Theorem cnextucn 24648

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 2763	. 2 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2763	. 2 ⊢ ∪ 𝐾 = ∪ 𝐾
3		cnextucn.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ TopSp)
4		cnextucn.j	. . . 4 ⊢ 𝐽 = (TopOpen‘𝑉)
5	4	tpstop 23282	. . 3 ⊢ (𝑉 ∈ TopSp → 𝐽 ∈ Top)
6	3, 5	syl 18	. 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 23281	. . . . 5 ⊢ (𝑊 ∈ TopSp → 𝑌 = ∪ 𝐾)
13	9, 12	syl 18	. . . 4 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
14	13	feq3d 6713	. . 3 ⊢ (𝜑 → (𝐹:𝐴⟶𝑌 ↔ 𝐹:𝐴⟶∪ 𝐾))
15	8, 14	mpbid 235	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶∪ 𝐾)
16		cnextucn.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
17		cnextucn.x	. . . . 5 ⊢ 𝑋 = (Base‘𝑉)
18	17, 4	tpsuni 23281	. . . 4 ⊢ (𝑉 ∈ TopSp → 𝑋 = ∪ 𝐽)
19	3, 18	syl 18	. . 3 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
20	16, 19	sseqtrd 3949	. 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
21		cnextucn.c	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
22	21, 19	eqtrd 2798	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = ∪ 𝐽)
23	10, 11	istps 23279	. . . . . 6 ⊢ (𝑊 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑌))
24	9, 23	sylib 221	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
25	24	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
26	19	eleq2d 2850	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽))
27	26	biimpar 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ 𝑋)
28	21	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((cls‘𝐽)‘𝐴) = 𝑋)
29	27, 28	eleqtrrd 2867	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
30		toptopon2 23263	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
31	6, 30	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
32		fveq2 6905	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝐽 → (TopOn‘𝑋) = (TopOn‘∪ 𝐽))
33	32	eleq2d 2850	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝐽 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)))
34	19, 33	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)))
35	31, 34	mpbird 260	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
36	35	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
37	16	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝐴 ⊆ 𝑋)
38		trnei 24238	. . . . . 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 1399	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
40	29, 39	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
41	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹:𝐴⟶𝑌)
42		flfval 24336	. . . 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 1399	. . 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 486	. . . 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 603	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘𝑈))
48		cnextucn.u	. . . . . 6 ⊢ 𝑈 = (UnifSt‘𝑊)
49	48	fveq2i 6908	. . . . 5 ⊢ (CauFilu‘𝑈) = (CauFilu‘(UnifSt‘𝑊))
50	47, 49	eleqtrdi 2874	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘(UnifSt‘𝑊)))
51	10	fvexi 6919	. . . . 5 ⊢ 𝑌 ∈ V
52		filfbas 24194	. . . . . 6 ⊢ ((((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
53	40, 52	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
54		fmfil 24290	. . . . 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 1496	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
56	10, 11	cuspcvg 24646	. . . 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 1399	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))) ≠ ∅)
58	43, 57	eqnetrd 3026	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝐽) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
59		cuspusp 24645	. . . 4 ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
60	44, 59	syl 18	. . 3 ⊢ (𝜑 → 𝑊 ∈ UnifSp)
61	11	uspreg 24619	. . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐾 ∈ Haus) → 𝐾 ∈ Reg)
62	60, 7, 61	syl2anc 596	. 2 ⊢ (𝜑 → 𝐾 ∈ Reg)
63	1, 2, 6, 7, 15, 20, 22, 58, 62	cnextcn 24413	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  {csn 4588  ∪ cuni 4870  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440  Basecbs 17413   ↾_t crest 17617  TopOpenctopn 17618  fBascfbas 21693  Topctop 23238  TopOnctopon 23255  TopSpctps 23277  clsccl 23363  neicnei 23442   Cn ccn 23569  Hauscha 23653  Regcreg 23654  Filcfil 24191   FilMap cfm 24279   fLim cflim 24280   fLimf cflf 24281  CnExtccnext 24405  UnifStcuss 24599  UnifSpcusp 24600  CauFil_uccfilu 24631  CUnifSpccusp 24642

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-1o 8501  df-2o 8502  df-map 8874  df-pm 8875  df-en 8999  df-fin 9002  df-fi 9427  df-rest 17619  df-topgen 17640  df-fbas 21702  df-fg 21703  df-top 23239  df-topon 23256  df-topsp 23278  df-bases 23291  df-cld 23364  df-ntr 23365  df-cls 23366  df-nei 23443  df-cn 23572  df-cnp 23573  df-haus 23660  df-reg 23661  df-tx 23908  df-fil 24192  df-fm 24284  df-flim 24285  df-flf 24286  df-cnext 24406  df-ust 24547  df-utop 24577  df-usp 24603  df-cusp 24643

This theorem is used by:  ucnextcn  24649