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Theorem blss2 17959
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 958 . 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 959 . 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 960 . 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 961 . . 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 8881 . 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 962 . . 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 8881 . 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 9211 . . 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 963 . . 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 17911 . . 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 1191 . 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 10560 . . . 4  |-  ( ( S  e.  RR  /\  R  e.  RR )  ->  ( S + e  - e R )  =  ( S  -  R
) )
136, 4, 12syl2anc 642 . . 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 4049 . 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 17958 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736    <_ cle 8868    - cmin 9037    - ecxne 10449   + ecxad 10450   * Metcxmt 16369   ballcbl 16371
This theorem is referenced by:  blhalf  17960  blss  17972  metdstri  18355  ssbnd  26512  totbndbnd  26513  heiborlem6  26540
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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