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Theorem pstmxmet 28702
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 eqid 2423 . . . . 5  |-  ( 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 ) } )
2 vex 3085 . . . . . . 7  |-  x  e. 
_V
3 vex 3085 . . . . . . 7  |-  y  e. 
_V
42, 3ab2rexex 6796 . . . . . 6  |-  { z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) }  e.  _V
54uniex 6599 . . . . 5  |-  U. {
z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) }  e.  _V
61, 5fnmpt2i 6874 . . . 4  |-  ( x  e.  ( X /.  .~  ) ,  y  e.  ( X /.  .~  )  |->  U. { z  |  E. a  e.  x  E. b  e.  y 
z  =  ( a D b ) } )  Fn  ( ( X /.  .~  )  X.  ( X /.  .~  ) )
7 pstmval.1 . . . . . 6  |-  .~  =  (~Met `  D )
87pstmval 28700 . . . . 5  |-  ( 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 ) } ) )
98fneq1d 5682 . . . 4  |-  ( 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 /.  .~  ) ) ) )
106, 9mpbiri 237 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  Fn  ( ( X /.  .~  )  X.  ( X /.  .~  ) ) )
11 simpllr 768 . . . . . . . . 9  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  x  =  [
a ]  .~  )
12 simpr 463 . . . . . . . . 9  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  y  =  [
b ]  .~  )
1311, 12oveq12d 6321 . . . . . . . 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 ]  .~  (pstoMet `  D
) [ b ]  .~  ) )
14 simp-5l 777 . . . . . . . . 9  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  D  e.  (PsMet `  X ) )
15 simp-4r 776 . . . . . . . . 9  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  a  e.  X
)
16 simplr 761 . . . . . . . . 9  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  b  e.  X
)
177pstmfval 28701 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  X  /\  b  e.  X )  ->  ( [ a ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  ( a D b ) )
1814, 15, 16, 17syl3anc 1265 . . . . . . . 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 ]  .~  )  =  ( a D b ) )
1913, 18eqtrd 2464 . . . . . . 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 )  =  ( a D b ) )
20 psmetf 21314 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
2114, 20syl 17 . . . . . . . 8  |-  ( ( ( ( ( ( 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* )
2221, 15, 16fovrnd 6453 . . . . . . 7  |-  ( ( ( ( ( ( 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* )
2319, 22eqeltrd 2511 . . . . . 6  |-  ( ( ( ( ( ( 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* )
24 elqsi 7423 . . . . . . . 8  |-  ( y  e.  ( X /.  .~  )  ->  E. b  e.  X  y  =  [ b ]  .~  )
2524ad2antll 734 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  E. b  e.  X  y  =  [ b ]  .~  )
2625ad2antrr 731 . . . . . 6  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  ( x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  ->  E. b  e.  X  y  =  [ b ]  .~  )
2723, 26r19.29a 2971 . . . . 5  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  ( x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  ->  ( x (pstoMet `  D ) y )  e.  RR* )
28 elqsi 7423 . . . . . 6  |-  ( x  e.  ( X /.  .~  )  ->  E. a  e.  X  x  =  [ a ]  .~  )
2928ad2antrl 733 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  E. a  e.  X  x  =  [ a ]  .~  )
3027, 29r19.29a 2971 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  ( x (pstoMet `  D ) y )  e.  RR* )
3130ralrimivva 2847 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  A. x  e.  ( X /.  .~  ) A. y  e.  ( X /.  .~  )
( x (pstoMet `  D
) y )  e. 
RR* )
32 ffnov 6412 . . 3  |-  ( (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* ) )
3310, 31, 32sylanbrc 669 . 2  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
) : ( ( X /.  .~  )  X.  ( X /.  .~  ) ) --> RR* )
34173expa 1206 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  ( [ a ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  ( a D b ) )
3534eqeq1d 2425 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0  <-> 
( a D b )  =  0 ) )
367breqi 4427 . . . . . . . . . . . 12  |-  ( a  .~  b  <->  a (~Met `  D ) b )
37 metidv 28697 . . . . . . . . . . . . 13  |-  ( ( D  e.  (PsMet `  X )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( a
(~Met `  D )
b  <->  ( a D b )  =  0 ) )
3837anassrs 653 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
a (~Met `  D
) b  <->  ( a D b )  =  0 ) )
3936, 38syl5bb 261 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
a  .~  b  <->  ( a D b )  =  0 ) )
40 metider 28699 . . . . . . . . . . . . . 14  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  Er  X )
4140ad2antrr 731 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (~Met `  D )  Er  X
)
42 ereq1 7376 . . . . . . . . . . . . . 14  |-  (  .~  =  (~Met `  D )  ->  (  .~  Er  X  <->  (~Met `  D )  Er  X
) )
437, 42ax-mp 5 . . . . . . . . . . . . 13  |-  (  .~  Er  X  <->  (~Met `  D )  Er  X )
4441, 43sylibr 216 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  .~  Er  X )
45 simplr 761 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  a  e.  X )
4644, 45erth 7414 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
a  .~  b  <->  [ a ]  .~  =  [ b ]  .~  ) )
4735, 39, 463bitr2d 285 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0  <->  [ a ]  .~  =  [ b ]  .~  ) )
4847adantllr 724 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  ( x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X )  /\  b  e.  X )  ->  (
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0  <->  [ a ]  .~  =  [ b ]  .~  ) )
4948adantlr 720 . . . . . . . 8  |-  ( ( ( ( ( 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 ]  .~  ) )
5049adantr 467 . . . . . . 7  |-  ( ( ( ( ( ( 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 ]  .~  ) )
5113eqeq1d 2425 . . . . . . 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  <-> 
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0 ) )
5211, 12eqeq12d 2445 . . . . . . 7  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X
)  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  ( x  =  y  <->  [ a ]  .~  =  [ b ]  .~  ) )
5350, 51, 523bitr4d 289 . . . . . 6  |-  ( ( ( ( ( ( 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 ) )
5453, 26r19.29a 2971 . . . . 5  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  ( x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  ->  ( ( x (pstoMet `  D )
y )  =  0  <-> 
x  =  y ) )
5554, 29r19.29a 2971 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  ( ( x (pstoMet `  D )
y )  =  0  <-> 
x  =  y ) )
56 simp-6l 779 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  D  e.  (PsMet `  X ) )
57 simplr 761 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  c  e.  X
)
58 simp-6r 780 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  a  e.  X
)
59 simp-4r 776 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  b  e.  X
)
60 psmettri2 21317 . . . . . . . . . . . . . 14  |-  ( ( D  e.  (PsMet `  X )  /\  (
c  e.  X  /\  a  e.  X  /\  b  e.  X )
)  ->  ( a D b )  <_ 
( ( c D a ) +e
( c D b ) ) )
6156, 57, 58, 59, 60syl13anc 1267 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( 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 ) ) )
62 simp-5r 778 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  x  =  [
a ]  .~  )
63 simpllr 768 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  y  =  [
b ]  .~  )
6462, 63oveq12d 6321 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( 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 ]  .~  ) )
6556, 58, 59, 17syl3anc 1265 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( 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 ) )
6664, 65eqtrd 2464 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( 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 ) )
67 simpr 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  z  =  [
c ]  .~  )
6867, 62oveq12d 6321 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( 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 ]  .~  ) )
697pstmfval 28701 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  (PsMet `  X )  /\  c  e.  X  /\  a  e.  X )  ->  ( [ c ]  .~  (pstoMet `  D ) [ a ]  .~  )  =  ( c D a ) )
7056, 57, 58, 69syl3anc 1265 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( 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 ) )
7168, 70eqtrd 2464 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( 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 ) )
7267, 63oveq12d 6321 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( 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 ]  .~  ) )
737pstmfval 28701 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  (PsMet `  X )  /\  c  e.  X  /\  b  e.  X )  ->  ( [ c ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  ( c D b ) )
7456, 57, 59, 73syl3anc 1265 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( 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 ) )
7572, 74eqtrd 2464 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( 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 ) )
7671, 75oveq12d 6321 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( 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 ) ) )
7761, 66, 763brtr4d 4452 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( 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 ) ) )
7877adantl6r 758 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( 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 ) ) )
79 elqsi 7423 . . . . . . . . . . . 12  |-  ( z  e.  ( X /.  .~  )  ->  E. c  e.  X  z  =  [ c ]  .~  )
8079ad5antlr 740 . . . . . . . . . . 11  |-  ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  z  e.  ( X /.  .~  ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  ->  E. c  e.  X  z  =  [ c ]  .~  )
8178, 80r19.29a 2971 . . . . . . . . . 10  |-  ( ( ( ( ( ( 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 ) ) )
8281adantl5r 757 . . . . . . . . 9  |-  ( ( ( ( ( ( ( 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 ) ) )
8324ad4antlr 738 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  y  e.  ( X /.  .~  )
)  /\  z  e.  ( X /.  .~  )
)  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  ->  E. b  e.  X  y  =  [ b ]  .~  )
8482, 83r19.29a 2971 . . . . . . . 8  |-  ( ( ( ( ( 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 ) ) )
8584adantl4r 756 . . . . . . 7  |-  ( ( ( ( ( ( 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 ) ) )
8628ad3antlr 736 . . . . . . 7  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  x  e.  ( X /.  .~  ) )  /\  y  e.  ( X /.  .~  )
)  /\  z  e.  ( X /.  .~  )
)  ->  E. a  e.  X  x  =  [ a ]  .~  )
8785, 86r19.29a 2971 . . . . . 6  |-  ( ( ( ( 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 ) ) )
8887ralrimiva 2840 . . . . 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 ) ) )
8988anasss 652 . . . 4  |-  ( ( 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 ) ) )
9055, 89jca 535 . . 3  |-  ( ( 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 ) ) ) )
9190ralrimivva 2847 . 2  |-  ( 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 ) ) ) )
92 elfvex 5906 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
93 qsexg 7427 . . 3  |-  ( X  e.  _V  ->  ( X /.  .~  )  e. 
_V )
94 isxmet 21331 . . 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 ) ) ) ) ) )
9592, 93, 943syl 18 . 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 ) ) ) ) ) )
9633, 91, 95mpbir2and 931 1  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  e.  ( *Met `  ( X /.  .~  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   {cab 2408   A.wral 2776   E.wrex 2777   _Vcvv 3082   U.cuni 4217   class class class wbr 4421    X. cxp 4849    Fn wfn 5594   -->wf 5595   ` cfv 5599  (class class class)co 6303    |-> cmpt2 6305    Er wer 7366   [cec 7367   /.cqs 7368   0cc0 9541   RR*cxr 9676    <_ cle 9678   +ecxad 11409  PsMetcpsmet 18947   *Metcxmt 18948  ~Metcmetid 28691  pstoMetcpstm 28692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-po 4772  df-so 4773  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-er 7369  df-ec 7371  df-qs 7375  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-2 10670  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-psmet 18955  df-xmet 18956  df-metid 28693  df-pstm 28694
This theorem is referenced by: (None)
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