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Theorem cnextucn 21917
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 2610 . 2 𝐽 = 𝐽
2 eqid 2610 . 2 𝐾 = 𝐾
3 cnextucn.v . . 3 (𝜑𝑉 ∈ TopSp)
4 cnextucn.j . . . 4 𝐽 = (TopOpen‘𝑉)
54tpstop 20554 . . 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 20553 . . . . 5 (𝑊 ∈ TopSp → 𝑌 = 𝐾)
139, 12syl 17 . . . 4 (𝜑𝑌 = 𝐾)
1413feq3d 5945 . . 3 (𝜑 → (𝐹:𝐴𝑌𝐹:𝐴 𝐾))
158, 14mpbid 221 . 2 (𝜑𝐹:𝐴 𝐾)
16 cnextucn.a . . 3 (𝜑𝐴𝑋)
17 cnextucn.x . . . . 5 𝑋 = (Base‘𝑉)
1817, 4tpsuni 20553 . . . 4 (𝑉 ∈ TopSp → 𝑋 = 𝐽)
193, 18syl 17 . . 3 (𝜑𝑋 = 𝐽)
2016, 19sseqtrd 3604 . 2 (𝜑𝐴 𝐽)
21 cnextucn.c . . 3 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
2221, 19eqtrd 2644 . 2 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐽)
2310, 11istps 20551 . . . . . 6 (𝑊 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑌))
249, 23sylib 207 . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
2524adantr 480 . . . 4 ((𝜑𝑥 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
2619eleq2d 2673 . . . . . . 7 (𝜑 → (𝑥𝑋𝑥 𝐽))
2726biimpar 501 . . . . . 6 ((𝜑𝑥 𝐽) → 𝑥𝑋)
2821adantr 480 . . . . . 6 ((𝜑𝑥 𝐽) → ((cls‘𝐽)‘𝐴) = 𝑋)
2927, 28eleqtrrd 2691 . . . . 5 ((𝜑𝑥 𝐽) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
301toptopon 20548 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
316, 30sylib 207 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
32 fveq2 6103 . . . . . . . . . 10 (𝑋 = 𝐽 → (TopOn‘𝑋) = (TopOn‘ 𝐽))
3332eleq2d 2673 . . . . . . . . 9 (𝑋 = 𝐽 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘ 𝐽)))
3419, 33syl 17 . . . . . . . 8 (𝜑 → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘ 𝐽)))
3531, 34mpbird 246 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
3635adantr 480 . . . . . 6 ((𝜑𝑥 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
3716adantr 480 . . . . . 6 ((𝜑𝑥 𝐽) → 𝐴𝑋)
38 trnei 21506 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋𝑥𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3936, 37, 27, 38syl3anc 1318 . . . . 5 ((𝜑𝑥 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
4029, 39mpbid 221 . . . 4 ((𝜑𝑥 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
418adantr 480 . . . 4 ((𝜑𝑥 𝐽) → 𝐹:𝐴𝑌)
42 flfval 21604 . . . 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 1318 . . 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 486 . . . . 5 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu𝑈))
48 cnextucn.u . . . . . 6 𝑈 = (UnifSt‘𝑊)
4948fveq2i 6106 . . . . 5 (CauFilu𝑈) = (CauFilu‘(UnifSt‘𝑊))
5047, 49syl6eleq 2698 . . . 4 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘(UnifSt‘𝑊)))
51 fvex 6113 . . . . . . 7 (Base‘𝑊) ∈ V
5210, 51eqeltri 2684 . . . . . 6 𝑌 ∈ V
5352a1i 11 . . . . 5 ((𝜑𝑥 𝐽) → 𝑌 ∈ V)
54 filfbas 21462 . . . . . 6 ((((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
5540, 54syl 17 . . . . 5 ((𝜑𝑥 𝐽) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
56 fmfil 21558 . . . . 5 ((𝑌 ∈ V ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴) ∧ 𝐹:𝐴𝑌) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
5753, 55, 41, 56syl3anc 1318 . . . 4 ((𝜑𝑥 𝐽) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (Fil‘𝑌))
5810, 11cuspcvg 21915 . . . 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))) ≠ ∅)
5945, 50, 57, 58syl3anc 1318 . . 3 ((𝜑𝑥 𝐽) → (𝐾 fLim ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))) ≠ ∅)
6043, 59eqnetrd 2849 . 2 ((𝜑𝑥 𝐽) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
61 cuspusp 21914 . . . 4 (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
6244, 61syl 17 . . 3 (𝜑𝑊 ∈ UnifSp)
6311uspreg 21888 . . 3 ((𝑊 ∈ UnifSp ∧ 𝐾 ∈ Haus) → 𝐾 ∈ Reg)
6462, 7, 63syl2anc 691 . 2 (𝜑𝐾 ∈ Reg)
651, 2, 6, 7, 15, 20, 22, 60, 64cnextcn 21681 1 (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  Vcvv 3173  wss 3540  c0 3874  {csn 4125   cuni 4372  wf 5800  cfv 5804  (class class class)co 6549  Basecbs 15695  t crest 15904  TopOpenctopn 15905  fBascfbas 19555  Topctop 20517  TopOnctopon 20518  TopSpctps 20519  clsccl 20632  neicnei 20711   Cn ccn 20838  Hauscha 20922  Regcreg 20923  Filcfil 21459   FilMap cfm 21547   fLim cflim 21548   fLimf cflf 21549  CnExtccnext 21673  UnifStcuss 21867  UnifSpcusp 21868  CauFiluccfilu 21900  CUnifSpccusp 21911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-fbas 19564  df-fg 19565  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-cn 20841  df-cnp 20842  df-haus 20929  df-reg 20930  df-tx 21175  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-cnext 21674  df-ust 21814  df-utop 21845  df-usp 21871  df-cusp 21912
This theorem is referenced by:  ucnextcn  21918
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