Theorem cnextucn 24201

Description: Extension by continuity. Proposition 11 of [BourbakiTop1] p. II.20. Given a topology 𝐽 on 𝑋, a subset 𝐴 dense in 𝑋, this states a condition for 𝐹 from 𝐴 to a space 𝑌 Hausdorff and complete to be extensible by continuity. (Contributed by Thierry Arnoux, 4-Dec-2017.)

Hypotheses

Ref	Expression
cnextucn.x	⊢ 𝑋 = (Base‘𝑉)
cnextucn.y	⊢ 𝑌 = (Base‘𝑊)
cnextucn.j	⊢ 𝐽 = (TopOpen‘𝑉)
cnextucn.k	⊢ 𝐾 = (TopOpen‘𝑊)
cnextucn.u	⊢ 𝑈 = (UnifSt‘𝑊)
cnextucn.v	⊢ (𝜑 → 𝑉 ∈ TopSp)
cnextucn.t	⊢ (𝜑 → 𝑊 ∈ TopSp)
cnextucn.w	⊢ (𝜑 → 𝑊 ∈ CUnifSp)
cnextucn.h	⊢ (𝜑 → 𝐾 ∈ Haus)
cnextucn.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
cnextucn.f	⊢ (𝜑 → 𝐹:𝐴⟶𝑌)
cnextucn.c	⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
cnextucn.l	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘𝑈))

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝜑,𝑥

Allowed substitution hints:   𝑈(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem cnextucn

Step	Hyp	Ref	Expression
1		eqid 2728	. 2 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2728	. 2 ⊢ ∪ 𝐾 = ∪ 𝐾
3		cnextucn.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ TopSp)
4		cnextucn.j	. . . 4 ⊢ 𝐽 = (TopOpen‘𝑉)
5	4	tpstop 22832	. . 3 ⊢ (𝑉 ∈ TopSp → 𝐽 ∈ Top)
6	3, 5	syl 17	. 2 ⊢ (𝜑 → 𝐽 ∈ Top)
7		cnextucn.h	. 2 ⊢ (𝜑 → 𝐾 ∈ Haus)
8		cnextucn.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝑌)
9		cnextucn.t	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ TopSp)
10		cnextucn.y	. . . . . 6 ⊢ 𝑌 = (Base‘𝑊)
11		cnextucn.k	. . . . . 6 ⊢ 𝐾 = (TopOpen‘𝑊)
12	10, 11	tpsuni 22831	. . . . 5 ⊢ (𝑊 ∈ TopSp → 𝑌 = ∪ 𝐾)
13	9, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
14	13	feq3d 6703	. . 3 ⊢ (𝜑 → (𝐹:𝐴⟶𝑌 ↔ 𝐹:𝐴⟶∪ 𝐾))
15	8, 14	mpbid 231	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶∪ 𝐾)
16		cnextucn.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
17		cnextucn.x	. . . . 5 ⊢ 𝑋 = (Base‘𝑉)
18	17, 4	tpsuni 22831	. . . 4 ⊢ (𝑉 ∈ TopSp → 𝑋 = ∪ 𝐽)
19	3, 18	syl 17	. . 3 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
20	16, 19	sseqtrd 4018	. 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
21		cnextucn.c	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
22	21, 19	eqtrd 2768	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = ∪ 𝐽)
23	10, 11	istps 22829	. . . . . 6 ⊢ (𝑊 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑌))
24	9, 23	sylib 217	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
25	24	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
26	19	eleq2d 2815	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽))
27	26	biimpar 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ 𝑋)
28	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((cls‘𝐽)‘𝐴) = 𝑋)
29	27, 28	eleqtrrd 2832	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
30		toptopon2 22813	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
31	6, 30	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
32		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝐽 → (TopOn‘𝑋) = (TopOn‘∪ 𝐽))
33	32	eleq2d 2815	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝐽 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)))
34	19, 33	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)))
35	31, 34	mpbird 257	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
36	35	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
37	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝐴 ⊆ 𝑋)
38		trnei 23789	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
39	36, 37, 27, 38	syl3anc 1369	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
40	29, 39	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
41	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹:𝐴⟶𝑌)
42		flfval 23887	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝑌) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))))
43	25, 40, 41, 42	syl3anc 1369	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))))
44		cnextucn.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ CUnifSp)
45	44	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝑊 ∈ CUnifSp)
46		cnextucn.l	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘𝑈))
47	27, 46	syldan 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘𝑈))
48		cnextucn.u	. . . . . 6 ⊢ 𝑈 = (UnifSt‘𝑊)
49	48	fveq2i 6894	. . . . 5 ⊢ (CauFilu‘𝑈) = (CauFilu‘(UnifSt‘𝑊))
50	47, 49	eleqtrdi 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘(UnifSt‘𝑊)))
51	10	fvexi 6905	. . . . 5 ⊢ 𝑌 ∈ V
52		filfbas 23745	. . . . . 6 ⊢ ((((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
53	40, 52	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
54		fmfil 23841	. . . . 5 ⊢ ((𝑌 ∈ V ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴) ∧ 𝐹:𝐴⟶𝑌) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
55	51, 53, 41, 54	mp3an2i 1463	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
56	10, 11	cuspcvg 24199	. . . 4 ⊢ ((𝑊 ∈ CUnifSp ∧ ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌)) → (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))) ≠ ∅)
57	45, 50, 55, 56	syl3anc 1369	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))) ≠ ∅)
58	43, 57	eqnetrd 3004	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
59		cuspusp 24198	. . . 4 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
60	44, 59	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ UnifSp)
61	11	uspreg 24172	. . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐾 ∈ Haus) → 𝐾 ∈ Reg)
62	60, 7, 61	syl2anc 583	. 2 ⊢ (𝜑 → 𝐾 ∈ Reg)
63	1, 2, 6, 7, 15, 20, 22, 58, 62	cnextcn 23964	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ≠ wne 2936  Vcvv 3470   ⊆ wss 3945  ∅c0 4318  {csn 4624  ∪ cuni 4903  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414  Basecbs 17173   ↾_t crest 17395  TopOpenctopn 17396  fBascfbas 21260  Topctop 22788  TopOnctopon 22805  TopSpctps 22827  clsccl 22915  neicnei 22994   Cn ccn 23121  Hauscha 23205  Regcreg 23206  Filcfil 23742   FilMap cfm 23830   fLim cflim 23831   fLimf cflf 23832  CnExtccnext 23956  UnifStcuss 24151  UnifSpcusp 24152  CauFil_uccfilu 24184  CUnifSpccusp 24195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-1o 8480  df-er 8718  df-map 8840  df-pm 8841  df-en 8958  df-fin 8961  df-fi 9428  df-rest 17397  df-topgen 17418  df-fbas 21269  df-fg 21270  df-top 22789  df-topon 22806  df-topsp 22828  df-bases 22842  df-cld 22916  df-ntr 22917  df-cls 22918  df-nei 22995  df-cn 23124  df-cnp 23125  df-haus 23212  df-reg 23213  df-tx 23459  df-fil 23743  df-fm 23835  df-flim 23836  df-flf 23837  df-cnext 23957  df-ust 24098  df-utop 24129  df-usp 24155  df-cusp 24196

This theorem is referenced by:  ucnextcn  24202