CauFilu‘𝑈))
70	26, 68, 69	syl2anc 586	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎))) ∈ (CauFilu‘𝑈))
71	42, 70	eqeltrd 2915	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘𝑈))
72	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 71	cnextucn 22914	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ⊆ wss 3938  ∅c0 4293  {csn 4569   class class class wbr 5068   ↦ cmpt 5148   × cxp 5555  ran crn 5558   “ cima 5560  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  Basecbs 16485   ↾_s cress 16486   ↾_t crest 16696  TopOpenctopn 16697  fBascfbas 20535  filGencfg 20536  TopOnctopon 21520  TopSpctps 21542  clsccl 21628  neicnei 21707   Cn ccn 21834  Hauscha 21918  Filcfil 22455   FilMap cfm 22543  CnExtccnext 22669  UnifOncust 22810  unifTopcutop 22841  UnifStcuss 22864  UnifSpcusp 22865   Cn_ucucn 22886  CauFil_uccfilu 22897  CUnifSpccusp 22908

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fi 8877  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-unif 16590  df-rest 16698  df-topgen 16719  df-fbas 20544  df-fg 20545  df-top 21504  df-topon 21521  df-topsp 21543  df-bases 21556  df-cld 21629  df-ntr 21630  df-cls 21631  df-nei 21708  df-cn 21837  df-cnp 21838  df-haus 21925  df-reg 21926  df-tx 22172  df-fil 22456  df-fm 22548  df-flim 22549  df-flf 22550  df-cnext 22670  df-ust 22811  df-utop 22842  df-uss 22867  df-usp 22868  df-ucn 22887  df-cfilu 22898  df-cusp 22909

This theorem is referenced by:  rrhcn  31240