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Theorem blres 18461
Description: A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)
Hypothesis
Ref Expression
blres.2  |-  C  =  ( D  |`  ( Y  X.  Y ) )
Assertion
Ref Expression
blres  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  ( X  i^i  Y )  /\  R  e.  RR* )  ->  ( P (
ball `  C ) R )  =  ( ( P ( ball `  D ) R )  i^i  Y ) )

Proof of Theorem blres
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 inss2 3562 . . . . . . . . . 10  |-  ( X  i^i  Y )  C_  Y
21sseli 3344 . . . . . . . . 9  |-  ( P  e.  ( X  i^i  Y )  ->  P  e.  Y )
3 blres.2 . . . . . . . . . . 11  |-  C  =  ( D  |`  ( Y  X.  Y ) )
43oveqi 6094 . . . . . . . . . 10  |-  ( P C x )  =  ( P ( D  |`  ( Y  X.  Y
) ) x )
5 ovres 6213 . . . . . . . . . 10  |-  ( ( P  e.  Y  /\  x  e.  Y )  ->  ( P ( D  |`  ( Y  X.  Y
) ) x )  =  ( P D x ) )
64, 5syl5eq 2480 . . . . . . . . 9  |-  ( ( P  e.  Y  /\  x  e.  Y )  ->  ( P C x )  =  ( P D x ) )
72, 6sylan 458 . . . . . . . 8  |-  ( ( P  e.  ( X  i^i  Y )  /\  x  e.  Y )  ->  ( P C x )  =  ( P D x ) )
87breq1d 4222 . . . . . . 7  |-  ( ( P  e.  ( X  i^i  Y )  /\  x  e.  Y )  ->  ( ( P C x )  <  R  <->  ( P D x )  <  R ) )
98anbi2d 685 . . . . . 6  |-  ( ( P  e.  ( X  i^i  Y )  /\  x  e.  Y )  ->  ( ( x  e.  X  /\  ( P C x )  < 
R )  <->  ( x  e.  X  /\  ( P D x )  < 
R ) ) )
109pm5.32da 623 . . . . 5  |-  ( P  e.  ( X  i^i  Y )  ->  ( (
x  e.  Y  /\  ( x  e.  X  /\  ( P C x )  <  R ) )  <->  ( x  e.  Y  /\  ( x  e.  X  /\  ( P D x )  < 
R ) ) ) )
11103ad2ant2 979 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  ( X  i^i  Y )  /\  R  e.  RR* )  ->  ( ( x  e.  Y  /\  (
x  e.  X  /\  ( P C x )  <  R ) )  <-> 
( x  e.  Y  /\  ( x  e.  X  /\  ( P D x )  <  R ) ) ) )
12 elin 3530 . . . . . . 7  |-  ( x  e.  ( X  i^i  Y )  <->  ( x  e.  X  /\  x  e.  Y ) )
13 ancom 438 . . . . . . 7  |-  ( ( x  e.  X  /\  x  e.  Y )  <->  ( x  e.  Y  /\  x  e.  X )
)
1412, 13bitri 241 . . . . . 6  |-  ( x  e.  ( X  i^i  Y )  <->  ( x  e.  Y  /\  x  e.  X ) )
1514anbi1i 677 . . . . 5  |-  ( ( x  e.  ( X  i^i  Y )  /\  ( P C x )  <  R )  <->  ( (
x  e.  Y  /\  x  e.  X )  /\  ( P C x )  <  R ) )
16 anass 631 . . . . 5  |-  ( ( ( x  e.  Y  /\  x  e.  X
)  /\  ( P C x )  < 
R )  <->  ( x  e.  Y  /\  (
x  e.  X  /\  ( P C x )  <  R ) ) )
1715, 16bitri 241 . . . 4  |-  ( ( x  e.  ( X  i^i  Y )  /\  ( P C x )  <  R )  <->  ( x  e.  Y  /\  (
x  e.  X  /\  ( P C x )  <  R ) ) )
18 ancom 438 . . . 4  |-  ( ( ( x  e.  X  /\  ( P D x )  <  R )  /\  x  e.  Y
)  <->  ( x  e.  Y  /\  ( x  e.  X  /\  ( P D x )  < 
R ) ) )
1911, 17, 183bitr4g 280 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  ( X  i^i  Y )  /\  R  e.  RR* )  ->  ( ( x  e.  ( X  i^i  Y )  /\  ( P C x )  < 
R )  <->  ( (
x  e.  X  /\  ( P D x )  <  R )  /\  x  e.  Y )
) )
20 xmetres 18394 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  ( D  |`  ( Y  X.  Y ) )  e.  ( * Met `  ( X  i^i  Y ) ) )
213, 20syl5eqel 2520 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  C  e.  ( * Met `  ( X  i^i  Y ) ) )
22 elbl 18418 . . . 4  |-  ( ( C  e.  ( * Met `  ( X  i^i  Y ) )  /\  P  e.  ( X  i^i  Y )  /\  R  e.  RR* )  ->  ( x  e.  ( P ( ball `  C ) R )  <-> 
( x  e.  ( X  i^i  Y )  /\  ( P C x )  <  R
) ) )
2321, 22syl3an1 1217 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  ( X  i^i  Y )  /\  R  e.  RR* )  ->  ( x  e.  ( P ( ball `  C ) R )  <-> 
( x  e.  ( X  i^i  Y )  /\  ( P C x )  <  R
) ) )
24 elin 3530 . . . 4  |-  ( x  e.  ( ( P ( ball `  D
) R )  i^i 
Y )  <->  ( x  e.  ( P ( ball `  D ) R )  /\  x  e.  Y
) )
25 inss1 3561 . . . . . . 7  |-  ( X  i^i  Y )  C_  X
2625sseli 3344 . . . . . 6  |-  ( P  e.  ( X  i^i  Y )  ->  P  e.  X )
27 elbl 18418 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( x  e.  ( P ( ball `  D
) R )  <->  ( x  e.  X  /\  ( P D x )  < 
R ) ) )
2826, 27syl3an2 1218 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  ( X  i^i  Y )  /\  R  e.  RR* )  ->  ( x  e.  ( P ( ball `  D ) R )  <-> 
( x  e.  X  /\  ( P D x )  <  R ) ) )
2928anbi1d 686 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  ( X  i^i  Y )  /\  R  e.  RR* )  ->  ( ( x  e.  ( P (
ball `  D ) R )  /\  x  e.  Y )  <->  ( (
x  e.  X  /\  ( P D x )  <  R )  /\  x  e.  Y )
) )
3024, 29syl5bb 249 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  ( X  i^i  Y )  /\  R  e.  RR* )  ->  ( x  e.  ( ( P (
ball `  D ) R )  i^i  Y
)  <->  ( ( x  e.  X  /\  ( P D x )  < 
R )  /\  x  e.  Y ) ) )
3119, 23, 303bitr4d 277 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  ( X  i^i  Y )  /\  R  e.  RR* )  ->  ( x  e.  ( P ( ball `  C ) R )  <-> 
x  e.  ( ( P ( ball `  D
) R )  i^i 
Y ) ) )
3231eqrdv 2434 1  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  ( X  i^i  Y )  /\  R  e.  RR* )  ->  ( P (
ball `  C ) R )  =  ( ( P ( ball `  D ) R )  i^i  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3319   class class class wbr 4212    X. cxp 4876    |` cres 4880   ` cfv 5454  (class class class)co 6081   RR*cxr 9119    < clt 9120   * Metcxmt 16686   ballcbl 16688
This theorem is referenced by:  metrest  18554  xrsmopn  18843  lebnumii  18991  blssp  26462  sstotbnd2  26483  blbnd  26496  ssbnd  26497
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020  df-xr 9124  df-psmet 16694  df-xmet 16695  df-bl 16697
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