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Theorem cnextucn 23678
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 2733 . 2 βˆͺ 𝐽 = βˆͺ 𝐽
2 eqid 2733 . 2 βˆͺ 𝐾 = βˆͺ 𝐾
3 cnextucn.v . . 3 (πœ‘ β†’ 𝑉 ∈ TopSp)
4 cnextucn.j . . . 4 𝐽 = (TopOpenβ€˜π‘‰)
54tpstop 22309 . . 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 22308 . . . . 5 (π‘Š ∈ TopSp β†’ π‘Œ = βˆͺ 𝐾)
139, 12syl 17 . . . 4 (πœ‘ β†’ π‘Œ = βˆͺ 𝐾)
1413feq3d 6659 . . 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 22308 . . . 4 (𝑉 ∈ TopSp β†’ 𝑋 = βˆͺ 𝐽)
193, 18syl 17 . . 3 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
2016, 19sseqtrd 3988 . 2 (πœ‘ β†’ 𝐴 βŠ† βˆͺ 𝐽)
21 cnextucn.c . . 3 (πœ‘ β†’ ((clsβ€˜π½)β€˜π΄) = 𝑋)
2221, 19eqtrd 2773 . 2 (πœ‘ β†’ ((clsβ€˜π½)β€˜π΄) = βˆͺ 𝐽)
2310, 11istps 22306 . . . . . 6 (π‘Š ∈ TopSp ↔ 𝐾 ∈ (TopOnβ€˜π‘Œ))
249, 23sylib 217 . . . . 5 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
2524adantr 482 . . . 4 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
2619eleq2d 2820 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝐽))
2726biimpar 479 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ π‘₯ ∈ 𝑋)
2821adantr 482 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π΄) = 𝑋)
2927, 28eleqtrrd 2837 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄))
30 toptopon2 22290 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
316, 30sylib 217 . . . . . . . 8 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
32 fveq2 6846 . . . . . . . . . 10 (𝑋 = βˆͺ 𝐽 β†’ (TopOnβ€˜π‘‹) = (TopOnβ€˜βˆͺ 𝐽))
3332eleq2d 2820 . . . . . . . . 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 482 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
3716adantr 482 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐴 βŠ† 𝑋)
38 trnei 23266 . . . . . 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 1372 . . . . 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 482 . . . 4 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ 𝐹:π΄βŸΆπ‘Œ)
42 flfval 23364 . . . 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 1372 . . 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 482 . . . 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 592 . . . . 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 6849 . . . . 5 (CauFiluβ€˜π‘ˆ) = (CauFiluβ€˜(UnifStβ€˜π‘Š))
5047, 49eleqtrdi 2844 . . . 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 6860 . . . . 5 π‘Œ ∈ V
52 filfbas 23222 . . . . . 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 23318 . . . . 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 1467 . . . 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 23676 . . . 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 1372 . . 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 3008 . 2 ((πœ‘ ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) β‰  βˆ…)
59 cuspusp 23675 . . . 4 (π‘Š ∈ CUnifSp β†’ π‘Š ∈ UnifSp)
6044, 59syl 17 . . 3 (πœ‘ β†’ π‘Š ∈ UnifSp)
6111uspreg 23649 . . 3 ((π‘Š ∈ UnifSp ∧ 𝐾 ∈ Haus) β†’ 𝐾 ∈ Reg)
6260, 7, 61syl2anc 585 . 2 (πœ‘ β†’ 𝐾 ∈ Reg)
631, 2, 6, 7, 15, 20, 22, 58, 62cnextcn 23441 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  Vcvv 3447   βŠ† wss 3914  βˆ…c0 4286  {csn 4590  βˆͺ cuni 4869  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091   β†Ύt crest 17310  TopOpenctopn 17311  fBascfbas 20807  Topctop 22265  TopOnctopon 22282  TopSpctps 22304  clsccl 22392  neicnei 22471   Cn ccn 22598  Hauscha 22682  Regcreg 22683  Filcfil 23219   FilMap cfm 23307   fLim cflim 23308   fLimf cflf 23309  CnExtccnext 23433  UnifStcuss 23628  UnifSpcusp 23629  CauFiluccfilu 23661  CUnifSpccusp 23672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-map 8773  df-pm 8774  df-en 8890  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-fbas 20816  df-fg 20817  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-cld 22393  df-ntr 22394  df-cls 22395  df-nei 22472  df-cn 22601  df-cnp 22602  df-haus 22689  df-reg 22690  df-tx 22936  df-fil 23220  df-fm 23312  df-flim 23313  df-flf 23314  df-cnext 23434  df-ust 23575  df-utop 23606  df-usp 23632  df-cusp 23673
This theorem is referenced by:  ucnextcn  23679
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