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Theorem xpsdsval 21176
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 2402 . . . . 5  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )
7 eqid 2402 . . . . 5  |-  (Scalar `  R )  =  (Scalar `  R )
8 eqid 2402 . . . . 5  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } ) )
91, 2, 3, 4, 5, 6, 7, 8xpsval 15186 . . . 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 15187 . . . 4  |-  ( ph  ->  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) ) )
116xpsff1o2 15185 . . . . 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 5811 . . . . 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 13 . . . 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 6306 . . . . 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 2402 . . . 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 21174 . . . . 5  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( *Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
23 ssid 3461 . . . . 5  |-  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )  C_  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )
24 xmetres2 21156 . . . . 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 660 . . . 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 6281 . . . . . 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 15181 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( A ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) B )  =  `' ( { A }  +c  { B } ) )
3027, 28, 29syl2anc 659 . . . . . 6  |-  ( ph  ->  ( A ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) B )  =  `' ( { A }  +c  { B } ) )
3126, 30syl5eqr 2457 . . . . 5  |-  ( ph  ->  ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) `
 <. A ,  B >. )  =  `' ( { A }  +c  { B } ) )
32 opelxpi 4855 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
3327, 28, 32syl2anc 659 . . . . . 6  |-  ( ph  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
34 f1of 5799 . . . . . . . 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 6008 . . . . . 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 17 . . . . 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 2491 . . . 4  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )
39 df-ov 6281 . . . . . 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 15181 . . . . . . 7  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  ( C ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) D )  =  `' ( { C }  +c  { D } ) )
4340, 41, 42syl2anc 659 . . . . . 6  |-  ( ph  ->  ( C ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) D )  =  `' ( { C }  +c  { D } ) )
4439, 43syl5eqr 2457 . . . . 5  |-  ( ph  ->  ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) `
 <. C ,  D >. )  =  `' ( { C }  +c  { D } ) )
45 opelxpi 4855 . . . . . . 7  |-  ( ( C  e.  X  /\  D  e.  Y )  -> 
<. C ,  D >.  e.  ( X  X.  Y
) )
4640, 41, 45syl2anc 659 . . . . . 6  |-  ( ph  -> 
<. C ,  D >.  e.  ( X  X.  Y
) )
4735ffvelrni 6008 . . . . . 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 17 . . . . 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 2491 . . . 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 21169 . . 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 6424 . . 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 2443 . 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 6165 . . . . 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 661 . . . 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 6165 . . . . 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 661 . . . 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 6296 . 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 2402 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )
61 fvex 5859 . . . . 5  |-  (Scalar `  R )  e.  _V
6261a1i 11 . . . 4  |-  ( ph  ->  (Scalar `  R )  e.  _V )
63 2on 7175 . . . . 5  |-  2o  e.  On
6463a1i 11 . . . 4  |-  ( ph  ->  2o  e.  On )
65 xpscfn 15173 . . . . 5  |-  ( ( R  e.  V  /\  S  e.  W )  ->  `' ( { R }  +c  { S }
)  Fn  2o )
664, 5, 65syl2anc 659 . . . 4  |-  ( ph  ->  `' ( { R }  +c  { S }
)  Fn  2o )
6738, 10eleqtrd 2492 . . . 4  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ( Base `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) ) )
6849, 10eleqtrd 2492 . . . 4  |-  ( ph  ->  `' ( { C }  +c  { D }
)  e.  ( Base `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) ) )
69 eqid 2402 . . . 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 15092 . . 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 7180 . . . . . . . . . . 11  |-  2o  =  { (/) ,  1o }
7271rexeqi 3009 . . . . . . . . . 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 4526 . . . . . . . . . . 11  |-  (/)  e.  _V
74 1on 7174 . . . . . . . . . . . 12  |-  1o  e.  On
7574elexi 3069 . . . . . . . . . . 11  |-  1o  e.  _V
76 fveq2 5849 . . . . . . . . . . . . . 14  |-  ( k  =  (/)  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  ( `' ( { R }  +c  { S } ) `  (/) ) )
7776fveq2d 5853 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  ( `' ( { R }  +c  { S } ) `  (/) ) ) )
78 fveq2 5849 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( `' ( { A }  +c  { B } ) `
 k )  =  ( `' ( { A }  +c  { B } ) `  (/) ) )
79 fveq2 5849 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( `' ( { C }  +c  { D } ) `
 k )  =  ( `' ( { C }  +c  { D } ) `  (/) ) )
8077, 78, 79oveq123d 6299 . . . . . . . . . . . 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 2416 . . . . . . . . . . 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 5849 . . . . . . . . . . . . . 14  |-  ( k  =  1o  ->  ( `' ( { R }  +c  { S }
) `  k )  =  ( `' ( { R }  +c  { S } ) `  1o ) )
8382fveq2d 5853 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  =  ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) )
84 fveq2 5849 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( `' ( { A }  +c  { B }
) `  k )  =  ( `' ( { A }  +c  { B } ) `  1o ) )
85 fveq2 5849 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( `' ( { C }  +c  { D }
) `  k )  =  ( `' ( { C }  +c  { D } ) `  1o ) )
8683, 84, 85oveq123d 6299 . . . . . . . . . . . 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 2416 . . . . . . . . . . 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 4026 . . . . . . . . . 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 15174 . . . . . . . . . . . . . . 15  |-  ( R  e.  V  ->  ( `' ( { R }  +c  { S }
) `  (/) )  =  R )
914, 90syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  (/) )  =  R )
9291fveq2d 5853 . . . . . . . . . . . . 13  |-  ( ph  ->  ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) )  =  ( dist `  R
) )
93 xpsc0 15174 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  ( `' ( { A }  +c  { B }
) `  (/) )  =  A )
9427, 93syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
95 xpsc0 15174 . . . . . . . . . . . . . 14  |-  ( C  e.  X  ->  ( `' ( { C }  +c  { D }
) `  (/) )  =  C )
9640, 95syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { C }  +c  { D } ) `  (/) )  =  C )
9792, 94, 96oveq123d 6299 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  (/) ) ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) ) ( `' ( { C }  +c  { D } ) `  (/) ) )  =  ( A (
dist `  R ) C ) )
9818oveqi 6291 . . . . . . . . . . . . 13  |-  ( A M C )  =  ( A ( (
dist `  R )  |`  ( X  X.  X
) ) C )
9927, 40ovresd 6424 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A ( (
dist `  R )  |`  ( X  X.  X
) ) C )  =  ( A (
dist `  R ) C ) )
10098, 99syl5eq 2455 . . . . . . . . . . . 12  |-  ( ph  ->  ( A M C )  =  ( A ( dist `  R
) C ) )
10197, 100eqtr4d 2446 . . . . . . . . . . 11  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  (/) ) ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) ) ( `' ( { C }  +c  { D } ) `  (/) ) )  =  ( A M C ) )
102101eqeq2d 2416 . . . . . . . . . 10  |-  ( ph  ->  ( x  =  ( ( `' ( { A }  +c  { B } ) `  (/) ) (
dist `  ( `' ( { R }  +c  { S } ) `  (/) ) ) ( `' ( { C }  +c  { D } ) `
 (/) ) )  <->  x  =  ( A M C ) ) )
103 xpsc1 15175 . . . . . . . . . . . . . . 15  |-  ( S  e.  W  ->  ( `' ( { R }  +c  { S }
) `  1o )  =  S )
1045, 103syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  1o )  =  S )
105104fveq2d 5853 . . . . . . . . . . . . 13  |-  ( ph  ->  ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
)  =  ( dist `  S ) )
106 xpsc1 15175 . . . . . . . . . . . . . 14  |-  ( B  e.  Y  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  B )
10728, 106syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) `  1o )  =  B )
108 xpsc1 15175 . . . . . . . . . . . . . 14  |-  ( D  e.  Y  ->  ( `' ( { C }  +c  { D }
) `  1o )  =  D )
10941, 108syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { C }  +c  { D } ) `  1o )  =  D )
110105, 107, 109oveq123d 6299 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) )  =  ( B ( dist `  S
) D ) )
11119oveqi 6291 . . . . . . . . . . . . 13  |-  ( B N D )  =  ( B ( (
dist `  S )  |`  ( Y  X.  Y
) ) D )
11228, 41ovresd 6424 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B ( (
dist `  S )  |`  ( Y  X.  Y
) ) D )  =  ( B (
dist `  S ) D ) )
113111, 112syl5eq 2455 . . . . . . . . . . . 12  |-  ( ph  ->  ( B N D )  =  ( B ( dist `  S
) D ) )
114110, 113eqtr4d 2446 . . . . . . . . . . 11  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) )  =  ( B N D ) )
115114eqeq2d 2416 . . . . . . . . . 10  |-  ( ph  ->  ( x  =  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) )  <->  x  =  ( B N D ) ) )
116102, 115orbi12d 708 . . . . . . . . 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 3062 . . . . . . . . 9  |-  x  e. 
_V
119 eqid 2402 . . . . . . . . . 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 5070 . . . . . . . . 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 3990 . . . . . . . 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 2399 . . . . . 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 3596 . . . . 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 3587 . . . . 5  |-  ( { ( A M C ) ,  ( B N D ) }  u.  { 0 } )  =  ( { 0 }  u.  {
( A M C ) ,  ( B N D ) } )
127125, 126syl6eq 2459 . . . 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 7939 . . 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 9670 . . . . . 6  |-  0  e.  RR*
130129a1i 11 . . . . 5  |-  ( ph  ->  0  e.  RR* )
131130snssd 4117 . . . 4  |-  ( ph  ->  { 0 }  C_  RR* )
132 xmetcl 21126 . . . . . 6  |-  ( ( M  e.  ( *Met `  X )  /\  A  e.  X  /\  C  e.  X
)  ->  ( A M C )  e.  RR* )
13320, 27, 40, 132syl3anc 1230 . . . . 5  |-  ( ph  ->  ( A M C )  e.  RR* )
134 xmetcl 21126 . . . . . 6  |-  ( ( N  e.  ( *Met `  Y )  /\  B  e.  Y  /\  D  e.  Y
)  ->  ( B N D )  e.  RR* )
13521, 28, 41, 134syl3anc 1230 . . . . 5  |-  ( ph  ->  ( B N D )  e.  RR* )
136 prssi 4128 . . . . 5  |-  ( ( ( A M C )  e.  RR*  /\  ( B N D )  e. 
RR* )  ->  { ( A M C ) ,  ( B N D ) }  C_  RR* )
137133, 135, 136syl2anc 659 . . . 4  |-  ( ph  ->  { ( A M C ) ,  ( B N D ) }  C_  RR* )
138 xrltso 11400 . . . . . 6  |-  <  Or  RR*
139 supsn 7964 . . . . . 6  |-  ( (  <  Or  RR*  /\  0  e.  RR* )  ->  sup ( { 0 } ,  RR* ,  <  )  =  0 )
140138, 129, 139mp2an 670 . . . . 5  |-  sup ( { 0 } ,  RR* ,  <  )  =  0
141 supxrcl 11559 . . . . . . 7  |-  ( { ( A M C ) ,  ( B N D ) } 
C_  RR*  ->  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  )  e.  RR* )
142137, 141syl 17 . . . . . 6  |-  ( ph  ->  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  )  e. 
RR* )
143 xmetge0 21139 . . . . . . 7  |-  ( ( M  e.  ( *Met `  X )  /\  A  e.  X  /\  C  e.  X
)  ->  0  <_  ( A M C ) )
14420, 27, 40, 143syl3anc 1230 . . . . . 6  |-  ( ph  ->  0  <_  ( A M C ) )
145 ovex 6306 . . . . . . . 8  |-  ( A M C )  e. 
_V
146145prid1 4080 . . . . . . 7  |-  ( A M C )  e. 
{ ( A M C ) ,  ( B N D ) }
147 supxrub 11569 . . . . . . 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 660 . . . . . 6  |-  ( ph  ->  ( A M C )  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
149130, 133, 142, 144, 148xrletrd 11418 . . . . 5  |-  ( ph  ->  0  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
150140, 149syl5eqbr 4428 . . . 4  |-  ( ph  ->  sup ( { 0 } ,  RR* ,  <  )  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
151 supxrun 11560 . . . 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 1230 . . 3  |-  ( ph  ->  sup ( ( { 0 }  u.  {
( A M C ) ,  ( B N D ) } ) ,  RR* ,  <  )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  )
)
15370, 128, 1523eqtrd 2447 . 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 2451 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 366    = wceq 1405    e. wcel 1842   E.wrex 2755   _Vcvv 3059    u. cun 3412    C_ wss 3414   (/)c0 3738   {csn 3972   {cpr 3974   <.cop 3978   class class class wbr 4395    |-> cmpt 4453    Or wor 4743    X. cxp 4821   `'ccnv 4822   ran crn 4824    |` cres 4825   Oncon0 5410    Fn wfn 5564   -->wf 5565   -1-1-onto->wf1o 5568   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   1oc1o 7160   2oc2o 7161   supcsup 7934    +c ccda 8579   0cc0 9522   RR*cxr 9657    < clt 9658    <_ cle 9659   Basecbs 14841  Scalarcsca 14912   distcds 14918   X_scprds 15060    X.s cxps 15120   *Metcxmt 18723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-icc 11589  df-fz 11727  df-fzo 11855  df-seq 12152  df-hash 12453  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-hom 14933  df-cco 14934  df-0g 15056  df-gsum 15057  df-prds 15062  df-xrs 15116  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-xmet 18732
This theorem is referenced by:  tmsxpsval  21333
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