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Theorem psmet0 12391
Description: The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
psmet0  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X )  ->  ( A D A )  =  0 )

Proof of Theorem psmet0
Dummy variables  a  b  c  d  e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-psmet 12051 . . . . . . . . 9  |- PsMet  =  ( d  e.  _V  |->  { e  e.  ( RR*  ^m  ( d  X.  d
) )  |  A. a  e.  d  (
( a e a )  =  0  /\ 
A. b  e.  d 
A. c  e.  d  ( a e b )  <_  ( (
c e a ) +e ( c e b ) ) ) } )
21mptrcl 5469 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
3 ispsmet 12387 . . . . . . . 8  |-  ( X  e.  _V  ->  ( D  e.  (PsMet `  X
)  <->  ( D :
( X  X.  X
) --> RR*  /\  A. a  e.  X  ( (
a D a )  =  0  /\  A. b  e.  X  A. c  e.  X  (
a D b )  <_  ( ( c D a ) +e ( c D b ) ) ) ) ) )
42, 3syl 14 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  ( D  e.  (PsMet `  X )  <->  ( D : ( X  X.  X ) --> RR* 
/\  A. a  e.  X  ( ( a D a )  =  0  /\  A. b  e.  X  A. c  e.  X  ( a D b )  <_  (
( c D a ) +e ( c D b ) ) ) ) ) )
54ibi 175 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( D : ( X  X.  X ) --> RR*  /\  A. a  e.  X  (
( a D a )  =  0  /\ 
A. b  e.  X  A. c  e.  X  ( a D b )  <_  ( (
c D a ) +e ( c D b ) ) ) ) )
65simprd 113 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  A. a  e.  X  ( (
a D a )  =  0  /\  A. b  e.  X  A. c  e.  X  (
a D b )  <_  ( ( c D a ) +e ( c D b ) ) ) )
76r19.21bi 2495 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  ->  (
( a D a )  =  0  /\ 
A. b  e.  X  A. c  e.  X  ( a D b )  <_  ( (
c D a ) +e ( c D b ) ) ) )
87simpld 111 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  ->  (
a D a )  =  0 )
98ralrimiva 2480 . 2  |-  ( D  e.  (PsMet `  X
)  ->  A. a  e.  X  ( a D a )  =  0 )
10 id 19 . . . . 5  |-  ( a  =  A  ->  a  =  A )
1110, 10oveq12d 5758 . . . 4  |-  ( a  =  A  ->  (
a D a )  =  ( A D A ) )
1211eqeq1d 2124 . . 3  |-  ( a  =  A  ->  (
( a D a )  =  0  <->  ( A D A )  =  0 ) )
1312rspcv 2757 . 2  |-  ( A  e.  X  ->  ( A. a  e.  X  ( a D a )  =  0  -> 
( A D A )  =  0 ) )
149, 13mpan9 277 1  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X )  ->  ( A D A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   A.wral 2391   {crab 2395   _Vcvv 2658   class class class wbr 3897    X. cxp 4505   -->wf 5087   ` cfv 5091  (class class class)co 5740    ^m cmap 6508   0cc0 7584   RR*cxr 7763    <_ cle 7765   +ecxad 9497  PsMetcpsmet 12043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-map 6510  df-pnf 7766  df-mnf 7767  df-xr 7768  df-psmet 12051
This theorem is referenced by:  psmetsym  12393  psmetge0  12395  psmetres2  12397  distspace  12399  xblcntrps  12477  ssblps  12489
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