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Theorem ssblex 17974
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 10402 . . 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 3601 . . 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 10390 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR )
72rpred 10390 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR )
81rpred 10390 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR )
93rpred 10390 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR )
10 min1 10517 . . . 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 10393 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
0  <  R )
13 halfpos 9942 . . . . 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 8974 . 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 10391 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR* )
193rpxrd 10391 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR* )
20 min2 10518 . . . 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 17971 . . 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 4026 . . . 4  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( x  <  R  <->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <  R )
)
25 oveq2 5866 . . . . 5  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( P ( ball `  D ) x )  =  ( P (
ball `  D ) if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
) ) )
2625sseq1d 3205 . . . 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 2884 . 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 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ifcif 3565   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   RR*cxr 8866    < clt 8867    <_ cle 8868    / cdiv 9423   2c2 9795   RR+crp 10354   * Metcxmt 16369   ballcbl 16371
This theorem is referenced by:  mopni3  18040
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-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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-reu 2550  df-rmo 2551  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-riota 6304  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-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-xmet 16373  df-bl 16375
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