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

Theorem lpbl 18535
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 961 . . . 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 18472 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Top )
41, 3syl 16 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  J  e.  Top )
5 simpl2 962 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  S  C_  X )
62mopnuni 18473 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
71, 6syl 16 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  X  =  U. J )
85, 7sseqtrd 3386 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  S  C_  U. J )
9 eqid 2438 . . . . . . . 8  |-  U. J  =  U. J
109lpss 17208 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( ( limPt `  J ) `  S
)  C_  U. J )
114, 8, 10syl2anc 644 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  -> 
( ( limPt `  J
) `  S )  C_ 
U. J )
12 simpl3 963 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  P  e.  ( ( limPt `  J ) `  S ) )
1311, 12sseldd 3351 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  P  e.  U. J )
1413, 7eleqtrrd 2515 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  P  e.  X )
15 simpr 449 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  P  e.  (
( limPt `  J ) `  S ) )  /\  R  e.  RR+ )  ->  R  e.  RR+ )
162blnei 18534 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR+ )  ->  ( P ( ball `  D ) R )  e.  ( ( nei `  J ) `  { P } ) )
171, 14, 15, 16syl3anc 1185 . . 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 17211 . . . . 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 1185 . . . 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 203 . . 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 3537 . . . . 5  |-  ( x  =  ( P (
ball `  D ) R )  ->  (
x  i^i  ( S  \  { P } ) )  =  ( ( P ( ball `  D
) R )  i^i  ( S  \  { P } ) ) )
2221neeq1d 2616 . . . 4  |-  ( x  =  ( P (
ball `  D ) R )  ->  (
( x  i^i  ( S  \  { P }
) )  =/=  (/)  <->  ( ( P ( ball `  D
) R )  i^i  ( S  \  { P } ) )  =/=  (/) ) )
2322rspcva 3052 . . 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 644 . 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 3532 . . . . 5  |-  ( x  e.  ( ( P ( ball `  D
) R )  i^i  ( S  \  { P } ) )  <->  ( x  e.  ( P ( ball `  D ) R )  /\  x  e.  ( S  \  { P } ) ) )
26 eldifi 3471 . . . . . . 7  |-  ( x  e.  ( S  \  { P } )  ->  x  e.  S )
2726anim2i 554 . . . . . 6  |-  ( ( x  e.  ( P ( ball `  D
) R )  /\  x  e.  ( S  \  { P } ) )  ->  ( x  e.  ( P ( ball `  D ) R )  /\  x  e.  S
) )
2827ancomd 440 . . . . 5  |-  ( ( x  e.  ( P ( ball `  D
) R )  /\  x  e.  ( S  \  { P } ) )  ->  ( x  e.  S  /\  x  e.  ( P ( ball `  D ) R ) ) )
2925, 28sylbi 189 . . . 4  |-  ( x  e.  ( ( P ( ball `  D
) R )  i^i  ( S  \  { P } ) )  -> 
( x  e.  S  /\  x  e.  ( P ( ball `  D
) R ) ) )
3029eximi 1586 . . 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 3639 . . 3  |-  ( ( ( P ( ball `  D ) R )  i^i  ( S  \  { P } ) )  =/=  (/)  <->  E. x  x  e.  ( ( P (
ball `  D ) R )  i^i  ( S  \  { P }
) ) )
32 df-rex 2713 . . 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 259 . 2  |-  ( ( ( P ( ball `  D ) R )  i^i  ( S  \  { P } ) )  =/=  (/)  ->  E. x  e.  S  x  e.  ( P ( ball `  D
) R ) )
3424, 33syl 16 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 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816   U.cuni 4017   ` cfv 5456  (class class class)co 6083   RR+crp 10614   * Metcxmt 16688   ballcbl 16690   MetOpencmopn 16693   Topctop 16960   neicnei 17163   limPtclp 17200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-topgen 13669  df-psmet 16696  df-xmet 16697  df-bl 16699  df-mopn 16700  df-top 16965  df-bases 16967  df-topon 16968  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202
  Copyright terms: Public domain W3C validator