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Theorem xpsdsval 20612
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 2460 . . . . 5  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )
7 eqid 2460 . . . . 5  |-  (Scalar `  R )  =  (Scalar `  R )
8 eqid 2460 . . . . 5  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } ) )
91, 2, 3, 4, 5, 6, 7, 8xpsval 14816 . . . 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 14817 . . . 4  |-  ( ph  ->  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) ) )
116xpsff1o2 14815 . . . . 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 5819 . . . . 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 6300 . . . . 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 2460 . . . 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 20610 . . . . 5  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( *Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
23 ssid 3516 . . . . 5  |-  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )  C_  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )
24 xmetres2 20592 . . . . 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 6278 . . . . . 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 14811 . . . . . . 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 2515 . . . . 5  |-  ( ph  ->  ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) `
 <. A ,  B >. )  =  `' ( { A }  +c  { B } ) )
32 opelxpi 5023 . . . . . . 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 5807 . . . . . . . 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 6011 . . . . . 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 2549 . . . 4  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )
39 df-ov 6278 . . . . . 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 14811 . . . . . . 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 2515 . . . . 5  |-  ( ph  ->  ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) `
 <. C ,  D >. )  =  `' ( { C }  +c  { D } ) )
45 opelxpi 5023 . . . . . . 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 6011 . . . . . 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 2549 . . . 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 20605 . . 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 6418 . . 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 2501 . 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 6163 . . . . 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 6163 . . . . 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 6293 . 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 2460 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )
61 fvex 5867 . . . . 5  |-  (Scalar `  R )  e.  _V
6261a1i 11 . . . 4  |-  ( ph  ->  (Scalar `  R )  e.  _V )
63 2on 7128 . . . . 5  |-  2o  e.  On
6463a1i 11 . . . 4  |-  ( ph  ->  2o  e.  On )
65 xpscfn 14803 . . . . 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 2550 . . . 4  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ( Base `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) ) )
6849, 10eleqtrd 2550 . . . 4  |-  ( ph  ->  `' ( { C }  +c  { D }
)  e.  ( Base `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) ) )
69 eqid 2460 . . . 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 14722 . . 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 7133 . . . . . . . . . . 11  |-  2o  =  { (/) ,  1o }
7271rexeqi 3056 . . . . . . . . . 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 4570 . . . . . . . . . . 11  |-  (/)  e.  _V
74 1on 7127 . . . . . . . . . . . 12  |-  1o  e.  On
7574elexi 3116 . . . . . . . . . . 11  |-  1o  e.  _V
76 fveq2 5857 . . . . . . . . . . . . . 14  |-  ( k  =  (/)  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  ( `' ( { R }  +c  { S } ) `  (/) ) )
7776fveq2d 5861 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  ( `' ( { R }  +c  { S } ) `  (/) ) ) )
78 fveq2 5857 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( `' ( { A }  +c  { B } ) `
 k )  =  ( `' ( { A }  +c  { B } ) `  (/) ) )
79 fveq2 5857 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( `' ( { C }  +c  { D } ) `
 k )  =  ( `' ( { C }  +c  { D } ) `  (/) ) )
8077, 78, 79oveq123d 6296 . . . . . . . . . . . 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 2474 . . . . . . . . . . 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 5857 . . . . . . . . . . . . . 14  |-  ( k  =  1o  ->  ( `' ( { R }  +c  { S }
) `  k )  =  ( `' ( { R }  +c  { S } ) `  1o ) )
8382fveq2d 5861 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  =  ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) )
84 fveq2 5857 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( `' ( { A }  +c  { B }
) `  k )  =  ( `' ( { A }  +c  { B } ) `  1o ) )
85 fveq2 5857 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( `' ( { C }  +c  { D }
) `  k )  =  ( `' ( { C }  +c  { D } ) `  1o ) )
8683, 84, 85oveq123d 6296 . . . . . . . . . . . 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 2474 . . . . . . . . . . 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 4074 . . . . . . . . . 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 14804 . . . . . . . . . . . . . . 15  |-  ( R  e.  V  ->  ( `' ( { R }  +c  { S }
) `  (/) )  =  R )
914, 90syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  (/) )  =  R )
9291fveq2d 5861 . . . . . . . . . . . . 13  |-  ( ph  ->  ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) )  =  ( dist `  R
) )
93 xpsc0 14804 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  ( `' ( { A }  +c  { B }
) `  (/) )  =  A )
9427, 93syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
95 xpsc0 14804 . . . . . . . . . . . . . 14  |-  ( C  e.  X  ->  ( `' ( { C }  +c  { D }
) `  (/) )  =  C )
9640, 95syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { C }  +c  { D } ) `  (/) )  =  C )
9792, 94, 96oveq123d 6296 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  (/) ) ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) ) ( `' ( { C }  +c  { D } ) `  (/) ) )  =  ( A (
dist `  R ) C ) )
9818oveqi 6288 . . . . . . . . . . . . 13  |-  ( A M C )  =  ( A ( (
dist `  R )  |`  ( X  X.  X
) ) C )
9927, 40ovresd 6418 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A ( (
dist `  R )  |`  ( X  X.  X
) ) C )  =  ( A (
dist `  R ) C ) )
10098, 99syl5eq 2513 . . . . . . . . . . . 12  |-  ( ph  ->  ( A M C )  =  ( A ( dist `  R
) C ) )
10197, 100eqtr4d 2504 . . . . . . . . . . 11  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  (/) ) ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) ) ( `' ( { C }  +c  { D } ) `  (/) ) )  =  ( A M C ) )
102101eqeq2d 2474 . . . . . . . . . 10  |-  ( ph  ->  ( x  =  ( ( `' ( { A }  +c  { B } ) `  (/) ) (
dist `  ( `' ( { R }  +c  { S } ) `  (/) ) ) ( `' ( { C }  +c  { D } ) `
 (/) ) )  <->  x  =  ( A M C ) ) )
103 xpsc1 14805 . . . . . . . . . . . . . . 15  |-  ( S  e.  W  ->  ( `' ( { R }  +c  { S }
) `  1o )  =  S )
1045, 103syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  1o )  =  S )
105104fveq2d 5861 . . . . . . . . . . . . 13  |-  ( ph  ->  ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
)  =  ( dist `  S ) )
106 xpsc1 14805 . . . . . . . . . . . . . 14  |-  ( B  e.  Y  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  B )
10728, 106syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) `  1o )  =  B )
108 xpsc1 14805 . . . . . . . . . . . . . 14  |-  ( D  e.  Y  ->  ( `' ( { C }  +c  { D }
) `  1o )  =  D )
10941, 108syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { C }  +c  { D } ) `  1o )  =  D )
110105, 107, 109oveq123d 6296 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) )  =  ( B ( dist `  S
) D ) )
11119oveqi 6288 . . . . . . . . . . . . 13  |-  ( B N D )  =  ( B ( (
dist `  S )  |`  ( Y  X.  Y
) ) D )
11228, 41ovresd 6418 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B ( (
dist `  S )  |`  ( Y  X.  Y
) ) D )  =  ( B (
dist `  S ) D ) )
113111, 112syl5eq 2513 . . . . . . . . . . . 12  |-  ( ph  ->  ( B N D )  =  ( B ( dist `  S
) D ) )
114110, 113eqtr4d 2504 . . . . . . . . . . 11  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) )  =  ( B N D ) )
115114eqeq2d 2474 . . . . . . . . . 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 3109 . . . . . . . . 9  |-  x  e. 
_V
119 eqid 2460 . . . . . . . . . 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 5240 . . . . . . . . 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 4038 . . . . . . . 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 2457 . . . . . 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 3650 . . . . 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 3641 . . . . 5  |-  ( { ( A M C ) ,  ( B N D ) }  u.  { 0 } )  =  ( { 0 }  u.  {
( A M C ) ,  ( B N D ) } )
127125, 126syl6eq 2517 . . . 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 7895 . . 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 9629 . . . . . 6  |-  0  e.  RR*
130129a1i 11 . . . . 5  |-  ( ph  ->  0  e.  RR* )
131130snssd 4165 . . . 4  |-  ( ph  ->  { 0 }  C_  RR* )
132 xmetcl 20562 . . . . . 6  |-  ( ( M  e.  ( *Met `  X )  /\  A  e.  X  /\  C  e.  X
)  ->  ( A M C )  e.  RR* )
13320, 27, 40, 132syl3anc 1223 . . . . 5  |-  ( ph  ->  ( A M C )  e.  RR* )
134 xmetcl 20562 . . . . . 6  |-  ( ( N  e.  ( *Met `  Y )  /\  B  e.  Y  /\  D  e.  Y
)  ->  ( B N D )  e.  RR* )
13521, 28, 41, 134syl3anc 1223 . . . . 5  |-  ( ph  ->  ( B N D )  e.  RR* )
136 prssi 4176 . . . . 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 11336 . . . . . 6  |-  <  Or  RR*
139 supsn 7919 . . . . . 6  |-  ( (  <  Or  RR*  /\  0  e.  RR* )  ->  sup ( { 0 } ,  RR* ,  <  )  =  0 )
140138, 129, 139mp2an 672 . . . . 5  |-  sup ( { 0 } ,  RR* ,  <  )  =  0
141 supxrcl 11495 . . . . . . 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 20575 . . . . . . 7  |-  ( ( M  e.  ( *Met `  X )  /\  A  e.  X  /\  C  e.  X
)  ->  0  <_  ( A M C ) )
14420, 27, 40, 143syl3anc 1223 . . . . . 6  |-  ( ph  ->  0  <_  ( A M C ) )
145 ovex 6300 . . . . . . . 8  |-  ( A M C )  e. 
_V
146145prid1 4128 . . . . . . 7  |-  ( A M C )  e. 
{ ( A M C ) ,  ( B N D ) }
147 supxrub 11505 . . . . . . 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 11354 . . . . 5  |-  ( ph  ->  0  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
150140, 149syl5eqbr 4473 . . . 4  |-  ( ph  ->  sup ( { 0 } ,  RR* ,  <  )  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
151 supxrun 11496 . . . 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 1223 . . 3  |-  ( ph  ->  sup ( ( { 0 }  u.  {
( A M C ) ,  ( B N D ) } ) ,  RR* ,  <  )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  )
)
15370, 128, 1523eqtrd 2505 . 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 2509 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 1374    e. wcel 1762   E.wrex 2808   _Vcvv 3106    u. cun 3467    C_ wss 3469   (/)c0 3778   {csn 4020   {cpr 4022   <.cop 4026   class class class wbr 4440    |-> cmpt 4498    Or wor 4792   Oncon0 4871    X. cxp 4990   `'ccnv 4991   ran crn 4993    |` cres 4994    Fn wfn 5574   -->wf 5575   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   1oc1o 7113   2oc2o 7114   supcsup 7889    +c ccda 8536   0cc0 9481   RR*cxr 9616    < clt 9617    <_ cle 9618   Basecbs 14479  Scalarcsca 14547   distcds 14553   X_scprds 14690    X.s cxps 14750   *Metcxmt 18167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-hom 14568  df-cco 14569  df-0g 14686  df-gsum 14687  df-prds 14692  df-xrs 14746  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-xmet 18176
This theorem is referenced by:  tmsxpsval  20769
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