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Theorem blin 21141
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 21051 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  x  e.  X
)  ->  ( P D x )  e. 
RR* )
213expa 1196 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  x  e.  X )  ->  ( P D x )  e. 
RR* )
32adantlr 714 . . . . 5  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  ( P D x )  e.  RR* )
4 simplrl 761 . . . . 5  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  R  e.  RR* )
5 simplrr 762 . . . . 5  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  S  e.  RR* )
6 xrltmin 11408 . . . . 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 1228 . . . 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 641 . . 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 3986 . . . 4  |-  ( ( R  e.  RR*  /\  S  e.  RR* )  ->  if ( R  <_  S ,  R ,  S )  e.  RR* )
10 elbl 21108 . . . . 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 1196 . . . 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 474 . . 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 21108 . . . . . . 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 1196 . . . . . 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 716 . . . . 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 21108 . . . . . . 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 1196 . . . . . 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 715 . . . . 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 710 . . . 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 3683 . . . 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 828 . . . 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 288 . . 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 286 . 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 2454 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    i^i cin 3470   ifcif 3944   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   RR*cxr 9644    < clt 9645    <_ cle 9646   *Metcxmt 18621   ballcbl 18623
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-pre-lttri 9583  ax-pre-lttrn 9584
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-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-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-psmet 18629  df-xmet 18630  df-bl 18632
This theorem is referenced by:  blin2  21149
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