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Theorem psmet0 20639
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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5893 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
2 ispsmet 20635 . . . . . . . 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 ) ) ) ) ) )
31, 2syl 16 . . . . . . 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 ) ) ) ) ) )
43ibi 241 . . . . . 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 ) ) ) ) )
54simprd 463 . . . . 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 ) ) ) )
65r19.21bi 2833 . . . 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 ) ) ) )
76simpld 459 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  ->  (
a D a )  =  0 )
87ralrimiva 2878 . 2  |-  ( D  e.  (PsMet `  X
)  ->  A. a  e.  X  ( a D a )  =  0 )
9 id 22 . . . . 5  |-  ( a  =  A  ->  a  =  A )
109, 9oveq12d 6303 . . . 4  |-  ( a  =  A  ->  (
a D a )  =  ( A D A ) )
1110eqeq1d 2469 . . 3  |-  ( a  =  A  ->  (
( a D a )  =  0  <->  ( A D A )  =  0 ) )
1211rspcv 3210 . 2  |-  ( A  e.  X  ->  ( A. a  e.  X  ( a D a )  =  0  -> 
( A D A )  =  0 ) )
138, 12mpan9 469 1  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X )  ->  ( A D A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   class class class wbr 4447    X. cxp 4997   -->wf 5584   ` cfv 5588  (class class class)co 6285   0cc0 9493   RR*cxr 9628    <_ cle 9630   +ecxad 11317  PsMetcpsmet 18213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-map 7423  df-xr 9633  df-psmet 18222
This theorem is referenced by:  psmetsym  20641  psmetge0  20643  psmetres2  20645  xblcntrps  20740  ssblps  20752  metustid  20890  metider  27624  pstmfval  27626
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