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Theorem blss2ps 20111
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 991 . 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 992 . 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 993 . 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 994 . . 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 9545 . 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 995 . . 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 9545 . 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 9888 . . 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 996 . . 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 20024 . . 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 1228 . 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 11315 . . . 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 4427 . 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 20109 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 965    = wceq 1370    e. wcel 1758    C_ wss 3437   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   RRcr 9393    <_ cle 9531    - cmin 9707    -ecxne 11198   +ecxad 11199  PsMetcpsmet 17926   ballcbl 17929
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 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-2 10492  df-rp 11104  df-xneg 11201  df-xadd 11202  df-xmul 11203  df-psmet 17935  df-bl 17938
This theorem is referenced by:  blssps  20132
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