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Theorem cnextucn 24195
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 2727 . 2 βˆͺ 𝐽 = βˆͺ 𝐽
2 eqid 2727 . 2 βˆͺ 𝐾 = βˆͺ 𝐾
3 cnextucn.v . . 3 (πœ‘ β†’ 𝑉 ∈ TopSp)
4 cnextucn.j . . . 4 𝐽 = (TopOpenβ€˜π‘‰)
54tpstop 22826 . . 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 22825 . . . . 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 22825 . . . 4 (𝑉 ∈ TopSp β†’ 𝑋 = βˆͺ 𝐽)
193, 18syl 17 . . 3 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
2016, 19sseqtrd 4018 . 2 (πœ‘ β†’ 𝐴 βŠ† βˆͺ 𝐽)
21 cnextucn.c . . 3 (πœ‘ β†’ ((clsβ€˜π½)β€˜π΄) = 𝑋)
2221, 19eqtrd 2767 . 2 (πœ‘ β†’ ((clsβ€˜π½)β€˜π΄) = βˆͺ 𝐽)
2310, 11istps 22823 . . . . . 6 (π‘Š ∈ TopSp ↔ 𝐾 ∈ (TopOnβ€˜π‘Œ))
249, 23sylib 217 . . . . 5 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
2524adantr 480 . . . 4 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
2619eleq2d 2814 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝐽))
2726biimpar 477 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ π‘₯ ∈ 𝑋)
2821adantr 480 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π΄) = 𝑋)
2927, 28eleqtrrd 2831 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄))
30 toptopon2 22807 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
316, 30sylib 217 . . . . . . . 8 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
32 fveq2 6891 . . . . . . . . . 10 (𝑋 = βˆͺ 𝐽 β†’ (TopOnβ€˜π‘‹) = (TopOnβ€˜βˆͺ 𝐽))
3332eleq2d 2814 . . . . . . . . 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 257 . . . . . . 7 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
3635adantr 480 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
3716adantr 480 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐴 βŠ† 𝑋)
38 trnei 23783 . . . . . 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 480 . . . 4 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐹:π΄βŸΆπ‘Œ)
42 flfval 23881 . . . 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 480 . . . 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 590 . . . . 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 6894 . . . . 5 (CauFiluβ€˜π‘ˆ) = (CauFiluβ€˜(UnifStβ€˜π‘Š))
5047, 49eleqtrdi 2838 . . . 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 6905 . . . . 5 π‘Œ ∈ V
52 filfbas 23739 . . . . . 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 23835 . . . . 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 1463 . . . 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 24193 . . . 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 3003 . 2 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) β‰  βˆ…)
59 cuspusp 24192 . . . 4 (π‘Š ∈ CUnifSp β†’ π‘Š ∈ UnifSp)
6044, 59syl 17 . . 3 (πœ‘ β†’ π‘Š ∈ UnifSp)
6111uspreg 24166 . . 3 ((π‘Š ∈ UnifSp ∧ 𝐾 ∈ Haus) β†’ 𝐾 ∈ Reg)
6260, 7, 61syl2anc 583 . 2 (πœ‘ β†’ 𝐾 ∈ Reg)
631, 2, 6, 7, 15, 20, 22, 58, 62cnextcn 23958 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 395   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  Vcvv 3469   βŠ† wss 3944  βˆ…c0 4318  {csn 4624  βˆͺ cuni 4903  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414  Basecbs 17171   β†Ύt crest 17393  TopOpenctopn 17394  fBascfbas 21254  Topctop 22782  TopOnctopon 22799  TopSpctps 22821  clsccl 22909  neicnei 22988   Cn ccn 23115  Hauscha 23199  Regcreg 23200  Filcfil 23736   FilMap cfm 23824   fLim cflim 23825   fLimf cflf 23826  CnExtccnext 23950  UnifStcuss 24145  UnifSpcusp 24146  CauFiluccfilu 24178  CUnifSpccusp 24189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-1o 8480  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-fin 8959  df-fi 9426  df-rest 17395  df-topgen 17416  df-fbas 21263  df-fg 21264  df-top 22783  df-topon 22800  df-topsp 22822  df-bases 22836  df-cld 22910  df-ntr 22911  df-cls 22912  df-nei 22989  df-cn 23118  df-cnp 23119  df-haus 23206  df-reg 23207  df-tx 23453  df-fil 23737  df-fm 23829  df-flim 23830  df-flf 23831  df-cnext 23951  df-ust 24092  df-utop 24123  df-usp 24149  df-cusp 24190
This theorem is referenced by:  ucnextcn  24196
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