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Theorem blsscls2 18536
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 1002 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  R  <  T )
3 xmetcl 18363 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  z  e.  X
)  ->  ( P D z )  e. 
RR* )
433expa 1154 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  z  e.  X )  ->  ( P D z )  e. 
RR* )
54adantlr 697 . . . . . . 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 1000 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  R  e.  RR* )
7 simplr2 1001 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  T  e.  RR* )
8 xrlelttr 10748 . . . . . . . 8  |-  ( ( ( P D z )  e.  RR*  /\  R  e.  RR*  /\  T  e. 
RR* )  ->  (
( ( P D z )  <_  R  /\  R  <  T )  ->  ( P D z )  <  T
) )
98exp3acom23 1382 . . . . . . 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 1185 . . . . . 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 15 . . . . 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 959 . . . . . . 7  |-  ( ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T )  ->  T  e.  RR* )
13 elbl2 18422 . . . . . . . 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 801 . . . . . . 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 635 . . . . . 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 631 . . . . 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 227 . . . 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 2791 . . 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 3422 . . 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 205 . 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 3394 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711    C_ wss 3322   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   RR*cxr 9121    < clt 9122    <_ cle 9123   * Metcxmt 16688   ballcbl 16690   MetOpencmopn 16693
This theorem is referenced by:  blcld  18537  blsscls  18539  ubthlem1  22374
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-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-pre-lttri 9066  ax-pre-lttrn 9067
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-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  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-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-psmet 16696  df-xmet 16697  df-bl 16699
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