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Theorem ssblex 18459
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 734 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR+ )
21rphalfcld 10661 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR+ )
3 simprr 735 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR+ )
4 ifcl 3776 . . 3  |-  ( ( ( R  /  2
)  e.  RR+  /\  S  e.  RR+ )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR+ )
52, 3, 4syl2anc 644 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR+ )
65rpred 10649 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR )
72rpred 10649 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR )
81rpred 10649 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR )
93rpred 10649 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR )
10 min1 10777 . . . 4  |-  ( ( ( R  /  2
)  e.  RR  /\  S  e.  RR )  ->  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  <_  ( R  /  2 ) )
117, 9, 10syl2anc 644 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <_  ( R  /  2 ) )
121rpgt0d 10652 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
0  <  R )
13 halfpos 10199 . . . . 5  |-  ( R  e.  RR  ->  (
0  <  R  <->  ( R  /  2 )  < 
R ) )
148, 13syl 16 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( 0  <  R  <->  ( R  /  2 )  <  R ) )
1512, 14mpbid 203 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  <  R )
166, 7, 8, 11, 15lelttrd 9229 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <  R )
17 simpl 445 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( D  e.  ( * Met `  X
)  /\  P  e.  X ) )
185rpxrd 10650 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR* )
193rpxrd 10650 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR* )
20 min2 10778 . . . 4  |-  ( ( ( R  /  2
)  e.  RR  /\  S  e.  RR )  ->  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  <_  S
)
217, 9, 20syl2anc 644 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <_  S )
22 ssbl 18454 . . 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 1190 . 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 4216 . . . 4  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( x  <  R  <->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <  R )
)
25 oveq2 6090 . . . . 5  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( P ( ball `  D ) x )  =  ( P (
ball `  D ) if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
) ) )
2625sseq1d 3376 . . . 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 693 . . 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 3053 . 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 1183 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2707    C_ wss 3321   ifcif 3740   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   RRcr 8990   0cc0 8991   RR*cxr 9120    < clt 9121    <_ cle 9122    / cdiv 9678   2c2 10050   RR+crp 10613   * Metcxmt 16687   ballcbl 16689
This theorem is referenced by:  mopni3  18525
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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-2 10059  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-psmet 16695  df-xmet 16696  df-bl 16698
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