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Theorem xmeteq0 21402
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 5914 . . . . . . 7  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
2 isxmet 21388 . . . . . . 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 249 . . . . 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 469 . . . 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 463 . . . . . 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 2793 . . . . 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 2793 . . . 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 6322 . . . . . 6  |-  ( x  =  A  ->  (
x D y )  =  ( A D y ) )
1110eqeq1d 2464 . . . . 5  |-  ( x  =  A  ->  (
( x D y )  =  0  <->  ( A D y )  =  0 ) )
12 eqeq1 2466 . . . . 5  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
1311, 12bibi12d 327 . . . 4  |-  ( x  =  A  ->  (
( ( x D y )  =  0  <-> 
x  =  y )  <-> 
( ( A D y )  =  0  <-> 
A  =  y ) ) )
14 oveq2 6323 . . . . . 6  |-  ( y  =  B  ->  ( A D y )  =  ( A D B ) )
1514eqeq1d 2464 . . . . 5  |-  ( y  =  B  ->  (
( A D y )  =  0  <->  ( A D B )  =  0 ) )
16 eqeq2 2473 . . . . 5  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
1715, 16bibi12d 327 . . . 4  |-  ( y  =  B  ->  (
( ( A D y )  =  0  <-> 
A  =  y )  <-> 
( ( A D B )  =  0  <-> 
A  =  B ) ) )
1813, 17rspc2v 3171 . . 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 31 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
( A  e.  X  /\  B  e.  X
)  ->  ( ( A D B )  =  0  <->  A  =  B
) ) )
20193impib 1213 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 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   A.wral 2749   class class class wbr 4416    X. cxp 4851   dom cdm 4853   -->wf 5597   ` cfv 5601  (class class class)co 6315   0cc0 9565   RR*cxr 9700    <_ cle 9702   +ecxad 11436   *Metcxmt 19004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-map 7500  df-xr 9705  df-xmet 19012
This theorem is referenced by:  meteq0  21403  xmet0  21406  xmetgt0  21422  xmetres2  21425  prdsxmetlem  21432  imasf1oxmet  21439  xblss2  21466  xmseq0  21528  comet  21577
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