Theorem ucnextcn 21918

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 21878	. . . . . 6 ⊢ ((𝑉 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
15	12, 10, 14	syl2anc 691	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (UnifOn‘𝐴))
16		cuspusp 21914	. . . . . . . 8 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
17	8, 16	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ UnifSp)
18	2, 5, 4	isusp 21875	. . . . . . 7 ⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑌) ∧ 𝐾 = (unifTop‘𝑈)))
19	17, 18	sylib 207	. . . . . 6 ⊢ (𝜑 → (𝑈 ∈ (UnifOn‘𝑌) ∧ 𝐾 = (unifTop‘𝑈)))
20	19	simpld 474	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑌))
21		isucn 21892	. . . . 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 691	. . . 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 221	. . 3 ⊢ (𝜑 → (𝐹:𝐴⟶𝑌 ∧ ∀𝑤 ∈ 𝑈 ∃𝑣 ∈ 𝑇 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑣𝑧 → (𝐹‘𝑦)𝑤(𝐹‘𝑧))))
24	23	simpld 474	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝑌)
25		ucnextcn.c	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
26	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑈 ∈ (UnifOn‘𝑌))
27	26	elfvexd 6132	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ V)
28		simpr 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
29	25	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((cls‘𝐽)‘𝐴) = 𝑋)
30	28, 29	eleqtrrd 2691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
31	1, 3	istps 20551	. . . . . . . . 9 ⊢ (𝑉 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
32	6, 31	sylib 207	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
33	32	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ⊆ 𝑋)
35		trnei 21506	. . . . . . 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 1318	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
37	30, 36	mpbid 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
38		filfbas 21462	. . . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹:𝐴⟶𝑌)
41		fmval 21557	. . . 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 1318	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎))))
43	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
44	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹 ∈ (𝑇 Cnu𝑈))
45		ucnextcn.s	. . . . . . . . . . 11 ⊢ 𝑆 = (UnifSt‘𝑉)
46	1, 45, 3	isusp 21875	. . . . . . . . . 10 ⊢ (𝑉 ∈ UnifSp ↔ (𝑆 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑆)))
47	12, 46	sylib 207	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑆)))
48	47	simpld 474	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (UnifOn‘𝑋))
49	48	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ (UnifOn‘𝑋))
50	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑉 ∈ UnifSp)
51	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑉 ∈ TopSp)
52	1, 3, 45	neipcfilu 21910	. . . . . . . 8 ⊢ ((𝑉 ∈ UnifSp ∧ 𝑉 ∈ TopSp ∧ 𝑥 ∈ 𝑋) → ((nei‘𝐽)‘{𝑥}) ∈ (CauFilu‘𝑆))
53	50, 51, 28, 52	syl3anc 1318	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((nei‘𝐽)‘{𝑥}) ∈ (CauFilu‘𝑆))
54		0nelfb 21445	. . . . . . . 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 21908	. . . . . . 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 1323	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu‘(𝑆 ↾t (𝐴 × 𝐴))))
58	43	elfvexd 6132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ V)
59		ressuss 21877	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (UnifSt‘(𝑉 ↾s 𝐴)) = ((UnifSt‘𝑉) ↾t (𝐴 × 𝐴)))
60	45	oveq1i 6559	. . . . . . . . 9 ⊢ (𝑆 ↾t (𝐴 × 𝐴)) = ((UnifSt‘𝑉) ↾t (𝐴 × 𝐴))
61	59, 13, 60	3eqtr4g 2669	. . . . . . . 8 ⊢ (𝐴 ∈ V → 𝑇 = (𝑆 ↾t (𝐴 × 𝐴)))
62	61	fveq2d 6107	. . . . . . 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 2691	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu‘𝑇))
65		imaeq2 5381	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝐹 “ 𝑎) = (𝐹 “ 𝑏))
66	65	cbvmptv 4678	. . . . . 6 ⊢ (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎)) = (𝑏 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑏))
67	66	rneqi 5273	. . . . 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 21906	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎)) ∈ (CauFilu‘𝑈))
69		cfilufg 21907	. . . 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 691	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹 “ 𝑎))) ∈ (CauFilu‘𝑈))
71	42, 70	eqeltrd 2688	. 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 21917	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ran crn 5039   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾_s cress 15696   ↾_t crest 15904  TopOpenctopn 15905  fBascfbas 19555  filGencfg 19556  TopOnctopon 20518  TopSpctps 20519  clsccl 20632  neicnei 20711   Cn ccn 20838  Hauscha 20922  Filcfil 21459   FilMap cfm 21547  CnExtccnext 21673  UnifOncust 21813  unifTopcutop 21844  UnifStcuss 21867  UnifSpcusp 21868   Cn_ucucn 21889  CauFil_uccfilu 21900  CUnifSpccusp 21911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-unif 15792  df-rest 15906  df-topgen 15927  df-fbas 19564  df-fg 19565  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-cn 20841  df-cnp 20842  df-haus 20929  df-reg 20930  df-tx 21175  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-cnext 21674  df-ust 21814  df-utop 21845  df-uss 21870  df-usp 21871  df-ucn 21890  df-cfilu 21901  df-cusp 21912

This theorem is referenced by:  rrhcn  29369