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Theorem cnextucn 22839
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 2818 . 2 𝐽 = 𝐽
2 eqid 2818 . 2 𝐾 = 𝐾
3 cnextucn.v . . 3 (𝜑𝑉 ∈ TopSp)
4 cnextucn.j . . . 4 𝐽 = (TopOpen‘𝑉)
54tpstop 21473 . . 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 21472 . . . . 5 (𝑊 ∈ TopSp → 𝑌 = 𝐾)
139, 12syl 17 . . . 4 (𝜑𝑌 = 𝐾)
1413feq3d 6494 . . 3 (𝜑 → (𝐹:𝐴𝑌𝐹:𝐴 𝐾))
158, 14mpbid 233 . 2 (𝜑𝐹:𝐴 𝐾)
16 cnextucn.a . . 3 (𝜑𝐴𝑋)
17 cnextucn.x . . . . 5 𝑋 = (Base‘𝑉)
1817, 4tpsuni 21472 . . . 4 (𝑉 ∈ TopSp → 𝑋 = 𝐽)
193, 18syl 17 . . 3 (𝜑𝑋 = 𝐽)
2016, 19sseqtrd 4004 . 2 (𝜑𝐴 𝐽)
21 cnextucn.c . . 3 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
2221, 19eqtrd 2853 . 2 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐽)
2310, 11istps 21470 . . . . . 6 (𝑊 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑌))
249, 23sylib 219 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
2524adantr 481 . . . 4 ((𝜑𝑥 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
2619eleq2d 2895 . . . . . . 7 (𝜑 → (𝑥𝑋𝑥 𝐽))
2726biimpar 478 . . . . . 6 ((𝜑𝑥 𝐽) → 𝑥𝑋)
2821adantr 481 . . . . . 6 ((𝜑𝑥 𝐽) → ((cls‘𝐽)‘𝐴) = 𝑋)
2927, 28eleqtrrd 2913 . . . . 5 ((𝜑𝑥 𝐽) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
30 toptopon2 21454 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
316, 30sylib 219 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
32 fveq2 6663 . . . . . . . . . 10 (𝑋 = 𝐽 → (TopOn‘𝑋) = (TopOn‘ 𝐽))
3332eleq2d 2895 . . . . . . . . 9 (𝑋 = 𝐽 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘ 𝐽)))
3419, 33syl 17 . . . . . . . 8 (𝜑 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘ 𝐽)))
3531, 34mpbird 258 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
3635adantr 481 . . . . . 6 ((𝜑𝑥 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
3716adantr 481 . . . . . 6 ((𝜑𝑥 𝐽) → 𝐴𝑋)
38 trnei 22428 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋𝑥𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3936, 37, 27, 38syl3anc 1363 . . . . 5 ((𝜑𝑥 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
4029, 39mpbid 233 . . . 4 ((𝜑𝑥 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
418adantr 481 . . . 4 ((𝜑𝑥 𝐽) → 𝐹:𝐴𝑌)
42 flfval 22526 . . . 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 1363 . . 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 481 . . . 4 ((𝜑𝑥 𝐽) → 𝑊 ∈ CUnifSp)
46 cnextucn.l . . . . . 6 ((𝜑𝑥𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu𝑈))
4727, 46syldan 591 . . . . 5 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu𝑈))
48 cnextucn.u . . . . . 6 𝑈 = (UnifSt‘𝑊)
4948fveq2i 6666 . . . . 5 (CauFilu𝑈) = (CauFilu‘(UnifSt‘𝑊))
5047, 49eleqtrdi 2920 . . . 4 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘(UnifSt‘𝑊)))
5110fvexi 6677 . . . . 5 𝑌 ∈ V
52 filfbas 22384 . . . . . 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 22480 . . . . 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 1457 . . . 4 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
5610, 11cuspcvg 22837 . . . 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 1363 . . 3 ((𝜑𝑥 𝐽) → (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))) ≠ ∅)
5843, 57eqnetrd 3080 . 2 ((𝜑𝑥 𝐽) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
59 cuspusp 22836 . . . 4 (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
6044, 59syl 17 . . 3 (𝜑𝑊 ∈ UnifSp)
6111uspreg 22810 . . 3 ((𝑊 ∈ UnifSp ∧ 𝐾 ∈ Haus) → 𝐾 ∈ Reg)
6260, 7, 61syl2anc 584 . 2 (𝜑𝐾 ∈ Reg)
631, 2, 6, 7, 15, 20, 22, 58, 62cnextcn 22603 1 (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  Vcvv 3492  wss 3933  c0 4288  {csn 4557   cuni 4830  wf 6344  cfv 6348  (class class class)co 7145  Basecbs 16471  t crest 16682  TopOpenctopn 16683  fBascfbas 20461  Topctop 21429  TopOnctopon 21446  TopSpctps 21468  clsccl 21554  neicnei 21633   Cn ccn 21760  Hauscha 21844  Regcreg 21845  Filcfil 22381   FilMap cfm 22469   fLim cflim 22470   fLimf cflf 22471  CnExtccnext 22595  UnifStcuss 22789  UnifSpcusp 22790  CauFiluccfilu 22822  CUnifSpccusp 22833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-fin 8501  df-fi 8863  df-rest 16684  df-topgen 16705  df-fbas 20470  df-fg 20471  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-cn 21763  df-cnp 21764  df-haus 21851  df-reg 21852  df-tx 22098  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-cnext 22596  df-ust 22736  df-utop 22767  df-usp 22793  df-cusp 22834
This theorem is referenced by:  ucnextcn  22840
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