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Theorem ucnextcn 24253
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 24212 . . . . . 6 ((𝑉 ∈ UnifSp ∧ 𝐴𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
1512, 10, 14syl2anc 582 . . . . 5 (𝜑𝑇 ∈ (UnifOn‘𝐴))
16 cuspusp 24249 . . . . . . . 8 (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
178, 16syl 17 . . . . . . 7 (𝜑𝑊 ∈ UnifSp)
182, 5, 4isusp 24210 . . . . . . 7 (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑌) ∧ 𝐾 = (unifTop‘𝑈)))
1917, 18sylib 217 . . . . . 6 (𝜑 → (𝑈 ∈ (UnifOn‘𝑌) ∧ 𝐾 = (unifTop‘𝑈)))
2019simpld 493 . . . . 5 (𝜑𝑈 ∈ (UnifOn‘𝑌))
21 isucn 24227 . . . . 5 ((𝑇 ∈ (UnifOn‘𝐴) ∧ 𝑈 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑇 Cnu𝑈) ↔ (𝐹:𝐴𝑌 ∧ ∀𝑤𝑈𝑣𝑇𝑦𝐴𝑧𝐴 (𝑦𝑣𝑧 → (𝐹𝑦)𝑤(𝐹𝑧)))))
2215, 20, 21syl2anc 582 . . . 4 (𝜑 → (𝐹 ∈ (𝑇 Cnu𝑈) ↔ (𝐹:𝐴𝑌 ∧ ∀𝑤𝑈𝑣𝑇𝑦𝐴𝑧𝐴 (𝑦𝑣𝑧 → (𝐹𝑦)𝑤(𝐹𝑧)))))
2311, 22mpbid 231 . . 3 (𝜑 → (𝐹:𝐴𝑌 ∧ ∀𝑤𝑈𝑣𝑇𝑦𝐴𝑧𝐴 (𝑦𝑣𝑧 → (𝐹𝑦)𝑤(𝐹𝑧))))
2423simpld 493 . 2 (𝜑𝐹:𝐴𝑌)
25 ucnextcn.c . 2 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
2620adantr 479 . . . . 5 ((𝜑𝑥𝑋) → 𝑈 ∈ (UnifOn‘𝑌))
2726elfvexd 6935 . . . 4 ((𝜑𝑥𝑋) → 𝑌 ∈ V)
28 simpr 483 . . . . . . 7 ((𝜑𝑥𝑋) → 𝑥𝑋)
2925adantr 479 . . . . . . 7 ((𝜑𝑥𝑋) → ((cls‘𝐽)‘𝐴) = 𝑋)
3028, 29eleqtrrd 2828 . . . . . 6 ((𝜑𝑥𝑋) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
311, 3istps 22880 . . . . . . . . 9 (𝑉 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
326, 31sylib 217 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘𝑋))
3332adantr 479 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑋))
3410adantr 479 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐴𝑋)
35 trnei 23840 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋𝑥𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3633, 34, 28, 35syl3anc 1368 . . . . . 6 ((𝜑𝑥𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3730, 36mpbid 231 . . . . 5 ((𝜑𝑥𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
38 filfbas 23796 . . . . 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 479 . . . 4 ((𝜑𝑥𝑋) → 𝐹:𝐴𝑌)
41 fmval 23891 . . . 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 1368 . . 3 ((𝜑𝑥𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎))))
4315adantr 479 . . . . 5 ((𝜑𝑥𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
4411adantr 479 . . . . 5 ((𝜑𝑥𝑋) → 𝐹 ∈ (𝑇 Cnu𝑈))
45 ucnextcn.s . . . . . . . . . . 11 𝑆 = (UnifSt‘𝑉)
461, 45, 3isusp 24210 . . . . . . . . . 10 (𝑉 ∈ UnifSp ↔ (𝑆 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑆)))
4712, 46sylib 217 . . . . . . . . 9 (𝜑 → (𝑆 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑆)))
4847simpld 493 . . . . . . . 8 (𝜑𝑆 ∈ (UnifOn‘𝑋))
4948adantr 479 . . . . . . 7 ((𝜑𝑥𝑋) → 𝑆 ∈ (UnifOn‘𝑋))
5012adantr 479 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝑉 ∈ UnifSp)
516adantr 479 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝑉 ∈ TopSp)
521, 3, 45neipcfilu 24245 . . . . . . . 8 ((𝑉 ∈ UnifSp ∧ 𝑉 ∈ TopSp ∧ 𝑥𝑋) → ((nei‘𝐽)‘{𝑥}) ∈ (CauFilu𝑆))
5350, 51, 28, 52syl3anc 1368 . . . . . . 7 ((𝜑𝑥𝑋) → ((nei‘𝐽)‘{𝑥}) ∈ (CauFilu𝑆))
54 0nelfb 23779 . . . . . . . 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 24243 . . . . . . 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 1372 . . . . . 6 ((𝜑𝑥𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu‘(𝑆t (𝐴 × 𝐴))))
5843elfvexd 6935 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐴 ∈ V)
59 ressuss 24211 . . . . . . . . 9 (𝐴 ∈ V → (UnifSt‘(𝑉s 𝐴)) = ((UnifSt‘𝑉) ↾t (𝐴 × 𝐴)))
6045oveq1i 7429 . . . . . . . . 9 (𝑆t (𝐴 × 𝐴)) = ((UnifSt‘𝑉) ↾t (𝐴 × 𝐴))
6159, 13, 603eqtr4g 2790 . . . . . . . 8 (𝐴 ∈ V → 𝑇 = (𝑆t (𝐴 × 𝐴)))
6261fveq2d 6900 . . . . . . 7 (𝐴 ∈ V → (CauFilu𝑇) = (CauFilu‘(𝑆t (𝐴 × 𝐴))))
6358, 62syl 17 . . . . . 6 ((𝜑𝑥𝑋) → (CauFilu𝑇) = (CauFilu‘(𝑆t (𝐴 × 𝐴))))
6457, 63eleqtrrd 2828 . . . . 5 ((𝜑𝑥𝑋) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (CauFilu𝑇))
65 imaeq2 6060 . . . . . . 7 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
6665cbvmptv 5262 . . . . . 6 (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎)) = (𝑏 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑏))
6766rneqi 5939 . . . . 5 ran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎)) = ran (𝑏 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑏))
6843, 26, 44, 64, 67fmucnd 24241 . . . 4 ((𝜑𝑥𝑋) → ran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎)) ∈ (CauFilu𝑈))
69 cfilufg 24242 . . . 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 582 . . 3 ((𝜑𝑥𝑋) → (𝑌filGenran (𝑎 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↦ (𝐹𝑎))) ∈ (CauFilu𝑈))
7142, 70eqeltrd 2825 . 2 ((𝜑𝑥𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu𝑈))
721, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 71cnextucn 24252 1 (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  Vcvv 3461  wss 3944  c0 4322  {csn 4630   class class class wbr 5149  cmpt 5232   × cxp 5676  ran crn 5679  cima 5681  wf 6545  cfv 6549  (class class class)co 7419  Basecbs 17183  s cress 17212  t crest 17405  TopOpenctopn 17406  fBascfbas 21284  filGencfg 21285  TopOnctopon 22856  TopSpctps 22878  clsccl 22966  neicnei 23045   Cn ccn 23172  Hauscha 23256  Filcfil 23793   FilMap cfm 23881  CnExtccnext 24007  UnifOncust 24148  unifTopcutop 24179  UnifStcuss 24202  UnifSpcusp 24203   Cnucucn 24224  CauFiluccfilu 24235  CUnifSpccusp 24246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fi 9436  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-unif 17259  df-rest 17407  df-topgen 17428  df-fbas 21293  df-fg 21294  df-top 22840  df-topon 22857  df-topsp 22879  df-bases 22893  df-cld 22967  df-ntr 22968  df-cls 22969  df-nei 23046  df-cn 23175  df-cnp 23176  df-haus 23263  df-reg 23264  df-tx 23510  df-fil 23794  df-fm 23886  df-flim 23887  df-flf 23888  df-cnext 24008  df-ust 24149  df-utop 24180  df-uss 24205  df-usp 24206  df-ucn 24225  df-cfilu 24236  df-cusp 24247
This theorem is referenced by:  rrhcn  33729
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