Theorem ucnextcn 22485

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 22445	. . . . . 6 ⊢ ((𝑉 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
15	12, 10, 14	syl2anc 579	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (UnifOn‘𝐴))
16		cuspusp 22481	. . . . . . . 8 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
17	8, 16	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ UnifSp)
18	2, 5, 4	isusp 22442	. . . . . . 7 ⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑌) ∧ 𝐾 = (unifTop‘𝑈)))
19	17, 18	sylib 210	. . . . . 6 ⊢ (𝜑 → (𝑈 ∈ (UnifOn‘𝑌) ∧ 𝐾 = (unifTop‘𝑈)))
20	19	simpld 490	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑌))
21		isucn 22459	. . . . 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 579	. . . 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 224	. . 3 ⊢ (𝜑 → (𝐹:𝐴⟶𝑌 ∧ ∀𝑤 ∈ 𝑈 ∃𝑣 ∈ 𝑇 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑣𝑧 → (𝐹‘𝑦)𝑤(𝐹‘𝑧))))
24	23	simpld 490	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝑌)
25		ucnextcn.c	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
26	20	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑈 ∈ (UnifOn‘𝑌))
27	26	elfvexd 6472	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ V)
28		simpr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
29	25	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((cls‘𝐽)‘𝐴) = 𝑋)
30	28, 29	eleqtrrd 2909	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
31	1, 3	istps 21116	. . . . . . . . 9 ⊢ (𝑉 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
32	6, 31	sylib 210	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
33	32	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34	10	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ⊆ 𝑋)
35		trnei 22073	. . . . . . 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 1494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
37	30, 36	mpbid 224	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
38		filfbas 22029	. . . . 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 474	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹:𝐴⟶𝑌)
41		fmval 22124	. . . 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 1494	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎))))
43	15	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
44	11	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹 ∈ (𝑇 Cnu𝑈))
45		ucnextcn.s	. . . . . . . . . . 11 ⊢ 𝑆 = (UnifSt‘𝑉)
46	1, 45, 3	isusp 22442	. . . . . . . . . 10 ⊢ (𝑉 ∈ UnifSp ↔ (𝑆 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑆)))
47	12, 46	sylib 210	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑆)))
48	47	simpld 490	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (UnifOn‘𝑋))
49	48	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ (UnifOn‘𝑋))
50	12	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑉 ∈ UnifSp)
51	6	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑉 ∈ TopSp)
52	1, 3, 45	neipcfilu 22477	. . . . . . . 8 ⊢ ((𝑉 ∈ UnifSp ∧ 𝑉 ∈ TopSp ∧ 𝑥 ∈ 𝑋) → ((nei‘𝐽)‘{𝑥}) ∈ (CauFilu‘𝑆))
53	50, 51, 28, 52	syl3anc 1494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((nei‘𝐽)‘{𝑥}) ∈ (CauFilu‘𝑆))
54		0nelfb 22012	. . . . . . . 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 22475	. . . . . . 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 1498	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu‘(𝑆 ↾t (𝐴 × 𝐴))))
58	43	elfvexd 6472	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ V)
59		ressuss 22444	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (UnifSt‘(𝑉 ↾s 𝐴)) = ((UnifSt‘𝑉) ↾t (𝐴 × 𝐴)))
60	45	oveq1i 6920	. . . . . . . . 9 ⊢ (𝑆 ↾t (𝐴 × 𝐴)) = ((UnifSt‘𝑉) ↾t (𝐴 × 𝐴))
61	59, 13, 60	3eqtr4g 2886	. . . . . . . 8 ⊢ (𝐴 ∈ V → 𝑇 = (𝑆 ↾t (𝐴 × 𝐴)))
62	61	fveq2d 6441	. . . . . . 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 2909	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu‘𝑇))
65		imaeq2 5707	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝐹 “ 𝑎) = (𝐹 “ 𝑏))
66	65	cbvmptv 4975	. . . . . 6 ⊢ (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎)) = (𝑏 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑏))
67	66	rneqi 5588	. . . . 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 22473	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎)) ∈ (CauFilu‘𝑈))
69		cfilufg 22474	. . . 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 579	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎))) ∈ (CauFilu‘𝑈))
71	42, 70	eqeltrd 2906	. 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 22484	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∀wral 3117  ∃wrex 3118  Vcvv 3414   ⊆ wss 3798  ∅c0 4146  {csn 4399   class class class wbr 4875   ↦ cmpt 4954   × cxp 5344  ran crn 5347   “ cima 5349  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910  Basecbs 16229   ↾_s cress 16230   ↾_t crest 16441  TopOpenctopn 16442  fBascfbas 20101  filGencfg 20102  TopOnctopon 21092  TopSpctps 21114  clsccl 21200  neicnei 21279   Cn ccn 21406  Hauscha 21490  Filcfil 22026   FilMap cfm 22114  CnExtccnext 22240  UnifOncust 22380  unifTopcutop 22411  UnifStcuss 22434  UnifSpcusp 22435   Cn_ucucn 22456  CauFil_uccfilu 22467  CUnifSpccusp 22478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-fi 8592  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-3 11422  df-4 11423  df-5 11424  df-6 11425  df-7 11426  df-8 11427  df-9 11428  df-n0 11626  df-z 11712  df-dec 11829  df-ndx 16232  df-slot 16233  df-base 16235  df-sets 16236  df-ress 16237  df-unif 16335  df-rest 16443  df-topgen 16464  df-fbas 20110  df-fg 20111  df-top 21076  df-topon 21093  df-topsp 21115  df-bases 21128  df-cld 21201  df-ntr 21202  df-cls 21203  df-nei 21280  df-cn 21409  df-cnp 21410  df-haus 21497  df-reg 21498  df-tx 21743  df-fil 22027  df-fm 22119  df-flim 22120  df-flf 22121  df-cnext 22241  df-ust 22381  df-utop 22412  df-uss 22437  df-usp 22438  df-ucn 22457  df-cfilu 22468  df-cusp 22479

This theorem is referenced by:  rrhcn  30582