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Theorem cnextucn 24326
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 2734 . 2 𝐽 = 𝐽
2 eqid 2734 . 2 𝐾 = 𝐾
3 cnextucn.v . . 3 (𝜑𝑉 ∈ TopSp)
4 cnextucn.j . . . 4 𝐽 = (TopOpen‘𝑉)
54tpstop 22957 . . 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 22956 . . . . 5 (𝑊 ∈ TopSp → 𝑌 = 𝐾)
139, 12syl 17 . . . 4 (𝜑𝑌 = 𝐾)
1413feq3d 6733 . . 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 22956 . . . 4 (𝑉 ∈ TopSp → 𝑋 = 𝐽)
193, 18syl 17 . . 3 (𝜑𝑋 = 𝐽)
2016, 19sseqtrd 4043 . 2 (𝜑𝐴 𝐽)
21 cnextucn.c . . 3 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
2221, 19eqtrd 2774 . 2 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐽)
2310, 11istps 22954 . . . . . 6 (𝑊 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑌))
249, 23sylib 218 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
2524adantr 480 . . . 4 ((𝜑𝑥 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
2619eleq2d 2824 . . . . . . 7 (𝜑 → (𝑥𝑋𝑥 𝐽))
2726biimpar 477 . . . . . 6 ((𝜑𝑥 𝐽) → 𝑥𝑋)
2821adantr 480 . . . . . 6 ((𝜑𝑥 𝐽) → ((cls‘𝐽)‘𝐴) = 𝑋)
2927, 28eleqtrrd 2841 . . . . 5 ((𝜑𝑥 𝐽) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
30 toptopon2 22938 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
316, 30sylib 218 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
32 fveq2 6919 . . . . . . . . . 10 (𝑋 = 𝐽 → (TopOn‘𝑋) = (TopOn‘ 𝐽))
3332eleq2d 2824 . . . . . . . . 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 23914 . . . . . 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 24012 . . . 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 6922 . . . . 5 (CauFilu𝑈) = (CauFilu‘(UnifSt‘𝑊))
5047, 49eleqtrdi 2848 . . . 4 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘(UnifSt‘𝑊)))
5110fvexi 6933 . . . . 5 𝑌 ∈ V
52 filfbas 23870 . . . . . 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 23966 . . . . 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 24324 . . . 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 3010 . 2 ((𝜑𝑥 𝐽) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
59 cuspusp 24323 . . . 4 (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
6044, 59syl 17 . . 3 (𝜑𝑊 ∈ UnifSp)
6111uspreg 24297 . . 3 ((𝑊 ∈ UnifSp ∧ 𝐾 ∈ Haus) → 𝐾 ∈ Reg)
6260, 7, 61syl2anc 583 . 2 (𝜑𝐾 ∈ Reg)
631, 2, 6, 7, 15, 20, 22, 58, 62cnextcn 24089 1 (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2103  wne 2942  Vcvv 3482  wss 3970  c0 4347  {csn 4648   cuni 4931  wf 6568  cfv 6572  (class class class)co 7445  Basecbs 17253  t crest 17475  TopOpenctopn 17476  fBascfbas 21370  Topctop 22913  TopOnctopon 22930  TopSpctps 22952  clsccl 23040  neicnei 23119   Cn ccn 23246  Hauscha 23330  Regcreg 23331  Filcfil 23867   FilMap cfm 23955   fLim cflim 23956   fLimf cflf 23957  CnExtccnext 24081  UnifStcuss 24276  UnifSpcusp 24277  CauFiluccfilu 24309  CUnifSpccusp 24320
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-iin 5022  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-1o 8518  df-2o 8519  df-map 8882  df-pm 8883  df-en 9000  df-fin 9003  df-fi 9476  df-rest 17477  df-topgen 17498  df-fbas 21379  df-fg 21380  df-top 22914  df-topon 22931  df-topsp 22953  df-bases 22967  df-cld 23041  df-ntr 23042  df-cls 23043  df-nei 23120  df-cn 23249  df-cnp 23250  df-haus 23337  df-reg 23338  df-tx 23584  df-fil 23868  df-fm 23960  df-flim 23961  df-flf 23962  df-cnext 24082  df-ust 24223  df-utop 24254  df-usp 24280  df-cusp 24321
This theorem is referenced by:  ucnextcn  24327
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