Theorem cnextucn 23363

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 2738	. 2 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2738	. 2 ⊢ ∪ 𝐾 = ∪ 𝐾
3		cnextucn.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ TopSp)
4		cnextucn.j	. . . 4 ⊢ 𝐽 = (TopOpen‘𝑉)
5	4	tpstop 21994	. . 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 21993	. . . . 5 ⊢ (𝑊 ∈ TopSp → 𝑌 = ∪ 𝐾)
13	9, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
14	13	feq3d 6571	. . 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 21993	. . . 4 ⊢ (𝑉 ∈ TopSp → 𝑋 = ∪ 𝐽)
19	3, 18	syl 17	. . 3 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
20	16, 19	sseqtrd 3957	. 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
21		cnextucn.c	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
22	21, 19	eqtrd 2778	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = ∪ 𝐽)
23	10, 11	istps 21991	. . . . . 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 2824	. . . . . . 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 2842	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
30		toptopon2 21975	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
31	6, 30	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
32		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝐽 → (TopOn‘𝑋) = (TopOn‘∪ 𝐽))
33	32	eleq2d 2824	. . . . . . . . 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 256	. . . . . . 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 22951	. . . . . 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 23049	. . . 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 6759	. . . . 5 ⊢ (CauFilu‘𝑈) = (CauFilu‘(UnifSt‘𝑊))
50	47, 49	eleqtrdi 2849	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘(UnifSt‘𝑊)))
51	10	fvexi 6770	. . . . 5 ⊢ 𝑌 ∈ V
52		filfbas 22907	. . . . . 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 23003	. . . . 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 1464	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
56	10, 11	cuspcvg 23361	. . . 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 3010	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
59		cuspusp 23360	. . . 4 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
60	44, 59	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ UnifSp)
61	11	uspreg 23334	. . 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 23126	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∪ cuni 4836  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Basecbs 16840   ↾_t crest 17048  TopOpenctopn 17049  fBascfbas 20498  Topctop 21950  TopOnctopon 21967  TopSpctps 21989  clsccl 22077  neicnei 22156   Cn ccn 22283  Hauscha 22367  Regcreg 22368  Filcfil 22904   FilMap cfm 22992   fLim cflim 22993   fLimf cflf 22994  CnExtccnext 23118  UnifStcuss 23313  UnifSpcusp 23314  CauFil_uccfilu 23346  CUnifSpccusp 23357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-fin 8695  df-fi 9100  df-rest 17050  df-topgen 17071  df-fbas 20507  df-fg 20508  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-cld 22078  df-ntr 22079  df-cls 22080  df-nei 22157  df-cn 22286  df-cnp 22287  df-haus 22374  df-reg 22375  df-tx 22621  df-fil 22905  df-fm 22997  df-flim 22998  df-flf 22999  df-cnext 23119  df-ust 23260  df-utop 23291  df-usp 23317  df-cusp 23358

This theorem is referenced by:  ucnextcn  23364