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Theorem pstmxmet 28030
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 2382 . . . . 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 3037 . . . . . . 7  |-  x  e. 
_V
3 vex 3037 . . . . . . 7  |-  y  e. 
_V
42, 3ab2rexex 6690 . . . . . 6  |-  { z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) }  e.  _V
54uniex 6495 . . . . 5  |-  U. {
z  |  E. a  e.  x  E. b  e.  y  z  =  ( a D b ) }  e.  _V
61, 5fnmpt2i 6768 . . . 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 28028 . . . . 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 5579 . . . 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 233 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  Fn  ( ( X /.  .~  )  X.  ( X /.  .~  ) ) )
11 simpllr 758 . . . . . . . . 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 459 . . . . . . . . 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 6214 . . . . . . . 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 767 . . . . . . . . 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 766 . . . . . . . . 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 753 . . . . . . . . 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 28029 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  X  /\  b  e.  X )  ->  ( [ a ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  ( a D b ) )
1814, 15, 16, 17syl3anc 1226 . . . . . . . 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 2423 . . . . . . 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 20895 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
2114, 20syl 16 . . . . . . . 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 6346 . . . . . . 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 2470 . . . . . 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 7283 . . . . . . . 8  |-  ( y  e.  ( X /.  .~  )  ->  E. b  e.  X  y  =  [ b ]  .~  )
2524ad2antll 726 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  E. b  e.  X  y  =  [ b ]  .~  )
2625ad2antrr 723 . . . . . 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 2924 . . . . 5  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  ( x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  ->  ( x (pstoMet `  D ) y )  e.  RR* )
28 elqsi 7283 . . . . . 6  |-  ( x  e.  ( X /.  .~  )  ->  E. a  e.  X  x  =  [ a ]  .~  )
2928ad2antrl 725 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  E. a  e.  X  x  =  [ a ]  .~  )
3027, 29r19.29a 2924 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  ( x (pstoMet `  D ) y )  e.  RR* )
3130ralrimivva 2803 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  A. x  e.  ( X /.  .~  ) A. y  e.  ( X /.  .~  )
( x (pstoMet `  D
) y )  e. 
RR* )
32 ffnov 6305 . . 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 662 . 2  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
) : ( ( X /.  .~  )  X.  ( X /.  .~  ) ) --> RR* )
34173expa 1194 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  ( [ a ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  ( a D b ) )
3534eqeq1d 2384 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0  <-> 
( a D b )  =  0 ) )
367breqi 4373 . . . . . . . . . . . 12  |-  ( a  .~  b  <->  a (~Met `  D ) b )
37 metidv 28025 . . . . . . . . . . . . 13  |-  ( ( D  e.  (PsMet `  X )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( a
(~Met `  D )
b  <->  ( a D b )  =  0 ) )
3837anassrs 646 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
a (~Met `  D
) b  <->  ( a D b )  =  0 ) )
3936, 38syl5bb 257 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
a  .~  b  <->  ( a D b )  =  0 ) )
40 metider 28027 . . . . . . . . . . . . . 14  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  Er  X )
4140ad2antrr 723 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (~Met `  D )  Er  X
)
42 ereq1 7236 . . . . . . . . . . . . . 14  |-  (  .~  =  (~Met `  D )  ->  (  .~  Er  X  <->  (~Met `  D )  Er  X
) )
437, 42ax-mp 5 . . . . . . . . . . . . 13  |-  (  .~  Er  X  <->  (~Met `  D )  Er  X )
4441, 43sylibr 212 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  .~  Er  X )
45 simplr 753 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  a  e.  X )
4644, 45erth 7274 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
a  .~  b  <->  [ a ]  .~  =  [ b ]  .~  ) )
4735, 39, 463bitr2d 281 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  b  e.  X )  ->  (
( [ a ]  .~  (pstoMet `  D
) [ b ]  .~  )  =  0  <->  [ a ]  .~  =  [ b ]  .~  ) )
4847adantllr 716 . . . . . . . . 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 712 . . . . . . . 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 463 . . . . . . 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 2384 . . . . . . 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 2404 . . . . . . 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 285 . . . . . 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 2924 . . . . 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 2924 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  ( X /.  .~  )  /\  y  e.  ( X /.  .~  ) ) )  ->  ( ( x (pstoMet `  D )
y )  =  0  <-> 
x  =  y ) )
56 simp-6l 769 . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . 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 20898 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  x  =  [
a ]  .~  )
63 simpllr 758 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  y  =  [
b ]  .~  )
6462, 63oveq12d 6214 . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . . 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 2423 . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  /\  x  =  [ a ]  .~  )  /\  b  e.  X
)  /\  y  =  [ b ]  .~  )  /\  c  e.  X
)  /\  z  =  [ c ]  .~  )  ->  z  =  [
c ]  .~  )
6867, 62oveq12d 6214 . . . . . . . . . . . . . . 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 28029 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  (PsMet `  X )  /\  c  e.  X  /\  a  e.  X )  ->  ( [ c ]  .~  (pstoMet `  D ) [ a ]  .~  )  =  ( c D a ) )
7056, 57, 58, 69syl3anc 1226 . . . . . . . . . . . . . . 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 2423 . . . . . . . . . . . . . 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 6214 . . . . . . . . . . . . . . 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 28029 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  (PsMet `  X )  /\  c  e.  X  /\  b  e.  X )  ->  ( [ c ]  .~  (pstoMet `  D ) [ b ]  .~  )  =  ( c D b ) )
7456, 57, 59, 73syl3anc 1226 . . . . . . . . . . . . . . 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 2423 . . . . . . . . . . . . . 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 6214 . . . . . . . . . . . . 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 4397 . . . . . . . . . . . 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 750 . . . . . . . . . . 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 7283 . . . . . . . . . . . 12  |-  ( z  e.  ( X /.  .~  )  ->  E. c  e.  X  z  =  [ c ]  .~  )
8079ad5antlr 732 . . . . . . . . . . 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 2924 . . . . . . . . . 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 749 . . . . . . . . 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 730 . . . . . . . . 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 2924 . . . . . . . 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 748 . . . . . . 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 728 . . . . . . 7  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  x  e.  ( X /.  .~  ) )  /\  y  e.  ( X /.  .~  )
)  /\  z  e.  ( X /.  .~  )
)  ->  E. a  e.  X  x  =  [ a ]  .~  )
8785, 86r19.29a 2924 . . . . . 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 2796 . . . . 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 645 . . . 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 530 . . 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 2803 . 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 5801 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
93 qsexg 7287 . . 3  |-  ( X  e.  _V  ->  ( X /.  .~  )  e. 
_V )
94 isxmet 20912 . . 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 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 ) ) ) ) ) )
9633, 91, 95mpbir2and 920 1  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  e.  ( *Met `  ( X /.  .~  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   {cab 2367   A.wral 2732   E.wrex 2733   _Vcvv 3034   U.cuni 4163   class class class wbr 4367    X. cxp 4911    Fn wfn 5491   -->wf 5492   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198    Er wer 7226   [cec 7227   /.cqs 7228   0cc0 9403   RR*cxr 9538    <_ cle 9540   +ecxad 11237  PsMetcpsmet 18515   *Metcxmt 18516  ~Metcmetid 28019  pstoMetcpstm 28020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-er 7229  df-ec 7231  df-qs 7235  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-2 10511  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-psmet 18524  df-xmet 18525  df-metid 28021  df-pstm 28022
This theorem is referenced by: (None)
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