Theorem ucnextcn 22910

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 22870	. . . . . 6 ⊢ ((𝑉 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
15	12, 10, 14	syl2anc 587	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (UnifOn‘𝐴))
16		cuspusp 22906	. . . . . . . 8 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
17	8, 16	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ UnifSp)
18	2, 5, 4	isusp 22867	. . . . . . 7 ⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑌) ∧ 𝐾 = (unifTop‘𝑈)))
19	17, 18	sylib 221	. . . . . 6 ⊢ (𝜑 → (𝑈 ∈ (UnifOn‘𝑌) ∧ 𝐾 = (unifTop‘𝑈)))
20	19	simpld 498	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑌))
21		isucn 22884	. . . . 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 587	. . . 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 235	. . 3 ⊢ (𝜑 → (𝐹:𝐴⟶𝑌 ∧ ∀𝑤 ∈ 𝑈 ∃𝑣 ∈ 𝑇 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑣𝑧 → (𝐹‘𝑦)𝑤(𝐹‘𝑧))))
24	23	simpld 498	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝑌)
25		ucnextcn.c	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
26	20	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑈 ∈ (UnifOn‘𝑌))
27	26	elfvexd 6679	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ V)
28		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
29	25	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((cls‘𝐽)‘𝐴) = 𝑋)
30	28, 29	eleqtrrd 2893	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
31	1, 3	istps 21539	. . . . . . . . 9 ⊢ (𝑉 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
32	6, 31	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
33	32	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34	10	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ⊆ 𝑋)
35		trnei 22497	. . . . . . 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 1368	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
37	30, 36	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
38		filfbas 22453	. . . . 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹:𝐴⟶𝑌)
41		fmval 22548	. . . 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 1368	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎))))
43	15	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
44	11	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹 ∈ (𝑇 Cnu𝑈))
45		ucnextcn.s	. . . . . . . . . . 11 ⊢ 𝑆 = (UnifSt‘𝑉)
46	1, 45, 3	isusp 22867	. . . . . . . . . 10 ⊢ (𝑉 ∈ UnifSp ↔ (𝑆 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑆)))
47	12, 46	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑆)))
48	47	simpld 498	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (UnifOn‘𝑋))
49	48	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ (UnifOn‘𝑋))
50	12	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑉 ∈ UnifSp)
51	6	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑉 ∈ TopSp)
52	1, 3, 45	neipcfilu 22902	. . . . . . . 8 ⊢ ((𝑉 ∈ UnifSp ∧ 𝑉 ∈ TopSp ∧ 𝑥 ∈ 𝑋) → ((nei‘𝐽)‘{𝑥}) ∈ (CauFilu‘𝑆))
53	50, 51, 28, 52	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((nei‘𝐽)‘{𝑥}) ∈ (CauFilu‘𝑆))
54		0nelfb 22436	. . . . . . . 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 22900	. . . . . . 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 1372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu‘(𝑆 ↾t (𝐴 × 𝐴))))
58	43	elfvexd 6679	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ V)
59		ressuss 22869	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (UnifSt‘(𝑉 ↾s 𝐴)) = ((UnifSt‘𝑉) ↾t (𝐴 × 𝐴)))
60	45	oveq1i 7145	. . . . . . . . 9 ⊢ (𝑆 ↾t (𝐴 × 𝐴)) = ((UnifSt‘𝑉) ↾t (𝐴 × 𝐴))
61	59, 13, 60	3eqtr4g 2858	. . . . . . . 8 ⊢ (𝐴 ∈ V → 𝑇 = (𝑆 ↾t (𝐴 × 𝐴)))
62	61	fveq2d 6649	. . . . . . 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 2893	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu‘𝑇))
65		imaeq2 5892	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝐹 “ 𝑎) = (𝐹 “ 𝑏))
66	65	cbvmptv 5133	. . . . . 6 ⊢ (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎)) = (𝑏 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑏))
67	66	rneqi 5771	. . . . 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 22898	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎)) ∈ (CauFilu‘𝑈))
69		cfilufg 22899	. . . 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 587	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎))) ∈ (CauFilu‘𝑈))
71	42, 70	eqeltrd 2890	. 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 22909	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  {csn 4525   class class class wbr 5030   ↦ cmpt 5110   × cxp 5517  ran crn 5520   “ cima 5522  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  Basecbs 16475   ↾_s cress 16476   ↾_t crest 16686  TopOpenctopn 16687  fBascfbas 20079  filGencfg 20080  TopOnctopon 21515  TopSpctps 21537  clsccl 21623  neicnei 21702   Cn ccn 21829  Hauscha 21913  Filcfil 22450   FilMap cfm 22538  CnExtccnext 22664  UnifOncust 22805  unifTopcutop 22836  UnifStcuss 22859  UnifSpcusp 22860   Cn_ucucn 22881  CauFil_uccfilu 22892  CUnifSpccusp 22903

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-unif 16580  df-rest 16688  df-topgen 16709  df-fbas 20088  df-fg 20089  df-top 21499  df-topon 21516  df-topsp 21538  df-bases 21551  df-cld 21624  df-ntr 21625  df-cls 21626  df-nei 21703  df-cn 21832  df-cnp 21833  df-haus 21920  df-reg 21921  df-tx 22167  df-fil 22451  df-fm 22543  df-flim 22544  df-flf 22545  df-cnext 22665  df-ust 22806  df-utop 22837  df-uss 22862  df-usp 22863  df-ucn 22882  df-cfilu 22893  df-cusp 22904

This theorem is referenced by:  rrhcn  31348