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Theorem blss2 14304
Description: One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
blss2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  -> 
( P ( ball `  D ) R ) 
C_  ( Q (
ball `  D ) S ) )

Proof of Theorem blss2
StepHypRef Expression
1 simpl1 1002 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  D  e.  ( *Met `  X ) )
2 simpl2 1003 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  P  e.  X )
3 simpl3 1004 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  Q  e.  X )
4 simpr1 1005 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  R  e.  RR )
54rexrd 8025 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  R  e.  RR* )
6 simpr2 1006 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  S  e.  RR )
76rexrd 8025 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  S  e.  RR* )
86, 4resubcld 8356 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  -> 
( S  -  R
)  e.  RR )
9 simpr3 1007 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  -> 
( P D Q )  <_  ( S  -  R ) )
10 xmetlecl 14264 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( P  e.  X  /\  Q  e.  X )  /\  (
( S  -  R
)  e.  RR  /\  ( P D Q )  <_  ( S  -  R ) ) )  ->  ( P D Q )  e.  RR )
111, 2, 3, 8, 9, 10syl122anc 1258 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  -> 
( P D Q )  e.  RR )
12 rexsub 9871 . . . 4  |-  ( ( S  e.  RR  /\  R  e.  RR )  ->  ( S +e  -e R )  =  ( S  -  R
) )
136, 4, 12syl2anc 411 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  -> 
( S +e  -e R )  =  ( S  -  R
) )
149, 13breqtrrd 4046 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  -> 
( P D Q )  <_  ( S +e  -e R ) )
151, 2, 3, 5, 7, 11, 14xblss2 14302 1  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  -> 
( P ( ball `  D ) R ) 
C_  ( Q (
ball `  D ) S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160    C_ wss 3144   class class class wbr 4018   ` cfv 5231  (class class class)co 5891   RRcr 7828    <_ cle 8011    - cmin 8146    -ecxne 9787   +ecxad 9788   *Metcxmet 13810   ballcbl 13812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-map 6668  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-2 8996  df-xneg 9790  df-xadd 9791  df-psmet 13817  df-xmet 13818  df-bl 13820
This theorem is referenced by:  blhalf  14305  blss  14325
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