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Theorem ucnextcn 22031
Description: Extension by continuity. Theorem 2 of [BourbakiTop1] p. II.20. Given an uniform space on a set 𝑋, a subset 𝐴 dense in 𝑋, and a function 𝐹 uniformly continuous from 𝐴 to 𝑌, that function can be extended by continuity to the whole 𝑋, and its extension is uniformly continuous. (Contributed by Thierry Arnoux, 25-Jan-2018.)
Hypotheses
Ref Expression
ucnextcn.x 𝑋 = (Base‘𝑉)
ucnextcn.y 𝑌 = (Base‘𝑊)
ucnextcn.j 𝐽 = (TopOpen‘𝑉)
ucnextcn.k 𝐾 = (TopOpen‘𝑊)
ucnextcn.s 𝑆 = (UnifSt‘𝑉)
ucnextcn.t 𝑇 = (UnifSt‘(𝑉s 𝐴))
ucnextcn.u 𝑈 = (UnifSt‘𝑊)
ucnextcn.v (𝜑𝑉 ∈ TopSp)
ucnextcn.r (𝜑𝑉 ∈ UnifSp)
ucnextcn.w (𝜑𝑊 ∈ TopSp)
ucnextcn.z (𝜑𝑊 ∈ CUnifSp)
ucnextcn.h (𝜑𝐾 ∈ Haus)
ucnextcn.a (𝜑𝐴𝑋)
ucnextcn.f (𝜑𝐹 ∈ (𝑇 Cnu𝑈))
ucnextcn.c (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
Assertion
Ref Expression
ucnextcn (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Proof of Theorem ucnextcn
Dummy variables 𝑎 𝑏 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ucnextcn.x . 2 𝑋 = (Base‘𝑉)
2 ucnextcn.y . 2 𝑌 = (Base‘𝑊)
3 ucnextcn.j . 2 𝐽 = (TopOpen‘𝑉)
4 ucnextcn.k . 2 𝐾 = (TopOpen‘𝑊)
5 ucnextcn.u . 2 𝑈 = (UnifSt‘𝑊)
6 ucnextcn.v . 2 (𝜑𝑉 ∈ TopSp)
7 ucnextcn.w . 2 (𝜑𝑊 ∈ TopSp)
8 ucnextcn.z . 2 (𝜑𝑊 ∈ CUnifSp)
9 ucnextcn.h . 2 (𝜑𝐾 ∈ Haus)
10 ucnextcn.a . 2 (𝜑𝐴𝑋)
11 ucnextcn.f . . . 4 (𝜑𝐹 ∈ (𝑇 Cnu𝑈))
12 ucnextcn.r . . . . . 6 (𝜑𝑉 ∈ UnifSp)
13 ucnextcn.t . . . . . . 7 𝑇 = (UnifSt‘(𝑉s 𝐴))
141, 13ressust 21991 . . . . . 6 ((𝑉 ∈ UnifSp ∧ 𝐴𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
1512, 10, 14syl2anc 692 . . . . 5 (𝜑𝑇 ∈ (UnifOn‘𝐴))
16 cuspusp 22027 . . . . . . . 8 (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
178, 16syl 17 . . . . . . 7 (𝜑𝑊 ∈ UnifSp)
182, 5, 4isusp 21988 . . . . . . 7 (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑌) ∧ 𝐾 = (unifTop‘𝑈)))
1917, 18sylib 208 . . . . . 6 (𝜑 → (𝑈 ∈ (UnifOn‘𝑌) ∧ 𝐾 = (unifTop‘𝑈)))
2019simpld 475 . . . . 5 (𝜑𝑈 ∈ (UnifOn‘𝑌))
21 isucn 22005 . . . . 5 ((𝑇 ∈ (UnifOn‘𝐴) ∧ 𝑈 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑇 Cnu𝑈) ↔ (𝐹:𝐴𝑌 ∧ ∀𝑤𝑈𝑣𝑇𝑦𝐴𝑧𝐴 (𝑦𝑣𝑧 → (𝐹𝑦)𝑤(𝐹𝑧)))))
2215, 20, 21syl2anc 692 . . . 4 (𝜑 → (𝐹 ∈ (𝑇 Cnu𝑈) ↔ (𝐹:𝐴𝑌 ∧ ∀𝑤𝑈𝑣𝑇𝑦𝐴𝑧𝐴 (𝑦𝑣𝑧 → (𝐹𝑦)𝑤(𝐹𝑧)))))
2311, 22mpbid 222 . . 3 (𝜑 → (𝐹:𝐴𝑌 ∧ ∀𝑤𝑈𝑣𝑇𝑦𝐴𝑧𝐴 (𝑦𝑣𝑧 → (𝐹𝑦)𝑤(𝐹𝑧))))
2423simpld 475 . 2 (𝜑𝐹:𝐴𝑌)
25 ucnextcn.c . 2 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
2620adantr 481 . . . . 5 ((𝜑𝑥𝑋) → 𝑈 ∈ (UnifOn‘𝑌))
2726elfvexd 6184 . . . 4 ((𝜑𝑥𝑋) → 𝑌 ∈ V)
28 simpr 477 . . . . . . 7 ((𝜑𝑥𝑋) → 𝑥𝑋)
2925adantr 481 . . . . . . 7 ((𝜑𝑥𝑋) → ((cls‘𝐽)‘𝐴) = 𝑋)
3028, 29eleqtrrd 2701 . . . . . 6 ((𝜑𝑥𝑋) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
311, 3istps 20662 . . . . . . . . 9 (𝑉 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
326, 31sylib 208 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘𝑋))
3332adantr 481 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑋))
3410adantr 481 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐴𝑋)
35 trnei 21619 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋𝑥𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3633, 34, 28, 35syl3anc 1323 . . . . . 6 ((𝜑𝑥𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3730, 36mpbid 222 . . . . 5 ((𝜑𝑥𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
38 filfbas 21575 . . . . 5 ((((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
3937, 38syl 17 . . . 4 ((𝜑𝑥𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴))
4024adantr 481 . . . 4 ((𝜑𝑥𝑋) → 𝐹:𝐴𝑌)
41 fmval 21670 . . . 4 ((𝑌 ∈ V ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴) ∧ 𝐹:𝐴𝑌) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎))))
4227, 39, 40, 41syl3anc 1323 . . 3 ((𝜑𝑥𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎))))
4315adantr 481 . . . . 5 ((𝜑𝑥𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
4411adantr 481 . . . . 5 ((𝜑𝑥𝑋) → 𝐹 ∈ (𝑇 Cnu𝑈))
45 ucnextcn.s . . . . . . . . . . 11 𝑆 = (UnifSt‘𝑉)
461, 45, 3isusp 21988 . . . . . . . . . 10 (𝑉 ∈ UnifSp ↔ (𝑆 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑆)))
4712, 46sylib 208 . . . . . . . . 9 (𝜑 → (𝑆 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑆)))
4847simpld 475 . . . . . . . 8 (𝜑𝑆 ∈ (UnifOn‘𝑋))
4948adantr 481 . . . . . . 7 ((𝜑𝑥𝑋) → 𝑆 ∈ (UnifOn‘𝑋))
5012adantr 481 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝑉 ∈ UnifSp)
516adantr 481 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝑉 ∈ TopSp)
521, 3, 45neipcfilu 22023 . . . . . . . 8 ((𝑉 ∈ UnifSp ∧ 𝑉 ∈ TopSp ∧ 𝑥𝑋) → ((nei‘𝐽)‘{𝑥}) ∈ (CauFilu𝑆))
5350, 51, 28, 52syl3anc 1323 . . . . . . 7 ((𝜑𝑥𝑋) → ((nei‘𝐽)‘{𝑥}) ∈ (CauFilu𝑆))
54 0nelfb 21558 . . . . . . . 8 ((((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (fBas‘𝐴) → ¬ ∅ ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))
5539, 54syl 17 . . . . . . 7 ((𝜑𝑥𝑋) → ¬ ∅ ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))
56 trcfilu 22021 . . . . . . 7 ((𝑆 ∈ (UnifOn‘𝑋) ∧ (((nei‘𝐽)‘{𝑥}) ∈ (CauFilu𝑆) ∧ ¬ ∅ ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∧ 𝐴𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu‘(𝑆t (𝐴 × 𝐴))))
5749, 53, 55, 34, 56syl121anc 1328 . . . . . 6 ((𝜑𝑥𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu‘(𝑆t (𝐴 × 𝐴))))
5843elfvexd 6184 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐴 ∈ V)
59 ressuss 21990 . . . . . . . . 9 (𝐴 ∈ V → (UnifSt‘(𝑉s 𝐴)) = ((UnifSt‘𝑉) ↾t (𝐴 × 𝐴)))
6045oveq1i 6620 . . . . . . . . 9 (𝑆t (𝐴 × 𝐴)) = ((UnifSt‘𝑉) ↾t (𝐴 × 𝐴))
6159, 13, 603eqtr4g 2680 . . . . . . . 8 (𝐴 ∈ V → 𝑇 = (𝑆t (𝐴 × 𝐴)))
6261fveq2d 6157 . . . . . . 7 (𝐴 ∈ V → (CauFilu𝑇) = (CauFilu‘(𝑆t (𝐴 × 𝐴))))
6358, 62syl 17 . . . . . 6 ((𝜑𝑥𝑋) → (CauFilu𝑇) = (CauFilu‘(𝑆t (𝐴 × 𝐴))))
6457, 63eleqtrrd 2701 . . . . 5 ((𝜑𝑥𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu𝑇))
65 imaeq2 5426 . . . . . . 7 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
6665cbvmptv 4715 . . . . . 6 (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎)) = (𝑏 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑏))
6766rneqi 5317 . . . . 5 ran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎)) = ran (𝑏 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑏))
6843, 26, 44, 64, 67fmucnd 22019 . . . 4 ((𝜑𝑥𝑋) → ran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎)) ∈ (CauFilu𝑈))
69 cfilufg 22020 . . . 4 ((𝑈 ∈ (UnifOn‘𝑌) ∧ ran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎)) ∈ (CauFilu𝑈)) → (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎))) ∈ (CauFilu𝑈))
7026, 68, 69syl2anc 692 . . 3 ((𝜑𝑥𝑋) → (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎))) ∈ (CauFilu𝑈))
7142, 70eqeltrd 2698 . 2 ((𝜑𝑥𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu𝑈))
721, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 71cnextucn 22030 1 (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3189  wss 3559  c0 3896  {csn 4153   class class class wbr 4618  cmpt 4678   × cxp 5077  ran crn 5080  cima 5082  wf 5848  cfv 5852  (class class class)co 6610  Basecbs 15792  s cress 15793  t crest 16013  TopOpenctopn 16014  fBascfbas 19666  filGencfg 19667  TopOnctopon 20647  TopSpctps 20660  clsccl 20745  neicnei 20824   Cn ccn 20951  Hauscha 21035  Filcfil 21572   FilMap cfm 21660  CnExtccnext 21786  UnifOncust 21926  unifTopcutop 21957  UnifStcuss 21980  UnifSpcusp 21981   Cnucucn 22002  CauFiluccfilu 22013  CUnifSpccusp 22024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fi 8269  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-unif 15897  df-rest 16015  df-topgen 16036  df-fbas 19675  df-fg 19676  df-top 20631  df-topon 20648  df-topsp 20661  df-bases 20674  df-cld 20746  df-ntr 20747  df-cls 20748  df-nei 20825  df-cn 20954  df-cnp 20955  df-haus 21042  df-reg 21043  df-tx 21288  df-fil 21573  df-fm 21665  df-flim 21666  df-flf 21667  df-cnext 21787  df-ust 21927  df-utop 21958  df-uss 21983  df-usp 21984  df-ucn 22003  df-cfilu 22014  df-cusp 22025
This theorem is referenced by:  rrhcn  29847
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