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Theorem ssblex 17990
Description: A nested ball exists whose radius is less than any desired amount. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
Assertion
Ref Expression
ssblex  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  E. x  e.  RR+  (
x  <  R  /\  ( P ( ball `  D
) x )  C_  ( P ( ball `  D
) S ) ) )
Distinct variable groups:    x, D    x, R    x, P    x, S    x, X

Proof of Theorem ssblex
StepHypRef Expression
1 simprl 732 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR+ )
21rphalfcld 10418 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR+ )
3 simprr 733 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR+ )
4 ifcl 3614 . . 3  |-  ( ( ( R  /  2
)  e.  RR+  /\  S  e.  RR+ )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR+ )
52, 3, 4syl2anc 642 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR+ )
65rpred 10406 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR )
72rpred 10406 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR )
81rpred 10406 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR )
93rpred 10406 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR )
10 min1 10533 . . . 4  |-  ( ( ( R  /  2
)  e.  RR  /\  S  e.  RR )  ->  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  <_  ( R  /  2 ) )
117, 9, 10syl2anc 642 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <_  ( R  /  2 ) )
121rpgt0d 10409 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
0  <  R )
13 halfpos 9958 . . . . 5  |-  ( R  e.  RR  ->  (
0  <  R  <->  ( R  /  2 )  < 
R ) )
148, 13syl 15 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( 0  <  R  <->  ( R  /  2 )  <  R ) )
1512, 14mpbid 201 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  <  R )
166, 7, 8, 11, 15lelttrd 8990 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <  R )
17 simpl 443 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( D  e.  ( * Met `  X
)  /\  P  e.  X ) )
185rpxrd 10407 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR* )
193rpxrd 10407 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR* )
20 min2 10534 . . . 4  |-  ( ( ( R  /  2
)  e.  RR  /\  S  e.  RR )  ->  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  <_  S
)
217, 9, 20syl2anc 642 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <_  S )
22 ssbl 17987 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR*  /\  S  e.  RR* )  /\  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <_  S )  ->  ( P ( ball `  D ) if ( ( R  /  2
)  <_  S , 
( R  /  2
) ,  S ) )  C_  ( P
( ball `  D ) S ) )
2317, 18, 19, 21, 22syl121anc 1187 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( P ( ball `  D ) if ( ( R  /  2
)  <_  S , 
( R  /  2
) ,  S ) )  C_  ( P
( ball `  D ) S ) )
24 breq1 4042 . . . 4  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( x  <  R  <->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <  R )
)
25 oveq2 5882 . . . . 5  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( P ( ball `  D ) x )  =  ( P (
ball `  D ) if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
) ) )
2625sseq1d 3218 . . . 4  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( ( P (
ball `  D )
x )  C_  ( P ( ball `  D
) S )  <->  ( P
( ball `  D ) if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
) )  C_  ( P ( ball `  D
) S ) ) )
2724, 26anbi12d 691 . . 3  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( ( x  < 
R  /\  ( P
( ball `  D )
x )  C_  ( P ( ball `  D
) S ) )  <-> 
( if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  < 
R  /\  ( P
( ball `  D ) if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
) )  C_  ( P ( ball `  D
) S ) ) ) )
2827rspcev 2897 . 2  |-  ( ( if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  e.  RR+  /\  ( if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  < 
R  /\  ( P
( ball `  D ) if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
) )  C_  ( P ( ball `  D
) S ) ) )  ->  E. x  e.  RR+  ( x  < 
R  /\  ( P
( ball `  D )
x )  C_  ( P ( ball `  D
) S ) ) )
295, 16, 23, 28syl12anc 1180 1  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  E. x  e.  RR+  (
x  <  R  /\  ( P ( ball `  D
) x )  C_  ( P ( ball `  D
) S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   ifcif 3578   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   RR*cxr 8882    < clt 8883    <_ cle 8884    / cdiv 9439   2c2 9811   RR+crp 10370   * Metcxmt 16385   ballcbl 16387
This theorem is referenced by:  mopni3  18056
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-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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-reu 2563  df-rmo 2564  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-riota 6320  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-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-xmet 16389  df-bl 16391
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