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Theorem blsscls2 18050
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 17896 . . . . . . . . 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 10487 . . . . . . . 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 17950 . . . . . . . 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 2626 . . 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 3250 . . 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 3222 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 1623    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RR*cxr 8866    < clt 8867    <_ cle 8868   * Metcxmt 16369   ballcbl 16371   MetOpencmopn 16372
This theorem is referenced by:  blcld  18051  blsscls  18053  ubthlem1  21449
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  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-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-xmet 16373  df-bl 16375
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