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Theorem blss2ps 21031
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.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blss2ps  |-  ( ( ( D  e.  (PsMet `  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 blss2ps
StepHypRef Expression
1 simpl1 999 . 2  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  Q  e.  X )  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  D  e.  (PsMet `  X
) )
2 simpl2 1000 . 2  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  Q  e.  X )  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  P  e.  X )
3 simpl3 1001 . 2  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  Q  e.  X )  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  Q  e.  X )
4 simpr1 1002 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  Q  e.  X )  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  R  e.  RR )
54rexrd 9660 . 2  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  Q  e.  X )  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  R  e.  RR* )
6 simpr2 1003 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  Q  e.  X )  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  S  e.  RR )
76rexrd 9660 . 2  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  Q  e.  X )  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  ->  S  e.  RR* )
86, 4resubcld 10008 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  Q  e.  X )  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  -> 
( S  -  R
)  e.  RR )
9 simpr3 1004 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  Q  e.  X )  /\  ( R  e.  RR  /\  S  e.  RR  /\  ( P D Q )  <_ 
( S  -  R
) ) )  -> 
( P D Q )  <_  ( S  -  R ) )
10 psmetlecl 20944 . . 3  |-  ( ( D  e.  (PsMet `  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 1237 . 2  |-  ( ( ( D  e.  (PsMet `  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 11457 . . . 4  |-  ( ( S  e.  RR  /\  R  e.  RR )  ->  ( S +e  -e R )  =  ( S  -  R
) )
136, 4, 12syl2anc 661 . . 3  |-  ( ( ( D  e.  (PsMet `  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 4482 . 2  |-  ( ( ( D  e.  (PsMet `  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, 14xblss2ps 21029 1  |-  ( ( ( D  e.  (PsMet `  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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    C_ wss 3471   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   RRcr 9508    <_ cle 9646    - cmin 9824    -ecxne 11340   +ecxad 11341  PsMetcpsmet 18528   ballcbl 18531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-2 10615  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-psmet 18537  df-bl 18540
This theorem is referenced by:  blssps  21052
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