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Theorem blsscls2 18066
Description: A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)
Hypotheses
Ref Expression
mopni.1  |-  J  =  ( MetOpen `  D )
blcld.3  |-  S  =  { z  e.  X  |  ( P D z )  <_  R }
Assertion
Ref Expression
blsscls2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  ->  S  C_  ( P (
ball `  D ) T ) )
Distinct variable groups:    z, D    z, R    z, P    z, T    z, X
Allowed substitution hints:    S( z)    J( z)

Proof of Theorem blsscls2
StepHypRef Expression
1 blcld.3 . 2  |-  S  =  { z  e.  X  |  ( P D z )  <_  R }
2 simplr3 999 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  R  <  T )
3 xmetcl 17912 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  z  e.  X
)  ->  ( P D z )  e. 
RR* )
433expa 1151 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  z  e.  X )  ->  ( P D z )  e. 
RR* )
54adantlr 695 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  ( P D z )  e.  RR* )
6 simplr1 997 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  R  e.  RR* )
7 simplr2 998 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  T  e.  RR* )
8 xrlelttr 10503 . . . . . . . 8  |-  ( ( ( P D z )  e.  RR*  /\  R  e.  RR*  /\  T  e. 
RR* )  ->  (
( ( P D z )  <_  R  /\  R  <  T )  ->  ( P D z )  <  T
) )
98exp3acom23 1362 . . . . . . 7  |-  ( ( ( P D z )  e.  RR*  /\  R  e.  RR*  /\  T  e. 
RR* )  ->  ( R  <  T  ->  (
( P D z )  <_  R  ->  ( P D z )  <  T ) ) )
105, 6, 7, 9syl3anc 1182 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  ( R  <  T  ->  ( ( P D z )  <_  R  ->  ( P D z )  <  T ) ) )
112, 10mpd 14 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  ( ( P D z )  <_  R  ->  ( P D z )  <  T ) )
12 simp2 956 . . . . . . 7  |-  ( ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T )  ->  T  e.  RR* )
13 elbl2 17966 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  T  e.  RR* )  /\  ( P  e.  X  /\  z  e.  X ) )  -> 
( z  e.  ( P ( ball `  D
) T )  <->  ( P D z )  < 
T ) )
1413an4s 799 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( T  e.  RR*  /\  z  e.  X ) )  -> 
( z  e.  ( P ( ball `  D
) T )  <->  ( P D z )  < 
T ) )
1512, 14sylanr1 633 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  (
( R  e.  RR*  /\  T  e.  RR*  /\  R  <  T )  /\  z  e.  X ) )  -> 
( z  e.  ( P ( ball `  D
) T )  <->  ( P D z )  < 
T ) )
1615anassrs 629 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  ( z  e.  ( P ( ball `  D
) T )  <->  ( P D z )  < 
T ) )
1711, 16sylibrd 225 . . . 4  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  ( ( P D z )  <_  R  ->  z  e.  ( P ( ball `  D
) T ) ) )
1817ralrimiva 2639 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  ->  A. z  e.  X  ( ( P D z )  <_  R  ->  z  e.  ( P ( ball `  D
) T ) ) )
19 rabss 3263 . . 3  |-  ( { z  e.  X  | 
( P D z )  <_  R }  C_  ( P ( ball `  D ) T )  <->  A. z  e.  X  ( ( P D z )  <_  R  ->  z  e.  ( P ( ball `  D
) T ) ) )
2018, 19sylibr 203 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  ->  { z  e.  X  |  ( P D z )  <_  R }  C_  ( P (
ball `  D ) T ) )
211, 20syl5eqss 3235 1  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  ->  S  C_  ( P (
ball `  D ) T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RR*cxr 8882    < clt 8883    <_ cle 8884   * Metcxmt 16385   ballcbl 16387   MetOpencmopn 16388
This theorem is referenced by:  blcld  18067  blsscls  18069  ubthlem1  21465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-xmet 16389  df-bl 16391
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