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Theorem ssblex 20145
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 755 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR+ )
21rphalfcld 11154 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR+ )
3 simprr 756 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR+ )
4 ifcl 3942 . . 3  |-  ( ( ( R  /  2
)  e.  RR+  /\  S  e.  RR+ )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR+ )
52, 3, 4syl2anc 661 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR+ )
65rpred 11142 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR )
72rpred 11142 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR )
81rpred 11142 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR )
93rpred 11142 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR )
10 min1 11275 . . . 4  |-  ( ( ( R  /  2
)  e.  RR  /\  S  e.  RR )  ->  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  <_  ( R  /  2 ) )
117, 9, 10syl2anc 661 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <_  ( R  /  2 ) )
121rpgt0d 11145 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
0  <  R )
13 halfpos 10670 . . . . 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 210 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  <  R )
166, 7, 8, 11, 15lelttrd 9644 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <  R )
17 simpl 457 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( D  e.  ( *Met `  X
)  /\  P  e.  X ) )
185rpxrd 11143 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR* )
193rpxrd 11143 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR* )
20 min2 11276 . . . 4  |-  ( ( ( R  /  2
)  e.  RR  /\  S  e.  RR )  ->  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  <_  S
)
217, 9, 20syl2anc 661 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <_  S )
22 ssbl 20140 . . 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 1224 . 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 4406 . . . 4  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( x  <  R  <->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <  R )
)
25 oveq2 6211 . . . . 5  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( P ( ball `  D ) x )  =  ( P (
ball `  D ) if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
) ) )
2625sseq1d 3494 . . . 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 710 . . 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 3179 . 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 1217 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800    C_ wss 3439   ifcif 3902   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   RRcr 9396   0cc0 9397   RR*cxr 9532    < clt 9533    <_ cle 9534    / cdiv 10108   2c2 10486   RR+crp 11106   *Metcxmt 17936   ballcbl 17938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-2 10495  df-rp 11107  df-xneg 11204  df-xadd 11205  df-xmul 11206  df-psmet 17944  df-xmet 17945  df-bl 17947
This theorem is referenced by:  mopni3  20211
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