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Theorem blin 18443
Description: The intersection of two balls with the same center is the smaller of them. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
Assertion
Ref Expression
blin  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( ( P (
ball `  D ) R )  i^i  ( P ( ball `  D
) S ) )  =  ( P (
ball `  D ) if ( R  <_  S ,  R ,  S ) ) )

Proof of Theorem blin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 xmetcl 18353 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  x  e.  X
)  ->  ( P D x )  e. 
RR* )
213expa 1153 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  x  e.  X )  ->  ( P D x )  e. 
RR* )
32adantlr 696 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  ( P D x )  e.  RR* )
4 simplrl 737 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  R  e.  RR* )
5 simplrr 738 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  S  e.  RR* )
6 xrltmin 10762 . . . . 5  |-  ( ( ( P D x )  e.  RR*  /\  R  e.  RR*  /\  S  e. 
RR* )  ->  (
( P D x )  <  if ( R  <_  S ,  R ,  S )  <->  ( ( P D x )  <  R  /\  ( P D x )  <  S ) ) )
73, 4, 5, 6syl3anc 1184 . . . 4  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  ( ( P D x )  <  if ( R  <_  S ,  R ,  S )  <->  ( ( P D x )  <  R  /\  ( P D x )  <  S ) ) )
87pm5.32da 623 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( ( x  e.  X  /\  ( P D x )  < 
if ( R  <_  S ,  R ,  S ) )  <->  ( x  e.  X  /\  (
( P D x )  <  R  /\  ( P D x )  <  S ) ) ) )
9 ifcl 3767 . . . 4  |-  ( ( R  e.  RR*  /\  S  e.  RR* )  ->  if ( R  <_  S ,  R ,  S )  e.  RR* )
10 elbl 18410 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  if ( R  <_  S ,  R ,  S )  e.  RR* )  ->  ( x  e.  ( P ( ball `  D ) if ( R  <_  S ,  R ,  S )
)  <->  ( x  e.  X  /\  ( P D x )  < 
if ( R  <_  S ,  R ,  S ) ) ) )
11103expa 1153 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  if ( R  <_  S ,  R ,  S )  e.  RR* )  ->  (
x  e.  ( P ( ball `  D
) if ( R  <_  S ,  R ,  S ) )  <->  ( x  e.  X  /\  ( P D x )  < 
if ( R  <_  S ,  R ,  S ) ) ) )
129, 11sylan2 461 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( x  e.  ( P ( ball `  D
) if ( R  <_  S ,  R ,  S ) )  <->  ( x  e.  X  /\  ( P D x )  < 
if ( R  <_  S ,  R ,  S ) ) ) )
13 elbl 18410 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( x  e.  ( P ( ball `  D
) R )  <->  ( x  e.  X  /\  ( P D x )  < 
R ) ) )
14133expa 1153 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  R  e.  RR* )  ->  (
x  e.  ( P ( ball `  D
) R )  <->  ( x  e.  X  /\  ( P D x )  < 
R ) ) )
1514adantrr 698 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( x  e.  ( P ( ball `  D
) R )  <->  ( x  e.  X  /\  ( P D x )  < 
R ) ) )
16 elbl 18410 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  ( x  e.  ( P ( ball `  D
) S )  <->  ( x  e.  X  /\  ( P D x )  < 
S ) ) )
17163expa 1153 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  S  e.  RR* )  ->  (
x  e.  ( P ( ball `  D
) S )  <->  ( x  e.  X  /\  ( P D x )  < 
S ) ) )
1817adantrl 697 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( x  e.  ( P ( ball `  D
) S )  <->  ( x  e.  X  /\  ( P D x )  < 
S ) ) )
1915, 18anbi12d 692 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( ( x  e.  ( P ( ball `  D ) R )  /\  x  e.  ( P ( ball `  D
) S ) )  <-> 
( ( x  e.  X  /\  ( P D x )  < 
R )  /\  (
x  e.  X  /\  ( P D x )  <  S ) ) ) )
20 elin 3522 . . . 4  |-  ( x  e.  ( ( P ( ball `  D
) R )  i^i  ( P ( ball `  D ) S ) )  <->  ( x  e.  ( P ( ball `  D ) R )  /\  x  e.  ( P ( ball `  D
) S ) ) )
21 anandi 802 . . . 4  |-  ( ( x  e.  X  /\  ( ( P D x )  <  R  /\  ( P D x )  <  S ) )  <->  ( ( x  e.  X  /\  ( P D x )  < 
R )  /\  (
x  e.  X  /\  ( P D x )  <  S ) ) )
2219, 20, 213bitr4g 280 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( x  e.  ( ( P ( ball `  D ) R )  i^i  ( P (
ball `  D ) S ) )  <->  ( x  e.  X  /\  (
( P D x )  <  R  /\  ( P D x )  <  S ) ) ) )
238, 12, 223bitr4rd 278 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( x  e.  ( ( P ( ball `  D ) R )  i^i  ( P (
ball `  D ) S ) )  <->  x  e.  ( P ( ball `  D
) if ( R  <_  S ,  R ,  S ) ) ) )
2423eqrdv 2433 1  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  -> 
( ( P (
ball `  D ) R )  i^i  ( P ( ball `  D
) S ) )  =  ( P (
ball `  D ) if ( R  <_  S ,  R ,  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3311   ifcif 3731   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   RR*cxr 9111    < clt 9112    <_ cle 9113   * Metcxmt 16678   ballcbl 16680
This theorem is referenced by:  blin2  18451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-psmet 16686  df-xmet 16687  df-bl 16689
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