MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  stdbdmetval Unicode version

Theorem stdbdmetval 18060
Description: Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypothesis
Ref Expression
stdbdmet.1  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )
Assertion
Ref Expression
stdbdmetval  |-  ( ( R  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  if ( ( A C B )  <_  R , 
( A C B ) ,  R ) )
Distinct variable groups:    x, y, A    x, C, y    x, B, y    x, R, y   
x, X, y
Allowed substitution hints:    D( x, y)    V( x, y)

Proof of Theorem stdbdmetval
StepHypRef Expression
1 ovex 5883 . . . 4  |-  ( A C B )  e. 
_V
2 ifexg 3624 . . . 4  |-  ( ( ( A C B )  e.  _V  /\  R  e.  V )  ->  if ( ( A C B )  <_  R ,  ( A C B ) ,  R
)  e.  _V )
31, 2mpan 651 . . 3  |-  ( R  e.  V  ->  if ( ( A C B )  <_  R ,  ( A C B ) ,  R
)  e.  _V )
4 oveq12 5867 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x C y )  =  ( A C B ) )
54breq1d 4033 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x C y )  <_  R  <->  ( A C B )  <_  R ) )
6 eqidd 2284 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  R )
75, 4, 6ifbieq12d 3587 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R )  =  if ( ( A C B )  <_  R ,  ( A C B ) ,  R
) )
8 stdbdmet.1 . . . 4  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )
97, 8ovmpt2ga 5977 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  if ( ( A C B )  <_  R ,  ( A C B ) ,  R
)  e.  _V )  ->  ( A D B )  =  if ( ( A C B )  <_  R , 
( A C B ) ,  R ) )
103, 9syl3an3 1217 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  R  e.  V )  ->  ( A D B )  =  if ( ( A C B )  <_  R , 
( A C B ) ,  R ) )
11103comr 1159 1  |-  ( ( R  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  if ( ( A C B )  <_  R , 
( A C B ) ,  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   ifcif 3565   class class class wbr 4023  (class class class)co 5858    e. cmpt2 5860    <_ cle 8868
This theorem is referenced by:  stdbdbl  18063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
  Copyright terms: Public domain W3C validator