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Theorem blss2 18439
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 961 . 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 962 . 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 963 . 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 964 . . 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 9139 . 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 965 . . 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 9139 . 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 9470 . . 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 966 . . 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 18381 . . 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 1194 . 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 10824 . . . 4  |-  ( ( S  e.  RR  /\  R  e.  RR )  ->  ( S + e  - e R )  =  ( S  -  R
) )
136, 4, 12syl2anc 644 . . 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 4241 . 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 18437 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   RRcr 8994    <_ cle 9126    - cmin 9296    - ecxne 10712   + ecxad 10713   * Metcxmt 16691   ballcbl 16693
This theorem is referenced by:  blhalf  18440  blss  18460  metdstri  18886  ssbnd  26511  totbndbnd  26512  heiborlem6  26539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-2 10063  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-psmet 16699  df-xmet 16700  df-bl 16702
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