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Theorem pstmxmet 26464
Description: The metric induced by a pseudometric is a full-fledged metric on the equivalence classes of the metric identification. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
pstmval.1  |-  .~  =  (~Met `  D )
Assertion
Ref Expression
pstmxmet  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  e.  ( *Met `  ( X /.  .~  ) ) )

Proof of Theorem pstmxmet
Dummy variables  a 
b  c  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3075 . . . . . . . . 9  |-  x  e. 
_V
2 vex 3075 . . . . . . . . 9  |-  y  e. 
_V
31, 2ab2rexex 6673 . . . . . . . 8  |-  { z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) }  e.  _V
43uniex 6481 . . . . . . 7  |-  U. {
z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) }  e.  _V
54rgen2w 2896 . . . . . 6  |-  A. x  e.  ( X /.  .~  ) A. y  e.  ( X /.  .~  ) U. { z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) }  e.  _V
6 eqid 2452 . . . . . . 7  |-  ( x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y 
z  =  ( a D b ) } )  =  ( x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y 
z  =  ( a D b ) } )
76fnmpt2 6747 . . . . . 6  |-  ( A. x  e.  ( X /.  .~  ) A. y  e.  ( X /.  .~  ) U. { z  |  E. a  e.  x  E. b  e.  y 
z  =  ( a D b ) }  e.  _V  ->  (
x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) } )  Fn  ( ( X /.  .~  )  X.  ( X /.  .~  ) ) )
85, 7ax-mp 5 . . . . 5  |-  ( x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y 
z  =  ( a D b ) } )  Fn  ( ( X /.  .~  )  X.  ( X /.  .~  ) )
9 pstmval.1 . . . . . . 7  |-  .~  =  (~Met `  D )
109pstmval 26462 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  =  ( x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y 
z  =  ( a D b ) } ) )
1110fneq1d 5604 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( (pstoMet `  D )  Fn  (
( X /.  .~  )  X.  ( X /.  .~  ) )  <->  ( x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |-> 
U. { z  |  E. a  e.  x  E. b  e.  y 
z  =  ( a D b ) } )  Fn  ( ( X /.  .~  )  X.  ( X /.  .~  ) ) ) )
128, 11mpbiri 233 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  Fn  ( ( X /.  .~  )  X.  ( X /.  .~  ) ) )
13 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  x  =  [
a ]  .~  )
14 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  y  =  [
b ]  .~  )
1513, 14oveq12d 6213 . . . . . . . . 9  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  ( x (pstoMet `  D ) y )  =  ( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  ) )
16 simp-5l 767 . . . . . . . . . 10  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  D  e.  (PsMet `  X ) )
17 simp-4r 766 . . . . . . . . . 10  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  a  e.  X
)
18 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  b  e.  X
)
199pstmfval 26463 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  X  /\  b  e.  X )  ->  ( [ a ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  ( a D b ) )
2016, 17, 18, 19syl3anc 1219 . . . . . . . . 9  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  ( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  ( a D b ) )
2115, 20eqtrd 2493 . . . . . . . 8  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  ( x (pstoMet `  D ) y )  =  ( a D b ) )
22 psmetf 20009 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
2316, 22syl 16 . . . . . . . . 9  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  D : ( X  X.  X ) -->
RR* )
2423, 17, 18fovrnd 6340 . . . . . . . 8  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  ( a D b )  e.  RR* )
2521, 24eqeltrd 2540 . . . . . . 7  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  ( x (pstoMet `  D ) y )  e.  RR* )
26 elqsi 7259 . . . . . . . . 9  |-  ( y  e.  ( X /.  .~  )  ->  E. b  e.  X  y  =  [ b ]  .~  )
2726ad2antll 728 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  E. b  e.  X  y  =  [ b ]  .~  )
2827ad2antrr 725 . . . . . . 7  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  ( x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  ->  E. b  e.  X  y  =  [ b ]  .~  )
2925, 28r19.29a 2962 . . . . . 6  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  ( x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  ->  ( x (pstoMet `  D ) y )  e.  RR* )
30 elqsi 7259 . . . . . . 7  |-  ( x  e.  ( X /.  .~  )  ->  E. a  e.  X  x  =  [ a ]  .~  )
3130ad2antrl 727 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  E. a  e.  X  x  =  [ a ]  .~  )
3229, 31r19.29a 2962 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  ( x (pstoMet `  D ) y )  e.  RR* )
3332ralrimivva 2908 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  A. x  e.  ( X /.  .~  ) A. y  e.  ( X /.  .~  )
( x (pstoMet `  D
) y )  e. 
RR* )
34 ffnov 6299 . . . 4  |-  ( (pstoMet `  D ) : ( ( X /.  .~  )  X.  ( X /.  .~  ) ) --> RR*  <->  ( (pstoMet `  D )  Fn  (
( X /.  .~  )  X.  ( X /.  .~  ) )  /\  A. x  e.  ( X /.  .~  ) A. y  e.  ( X /.  .~  ) ( x (pstoMet `  D ) y )  e.  RR* ) )
3512, 33, 34sylanbrc 664 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
) : ( ( X /.  .~  )  X.  ( X /.  .~  ) ) --> RR* )
36 simpll 753 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  D  e.  (PsMet `  X )
)
37 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  a  e.  X )
38 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  b  e.  X )
3936, 37, 38, 19syl3anc 1219 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  ( [ a ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  ( a D b ) )
4039eqeq1d 2454 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0  <-> 
( a D b )  =  0 ) )
419breqi 4401 . . . . . . . . . . . . . 14  |-  ( a  .~  b  <->  a (~Met `  D ) b )
42 metidv 26459 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  (PsMet `  X )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( a
(~Met `  D )
b  <->  ( a D b )  =  0 ) )
4342anassrs 648 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
a (~Met `  D
) b  <->  ( a D b )  =  0 ) )
4441, 43syl5bb 257 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
a  .~  b  <->  ( a D b )  =  0 ) )
4540, 44bitr4d 256 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0  <-> 
a  .~  b )
)
46 metider 26461 . . . . . . . . . . . . . . 15  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  Er  X )
4736, 46syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (~Met `  D )  Er  X
)
48 ereq1 7213 . . . . . . . . . . . . . . 15  |-  (  .~  =  (~Met `  D )  ->  (  .~  Er  X  <->  (~Met `  D )  Er  X
) )
499, 48ax-mp 5 . . . . . . . . . . . . . 14  |-  (  .~  Er  X  <->  (~Met `  D )  Er  X )
5047, 49sylibr 212 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  .~  Er  X )
5150, 37erth 7250 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
a  .~  b  <->  [ a ]  .~  =  [ b ]  .~  ) )
5245, 51bitrd 253 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0  <->  [ a ]  .~  =  [ b ]  .~  ) )
5352adantllr 718 . . . . . . . . . 10  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  ( x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X )  /\  b  e.  X )  ->  (
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0  <->  [ a ]  .~  =  [ b ]  .~  ) )
5453adantlr 714 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  ( x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  )
) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  ->  ( ( [ a ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  0  <->  [ a ]  .~  =  [ b ]  .~  ) )
5554adantr 465 . . . . . . . 8  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  ( ( [ a ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  0  <->  [ a ]  .~  =  [ b ]  .~  ) )
5615eqeq1d 2454 . . . . . . . 8  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  ( ( x (pstoMet `  D )
y )  =  0  <-> 
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0 ) )
5713, 14eqeq12d 2474 . . . . . . . 8  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  ( x  =  y  <->  [ a ]  .~  =  [ b ]  .~  ) )
5855, 56, 573bitr4d 285 . . . . . . 7  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  ( ( x (pstoMet `  D )
y )  =  0  <-> 
x  =  y ) )
5958, 28r19.29a 2962 . . . . . 6  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  ( x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  ->  ( ( x (pstoMet `  D )
y )  =  0  <-> 
x  =  y ) )
6059, 31r19.29a 2962 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  ( ( x (pstoMet `  D )
y )  =  0  <-> 
x  =  y ) )
61 simp-6l 769 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  D  e.  (PsMet `  X ) )
62 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  c  e.  X
)
63 simp-6r 770 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  a  e.  X
)
64 simp-4r 766 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  b  e.  X
)
65 psmettri2 20012 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  (PsMet `  X )  /\  (
c  e.  X  /\  a  e.  X  /\  b  e.  X )
)  ->  ( a D b )  <_ 
( ( c D a ) +e
( c D b ) ) )
6661, 62, 63, 64, 65syl13anc 1221 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( a D b )  <_  (
( c D a ) +e ( c D b ) ) )
67 simp-5r 768 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  x  =  [
a ]  .~  )
68 simpllr 758 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  y  =  [
b ]  .~  )
6967, 68oveq12d 6213 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( x (pstoMet `  D ) y )  =  ( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  ) )
7061, 63, 64, 39syl21anc 1218 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  ( a D b ) )
7169, 70eqtrd 2493 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( x (pstoMet `  D ) y )  =  ( a D b ) )
72 simpr 461 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  z  =  [
c ]  .~  )
7372, 67oveq12d 6213 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( z (pstoMet `  D ) x )  =  ( [ c ]  .~  (pstoMet `  D
) [ a ]  .~  ) )
749pstmfval 26463 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  (PsMet `  X )  /\  c  e.  X  /\  a  e.  X )  ->  ( [ c ]  .~  (pstoMet `  D ) [ a ]  .~  )  =  ( c D a ) )
7561, 62, 63, 74syl3anc 1219 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( [ c ]  .~  (pstoMet `  D
) [ a ]  .~  )  =  ( c D a ) )
7673, 75eqtrd 2493 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( z (pstoMet `  D ) x )  =  ( c D a ) )
7772, 68oveq12d 6213 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( z (pstoMet `  D ) y )  =  ( [ c ]  .~  (pstoMet `  D
) [ b ]  .~  ) )
789pstmfval 26463 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  (PsMet `  X )  /\  c  e.  X  /\  b  e.  X )  ->  ( [ c ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  ( c D b ) )
7961, 62, 64, 78syl3anc 1219 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( [ c ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  ( c D b ) )
8077, 79eqtrd 2493 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( z (pstoMet `  D ) y )  =  ( c D b ) )
8176, 80oveq12d 6213 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) )  =  ( ( c D a ) +e ( c D b ) ) )
8271, 81breq12d 4408 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( ( x (pstoMet `  D )
y )  <_  (
( z (pstoMet `  D
) x ) +e ( z (pstoMet `  D ) y ) )  <->  ( a D b )  <_  (
( c D a ) +e ( c D b ) ) ) )
8366, 82mpbird 232 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( x (pstoMet `  D ) y )  <_  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) ) )
8483adantl6r 752 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  z  e.  ( X /.  .~  ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  ( x (pstoMet `  D ) y )  <_  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) ) )
85 elqsi 7259 . . . . . . . . . . . . 13  |-  ( z  e.  ( X /.  .~  )  ->  E. c  e.  X  z  =  [ c ]  .~  )
8685ad5antlr 734 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  z  e.  ( X /.  .~  ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  E. c  e.  X  z  =  [ c ]  .~  )
8784, 86r19.29a 2962 . . . . . . . . . . 11  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  z  e.  ( X /.  .~  ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  ( x (pstoMet `  D ) y )  <_  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) ) )
8887adantl5r 751 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  y  e.  ( X /.  .~  ) )  /\  z  e.  ( X /.  .~  ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  ( x (pstoMet `  D ) y )  <_  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) ) )
8926ad4antlr 732 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  y  e.  ( X /.  .~  )
)  /\  z  e.  ( X /.  .~  )
)  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  ->  E. b  e.  X  y  =  [ b ]  .~  )
9088, 89r19.29a 2962 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  y  e.  ( X /.  .~  )
)  /\  z  e.  ( X /.  .~  )
)  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  ->  ( x (pstoMet `  D ) y )  <_  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) ) )
9190adantl4r 750 . . . . . . . 8  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  x  e.  ( X /.  .~  ) )  /\  y  e.  ( X /.  .~  ) )  /\  z  e.  ( X /.  .~  ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  ->  ( x (pstoMet `  D ) y )  <_  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) ) )
9230ad3antlr 730 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  x  e.  ( X /.  .~  ) )  /\  y  e.  ( X /.  .~  )
)  /\  z  e.  ( X /.  .~  )
)  ->  E. a  e.  X  x  =  [ a ]  .~  )
9391, 92r19.29a 2962 . . . . . . 7  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  x  e.  ( X /.  .~  ) )  /\  y  e.  ( X /.  .~  )
)  /\  z  e.  ( X /.  .~  )
)  ->  ( x
(pstoMet `  D ) y )  <_  ( (
z (pstoMet `  D
) x ) +e ( z (pstoMet `  D ) y ) ) )
9493ralrimiva 2827 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  ( X /.  .~  ) )  /\  y  e.  ( X /.  .~  ) )  ->  A. z  e.  ( X /.  .~  ) ( x (pstoMet `  D ) y )  <_  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) ) )
9594anasss 647 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  A. z  e.  ( X /.  .~  )
( x (pstoMet `  D
) y )  <_ 
( ( z (pstoMet `  D ) x ) +e ( z (pstoMet `  D )
y ) ) )
9660, 95jca 532 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  ( ( ( x (pstoMet `  D
) y )  =  0  <->  x  =  y
)  /\  A. z  e.  ( X /.  .~  ) ( x (pstoMet `  D ) y )  <_  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) ) ) )
9796ralrimivva 2908 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  A. x  e.  ( X /.  .~  ) A. y  e.  ( X /.  .~  )
( ( ( x (pstoMet `  D )
y )  =  0  <-> 
x  =  y )  /\  A. z  e.  ( X /.  .~  ) ( x (pstoMet `  D ) y )  <_  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) ) ) )
9835, 97jca 532 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( (pstoMet `  D ) : ( ( X /.  .~  )  X.  ( X /.  .~  ) ) --> RR*  /\  A. x  e.  ( X /.  .~  ) A. y  e.  ( X /.  .~  ) ( ( ( x (pstoMet `  D
) y )  =  0  <->  x  =  y
)  /\  A. z  e.  ( X /.  .~  ) ( x (pstoMet `  D ) y )  <_  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) ) ) ) )
99 elfvex 5821 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
100 qsexg 7263 . . 3  |-  ( X  e.  _V  ->  ( X /.  .~  )  e. 
_V )
101 isxmet 20026 . . 3  |-  ( ( X /.  .~  )  e.  _V  ->  ( (pstoMet `  D )  e.  ( *Met `  ( X /.  .~  ) )  <-> 
( (pstoMet `  D
) : ( ( X /.  .~  )  X.  ( X /.  .~  ) ) --> RR*  /\  A. x  e.  ( X /.  .~  ) A. y  e.  ( X /.  .~  ) ( ( ( x (pstoMet `  D
) y )  =  0  <->  x  =  y
)  /\  A. z  e.  ( X /.  .~  ) ( x (pstoMet `  D ) y )  <_  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) ) ) ) ) )
10299, 100, 1013syl 20 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( (pstoMet `  D )  e.  ( *Met `  ( X /.  .~  ) )  <-> 
( (pstoMet `  D
) : ( ( X /.  .~  )  X.  ( X /.  .~  ) ) --> RR*  /\  A. x  e.  ( X /.  .~  ) A. y  e.  ( X /.  .~  ) ( ( ( x (pstoMet `  D
) y )  =  0  <->  x  =  y
)  /\  A. z  e.  ( X /.  .~  ) ( x (pstoMet `  D ) y )  <_  ( ( z (pstoMet `  D )
x ) +e
( z (pstoMet `  D
) y ) ) ) ) ) )
10398, 102mpbird 232 1  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  e.  ( *Met `  ( X /.  .~  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2437   A.wral 2796   E.wrex 2797   _Vcvv 3072   U.cuni 4194   class class class wbr 4395    X. cxp 4941    Fn wfn 5516   -->wf 5517   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197    Er wer 7203   [cec 7204   /.cqs 7205   0cc0 9388   RR*cxr 9523    <_ cle 9525   +ecxad 11193  PsMetcpsmet 17920   *Metcxmt 17921  ~Metcmetid 26453  pstoMetcpstm 26454
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-er 7206  df-ec 7208  df-qs 7212  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-2 10486  df-rp 11098  df-xneg 11195  df-xadd 11196  df-xmul 11197  df-psmet 17929  df-xmet 17930  df-metid 26455  df-pstm 26456
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator