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Theorem blsscls2 20737
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 1035 . . . . . 6  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  R  <  T )
3 xmetcl 20564 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  z  e.  X
)  ->  ( P D z )  e. 
RR* )
433expa 1191 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  z  e.  X )  ->  ( P D z )  e. 
RR* )
54adantlr 714 . . . . . . 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 1033 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  R  e.  RR* )
7 simplr2 1034 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T ) )  /\  z  e.  X )  ->  T  e.  RR* )
8 xrlelttr 11350 . . . . . . . 8  |-  ( ( ( P D z )  e.  RR*  /\  R  e.  RR*  /\  T  e. 
RR* )  ->  (
( ( P D z )  <_  R  /\  R  <  T )  ->  ( P D z )  <  T
) )
98expcomd 438 . . . . . . 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 1223 . . . . . 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 992 . . . . . . 7  |-  ( ( R  e.  RR*  /\  T  e.  RR*  /\  R  < 
T )  ->  T  e.  RR* )
13 elbl2 20623 . . . . . . . 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 823 . . . . . . 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 652 . . . . . 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 648 . . . . 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 234 . . . 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 2873 . . 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 3572 . . 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 212 . 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 3543 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809   {crab 2813    C_ wss 3471   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   RR*cxr 9618    < clt 9619    <_ cle 9620   *Metcxmt 18169   ballcbl 18171   MetOpencmopn 18174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-pre-lttri 9557  ax-pre-lttrn 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-psmet 18177  df-xmet 18178  df-bl 18180
This theorem is referenced by:  blcld  20738  blsscls  20740  ubthlem1  25450
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