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Theorem cnextucn 24335
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 (𝜑𝑉 ∈ TopSp)
cnextucn.t (𝜑𝑊 ∈ TopSp)
cnextucn.w (𝜑𝑊 ∈ CUnifSp)
cnextucn.h (𝜑𝐾 ∈ Haus)
cnextucn.a (𝜑𝐴𝑋)
cnextucn.f (𝜑𝐹:𝐴𝑌)
cnextucn.c (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
cnextucn.l ((𝜑𝑥𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu𝑈))
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
StepHypRef Expression
1 eqid 2740 . 2 𝐽 = 𝐽
2 eqid 2740 . 2 𝐾 = 𝐾
3 cnextucn.v . . 3 (𝜑𝑉 ∈ TopSp)
4 cnextucn.j . . . 4 𝐽 = (TopOpen‘𝑉)
54tpstop 22966 . . 3 (𝑉 ∈ TopSp → 𝐽 ∈ Top)
63, 5syl 17 . 2 (𝜑𝐽 ∈ Top)
7 cnextucn.h . 2 (𝜑𝐾 ∈ Haus)
8 cnextucn.f . . 3 (𝜑𝐹:𝐴𝑌)
9 cnextucn.t . . . . 5 (𝜑𝑊 ∈ TopSp)
10 cnextucn.y . . . . . 6 𝑌 = (Base‘𝑊)
11 cnextucn.k . . . . . 6 𝐾 = (TopOpen‘𝑊)
1210, 11tpsuni 22965 . . . . 5 (𝑊 ∈ TopSp → 𝑌 = 𝐾)
139, 12syl 17 . . . 4 (𝜑𝑌 = 𝐾)
1413feq3d 6736 . . 3 (𝜑 → (𝐹:𝐴𝑌𝐹:𝐴 𝐾))
158, 14mpbid 232 . 2 (𝜑𝐹:𝐴 𝐾)
16 cnextucn.a . . 3 (𝜑𝐴𝑋)
17 cnextucn.x . . . . 5 𝑋 = (Base‘𝑉)
1817, 4tpsuni 22965 . . . 4 (𝑉 ∈ TopSp → 𝑋 = 𝐽)
193, 18syl 17 . . 3 (𝜑𝑋 = 𝐽)
2016, 19sseqtrd 4049 . 2 (𝜑𝐴 𝐽)
21 cnextucn.c . . 3 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
2221, 19eqtrd 2780 . 2 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐽)
2310, 11istps 22963 . . . . . 6 (𝑊 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑌))
249, 23sylib 218 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
2524adantr 480 . . . 4 ((𝜑𝑥 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
2619eleq2d 2830 . . . . . . 7 (𝜑 → (𝑥𝑋𝑥 𝐽))
2726biimpar 477 . . . . . 6 ((𝜑𝑥 𝐽) → 𝑥𝑋)
2821adantr 480 . . . . . 6 ((𝜑𝑥 𝐽) → ((cls‘𝐽)‘𝐴) = 𝑋)
2927, 28eleqtrrd 2847 . . . . 5 ((𝜑𝑥 𝐽) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
30 toptopon2 22947 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
316, 30sylib 218 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
32 fveq2 6922 . . . . . . . . . 10 (𝑋 = 𝐽 → (TopOn‘𝑋) = (TopOn‘ 𝐽))
3332eleq2d 2830 . . . . . . . . 9 (𝑋 = 𝐽 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘ 𝐽)))
3419, 33syl 17 . . . . . . . 8 (𝜑 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘ 𝐽)))
3531, 34mpbird 257 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
3635adantr 480 . . . . . 6 ((𝜑𝑥 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
3716adantr 480 . . . . . 6 ((𝜑𝑥 𝐽) → 𝐴𝑋)
38 trnei 23923 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋𝑥𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3936, 37, 27, 38syl3anc 1371 . . . . 5 ((𝜑𝑥 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
4029, 39mpbid 232 . . . 4 ((𝜑𝑥 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
418adantr 480 . . . 4 ((𝜑𝑥 𝐽) → 𝐹:𝐴𝑌)
42 flfval 24021 . . . 4 ((𝐾 ∈ (TopOn‘𝑌) ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴𝑌) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))))
4325, 40, 41, 42syl3anc 1371 . . 3 ((𝜑𝑥 𝐽) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))))
44 cnextucn.w . . . . 5 (𝜑𝑊 ∈ CUnifSp)
4544adantr 480 . . . 4 ((𝜑𝑥 𝐽) → 𝑊 ∈ CUnifSp)
46 cnextucn.l . . . . . 6 ((𝜑𝑥𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu𝑈))
4727, 46syldan 590 . . . . 5 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu𝑈))
48 cnextucn.u . . . . . 6 𝑈 = (UnifSt‘𝑊)
4948fveq2i 6925 . . . . 5 (CauFilu𝑈) = (CauFilu‘(UnifSt‘𝑊))
5047, 49eleqtrdi 2854 . . . 4 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘(UnifSt‘𝑊)))
5110fvexi 6936 . . . . 5 𝑌 ∈ V
52 filfbas 23879 . . . . . 6 ((((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
5340, 52syl 17 . . . . 5 ((𝜑𝑥 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
54 fmfil 23975 . . . . 5 ((𝑌 ∈ V ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴) ∧ 𝐹:𝐴𝑌) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
5551, 53, 41, 54mp3an2i 1466 . . . 4 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
5610, 11cuspcvg 24333 . . . 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))) ≠ ∅)
5745, 50, 55, 56syl3anc 1371 . . 3 ((𝜑𝑥 𝐽) → (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))) ≠ ∅)
5843, 57eqnetrd 3014 . 2 ((𝜑𝑥 𝐽) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
59 cuspusp 24332 . . . 4 (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
6044, 59syl 17 . . 3 (𝜑𝑊 ∈ UnifSp)
6111uspreg 24306 . . 3 ((𝑊 ∈ UnifSp ∧ 𝐾 ∈ Haus) → 𝐾 ∈ Reg)
6260, 7, 61syl2anc 583 . 2 (𝜑𝐾 ∈ Reg)
631, 2, 6, 7, 15, 20, 22, 58, 62cnextcn 24098 1 (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2108  wne 2946  Vcvv 3488  wss 3976  c0 4352  {csn 4648   cuni 4931  wf 6571  cfv 6575  (class class class)co 7450  Basecbs 17260  t crest 17482  TopOpenctopn 17483  fBascfbas 21377  Topctop 22922  TopOnctopon 22939  TopSpctps 22961  clsccl 23049  neicnei 23128   Cn ccn 23255  Hauscha 23339  Regcreg 23340  Filcfil 23876   FilMap cfm 23964   fLim cflim 23965   fLimf cflf 23966  CnExtccnext 24090  UnifStcuss 24285  UnifSpcusp 24286  CauFiluccfilu 24318  CUnifSpccusp 24329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-om 7906  df-1st 8032  df-2nd 8033  df-1o 8524  df-2o 8525  df-map 8888  df-pm 8889  df-en 9006  df-fin 9009  df-fi 9482  df-rest 17484  df-topgen 17505  df-fbas 21386  df-fg 21387  df-top 22923  df-topon 22940  df-topsp 22962  df-bases 22976  df-cld 23050  df-ntr 23051  df-cls 23052  df-nei 23129  df-cn 23258  df-cnp 23259  df-haus 23346  df-reg 23347  df-tx 23593  df-fil 23877  df-fm 23969  df-flim 23970  df-flf 23971  df-cnext 24091  df-ust 24232  df-utop 24263  df-usp 24289  df-cusp 24330
This theorem is referenced by:  ucnextcn  24336
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