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Theorem xmeteq0 13862
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 13846 . . . . . . 7  |-  Rel  *Met
2 relelfvdm 5548 . . . . . . 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 13848 . . . . . 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 5882 . . . . . 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 5883 . . . . . 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 2855 . . 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 4004    X. cxp 4625   dom cdm 4627   Rel wrel 4632   -->wf 5213   ` cfv 5217  (class class class)co 5875   0cc0 7811   RR*cxr 7991    <_ cle 7993   +ecxad 9770   *Metcxmet 13443
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 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903
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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-map 6650  df-pnf 7994  df-mnf 7995  df-xr 7996  df-xmet 13451
This theorem is referenced by:  meteq0  13863  xmet0  13866  xmetres2  13882  xblss2  13908  xmseq0  13971  comet  14002  xmetxp  14010
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