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Theorem xmeteq0 20048
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 5828 . . . . . . 7  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
2 isxmet 20034 . . . . . . 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 16 . . . . . 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 463 . . . 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 457 . . . . . 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 2819 . . . . 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 2819 . . . 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 16 . . 3  |-  ( D  e.  ( *Met `  X )  ->  A. x  e.  X  A. y  e.  X  ( (
x D y )  =  0  <->  x  =  y ) )
10 oveq1 6210 . . . . . 6  |-  ( x  =  A  ->  (
x D y )  =  ( A D y ) )
1110eqeq1d 2456 . . . . 5  |-  ( x  =  A  ->  (
( x D y )  =  0  <->  ( A D y )  =  0 ) )
12 eqeq1 2458 . . . . 5  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
1311, 12bibi12d 321 . . . 4  |-  ( x  =  A  ->  (
( ( x D y )  =  0  <-> 
x  =  y )  <-> 
( ( A D y )  =  0  <-> 
A  =  y ) ) )
14 oveq2 6211 . . . . . 6  |-  ( y  =  B  ->  ( A D y )  =  ( A D B ) )
1514eqeq1d 2456 . . . . 5  |-  ( y  =  B  ->  (
( A D y )  =  0  <->  ( A D B )  =  0 ) )
16 eqeq2 2469 . . . . 5  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
1715, 16bibi12d 321 . . . 4  |-  ( y  =  B  ->  (
( ( A D y )  =  0  <-> 
A  =  y )  <-> 
( ( A D B )  =  0  <-> 
A  =  B ) ) )
1813, 17rspc2v 3186 . . 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 30 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
( A  e.  X  /\  B  e.  X
)  ->  ( ( A D B )  =  0  <->  A  =  B
) ) )
20193impib 1186 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   class class class wbr 4403    X. cxp 4949   dom cdm 4951   -->wf 5525   ` cfv 5529  (class class class)co 6203   0cc0 9396   RR*cxr 9531    <_ cle 9533   +ecxad 11201   *Metcxmt 17929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329  df-xr 9536  df-xmet 17938
This theorem is referenced by:  meteq0  20049  xmet0  20052  xmetgt0  20068  xmetres2  20071  prdsxmetlem  20078  imasf1oxmet  20085  xblss2  20112  xmseq0  20174  comet  20223
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