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Theorem psmet0 12423
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 12083 . . . . . . . . 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 5471 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
3 ispsmet 12419 . . . . . . . 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 2497 . . . 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 2482 . 2  |-  ( D  e.  (PsMet `  X
)  ->  A. a  e.  X  ( a D a )  =  0 )
10 id 19 . . . . 5  |-  ( a  =  A  ->  a  =  A )
1110, 10oveq12d 5760 . . . 4  |-  ( a  =  A  ->  (
a D a )  =  ( A D A ) )
1211eqeq1d 2126 . . 3  |-  ( a  =  A  ->  (
( a D a )  =  0  <->  ( A D A )  =  0 ) )
1312rspcv 2759 . 2  |-  ( A  e.  X  ->  ( A. a  e.  X  ( a D a )  =  0  -> 
( A D A )  =  0 ) )
149, 13mpan9 279 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 1316    e. wcel 1465   A.wral 2393   {crab 2397   _Vcvv 2660   class class class wbr 3899    X. cxp 4507   -->wf 5089   ` cfv 5093  (class class class)co 5742    ^m cmap 6510   0cc0 7588   RR*cxr 7767    <_ cle 7769   +ecxad 9525  PsMetcpsmet 12075
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-map 6512  df-pnf 7770  df-mnf 7771  df-xr 7772  df-psmet 12083
This theorem is referenced by:  psmetsym  12425  psmetge0  12427  psmetres2  12429  distspace  12431  xblcntrps  12509  ssblps  12521
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