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Theorem cnextucn 24028
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 π‘Œ Hausdorff and complete to be extensible by continuity. (Contributed by Thierry Arnoux, 4-Dec-2017.)
Hypotheses
Ref Expression
cnextucn.x 𝑋 = (Baseβ€˜π‘‰)
cnextucn.y π‘Œ = (Baseβ€˜π‘Š)
cnextucn.j 𝐽 = (TopOpenβ€˜π‘‰)
cnextucn.k 𝐾 = (TopOpenβ€˜π‘Š)
cnextucn.u π‘ˆ = (UnifStβ€˜π‘Š)
cnextucn.v (πœ‘ β†’ 𝑉 ∈ TopSp)
cnextucn.t (πœ‘ β†’ π‘Š ∈ TopSp)
cnextucn.w (πœ‘ β†’ π‘Š ∈ CUnifSp)
cnextucn.h (πœ‘ β†’ 𝐾 ∈ Haus)
cnextucn.a (πœ‘ β†’ 𝐴 βŠ† 𝑋)
cnextucn.f (πœ‘ β†’ 𝐹:π΄βŸΆπ‘Œ)
cnextucn.c (πœ‘ β†’ ((clsβ€˜π½)β€˜π΄) = 𝑋)
cnextucn.l ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘Œ FilMap 𝐹)β€˜(((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴)) ∈ (CauFiluβ€˜π‘ˆ))
Assertion
Ref Expression
cnextucn (πœ‘ β†’ ((𝐽CnExt𝐾)β€˜πΉ) ∈ (𝐽 Cn 𝐾))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐹   π‘₯,𝐽   π‘₯,𝐾   πœ‘,π‘₯
Allowed substitution hints:   π‘ˆ(π‘₯)   𝑉(π‘₯)   π‘Š(π‘₯)   𝑋(π‘₯)   π‘Œ(π‘₯)

Proof of Theorem cnextucn
StepHypRef Expression
1 eqid 2730 . 2 βˆͺ 𝐽 = βˆͺ 𝐽
2 eqid 2730 . 2 βˆͺ 𝐾 = βˆͺ 𝐾
3 cnextucn.v . . 3 (πœ‘ β†’ 𝑉 ∈ TopSp)
4 cnextucn.j . . . 4 𝐽 = (TopOpenβ€˜π‘‰)
54tpstop 22659 . . 3 (𝑉 ∈ TopSp β†’ 𝐽 ∈ Top)
63, 5syl 17 . 2 (πœ‘ β†’ 𝐽 ∈ Top)
7 cnextucn.h . 2 (πœ‘ β†’ 𝐾 ∈ Haus)
8 cnextucn.f . . 3 (πœ‘ β†’ 𝐹:π΄βŸΆπ‘Œ)
9 cnextucn.t . . . . 5 (πœ‘ β†’ π‘Š ∈ TopSp)
10 cnextucn.y . . . . . 6 π‘Œ = (Baseβ€˜π‘Š)
11 cnextucn.k . . . . . 6 𝐾 = (TopOpenβ€˜π‘Š)
1210, 11tpsuni 22658 . . . . 5 (π‘Š ∈ TopSp β†’ π‘Œ = βˆͺ 𝐾)
139, 12syl 17 . . . 4 (πœ‘ β†’ π‘Œ = βˆͺ 𝐾)
1413feq3d 6703 . . 3 (πœ‘ β†’ (𝐹:π΄βŸΆπ‘Œ ↔ 𝐹:𝐴⟢βˆͺ 𝐾))
158, 14mpbid 231 . 2 (πœ‘ β†’ 𝐹:𝐴⟢βˆͺ 𝐾)
16 cnextucn.a . . 3 (πœ‘ β†’ 𝐴 βŠ† 𝑋)
17 cnextucn.x . . . . 5 𝑋 = (Baseβ€˜π‘‰)
1817, 4tpsuni 22658 . . . 4 (𝑉 ∈ TopSp β†’ 𝑋 = βˆͺ 𝐽)
193, 18syl 17 . . 3 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
2016, 19sseqtrd 4021 . 2 (πœ‘ β†’ 𝐴 βŠ† βˆͺ 𝐽)
21 cnextucn.c . . 3 (πœ‘ β†’ ((clsβ€˜π½)β€˜π΄) = 𝑋)
2221, 19eqtrd 2770 . 2 (πœ‘ β†’ ((clsβ€˜π½)β€˜π΄) = βˆͺ 𝐽)
2310, 11istps 22656 . . . . . 6 (π‘Š ∈ TopSp ↔ 𝐾 ∈ (TopOnβ€˜π‘Œ))
249, 23sylib 217 . . . . 5 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
2524adantr 479 . . . 4 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
2619eleq2d 2817 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝐽))
2726biimpar 476 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ π‘₯ ∈ 𝑋)
2821adantr 479 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π΄) = 𝑋)
2927, 28eleqtrrd 2834 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄))
30 toptopon2 22640 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
316, 30sylib 217 . . . . . . . 8 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
32 fveq2 6890 . . . . . . . . . 10 (𝑋 = βˆͺ 𝐽 β†’ (TopOnβ€˜π‘‹) = (TopOnβ€˜βˆͺ 𝐽))
3332eleq2d 2817 . . . . . . . . 9 (𝑋 = βˆͺ 𝐽 β†’ (𝐽 ∈ (TopOnβ€˜π‘‹) ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽)))
3419, 33syl 17 . . . . . . . 8 (πœ‘ β†’ (𝐽 ∈ (TopOnβ€˜π‘‹) ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽)))
3531, 34mpbird 256 . . . . . . 7 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
3635adantr 479 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
3716adantr 479 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐴 βŠ† 𝑋)
38 trnei 23616 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ↔ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (Filβ€˜π΄)))
3936, 37, 27, 38syl3anc 1369 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ↔ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (Filβ€˜π΄)))
4029, 39mpbid 231 . . . 4 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (Filβ€˜π΄))
418adantr 479 . . . 4 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐹:π΄βŸΆπ‘Œ)
42 flfval 23714 . . . 4 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (Filβ€˜π΄) ∧ 𝐹:π΄βŸΆπ‘Œ) β†’ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) = (𝐾 fLim ((π‘Œ FilMap 𝐹)β€˜(((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))))
4325, 40, 41, 42syl3anc 1369 . . 3 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) = (𝐾 fLim ((π‘Œ FilMap 𝐹)β€˜(((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))))
44 cnextucn.w . . . . 5 (πœ‘ β†’ π‘Š ∈ CUnifSp)
4544adantr 479 . . . 4 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ π‘Š ∈ CUnifSp)
46 cnextucn.l . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘Œ FilMap 𝐹)β€˜(((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴)) ∈ (CauFiluβ€˜π‘ˆ))
4727, 46syldan 589 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ((π‘Œ FilMap 𝐹)β€˜(((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴)) ∈ (CauFiluβ€˜π‘ˆ))
48 cnextucn.u . . . . . 6 π‘ˆ = (UnifStβ€˜π‘Š)
4948fveq2i 6893 . . . . 5 (CauFiluβ€˜π‘ˆ) = (CauFiluβ€˜(UnifStβ€˜π‘Š))
5047, 49eleqtrdi 2841 . . . 4 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ((π‘Œ FilMap 𝐹)β€˜(((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴)) ∈ (CauFiluβ€˜(UnifStβ€˜π‘Š)))
5110fvexi 6904 . . . . 5 π‘Œ ∈ V
52 filfbas 23572 . . . . . 6 ((((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (Filβ€˜π΄) β†’ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (fBasβ€˜π΄))
5340, 52syl 17 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (fBasβ€˜π΄))
54 fmfil 23668 . . . . 5 ((π‘Œ ∈ V ∧ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (fBasβ€˜π΄) ∧ 𝐹:π΄βŸΆπ‘Œ) β†’ ((π‘Œ FilMap 𝐹)β€˜(((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴)) ∈ (Filβ€˜π‘Œ))
5551, 53, 41, 54mp3an2i 1464 . . . 4 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ((π‘Œ FilMap 𝐹)β€˜(((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴)) ∈ (Filβ€˜π‘Œ))
5610, 11cuspcvg 24026 . . . 4 ((π‘Š ∈ CUnifSp ∧ ((π‘Œ FilMap 𝐹)β€˜(((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴)) ∈ (CauFiluβ€˜(UnifStβ€˜π‘Š)) ∧ ((π‘Œ FilMap 𝐹)β€˜(((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴)) ∈ (Filβ€˜π‘Œ)) β†’ (𝐾 fLim ((π‘Œ FilMap 𝐹)β€˜(((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))) β‰  βˆ…)
5745, 50, 55, 56syl3anc 1369 . . 3 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ (𝐾 fLim ((π‘Œ FilMap 𝐹)β€˜(((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))) β‰  βˆ…)
5843, 57eqnetrd 3006 . 2 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) β‰  βˆ…)
59 cuspusp 24025 . . . 4 (π‘Š ∈ CUnifSp β†’ π‘Š ∈ UnifSp)
6044, 59syl 17 . . 3 (πœ‘ β†’ π‘Š ∈ UnifSp)
6111uspreg 23999 . . 3 ((π‘Š ∈ UnifSp ∧ 𝐾 ∈ Haus) β†’ 𝐾 ∈ Reg)
6260, 7, 61syl2anc 582 . 2 (πœ‘ β†’ 𝐾 ∈ Reg)
631, 2, 6, 7, 15, 20, 22, 58, 62cnextcn 23791 1 (πœ‘ β†’ ((𝐽CnExt𝐾)β€˜πΉ) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  Vcvv 3472   βŠ† wss 3947  βˆ…c0 4321  {csn 4627  βˆͺ cuni 4907  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148   β†Ύt crest 17370  TopOpenctopn 17371  fBascfbas 21132  Topctop 22615  TopOnctopon 22632  TopSpctps 22654  clsccl 22742  neicnei 22821   Cn ccn 22948  Hauscha 23032  Regcreg 23033  Filcfil 23569   FilMap cfm 23657   fLim cflim 23658   fLimf cflf 23659  CnExtccnext 23783  UnifStcuss 23978  UnifSpcusp 23979  CauFiluccfilu 24011  CUnifSpccusp 24022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-1o 8468  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-fin 8945  df-fi 9408  df-rest 17372  df-topgen 17393  df-fbas 21141  df-fg 21142  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-cn 22951  df-cnp 22952  df-haus 23039  df-reg 23040  df-tx 23286  df-fil 23570  df-fm 23662  df-flim 23663  df-flf 23664  df-cnext 23784  df-ust 23925  df-utop 23956  df-usp 23982  df-cusp 24023
This theorem is referenced by:  ucnextcn  24029
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