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Theorem blin 21448
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 21358 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  x  e.  X
)  ->  ( P D x )  e. 
RR* )
213expa 1209 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  x  e.  X )  ->  ( P D x )  e. 
RR* )
32adantlr 722 . . . . 5  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  ( P D x )  e.  RR* )
4 simplrl 771 . . . . 5  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  R  e.  RR* )
5 simplrr 772 . . . . 5  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  /\  x  e.  X )  ->  S  e.  RR* )
6 xrltmin 11484 . . . . 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 1269 . . . 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 647 . . 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 3925 . . . 4  |-  ( ( R  e.  RR*  /\  S  e.  RR* )  ->  if ( R  <_  S ,  R ,  S )  e.  RR* )
10 elbl 21415 . . . . 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 1209 . . . 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 477 . . 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 21415 . . . . . . 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 1209 . . . . . 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 724 . . . . 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 21415 . . . . . . 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 1209 . . . . . 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 723 . . . . 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 718 . . . 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 3619 . . . 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 838 . . . 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 292 . . 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 290 . 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 2451 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 188    /\ wa 371    = wceq 1446    e. wcel 1889    i^i cin 3405   ifcif 3883   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   RR*cxr 9679    < clt 9680    <_ cle 9681   *Metcxmt 18967   ballcbl 18969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-pre-lttri 9618  ax-pre-lttrn 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-psmet 18974  df-xmet 18975  df-bl 18977
This theorem is referenced by:  blin2  21456
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