MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lpbl Unicode version

Theorem lpbl 18049
Description: Every ball around a limit point  P of a subset  S includes a member of  S (even if  P  e/  S). (Contributed by NM, 9-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypothesis
Ref Expression
mopni.1  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
lpbl  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  E. x  e.  S  x  e.  ( P
( ball `  D ) R ) )
Distinct variable groups:    x, D    x, J    x, R    x, S    x, P    x, X

Proof of Theorem lpbl
StepHypRef Expression
1 simpl1 958 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  D  e.  ( * Met `  X ) )
2 mopni.1 . . . . . . . . 9  |-  J  =  ( MetOpen `  D )
32mopntop 17986 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Top )
41, 3syl 15 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  J  e.  Top )
5 simpl2 959 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  S  C_  X )
62mopnuni 17987 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
71, 6syl 15 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  X  =  U. J )
85, 7sseqtrd 3214 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  S  C_  U. J )
9 eqid 2283 . . . . . . . 8  |-  U. J  =  U. J
109lpss 16874 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( ( limPt `  J ) `  S
)  C_  U. J )
114, 8, 10syl2anc 642 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  -> 
( ( limPt `  J
) `  S )  C_ 
U. J )
12 simpl3 960 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  P  e.  ( ( limPt `  J ) `  S ) )
1311, 12sseldd 3181 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  P  e.  U. J )
1413, 7eleqtrrd 2360 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  P  e.  X )
15 simpr 447 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  R  e.  RR+ )
162blnei 18048 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR+ )  ->  ( P ( ball `  D ) R )  e.  ( ( nei `  J ) `  { P } ) )
171, 14, 15, 16syl3anc 1182 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  -> 
( P ( ball `  D ) R )  e.  ( ( nei `  J ) `  { P } ) )
189islp2 16877 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  U. J  /\  P  e.  U. J )  ->  ( P  e.  ( ( limPt `  J
) `  S )  <->  A. x  e.  ( ( nei `  J ) `
 { P }
) ( x  i^i  ( S  \  { P } ) )  =/=  (/) ) )
194, 8, 13, 18syl3anc 1182 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  A. x  e.  ( ( nei `  J
) `  { P } ) ( x  i^i  ( S  \  { P } ) )  =/=  (/) ) )
2012, 19mpbid 201 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  A. x  e.  (
( nei `  J
) `  { P } ) ( x  i^i  ( S  \  { P } ) )  =/=  (/) )
21 ineq1 3363 . . . . 5  |-  ( x  =  ( P (
ball `  D ) R )  ->  (
x  i^i  ( S  \  { P } ) )  =  ( ( P ( ball `  D
) R )  i^i  ( S  \  { P } ) ) )
2221neeq1d 2459 . . . 4  |-  ( x  =  ( P (
ball `  D ) R )  ->  (
( x  i^i  ( S  \  { P }
) )  =/=  (/)  <->  ( ( P ( ball `  D
) R )  i^i  ( S  \  { P } ) )  =/=  (/) ) )
2322rspcva 2882 . . 3  |-  ( ( ( P ( ball `  D ) R )  e.  ( ( nei `  J ) `  { P } )  /\  A. x  e.  ( ( nei `  J ) `  { P } ) ( x  i^i  ( S 
\  { P }
) )  =/=  (/) )  -> 
( ( P (
ball `  D ) R )  i^i  ( S  \  { P }
) )  =/=  (/) )
2417, 20, 23syl2anc 642 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  -> 
( ( P (
ball `  D ) R )  i^i  ( S  \  { P }
) )  =/=  (/) )
25 elin 3358 . . . . 5  |-  ( x  e.  ( ( P ( ball `  D
) R )  i^i  ( S  \  { P } ) )  <->  ( x  e.  ( P ( ball `  D ) R )  /\  x  e.  ( S  \  { P } ) ) )
26 eldifi 3298 . . . . . . 7  |-  ( x  e.  ( S  \  { P } )  ->  x  e.  S )
2726anim2i 552 . . . . . 6  |-  ( ( x  e.  ( P ( ball `  D
) R )  /\  x  e.  ( S  \  { P } ) )  ->  ( x  e.  ( P ( ball `  D ) R )  /\  x  e.  S
) )
2827ancomd 438 . . . . 5  |-  ( ( x  e.  ( P ( ball `  D
) R )  /\  x  e.  ( S  \  { P } ) )  ->  ( x  e.  S  /\  x  e.  ( P ( ball `  D ) R ) ) )
2925, 28sylbi 187 . . . 4  |-  ( x  e.  ( ( P ( ball `  D
) R )  i^i  ( S  \  { P } ) )  -> 
( x  e.  S  /\  x  e.  ( P ( ball `  D
) R ) ) )
3029eximi 1563 . . 3  |-  ( E. x  x  e.  ( ( P ( ball `  D ) R )  i^i  ( S  \  { P } ) )  ->  E. x ( x  e.  S  /\  x  e.  ( P ( ball `  D ) R ) ) )
31 n0 3464 . . 3  |-  ( ( ( P ( ball `  D ) R )  i^i  ( S  \  { P } ) )  =/=  (/)  <->  E. x  x  e.  ( ( P (
ball `  D ) R )  i^i  ( S  \  { P }
) ) )
32 df-rex 2549 . . 3  |-  ( E. x  e.  S  x  e.  ( P (
ball `  D ) R )  <->  E. x
( x  e.  S  /\  x  e.  ( P ( ball `  D
) R ) ) )
3330, 31, 323imtr4i 257 . 2  |-  ( ( ( P ( ball `  D ) R )  i^i  ( S  \  { P } ) )  =/=  (/)  ->  E. x  e.  S  x  e.  ( P ( ball `  D
) R ) )
3424, 33syl 15 1  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  E. x  e.  S  x  e.  ( P
( ball `  D ) R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827   ` cfv 5255  (class class class)co 5858   RR+crp 10354   * Metcxmt 16369   ballcbl 16371   MetOpencmopn 16372   Topctop 16631   neicnei 16834   limPtclp 16866
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-topgen 13344  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868
  Copyright terms: Public domain W3C validator