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Theorem xpsdsval 17947
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 2285 . . . . 5  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )
7 eqid 2285 . . . . 5  |-  (Scalar `  R )  =  (Scalar `  R )
8 eqid 2285 . . . . 5  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  =  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } ) )
91, 2, 3, 4, 5, 6, 7, 8xpsval 13476 . . . 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 13477 . . . 4  |-  ( ph  ->  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) ) )
116xpsff1o2 13475 . . . . 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 5487 . . . . 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 11 . . . 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 5885 . . . . 5  |-  ( (Scalar `  R ) X_s `' ( { R }  +c  { S }
) )  e.  _V
1514a1i 10 . . . 4  |-  ( ph  ->  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) )  e.  _V )
16 eqid 2285 . . . 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 17945 . . . . 5  |-  ( ph  ->  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )  e.  ( * Met `  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) ) )
23 ssid 3199 . . . . 5  |-  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )  C_  ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) )
24 xmetres2 17927 . . . . 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 643 . . . 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 5863 . . . . . 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 13471 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( A ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) B )  =  `' ( { A }  +c  { B } ) )
3027, 28, 29syl2anc 642 . . . . . 6  |-  ( ph  ->  ( A ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) B )  =  `' ( { A }  +c  { B } ) )
3126, 30syl5eqr 2331 . . . . 5  |-  ( ph  ->  ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) `
 <. A ,  B >. )  =  `' ( { A }  +c  { B } ) )
32 opelxpi 4723 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
3327, 28, 32syl2anc 642 . . . . . 6  |-  ( ph  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
34 f1of 5474 . . . . . . . 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 8 . . . . . . 7  |-  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) : ( X  X.  Y ) --> ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )
3635ffvelrni 5666 . . . . . 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 15 . . . . 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 2360 . . . 4  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) ) )
39 df-ov 5863 . . . . . 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 13471 . . . . . . 7  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  ( C ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) D )  =  `' ( { C }  +c  { D } ) )
4340, 41, 42syl2anc 642 . . . . . 6  |-  ( ph  ->  ( C ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) D )  =  `' ( { C }  +c  { D } ) )
4439, 43syl5eqr 2331 . . . . 5  |-  ( ph  ->  ( ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) `
 <. C ,  D >. )  =  `' ( { C }  +c  { D } ) )
45 opelxpi 4723 . . . . . . 7  |-  ( ( C  e.  X  /\  D  e.  Y )  -> 
<. C ,  D >.  e.  ( X  X.  Y
) )
4640, 41, 45syl2anc 642 . . . . . 6  |-  ( ph  -> 
<. C ,  D >.  e.  ( X  X.  Y
) )
4735ffvelrni 5666 . . . . . 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 15 . . . . 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 2360 . . . 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 17940 . . 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 }
) ) )
51 ovres 5989 . . . 4  |-  ( ( `' ( { A }  +c  { B }
)  e.  ran  (
x  e.  X , 
y  e.  Y  |->  `' ( { x }  +c  { y } ) )  /\  `' ( { C }  +c  { D } )  e. 
ran  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
y } ) ) )  ->  ( `' ( { 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 } ) ) )
5238, 49, 51syl2anc 642 . . 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 } ) ) )
5350, 52eqtrd 2317 . 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 } ) ) )
54 f1ocnvfv 5796 . . . . 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 >. ) )
5511, 33, 54sylancr 644 . . . 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 >. ) )
5631, 55mpd 14 . . 3  |-  ( ph  ->  ( `' ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `
 `' ( { A }  +c  { B } ) )  = 
<. A ,  B >. )
57 f1ocnvfv 5796 . . . . 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 >. ) )
5811, 46, 57sylancr 644 . . . 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 >. ) )
5944, 58mpd 14 . . 3  |-  ( ph  ->  ( `' ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } ) ) `
 `' ( { C }  +c  { D } ) )  = 
<. C ,  D >. )
6056, 59oveq12d 5878 . 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 >. ) )
61 eqid 2285 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) )  =  ( Base `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )
62 fvex 5541 . . . . 5  |-  (Scalar `  R )  e.  _V
6362a1i 10 . . . 4  |-  ( ph  ->  (Scalar `  R )  e.  _V )
64 2on 6489 . . . . 5  |-  2o  e.  On
6564a1i 10 . . . 4  |-  ( ph  ->  2o  e.  On )
66 xpscfn 13463 . . . . 5  |-  ( ( R  e.  V  /\  S  e.  W )  ->  `' ( { R }  +c  { S }
)  Fn  2o )
674, 5, 66syl2anc 642 . . . 4  |-  ( ph  ->  `' ( { R }  +c  { S }
)  Fn  2o )
6838, 10eleqtrd 2361 . . . 4  |-  ( ph  ->  `' ( { A }  +c  { B }
)  e.  ( Base `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) ) )
6949, 10eleqtrd 2361 . . . 4  |-  ( ph  ->  `' ( { C }  +c  { D }
)  e.  ( Base `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) ) )
70 eqid 2285 . . . 4  |-  ( dist `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) )  =  ( dist `  (
(Scalar `  R ) X_s `' ( { R }  +c  { S } ) ) )
718, 61, 63, 65, 67, 68, 69, 70prdsdsval 13379 . . 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* ,  <  ) )
72 df2o3 6494 . . . . . . . . . . 11  |-  2o  =  { (/) ,  1o }
7372rexeqi 2743 . . . . . . . . . 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 )
) )
74 0ex 4152 . . . . . . . . . . 11  |-  (/)  e.  _V
75 1on 6488 . . . . . . . . . . . 12  |-  1o  e.  On
7675elexi 2799 . . . . . . . . . . 11  |-  1o  e.  _V
77 fveq2 5527 . . . . . . . . . . . . . 14  |-  ( k  =  (/)  ->  ( `' ( { R }  +c  { S } ) `
 k )  =  ( `' ( { R }  +c  { S } ) `  (/) ) )
7877fveq2d 5531 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k
) )  =  (
dist `  ( `' ( { R }  +c  { S } ) `  (/) ) ) )
79 fveq2 5527 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( `' ( { A }  +c  { B } ) `
 k )  =  ( `' ( { A }  +c  { B } ) `  (/) ) )
80 fveq2 5527 . . . . . . . . . . . . 13  |-  ( k  =  (/)  ->  ( `' ( { C }  +c  { D } ) `
 k )  =  ( `' ( { C }  +c  { D } ) `  (/) ) )
8178, 79, 80oveq123d 5881 . . . . . . . . . . . 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 } ) `
 (/) ) ) )
8281eqeq2d 2296 . . . . . . . . . . 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 } ) `
 (/) ) ) ) )
83 fveq2 5527 . . . . . . . . . . . . . 14  |-  ( k  =  1o  ->  ( `' ( { R }  +c  { S }
) `  k )  =  ( `' ( { R }  +c  { S } ) `  1o ) )
8483fveq2d 5531 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( dist `  ( `' ( { R }  +c  { S } ) `  k ) )  =  ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) )
85 fveq2 5527 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( `' ( { A }  +c  { B }
) `  k )  =  ( `' ( { A }  +c  { B } ) `  1o ) )
86 fveq2 5527 . . . . . . . . . . . . 13  |-  ( k  =  1o  ->  ( `' ( { C }  +c  { D }
) `  k )  =  ( `' ( { C }  +c  { D } ) `  1o ) )
8784, 85, 86oveq123d 5881 . . . . . . . . . . . 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 ) ) )
8887eqeq2d 2296 . . . . . . . . . . 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 ) ) ) )
8974, 76, 82, 88rexpr 3689 . . . . . . . . . 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 ) ) ) )
9073, 89bitri 240 . . . . . . . . 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 ) ) ) )
91 xpsc0 13464 . . . . . . . . . . . . . . 15  |-  ( R  e.  V  ->  ( `' ( { R }  +c  { S }
) `  (/) )  =  R )
924, 91syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  (/) )  =  R )
9392fveq2d 5531 . . . . . . . . . . . . 13  |-  ( ph  ->  ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) )  =  ( dist `  R
) )
94 xpsc0 13464 . . . . . . . . . . . . . 14  |-  ( A  e.  X  ->  ( `' ( { A }  +c  { B }
) `  (/) )  =  A )
9527, 94syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
96 xpsc0 13464 . . . . . . . . . . . . . 14  |-  ( C  e.  X  ->  ( `' ( { C }  +c  { D }
) `  (/) )  =  C )
9740, 96syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { C }  +c  { D } ) `  (/) )  =  C )
9893, 95, 97oveq123d 5881 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  (/) ) ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) ) ( `' ( { C }  +c  { D } ) `  (/) ) )  =  ( A (
dist `  R ) C ) )
9918oveqi 5873 . . . . . . . . . . . . 13  |-  ( A M C )  =  ( A ( (
dist `  R )  |`  ( X  X.  X
) ) C )
100 ovres 5989 . . . . . . . . . . . . . 14  |-  ( ( A  e.  X  /\  C  e.  X )  ->  ( A ( (
dist `  R )  |`  ( X  X.  X
) ) C )  =  ( A (
dist `  R ) C ) )
10127, 40, 100syl2anc 642 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A ( (
dist `  R )  |`  ( X  X.  X
) ) C )  =  ( A (
dist `  R ) C ) )
10299, 101syl5eq 2329 . . . . . . . . . . . 12  |-  ( ph  ->  ( A M C )  =  ( A ( dist `  R
) C ) )
10398, 102eqtr4d 2320 . . . . . . . . . . 11  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  (/) ) ( dist `  ( `' ( { R }  +c  { S }
) `  (/) ) ) ( `' ( { C }  +c  { D } ) `  (/) ) )  =  ( A M C ) )
104103eqeq2d 2296 . . . . . . . . . 10  |-  ( ph  ->  ( x  =  ( ( `' ( { A }  +c  { B } ) `  (/) ) (
dist `  ( `' ( { R }  +c  { S } ) `  (/) ) ) ( `' ( { C }  +c  { D } ) `
 (/) ) )  <->  x  =  ( A M C ) ) )
105 xpsc1 13465 . . . . . . . . . . . . . . 15  |-  ( S  e.  W  ->  ( `' ( { R }  +c  { S }
) `  1o )  =  S )
1065, 105syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( `' ( { R }  +c  { S } ) `  1o )  =  S )
107106fveq2d 5531 . . . . . . . . . . . . 13  |-  ( ph  ->  ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
)  =  ( dist `  S ) )
108 xpsc1 13465 . . . . . . . . . . . . . 14  |-  ( B  e.  Y  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  B )
10928, 108syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) `  1o )  =  B )
110 xpsc1 13465 . . . . . . . . . . . . . 14  |-  ( D  e.  Y  ->  ( `' ( { C }  +c  { D }
) `  1o )  =  D )
11141, 110syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( `' ( { C }  +c  { D } ) `  1o )  =  D )
112107, 109, 111oveq123d 5881 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) )  =  ( B ( dist `  S
) D ) )
11319oveqi 5873 . . . . . . . . . . . . 13  |-  ( B N D )  =  ( B ( (
dist `  S )  |`  ( Y  X.  Y
) ) D )
114 ovres 5989 . . . . . . . . . . . . . 14  |-  ( ( B  e.  Y  /\  D  e.  Y )  ->  ( B ( (
dist `  S )  |`  ( Y  X.  Y
) ) D )  =  ( B (
dist `  S ) D ) )
11528, 41, 114syl2anc 642 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B ( (
dist `  S )  |`  ( Y  X.  Y
) ) D )  =  ( B (
dist `  S ) D ) )
116113, 115syl5eq 2329 . . . . . . . . . . . 12  |-  ( ph  ->  ( B N D )  =  ( B ( dist `  S
) D ) )
117112, 116eqtr4d 2320 . . . . . . . . . . 11  |-  ( ph  ->  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) )  =  ( B N D ) )
118117eqeq2d 2296 . . . . . . . . . 10  |-  ( ph  ->  ( x  =  ( ( `' ( { A }  +c  { B } ) `  1o ) ( dist `  ( `' ( { R }  +c  { S }
) `  1o )
) ( `' ( { C }  +c  { D } ) `  1o ) )  <->  x  =  ( B N D ) ) )
119104, 118orbi12d 690 . . . . . . . . 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 ) ) ) )
12090, 119syl5bb 248 . . . . . . . 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 ) ) ) )
121 vex 2793 . . . . . . . . 9  |-  x  e. 
_V
122 eqid 2285 . . . . . . . . . 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 ) ) )
123122elrnmpt 4928 . . . . . . . . 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 )
) ) )
124121, 123ax-mp 8 . . . . . . . 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 ) ) )
125121elpr 3660 . . . . . . . 8  |-  ( x  e.  { ( A M C ) ,  ( B N D ) }  <->  ( x  =  ( A M C )  \/  x  =  ( B N D ) ) )
126120, 124, 1253bitr4g 279 . . . . . . 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 ) } ) )
127126eqrdv 2283 . . . . . 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 ) } )
128127uneq1d 3330 . . . . 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 } ) )
129 uncom 3321 . . . . 5  |-  ( { ( A M C ) ,  ( B N D ) }  u.  { 0 } )  =  ( { 0 }  u.  {
( A M C ) ,  ( B N D ) } )
130128, 129syl6eq 2333 . . . 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 ) } ) )
131130supeq1d 7201 . . 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* ,  <  )
)
132 0xr 8880 . . . . . 6  |-  0  e.  RR*
133132a1i 10 . . . . 5  |-  ( ph  ->  0  e.  RR* )
134133snssd 3762 . . . 4  |-  ( ph  ->  { 0 }  C_  RR* )
135 xmetcl 17898 . . . . . 6  |-  ( ( M  e.  ( * Met `  X )  /\  A  e.  X  /\  C  e.  X
)  ->  ( A M C )  e.  RR* )
13620, 27, 40, 135syl3anc 1182 . . . . 5  |-  ( ph  ->  ( A M C )  e.  RR* )
137 xmetcl 17898 . . . . . 6  |-  ( ( N  e.  ( * Met `  Y )  /\  B  e.  Y  /\  D  e.  Y
)  ->  ( B N D )  e.  RR* )
13821, 28, 41, 137syl3anc 1182 . . . . 5  |-  ( ph  ->  ( B N D )  e.  RR* )
139 prssi 3773 . . . . 5  |-  ( ( ( A M C )  e.  RR*  /\  ( B N D )  e. 
RR* )  ->  { ( A M C ) ,  ( B N D ) }  C_  RR* )
140136, 138, 139syl2anc 642 . . . 4  |-  ( ph  ->  { ( A M C ) ,  ( B N D ) }  C_  RR* )
141 xrltso 10477 . . . . . 6  |-  <  Or  RR*
142 supsn 7222 . . . . . 6  |-  ( (  <  Or  RR*  /\  0  e.  RR* )  ->  sup ( { 0 } ,  RR* ,  <  )  =  0 )
143141, 132, 142mp2an 653 . . . . 5  |-  sup ( { 0 } ,  RR* ,  <  )  =  0
144 supxrcl 10635 . . . . . . 7  |-  ( { ( A M C ) ,  ( B N D ) } 
C_  RR*  ->  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  )  e.  RR* )
145140, 144syl 15 . . . . . 6  |-  ( ph  ->  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  )  e. 
RR* )
146 xmetge0 17911 . . . . . . 7  |-  ( ( M  e.  ( * Met `  X )  /\  A  e.  X  /\  C  e.  X
)  ->  0  <_  ( A M C ) )
14720, 27, 40, 146syl3anc 1182 . . . . . 6  |-  ( ph  ->  0  <_  ( A M C ) )
148 ovex 5885 . . . . . . . 8  |-  ( A M C )  e. 
_V
149148prid1 3736 . . . . . . 7  |-  ( A M C )  e. 
{ ( A M C ) ,  ( B N D ) }
150 supxrub 10645 . . . . . . 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* ,  <  ) )
151140, 149, 150sylancl 643 . . . . . 6  |-  ( ph  ->  ( A M C )  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
152133, 136, 145, 147, 151xrletrd 10495 . . . . 5  |-  ( ph  ->  0  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
153143, 152syl5eqbr 4058 . . . 4  |-  ( ph  ->  sup ( { 0 } ,  RR* ,  <  )  <_  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
154 supxrun 10636 . . . 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* ,  <  ) )
155134, 140, 153, 154syl3anc 1182 . . 3  |-  ( ph  ->  sup ( ( { 0 }  u.  {
( A M C ) ,  ( B N D ) } ) ,  RR* ,  <  )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  )
)
15671, 131, 1553eqtrd 2321 . 2  |-  ( ph  ->  ( `' ( { A }  +c  { B } ) ( dist `  ( (Scalar `  R
) X_s `' ( { R }  +c  { S }
) ) ) `' ( { C }  +c  { D } ) )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
15753, 60, 1563eqtr3d 2325 1  |-  ( ph  ->  ( <. A ,  B >. P <. C ,  D >. )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    = wceq 1625    e. wcel 1686   E.wrex 2546   _Vcvv 2790    u. cun 3152    C_ wss 3154   (/)c0 3457   {csn 3642   {cpr 3643   <.cop 3645   class class class wbr 4025    e. cmpt 4079    Or wor 4315   Oncon0 4394    X. cxp 4689   `'ccnv 4690   ran crn 4692    |` cres 4693    Fn wfn 5252   -->wf 5253   -1-1-onto->wf1o 5256   ` cfv 5257  (class class class)co 5860    e. cmpt2 5862   1oc1o 6474   2oc2o 6475   supcsup 7195    +c ccda 7795   0cc0 8739   RR*cxr 8868    < clt 8869    <_ cle 8870   Basecbs 13150  Scalarcsca 13213   distcds 13219   X_scprds 13348    X.s cxps 13411   * Metcxmt 16371
This theorem is referenced by:  tmsxpsval  18086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-icc 10665  df-fz 10785  df-fzo 10873  df-seq 11049  df-hash 11340  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375
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