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Theorem xmeteq0 21133
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 elfvdm 5875 . . . . . . 7  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
2 isxmet 21119 . . . . . . 7  |-  ( 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 ) ) ) ) ) )
31, 2syl 17 . . . . . 6  |-  ( 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 ) ) ) ) ) )
43ibi 241 . . . . 5  |-  ( 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 ) ) ) ) )
54simprd 461 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  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 ) ) ) )
6 simpl 455 . . . . . 6  |-  ( ( ( ( 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 ) )
76ralimi 2797 . . . . 5  |-  ( 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. y  e.  X  ( (
x D y )  =  0  <->  x  =  y ) )
87ralimi 2797 . . . 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 ) )
95, 8syl 17 . . 3  |-  ( D  e.  ( *Met `  X )  ->  A. x  e.  X  A. y  e.  X  ( (
x D y )  =  0  <->  x  =  y ) )
10 oveq1 6285 . . . . . 6  |-  ( x  =  A  ->  (
x D y )  =  ( A D y ) )
1110eqeq1d 2404 . . . . 5  |-  ( x  =  A  ->  (
( x D y )  =  0  <->  ( A D y )  =  0 ) )
12 eqeq1 2406 . . . . 5  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
1311, 12bibi12d 319 . . . 4  |-  ( x  =  A  ->  (
( ( x D y )  =  0  <-> 
x  =  y )  <-> 
( ( A D y )  =  0  <-> 
A  =  y ) ) )
14 oveq2 6286 . . . . . 6  |-  ( y  =  B  ->  ( A D y )  =  ( A D B ) )
1514eqeq1d 2404 . . . . 5  |-  ( y  =  B  ->  (
( A D y )  =  0  <->  ( A D B )  =  0 ) )
16 eqeq2 2417 . . . . 5  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
1715, 16bibi12d 319 . . . 4  |-  ( y  =  B  ->  (
( ( A D y )  =  0  <-> 
A  =  y )  <-> 
( ( A D B )  =  0  <-> 
A  =  B ) ) )
1813, 17rspc2v 3169 . . 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 28 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
( A  e.  X  /\  B  e.  X
)  ->  ( ( A D B )  =  0  <->  A  =  B
) ) )
20193impib 1195 1  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A D B )  =  0  <->  A  =  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   class class class wbr 4395    X. cxp 4821   dom cdm 4823   -->wf 5565   ` cfv 5569  (class class class)co 6278   0cc0 9522   RR*cxr 9657    <_ cle 9659   +ecxad 11369   *Metcxmt 18723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7459  df-xr 9662  df-xmet 18732
This theorem is referenced by:  meteq0  21134  xmet0  21137  xmetgt0  21153  xmetres2  21156  prdsxmetlem  21163  imasf1oxmet  21170  xblss2  21197  xmseq0  21259  comet  21308
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