MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bl2in Structured version   Unicode version

Theorem bl2in 21069
Description: Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
bl2in  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  -> 
( ( P (
ball `  D ) R )  i^i  ( Q ( ball `  D
) R ) )  =  (/) )

Proof of Theorem bl2in
StepHypRef Expression
1 simpl1 997 . . 3  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  ->  D  e.  ( Met `  X ) )
2 metxmet 21003 . . 3  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
31, 2syl 16 . 2  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  ->  D  e.  ( *Met `  X ) )
4 simpl2 998 . 2  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  ->  P  e.  X )
5 simpl3 999 . 2  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  ->  Q  e.  X )
6 rexr 9628 . . 3  |-  ( R  e.  RR  ->  R  e.  RR* )
76ad2antrl 725 . 2  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  ->  R  e.  RR* )
8 simprl 754 . . . . 5  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  ->  R  e.  RR )
9 rexadd 11434 . . . . 5  |-  ( ( R  e.  RR  /\  R  e.  RR )  ->  ( R +e
R )  =  ( R  +  R ) )
108, 8, 9syl2anc 659 . . . 4  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  -> 
( R +e
R )  =  ( R  +  R ) )
118recnd 9611 . . . . 5  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  ->  R  e.  CC )
12112timesd 10777 . . . 4  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  -> 
( 2  x.  R
)  =  ( R  +  R ) )
1310, 12eqtr4d 2498 . . 3  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  -> 
( R +e
R )  =  ( 2  x.  R ) )
14 id 22 . . . . . 6  |-  ( R  e.  RR  ->  R  e.  RR )
15 metcl 21001 . . . . . 6  |-  ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X )  ->  ( P D Q )  e.  RR )
16 2re 10601 . . . . . . . 8  |-  2  e.  RR
17 2pos 10623 . . . . . . . 8  |-  0  <  2
1816, 17pm3.2i 453 . . . . . . 7  |-  ( 2  e.  RR  /\  0  <  2 )
19 lemuldiv2 10420 . . . . . . 7  |-  ( ( R  e.  RR  /\  ( P D Q )  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( 2  x.  R )  <_ 
( P D Q )  <->  R  <_  ( ( P D Q )  /  2 ) ) )
2018, 19mp3an3 1311 . . . . . 6  |-  ( ( R  e.  RR  /\  ( P D Q )  e.  RR )  -> 
( ( 2  x.  R )  <_  ( P D Q )  <->  R  <_  ( ( P D Q )  /  2 ) ) )
2114, 15, 20syl2anr 476 . . . . 5  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  R  e.  RR )  ->  ( ( 2  x.  R )  <_  ( P D Q )  <->  R  <_  ( ( P D Q )  /  2 ) ) )
2221biimprd 223 . . . 4  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  R  e.  RR )  ->  ( R  <_  ( ( P D Q )  / 
2 )  ->  (
2  x.  R )  <_  ( P D Q ) ) )
2322impr 617 . . 3  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  -> 
( 2  x.  R
)  <_  ( P D Q ) )
2413, 23eqbrtrd 4459 . 2  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  -> 
( R +e
R )  <_  ( P D Q ) )
25 bldisj 21067 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR*  /\  R  e. 
RR*  /\  ( R +e R )  <_  ( P D Q ) ) )  ->  ( ( P ( ball `  D
) R )  i^i  ( Q ( ball `  D ) R ) )  =  (/) )
263, 4, 5, 7, 7, 24, 25syl33anc 1241 1  |-  ( ( ( D  e.  ( Met `  X )  /\  P  e.  X  /\  Q  e.  X
)  /\  ( R  e.  RR  /\  R  <_ 
( ( P D Q )  /  2
) ) )  -> 
( ( P (
ball `  D ) R )  i^i  ( Q ( ball `  D
) R ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    i^i cin 3460   (/)c0 3783   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481    + caddc 9484    x. cmul 9486   RR*cxr 9616    < clt 9617    <_ cle 9618    / cdiv 10202   2c2 10581   +ecxad 11319   *Metcxmt 18598   Metcme 18599   ballcbl 18600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-2 10590  df-xneg 11321  df-xadd 11322  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator