ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xmeteq0 Unicode version

Theorem xmeteq0 13829
Description: The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmeteq0  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A D B )  =  0  <->  A  =  B
) )

Proof of Theorem xmeteq0
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xmetrel 13813 . . . . . . 7  |-  Rel  *Met
2 relelfvdm 5547 . . . . . . 7  |-  ( ( Rel  *Met  /\  D  e.  ( *Met `  X ) )  ->  X  e.  dom  *Met )
31, 2mpan 424 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
4 isxmet 13815 . . . . . 6  |-  ( X  e.  dom  *Met  ->  ( D  e.  ( *Met `  X
)  <->  ( D :
( X  X.  X
) --> RR*  /\  A. x  e.  X  A. y  e.  X  ( (
( x D y )  =  0  <->  x  =  y )  /\  A. z  e.  X  ( x D y )  <_  ( ( z D x ) +e ( z D y ) ) ) ) ) )
53, 4syl 14 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  ( D  e.  ( *Met `  X )  <->  ( D : ( X  X.  X ) --> RR*  /\  A. x  e.  X  A. y  e.  X  (
( ( x D y )  =  0  <-> 
x  =  y )  /\  A. z  e.  X  ( x D y )  <_  (
( z D x ) +e ( z D y ) ) ) ) ) )
65ibi 176 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  ( D : ( X  X.  X ) --> RR*  /\  A. x  e.  X  A. y  e.  X  (
( ( x D y )  =  0  <-> 
x  =  y )  /\  A. z  e.  X  ( x D y )  <_  (
( z D x ) +e ( z D y ) ) ) ) )
7 simpl 109 . . . . 5  |-  ( ( ( ( x D y )  =  0  <-> 
x  =  y )  /\  A. z  e.  X  ( x D y )  <_  (
( z D x ) +e ( z D y ) ) )  ->  (
( x D y )  =  0  <->  x  =  y ) )
872ralimi 2541 . . . 4  |-  ( A. x  e.  X  A. y  e.  X  (
( ( x D y )  =  0  <-> 
x  =  y )  /\  A. z  e.  X  ( x D y )  <_  (
( z D x ) +e ( z D y ) ) )  ->  A. x  e.  X  A. y  e.  X  ( (
x D y )  =  0  <->  x  =  y ) )
96, 8simpl2im 386 . . 3  |-  ( D  e.  ( *Met `  X )  ->  A. x  e.  X  A. y  e.  X  ( (
x D y )  =  0  <->  x  =  y ) )
10 oveq1 5881 . . . . . 6  |-  ( x  =  A  ->  (
x D y )  =  ( A D y ) )
1110eqeq1d 2186 . . . . 5  |-  ( x  =  A  ->  (
( x D y )  =  0  <->  ( A D y )  =  0 ) )
12 eqeq1 2184 . . . . 5  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
1311, 12bibi12d 235 . . . 4  |-  ( x  =  A  ->  (
( ( x D y )  =  0  <-> 
x  =  y )  <-> 
( ( A D y )  =  0  <-> 
A  =  y ) ) )
14 oveq2 5882 . . . . . 6  |-  ( y  =  B  ->  ( A D y )  =  ( A D B ) )
1514eqeq1d 2186 . . . . 5  |-  ( y  =  B  ->  (
( A D y )  =  0  <->  ( A D B )  =  0 ) )
16 eqeq2 2187 . . . . 5  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
1715, 16bibi12d 235 . . . 4  |-  ( y  =  B  ->  (
( ( A D y )  =  0  <-> 
A  =  y )  <-> 
( ( A D B )  =  0  <-> 
A  =  B ) ) )
1813, 17rspc2v 2854 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( ( x D y )  =  0  <->  x  =  y
)  ->  ( ( A D B )  =  0  <->  A  =  B
) ) )
199, 18syl5com 29 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
( A  e.  X  /\  B  e.  X
)  ->  ( ( A D B )  =  0  <->  A  =  B
) ) )
20193impib 1201 1  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A D B )  =  0  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   class class class wbr 4003    X. cxp 4624   dom cdm 4626   Rel wrel 4631   -->wf 5212   ` cfv 5216  (class class class)co 5874   0cc0 7810   RR*cxr 7990    <_ cle 7992   +ecxad 9769   *Metcxmet 13410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-map 6649  df-pnf 7993  df-mnf 7994  df-xr 7995  df-xmet 13418
This theorem is referenced by:  meteq0  13830  xmet0  13833  xmetres2  13849  xblss2  13875  xmseq0  13938  comet  13969  xmetxp  13977
  Copyright terms: Public domain W3C validator