MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpsdsval Structured version   Unicode version

Theorem xpsdsval 20757
Description: Value of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
Hypotheses
Ref Expression
xpsds.t  |-  T  =  ( R  X.s  S )
xpsds.x  |-  X  =  ( Base `  R
)
xpsds.y  |-  Y  =  ( Base `  S
)
xpsds.1  |-  ( ph  ->  R  e.  V )
xpsds.2  |-  ( ph  ->  S  e.  W )
xpsds.p  |-  P  =  ( dist `  T
)
xpsds.m  |-  M  =  ( ( dist `  R
)  |`  ( X  X.  X ) )
xpsds.n  |-  N  =  ( ( dist `  S
)  |`  ( Y  X.  Y ) )
xpsds.3  |-  ( ph  ->  M  e.  ( *Met `  X ) )
xpsds.4  |-  ( ph  ->  N  e.  ( *Met `  Y ) )
xpsds.a  |-  ( ph  ->  A  e.  X )
xpsds.b  |-  ( ph  ->  B  e.  Y )
xpsds.c  |-  ( ph  ->  C  e.  X )
xpsds.d  |-  ( ph  ->  D  e.  Y )
Assertion
Ref Expression
xpsdsval  |-  ( ph  ->  ( <. A ,  B >. P <. C ,  D >. )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )

Proof of Theorem xpsdsval
Dummy variables  x  k  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpsds.t . . . . 5  |-  T  =  ( R  X.s  S )
2 xpsds.x . . . . 5  |-  X  =  ( Base `  R
)
3 xpsds.y . . . . 5  |-  Y  =  ( Base `  S
)
4 xpsds.1 . . . . 5  |-  ( ph  ->  R  e.  V )
5 xpsds.2 . . . . 5  |-  ( ph  ->  S  e.  W )
6 eqid 2443 . . . . 5  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )
7 eqid 2443 . . . . 5  |-  (Scalar `  R )  =  (Scalar `  R )
8 eqid 2443 . . . . 5  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } ) )
91, 2, 3, 4, 5, 6, 7, 8xpsval 14846 . . . 4  |-  ( ph  ->  T  =  ( `' ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  "s  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) ) ) )
101, 2, 3, 4, 5, 6, 7, 8xpslem 14847 . . . 4  |-  ( ph  ->  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) ) )
116xpsff1o2 14845 . . . . 5  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) : ( X  X.  Y ) -1-1-onto-> ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )
12 f1ocnv 5818 . . . . 5  |-  ( ( x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) : ( X  X.  Y ) -1-1-onto-> ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )  ->  `' (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) : ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) -1-1-onto-> ( X  X.  Y
) )
1311, 12mp1i 12 . . . 4  |-  ( ph  ->  `' ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) : ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) -1-1-onto-> ( X  X.  Y ) )
14 ovex 6309 . . . . 5  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  e.  _V
1514a1i 11 . . . 4  |-  ( ph  ->  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) )  e.  _V )
16 eqid 2443 . . . 4  |-  ( (
dist `  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) ) )  |`  ( ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  X.  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )  =  ( ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  |`  ( ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  X.  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
17 xpsds.p . . . 4  |-  P  =  ( dist `  T
)
18 xpsds.m . . . . . 6  |-  M  =  ( ( dist `  R
)  |`  ( X  X.  X ) )
19 xpsds.n . . . . . 6  |-  N  =  ( ( dist `  S
)  |`  ( Y  X.  Y ) )
20 xpsds.3 . . . . . 6  |-  ( ph  ->  M  e.  ( *Met `  X ) )
21 xpsds.4 . . . . . 6  |-  ( ph  ->  N  e.  ( *Met `  Y ) )
221, 2, 3, 4, 5, 17, 18, 19, 20, 21xpsxmetlem 20755 . . . . 5  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( *Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
23 ssid 3508 . . . . 5  |-  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )  C_  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )
24 xmetres2 20737 . . . . 5  |-  ( ( ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( *Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )  /\  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  C_  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )  -> 
( ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  |`  ( ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  X.  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )  e.  ( *Met ` 
ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) ) )
2522, 23, 24sylancl 662 . . . 4  |-  ( ph  ->  ( ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  |`  ( ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  X.  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )  e.  ( *Met ` 
ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) ) )
26 df-ov 6284 . . . . . 6  |-  ( A ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) B )  =  ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `
 <. A ,  B >. )
27 xpsds.a . . . . . . 7  |-  ( ph  ->  A  e.  X )
28 xpsds.b . . . . . . 7  |-  ( ph  ->  B  e.  Y )
296xpsfval 14841 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( A ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) B )  =  `' ( { A }  +c  { B } ) )
3027, 28, 29syl2anc 661 . . . . . 6  |-  ( ph  ->  ( A ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) B )  =  `' ( { A }  +c  { B } ) )
3126, 30syl5eqr 2498 . . . . 5  |-  ( ph  ->  ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) `
 <. A ,  B >. )  =  `' ( { A }  +c  { B } ) )
32 opelxpi 5021 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
3327, 28, 32syl2anc 661 . . . . . 6  |-  ( ph  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
34 f1of 5806 . . . . . . . 8  |-  ( ( x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) : ( X  X.  Y ) -1-1-onto-> ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )  ->  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) : ( X  X.  Y ) --> ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )
3511, 34ax-mp 5 . . . . . . 7  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) : ( X  X.  Y ) --> ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )
3635ffvelrni 6015 . . . . . 6  |-  ( <. A ,  B >.  e.  ( X  X.  Y
)  ->  ( (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `  <. A ,  B >. )  e.  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )
3733, 36syl 16 . . . . 5  |-  ( ph  ->  ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) `
 <. A ,  B >. )  e.  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )
3831, 37eqeltrrd 2532 . . . 4  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )
39 df-ov 6284 . . . . . 6  |-  ( C ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) D )  =  ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `
 <. C ,  D >. )
40 xpsds.c . . . . . . 7  |-  ( ph  ->  C  e.  X )
41 xpsds.d . . . . . . 7  |-  ( ph  ->  D  e.  Y )
426xpsfval 14841 . . . . . . 7  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  ( C ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) D )  =  `' ( { C }  +c  { D } ) )
4340, 41, 42syl2anc 661 . . . . . 6  |-  ( ph  ->  ( C ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) D )  =  `' ( { C }  +c  { D } ) )
4439, 43syl5eqr 2498 . . . . 5  |-  ( ph  ->  ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) `
 <. C ,  D >. )  =  `' ( { C }  +c  { D } ) )
45 opelxpi 5021 . . . . . . 7  |-  ( ( C  e.  X  /\  D  e.  Y )  -> 
<. C ,  D >.  e.  ( X  X.  Y
) )
4640, 41, 45syl2anc 661 . . . . . 6  |-  ( ph  -> 
<. C ,  D >.  e.  ( X  X.  Y
) )
4735ffvelrni 6015 . . . . . 6  |-  ( <. C ,  D >.  e.  ( X  X.  Y
)  ->  ( (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `  <. C ,  D >. )  e.  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )
4846, 47syl 16 . . . . 5  |-  ( ph  ->  ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) `
 <. C ,  D >. )  e.  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )
4944, 48eqeltrrd 2532 . . . 4  |-  ( ph  ->  `' ( { C }  +c  { D }
)  e.  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )
509, 10, 13, 15, 16, 17, 25, 38, 49imasdsf1o 20750 . . 3  |-  ( ph  ->  ( ( `' ( x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `  `' ( { A }  +c  { B } ) ) P ( `' ( x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `  `' ( { C }  +c  { D } ) ) )  =  ( `' ( { A }  +c  { B } ) ( ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  |`  ( ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  X.  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) ) `' ( { C }  +c  { D }
) ) )
5138, 49ovresd 6428 . . 3  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) ( (
dist `  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) ) )  |`  ( ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  X.  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) ) `' ( { C }  +c  { D } ) )  =  ( `' ( { A }  +c  { B } ) (
dist `  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) ) ) `' ( { C }  +c  { D } ) ) )
5250, 51eqtrd 2484 . 2  |-  ( ph  ->  ( ( `' ( x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `  `' ( { A }  +c  { B } ) ) P ( `' ( x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `  `' ( { C }  +c  { D } ) ) )  =  ( `' ( { A }  +c  { B } ) ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) ) `' ( { C }  +c  { D } ) ) )
53 f1ocnvfv 6169 . . . . 5  |-  ( ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) : ( X  X.  Y ) -1-1-onto-> ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  /\  <. A ,  B >.  e.  ( X  X.  Y ) )  ->  ( (
( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `  <. A ,  B >. )  =  `' ( { A }  +c  { B }
)  ->  ( `' ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `  `' ( { A }  +c  { B } ) )  =  <. A ,  B >. ) )
5411, 33, 53sylancr 663 . . . 4  |-  ( ph  ->  ( ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `
 <. A ,  B >. )  =  `' ( { A }  +c  { B } )  -> 
( `' ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `
 `' ( { A }  +c  { B } ) )  = 
<. A ,  B >. ) )
5531, 54mpd 15 . . 3  |-  ( ph  ->  ( `' ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `
 `' ( { A }  +c  { B } ) )  = 
<. A ,  B >. )
56 f1ocnvfv 6169 . . . . 5  |-  ( ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) : ( X  X.  Y ) -1-1-onto-> ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  /\  <. C ,  D >.  e.  ( X  X.  Y ) )  ->  ( (
( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `  <. C ,  D >. )  =  `' ( { C }  +c  { D }
)  ->  ( `' ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `  `' ( { C }  +c  { D } ) )  =  <. C ,  D >. ) )
5711, 46, 56sylancr 663 . . . 4  |-  ( ph  ->  ( ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `
 <. C ,  D >. )  =  `' ( { C }  +c  { D } )  -> 
( `' ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `
 `' ( { C }  +c  { D } ) )  = 
<. C ,  D >. ) )
5844, 57mpd 15 . . 3  |-  ( ph  ->  ( `' ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `
 `' ( { C }  +c  { D } ) )  = 
<. C ,  D >. )
5955, 58oveq12d 6299 . 2  |-  ( ph  ->  ( ( `' ( x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `  `' ( { A }  +c  { B } ) ) P ( `' ( x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `  `' ( { C }  +c  { D } ) ) )  =  ( <. A ,  B >. P
<. C ,  D >. ) )
60 eqid 2443 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )
61 fvex 5866 . . . . 5  |-  (Scalar `  R )  e.  _V
6261a1i 11 . . . 4  |-  ( ph  ->  (Scalar `  R )  e.  _V )
63 2on 7140 . . . . 5  |-  2o  e.  On
6463a1i 11 . . . 4  |-  ( ph  ->  2o  e.  On )
65 xpscfn 14833 . . . . 5  |-  ( ( R  e.  V  /\  S  e.  W )  ->  `' ( { R }  +c  { S }
)  Fn  2o )
664, 5, 65syl2anc 661 . . . 4  |-  ( ph  ->  `' ( { R }  +c  { S }
)  Fn  2o )
6738, 10eleqtrd 2533 . . . 4  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ( Base `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) ) )
6849, 10eleqtrd 2533 . . . 4  |-  ( ph  ->  `' ( { C }  +c  { D }
)  e.  ( Base `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) ) )
69 eqid 2443 . . . 4  |-  ( dist `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) )  =  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )
708, 60, 62, 64, 66, 67, 68, 69prdsdsval 14752 . . 3  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) ( dist `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) ) `' ( { C }  +c  { D } ) )  =  sup (
( ran  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B } ) `
 k ) (
dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
) )  u.  {
0 } ) , 
RR* ,  <  ) )
71 df2o3 7145 . . . . . . . . . . 11  |-  2o  =  { (/) ,  1o }
7271rexeqi 3045 . . . . . . . . . 10  |-  ( E. k  e.  2o  x  =  ( ( `' ( { A }  +c  { B } ) `
 k ) (
dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
)  <->  E. k  e.  { (/)
,  1o } x  =  ( ( `' ( { A }  +c  { B } ) `
 k ) (
dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
) )
73 0ex 4567 . . . . . . . . . . 11  |-  (/)  e.  _V
74 1on 7139 . . . . . . . . . . . 12  |-  1o  e.  On
7574elexi 3105 . . . . . . . . . . 11  |-  1o  e.  _V
76 fveq2 5856 . . . . . . . . . . . . . 14  |-  ( k  =  (/)  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  ( `' ( { R }  +c  { S } ) `  (/) ) )
7776fveq2d 5860 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  ( `' ( { R }  +c  { S } ) `  (/) ) ) )
78 fveq2 5856 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( `' ( { A }  +c  { B } ) `
 k )  =  ( `' ( { A }  +c  { B } ) `  (/) ) )
79 fveq2 5856 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( `' ( { C }  +c  { D } ) `
 k )  =  ( `' ( { C }  +c  { D } ) `  (/) ) )
8077, 78, 79oveq123d 6302 . . . . . . . . . . . 12  |-  ( k  =  (/)  ->  ( ( `' ( { A }  +c  { B }
) `  k )
( dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
)  =  ( ( `' ( { A }  +c  { B }
) `  (/) ) (
dist `  ( `' ( { R }  +c  { S } ) `  (/) ) ) ( `' ( { C }  +c  { D } ) `
 (/) ) ) )
8180eqeq2d 2457 . . . . . . . . . . 11  |-  ( k  =  (/)  ->  ( x  =  ( ( `' ( { A }  +c  { B } ) `
 k ) (
dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
)  <->  x  =  (
( `' ( { A }  +c  { B } ) `  (/) ) (
dist `  ( `' ( { R }  +c  { S } ) `  (/) ) ) ( `' ( { C }  +c  { D } ) `
 (/) ) ) ) )
82 fveq2 5856 . . . . . . . . . . . . . 14  |-  ( k  =  1o  ->  ( `' ( { R }  +c  { S }
) `  k )  =  ( `' ( { R }  +c  { S } ) `  1o ) )
8382fveq2d 5860 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  =  ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) )
84 fveq2 5856 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( `' ( { A }  +c  { B }
) `  k )  =  ( `' ( { A }  +c  { B } ) `  1o ) )
85 fveq2 5856 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( `' ( { C }  +c  { D }
) `  k )  =  ( `' ( { C }  +c  { D } ) `  1o ) )
8683, 84, 85oveq123d 6302 . . . . . . . . . . . 12  |-  ( k  =  1o  ->  (
( `' ( { A }  +c  { B } ) `  k
) ( dist `  ( `' ( { R }  +c  { S }
) `  k )
) ( `' ( { C }  +c  { D } ) `  k ) )  =  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) ) )
8786eqeq2d 2457 . . . . . . . . . . 11  |-  ( k  =  1o  ->  (
x  =  ( ( `' ( { A }  +c  { B }
) `  k )
( dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
)  <->  x  =  (
( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) ) ) )
8873, 75, 81, 87rexpr 4068 . . . . . . . . . 10  |-  ( E. k  e.  { (/) ,  1o } x  =  ( ( `' ( { A }  +c  { B } ) `  k ) ( dist `  ( `' ( { R }  +c  { S } ) `  k
) ) ( `' ( { C }  +c  { D } ) `
 k ) )  <-> 
( x  =  ( ( `' ( { A }  +c  { B } ) `  (/) ) (
dist `  ( `' ( { R }  +c  { S } ) `  (/) ) ) ( `' ( { C }  +c  { D } ) `
 (/) ) )  \/  x  =  ( ( `' ( { A }  +c  { B }
) `  1o )
( dist `  ( `' ( { R }  +c  { S } ) `  1o ) ) ( `' ( { C }  +c  { D } ) `
 1o ) ) ) )
8972, 88bitri 249 . . . . . . . . 9  |-  ( E. k  e.  2o  x  =  ( ( `' ( { A }  +c  { B } ) `
 k ) (
dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
)  <->  ( x  =  ( ( `' ( { A }  +c  { B } ) `  (/) ) ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) ) ( `' ( { C }  +c  { D } ) `  (/) ) )  \/  x  =  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) ) ) )
90 xpsc0 14834 . . . . . . . . . . . . . . 15  |-  ( R  e.  V  ->  ( `' ( { R }  +c  { S }
) `  (/) )  =  R )
914, 90syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  (/) )  =  R )
9291fveq2d 5860 . . . . . . . . . . . . 13  |-  ( ph  ->  ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) )  =  ( dist `  R
) )
93 xpsc0 14834 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  ( `' ( { A }  +c  { B }
) `  (/) )  =  A )
9427, 93syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
95 xpsc0 14834 . . . . . . . . . . . . . 14  |-  ( C  e.  X  ->  ( `' ( { C }  +c  { D }
) `  (/) )  =  C )
9640, 95syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { C }  +c  { D } ) `  (/) )  =  C )
9792, 94, 96oveq123d 6302 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  (/) ) ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) ) ( `' ( { C }  +c  { D } ) `  (/) ) )  =  ( A (
dist `  R ) C ) )
9818oveqi 6294 . . . . . . . . . . . . 13  |-  ( A M C )  =  ( A ( (
dist `  R )  |`  ( X  X.  X
) ) C )
9927, 40ovresd 6428 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A ( (
dist `  R )  |`  ( X  X.  X
) ) C )  =  ( A (
dist `  R ) C ) )
10098, 99syl5eq 2496 . . . . . . . . . . . 12  |-  ( ph  ->  ( A M C )  =  ( A ( dist `  R
) C ) )
10197, 100eqtr4d 2487 . . . . . . . . . . 11  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  (/) ) ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) ) ( `' ( { C }  +c  { D } ) `  (/) ) )  =  ( A M C ) )
102101eqeq2d 2457 . . . . . . . . . 10  |-  ( ph  ->  ( x  =  ( ( `' ( { A }  +c  { B } ) `  (/) ) (
dist `  ( `' ( { R }  +c  { S } ) `  (/) ) ) ( `' ( { C }  +c  { D } ) `
 (/) ) )  <->  x  =  ( A M C ) ) )
103 xpsc1 14835 . . . . . . . . . . . . . . 15  |-  ( S  e.  W  ->  ( `' ( { R }  +c  { S }
) `  1o )  =  S )
1045, 103syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  1o )  =  S )
105104fveq2d 5860 . . . . . . . . . . . . 13  |-  ( ph  ->  ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
)  =  ( dist `  S ) )
106 xpsc1 14835 . . . . . . . . . . . . . 14  |-  ( B  e.  Y  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  B )
10728, 106syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) `  1o )  =  B )
108 xpsc1 14835 . . . . . . . . . . . . . 14  |-  ( D  e.  Y  ->  ( `' ( { C }  +c  { D }
) `  1o )  =  D )
10941, 108syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { C }  +c  { D } ) `  1o )  =  D )
110105, 107, 109oveq123d 6302 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) )  =  ( B ( dist `  S
) D ) )
11119oveqi 6294 . . . . . . . . . . . . 13  |-  ( B N D )  =  ( B ( (
dist `  S )  |`  ( Y  X.  Y
) ) D )
11228, 41ovresd 6428 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B ( (
dist `  S )  |`  ( Y  X.  Y
) ) D )  =  ( B (
dist `  S ) D ) )
113111, 112syl5eq 2496 . . . . . . . . . . . 12  |-  ( ph  ->  ( B N D )  =  ( B ( dist `  S
) D ) )
114110, 113eqtr4d 2487 . . . . . . . . . . 11  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) )  =  ( B N D ) )
115114eqeq2d 2457 . . . . . . . . . 10  |-  ( ph  ->  ( x  =  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) )  <->  x  =  ( B N D ) ) )
116102, 115orbi12d 709 . . . . . . . . 9  |-  ( ph  ->  ( ( x  =  ( ( `' ( { A }  +c  { B } ) `  (/) ) ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) ) ( `' ( { C }  +c  { D } ) `  (/) ) )  \/  x  =  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) ) )  <->  ( x  =  ( A M C )  \/  x  =  ( B N D ) ) ) )
11789, 116syl5bb 257 . . . . . . . 8  |-  ( ph  ->  ( E. k  e.  2o  x  =  ( ( `' ( { A }  +c  { B } ) `  k
) ( dist `  ( `' ( { R }  +c  { S }
) `  k )
) ( `' ( { C }  +c  { D } ) `  k ) )  <->  ( x  =  ( A M C )  \/  x  =  ( B N D ) ) ) )
118 vex 3098 . . . . . . . . 9  |-  x  e. 
_V
119 eqid 2443 . . . . . . . . . 10  |-  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B }
) `  k )
( dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
) )  =  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B } ) `  k
) ( dist `  ( `' ( { R }  +c  { S }
) `  k )
) ( `' ( { C }  +c  { D } ) `  k ) ) )
120119elrnmpt 5239 . . . . . . . . 9  |-  ( x  e.  _V  ->  (
x  e.  ran  (
k  e.  2o  |->  ( ( `' ( { A }  +c  { B } ) `  k
) ( dist `  ( `' ( { R }  +c  { S }
) `  k )
) ( `' ( { C }  +c  { D } ) `  k ) ) )  <->  E. k  e.  2o  x  =  ( ( `' ( { A }  +c  { B }
) `  k )
( dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
) ) )
121118, 120ax-mp 5 . . . . . . . 8  |-  ( x  e.  ran  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B }
) `  k )
( dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
) )  <->  E. k  e.  2o  x  =  ( ( `' ( { A }  +c  { B } ) `  k
) ( dist `  ( `' ( { R }  +c  { S }
) `  k )
) ( `' ( { C }  +c  { D } ) `  k ) ) )
122118elpr 4032 . . . . . . . 8  |-  ( x  e.  { ( A M C ) ,  ( B N D ) }  <->  ( x  =  ( A M C )  \/  x  =  ( B N D ) ) )
123117, 121, 1223bitr4g 288 . . . . . . 7  |-  ( ph  ->  ( x  e.  ran  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B } ) `  k ) ( dist `  ( `' ( { R }  +c  { S } ) `  k
) ) ( `' ( { C }  +c  { D } ) `
 k ) ) )  <->  x  e.  { ( A M C ) ,  ( B N D ) } ) )
124123eqrdv 2440 . . . . . 6  |-  ( ph  ->  ran  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B } ) `
 k ) (
dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
) )  =  {
( A M C ) ,  ( B N D ) } )
125124uneq1d 3642 . . . . 5  |-  ( ph  ->  ( ran  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B }
) `  k )
( dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
) )  u.  {
0 } )  =  ( { ( A M C ) ,  ( B N D ) }  u.  {
0 } ) )
126 uncom 3633 . . . . 5  |-  ( { ( A M C ) ,  ( B N D ) }  u.  { 0 } )  =  ( { 0 }  u.  {
( A M C ) ,  ( B N D ) } )
127125, 126syl6eq 2500 . . . 4  |-  ( ph  ->  ( ran  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B }
) `  k )
( dist `  ( `' ( { R }  +c  { S } ) `  k ) ) ( `' ( { C }  +c  { D }
) `  k )
) )  u.  {
0 } )  =  ( { 0 }  u.  { ( A M C ) ,  ( B N D ) } ) )
128127supeq1d 7908 . . 3  |-  ( ph  ->  sup ( ( ran  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B } ) `  k ) ( dist `  ( `' ( { R }  +c  { S } ) `  k
) ) ( `' ( { C }  +c  { D } ) `
 k ) ) )  u.  { 0 } ) ,  RR* ,  <  )  =  sup ( ( { 0 }  u.  { ( A M C ) ,  ( B N D ) } ) ,  RR* ,  <  )
)
129 0xr 9643 . . . . . 6  |-  0  e.  RR*
130129a1i 11 . . . . 5  |-  ( ph  ->  0  e.  RR* )
131130snssd 4160 . . . 4  |-  ( ph  ->  { 0 }  C_  RR* )
132 xmetcl 20707 . . . . . 6  |-  ( ( M  e.  ( *Met `  X )  /\  A  e.  X  /\  C  e.  X
)  ->  ( A M C )  e.  RR* )
13320, 27, 40, 132syl3anc 1229 . . . . 5  |-  ( ph  ->  ( A M C )  e.  RR* )
134 xmetcl 20707 . . . . . 6  |-  ( ( N  e.  ( *Met `  Y )  /\  B  e.  Y  /\  D  e.  Y
)  ->  ( B N D )  e.  RR* )
13521, 28, 41, 134syl3anc 1229 . . . . 5  |-  ( ph  ->  ( B N D )  e.  RR* )
136 prssi 4171 . . . . 5  |-  ( ( ( A M C )  e.  RR*  /\  ( B N D )  e. 
RR* )  ->  { ( A M C ) ,  ( B N D ) }  C_  RR* )
137133, 135, 136syl2anc 661 . . . 4  |-  ( ph  ->  { ( A M C ) ,  ( B N D ) }  C_  RR* )
138 xrltso 11356 . . . . . 6  |-  <  Or  RR*
139 supsn 7933 . . . . . 6  |-  ( (  <  Or  RR*  /\  0  e.  RR* )  ->  sup ( { 0 } ,  RR* ,  <  )  =  0 )
140138, 129, 139mp2an 672 . . . . 5  |-  sup ( { 0 } ,  RR* ,  <  )  =  0
141 supxrcl 11515 . . . . . . 7  |-  ( { ( A M C ) ,  ( B N D ) } 
C_  RR*  ->  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  )  e.  RR* )
142137, 141syl 16 . . . . . 6  |-  ( ph  ->  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  )  e. 
RR* )
143 xmetge0 20720 . . . . . . 7  |-  ( ( M  e.  ( *Met `  X )  /\  A  e.  X  /\  C  e.  X
)  ->  0  <_  ( A M C ) )
14420, 27, 40, 143syl3anc 1229 . . . . . 6  |-  ( ph  ->  0  <_  ( A M C ) )
145 ovex 6309 . . . . . . . 8  |-  ( A M C )  e. 
_V
146145prid1 4123 . . . . . . 7  |-  ( A M C )  e. 
{ ( A M C ) ,  ( B N D ) }
147 supxrub 11525 . . . . . . 7  |-  ( ( { ( A M C ) ,  ( B N D ) }  C_  RR*  /\  ( A M C )  e. 
{ ( A M C ) ,  ( B N D ) } )  ->  ( A M C )  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
148137, 146, 147sylancl 662 . . . . . 6  |-  ( ph  ->  ( A M C )  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
149130, 133, 142, 144, 148xrletrd 11374 . . . . 5  |-  ( ph  ->  0  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
150140, 149syl5eqbr 4470 . . . 4  |-  ( ph  ->  sup ( { 0 } ,  RR* ,  <  )  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
151 supxrun 11516 . . . 4  |-  ( ( { 0 }  C_  RR* 
/\  { ( A M C ) ,  ( B N D ) }  C_  RR*  /\  sup ( { 0 } ,  RR* ,  <  )  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )  ->  sup ( ( { 0 }  u.  { ( A M C ) ,  ( B N D ) } ) ,  RR* ,  <  )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
152131, 137, 150, 151syl3anc 1229 . . 3  |-  ( ph  ->  sup ( ( { 0 }  u.  {
( A M C ) ,  ( B N D ) } ) ,  RR* ,  <  )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  )
)
15370, 128, 1523eqtrd 2488 . 2  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) ( dist `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) ) `' ( { C }  +c  { D } ) )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
15452, 59, 1533eqtr3d 2492 1  |-  ( ph  ->  ( <. A ,  B >. P <. C ,  D >. )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1383    e. wcel 1804   E.wrex 2794   _Vcvv 3095    u. cun 3459    C_ wss 3461   (/)c0 3770   {csn 4014   {cpr 4016   <.cop 4020   class class class wbr 4437    |-> cmpt 4495    Or wor 4789   Oncon0 4868    X. cxp 4987   `'ccnv 4988   ran crn 4990    |` cres 4991    Fn wfn 5573   -->wf 5574   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   1oc1o 7125   2oc2o 7126   supcsup 7902    +c ccda 8550   0cc0 9495   RR*cxr 9630    < clt 9631    <_ cle 9632   Basecbs 14509  Scalarcsca 14577   distcds 14583   X_scprds 14720    X.s cxps 14780   *Metcxmt 18277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-icc 11545  df-fz 11682  df-fzo 11804  df-seq 12087  df-hash 12385  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-hom 14598  df-cco 14599  df-0g 14716  df-gsum 14717  df-prds 14722  df-xrs 14776  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-mulg 15934  df-cntz 16229  df-cmn 16674  df-xmet 18286
This theorem is referenced by:  tmsxpsval  20914
  Copyright terms: Public domain W3C validator