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Theorem blin 18345
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 18271 . . . . . . 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 10703 . . . . 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 3719 . . . 4  |-  ( ( R  e.  RR*  /\  S  e.  RR* )  ->  if ( R  <_  S ,  R ,  S )  e.  RR* )
10 elbl 18324 . . . . 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 18324 . . . . . . 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 18324 . . . . . . 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 3474 . . . 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 2386 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 1649    e. wcel 1717    i^i cin 3263   ifcif 3683   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   RR*cxr 9053    < clt 9054    <_ cle 9055   * Metcxmt 16613   ballcbl 16615
This theorem is referenced by:  blin2  18350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-pre-lttri 8998  ax-pre-lttrn 8999
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-xmet 16620  df-bl 16622
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