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Theorem pstmxmet 24245
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 2919 . . . . . . . . 9  |-  x  e. 
_V
2 vex 2919 . . . . . . . . 9  |-  y  e. 
_V
31, 2ab2rexex 6185 . . . . . . . 8  |-  { z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) }  e.  _V
43uniex 4664 . . . . . . 7  |-  U. {
z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) }  e.  _V
54rgen2w 2734 . . . . . 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 2404 . . . . . . 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 6378 . . . . . 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 8 . . . . 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 24243 . . . . . 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 5495 . . . . 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 225 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  Fn  ( ( X /.  .~  )  X.  ( X /.  .~  ) ) )
13 simpllr 736 . . . . . . . . . 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 448 . . . . . . . . . 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 6058 . . . . . . . . 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 745 . . . . . . . . . 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 744 . . . . . . . . . 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 732 . . . . . . . . . 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 24244 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  X  /\  b  e.  X )  ->  ( [ a ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  ( a D b ) )
2016, 17, 18, 19syl3anc 1184 . . . . . . . . 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 2436 . . . . . . . 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 18290 . . . . . . . . . 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 6177 . . . . . . . 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 2478 . . . . . . 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 6917 . . . . . . . . 9  |-  ( y  e.  ( X /.  .~  )  ->  E. b  e.  X  y  =  [ b ]  .~  )
2726ad2antll 710 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  E. b  e.  X  y  =  [ b ]  .~  )
2827ad2antrr 707 . . . . . . 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 2810 . . . . . 6  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  ( x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  ->  ( x (pstoMet `  D ) y )  e.  RR* )
30 elqsi 6917 . . . . . . 7  |-  ( x  e.  ( X /.  .~  )  ->  E. a  e.  X  x  =  [ a ]  .~  )
3130ad2antrl 709 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  E. a  e.  X  x  =  [ a ]  .~  )
3229, 31r19.29a 2810 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  ( x (pstoMet `  D ) y )  e.  RR* )
3332ralrimivva 2758 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  A. x  e.  ( X /.  .~  ) A. y  e.  ( X /.  .~  )
( x (pstoMet `  D
) y )  e. 
RR* )
34 ffnov 6133 . . . 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 646 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
) : ( ( X /.  .~  )  X.  ( X /.  .~  ) ) --> RR* )
36 simpll 731 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  D  e.  (PsMet `  X )
)
37 simplr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  a  e.  X )
38 simpr 448 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  b  e.  X )
3936, 37, 38, 19syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  ( [ a ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  ( a D b ) )
4039eqeq1d 2412 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0  <-> 
( a D b )  =  0 ) )
419breqi 4178 . . . . . . . . . . . . . 14  |-  ( a  .~  b  <->  a (~Met `  D ) b )
42 metidv 24240 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  (PsMet `  X )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( a
(~Met `  D )
b  <->  ( a D b )  =  0 ) )
4342anassrs 630 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
a (~Met `  D
) b  <->  ( a D b )  =  0 ) )
4441, 43syl5bb 249 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
a  .~  b  <->  ( a D b )  =  0 ) )
4540, 44bitr4d 248 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0  <-> 
a  .~  b )
)
46 metider 24242 . . . . . . . . . . . . . . 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 6871 . . . . . . . . . . . . . . 15  |-  (  .~  =  (~Met `  D )  ->  (  .~  Er  X  <->  (~Met `  D )  Er  X
) )
499, 48ax-mp 8 . . . . . . . . . . . . . 14  |-  (  .~  Er  X  <->  (~Met `  D )  Er  X )
5047, 49sylibr 204 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  .~  Er  X )
5150, 37erth 6908 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
a  .~  b  <->  [ a ]  .~  =  [ b ]  .~  ) )
5245, 51bitrd 245 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0  <->  [ a ]  .~  =  [ b ]  .~  ) )
5352adantllr 700 . . . . . . . . . 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 696 . . . . . . . . 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 452 . . . . . . . 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 2412 . . . . . . . 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 2418 . . . . . . . 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 277 . . . . . . 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 2810 . . . . . 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 2810 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  ( ( x (pstoMet `  D )
y )  =  0  <-> 
x  =  y ) )
61 simp-6l 747 . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . 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 748 . . . . . . . . . . . . . . 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 744 . . . . . . . . . . . . . . 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 18293 . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  x  =  [
a ]  .~  )
68 simpllr 736 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  y  =  [
b ]  .~  )
6967, 68oveq12d 6058 . . . . . . . . . . . . . . . 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 1183 . . . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  z  =  [
c ]  .~  )
7372, 67oveq12d 6058 . . . . . . . . . . . . . . . . 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 24244 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  (PsMet `  X )  /\  c  e.  X  /\  a  e.  X )  ->  ( [ c ]  .~  (pstoMet `  D ) [ a ]  .~  )  =  ( c D a ) )
7561, 62, 63, 74syl3anc 1184 . . . . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . . 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 6058 . . . . . . . . . . . . . . . . 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 24244 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  (PsMet `  X )  /\  c  e.  X  /\  b  e.  X )  ->  ( [ c ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  ( c D b ) )
7961, 62, 64, 78syl3anc 1184 . . . . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . . 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 6058 . . . . . . . . . . . . . . 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 4185 . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . 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 23912 . . . . . . . . . . . 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 6917 . . . . . . . . . . . . 13  |-  ( z  e.  ( X /.  .~  )  ->  E. c  e.  X  z  =  [ c ]  .~  )
8685ad5antlr 716 . . . . . . . . . . . 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 2810 . . . . . . . . . . 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 23911 . . . . . . . . . 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 714 . . . . . . . . . 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 2810 . . . . . . . . 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 23910 . . . . . . . 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 712 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  x  e.  ( X /.  .~  ) )  /\  y  e.  ( X /.  .~  )
)  /\  z  e.  ( X /.  .~  )
)  ->  E. a  e.  X  x  =  [ a ]  .~  )
9391, 92r19.29a 2810 . . . . . . 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 2749 . . . . . 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 629 . . . . 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 519 . . . 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 2758 . . 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 519 . 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 5717 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
100 qsexg 6921 . . 3  |-  ( X  e.  _V  ->  ( X /.  .~  )  e. 
_V )
101 isxmet 18307 . . 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 19 . 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 224 1  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  e.  ( * Met `  ( X /.  .~  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667   _Vcvv 2916   U.cuni 3975   class class class wbr 4172    X. cxp 4835    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042    Er wer 6861   [cec 6862   /.cqs 6863   0cc0 8946   RR*cxr 9075    <_ cle 9077   + ecxad 10664  PsMetcpsmet 16640   * Metcxmt 16641  ~Metcmetid 24234  pstoMetcpstm 24235
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-psmet 16649  df-xmet 16650  df-metid 24236  df-pstm 24237
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