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Theorem cnextucn 22477
 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 2825 . 2 𝐽 = 𝐽
2 eqid 2825 . 2 𝐾 = 𝐾
3 cnextucn.v . . 3 (𝜑𝑉 ∈ TopSp)
4 cnextucn.j . . . 4 𝐽 = (TopOpen‘𝑉)
54tpstop 21112 . . 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 21111 . . . . 5 (𝑊 ∈ TopSp → 𝑌 = 𝐾)
139, 12syl 17 . . . 4 (𝜑𝑌 = 𝐾)
1413feq3d 6265 . . 3 (𝜑 → (𝐹:𝐴𝑌𝐹:𝐴 𝐾))
158, 14mpbid 224 . 2 (𝜑𝐹:𝐴 𝐾)
16 cnextucn.a . . 3 (𝜑𝐴𝑋)
17 cnextucn.x . . . . 5 𝑋 = (Base‘𝑉)
1817, 4tpsuni 21111 . . . 4 (𝑉 ∈ TopSp → 𝑋 = 𝐽)
193, 18syl 17 . . 3 (𝜑𝑋 = 𝐽)
2016, 19sseqtrd 3866 . 2 (𝜑𝐴 𝐽)
21 cnextucn.c . . 3 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
2221, 19eqtrd 2861 . 2 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐽)
2310, 11istps 21109 . . . . . 6 (𝑊 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑌))
249, 23sylib 210 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
2524adantr 474 . . . 4 ((𝜑𝑥 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
2619eleq2d 2892 . . . . . . 7 (𝜑 → (𝑥𝑋𝑥 𝐽))
2726biimpar 471 . . . . . 6 ((𝜑𝑥 𝐽) → 𝑥𝑋)
2821adantr 474 . . . . . 6 ((𝜑𝑥 𝐽) → ((cls‘𝐽)‘𝐴) = 𝑋)
2927, 28eleqtrrd 2909 . . . . 5 ((𝜑𝑥 𝐽) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
301toptopon 21092 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
316, 30sylib 210 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
32 fveq2 6433 . . . . . . . . . 10 (𝑋 = 𝐽 → (TopOn‘𝑋) = (TopOn‘ 𝐽))
3332eleq2d 2892 . . . . . . . . 9 (𝑋 = 𝐽 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘ 𝐽)))
3419, 33syl 17 . . . . . . . 8 (𝜑 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘ 𝐽)))
3531, 34mpbird 249 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
3635adantr 474 . . . . . 6 ((𝜑𝑥 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
3716adantr 474 . . . . . 6 ((𝜑𝑥 𝐽) → 𝐴𝑋)
38 trnei 22066 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋𝑥𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3936, 37, 27, 38syl3anc 1496 . . . . 5 ((𝜑𝑥 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
4029, 39mpbid 224 . . . 4 ((𝜑𝑥 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
418adantr 474 . . . 4 ((𝜑𝑥 𝐽) → 𝐹:𝐴𝑌)
42 flfval 22164 . . . 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 1496 . . 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 474 . . . 4 ((𝜑𝑥 𝐽) → 𝑊 ∈ CUnifSp)
46 cnextucn.l . . . . . 6 ((𝜑𝑥𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu𝑈))
4727, 46syldan 587 . . . . 5 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu𝑈))
48 cnextucn.u . . . . . 6 𝑈 = (UnifSt‘𝑊)
4948fveq2i 6436 . . . . 5 (CauFilu𝑈) = (CauFilu‘(UnifSt‘𝑊))
5047, 49syl6eleq 2916 . . . 4 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘(UnifSt‘𝑊)))
5110fvexi 6447 . . . . . 6 𝑌 ∈ V
5251a1i 11 . . . . 5 ((𝜑𝑥 𝐽) → 𝑌 ∈ V)
53 filfbas 22022 . . . . . 6 ((((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
5440, 53syl 17 . . . . 5 ((𝜑𝑥 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
55 fmfil 22118 . . . . 5 ((𝑌 ∈ V ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴) ∧ 𝐹:𝐴𝑌) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
5652, 54, 41, 55syl3anc 1496 . . . 4 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
5710, 11cuspcvg 22475 . . . 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))) ≠ ∅)
5845, 50, 56, 57syl3anc 1496 . . 3 ((𝜑𝑥 𝐽) → (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))) ≠ ∅)
5943, 58eqnetrd 3066 . 2 ((𝜑𝑥 𝐽) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
60 cuspusp 22474 . . . 4 (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
6144, 60syl 17 . . 3 (𝜑𝑊 ∈ UnifSp)
6211uspreg 22448 . . 3 ((𝑊 ∈ UnifSp ∧ 𝐾 ∈ Haus) → 𝐾 ∈ Reg)
6361, 7, 62syl2anc 581 . 2 (𝜑𝐾 ∈ Reg)
641, 2, 6, 7, 15, 20, 22, 59, 63cnextcn 22241 1 (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166   ≠ wne 2999  Vcvv 3414   ⊆ wss 3798  ∅c0 4144  {csn 4397  ∪ cuni 4658  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  Basecbs 16222   ↾t crest 16434  TopOpenctopn 16435  fBascfbas 20094  Topctop 21068  TopOnctopon 21085  TopSpctps 21107  clsccl 21193  neicnei 21272   Cn ccn 21399  Hauscha 21483  Regcreg 21484  Filcfil 22019   FilMap cfm 22107   fLim cflim 22108   fLimf cflf 22109  CnExtccnext 22233  UnifStcuss 22427  UnifSpcusp 22428  CauFiluccfilu 22460  CUnifSpccusp 22471 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-en 8223  df-fin 8226  df-fi 8586  df-rest 16436  df-topgen 16457  df-fbas 20103  df-fg 20104  df-top 21069  df-topon 21086  df-topsp 21108  df-bases 21121  df-cld 21194  df-ntr 21195  df-cls 21196  df-nei 21273  df-cn 21402  df-cnp 21403  df-haus 21490  df-reg 21491  df-tx 21736  df-fil 22020  df-fm 22112  df-flim 22113  df-flf 22114  df-cnext 22234  df-ust 22374  df-utop 22405  df-usp 22431  df-cusp 22472 This theorem is referenced by:  ucnextcn  22478
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