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Theorem xmetxp 14010
Description: The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.)
Hypotheses
Ref Expression
xmetxp.p  |-  P  =  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  u
) M ( 1st `  v ) ) ,  ( ( 2nd `  u
) N ( 2nd `  v ) ) } ,  RR* ,  <  )
)
xmetxp.1  |-  ( ph  ->  M  e.  ( *Met `  X ) )
xmetxp.2  |-  ( ph  ->  N  e.  ( *Met `  Y ) )
Assertion
Ref Expression
xmetxp  |-  ( ph  ->  P  e.  ( *Met `  ( X  X.  Y ) ) )
Distinct variable groups:    u, M, v   
u, N, v    u, X, v    u, Y, v
Allowed substitution hints:    ph( v, u)    P( v, u)

Proof of Theorem xmetxp
Dummy variables  r  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xmetxp.1 . . . 4  |-  ( ph  ->  M  e.  ( *Met `  X ) )
2 eqid 2177 . . . . 5  |-  ( MetOpen `  M )  =  (
MetOpen `  M )
32mopnm 13951 . . . 4  |-  ( M  e.  ( *Met `  X )  ->  X  e.  ( MetOpen `  M )
)
41, 3syl 14 . . 3  |-  ( ph  ->  X  e.  ( MetOpen `  M ) )
5 xmetxp.2 . . . 4  |-  ( ph  ->  N  e.  ( *Met `  Y ) )
6 eqid 2177 . . . . 5  |-  ( MetOpen `  N )  =  (
MetOpen `  N )
76mopnm 13951 . . . 4  |-  ( N  e.  ( *Met `  Y )  ->  Y  e.  ( MetOpen `  N )
)
85, 7syl 14 . . 3  |-  ( ph  ->  Y  e.  ( MetOpen `  N ) )
9 xpexg 4741 . . 3  |-  ( ( X  e.  ( MetOpen `  M )  /\  Y  e.  ( MetOpen `  N )
)  ->  ( X  X.  Y )  e.  _V )
104, 8, 9syl2anc 411 . 2  |-  ( ph  ->  ( X  X.  Y
)  e.  _V )
111adantr 276 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  ->  M  e.  ( *Met `  X ) )
12 xp1st 6166 . . . . . . 7  |-  ( r  e.  ( X  X.  Y )  ->  ( 1st `  r )  e.  X )
1312ad2antrl 490 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( 1st `  r
)  e.  X )
14 xp1st 6166 . . . . . . 7  |-  ( s  e.  ( X  X.  Y )  ->  ( 1st `  s )  e.  X )
1514ad2antll 491 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( 1st `  s
)  e.  X )
16 xmetcl 13855 . . . . . 6  |-  ( ( M  e.  ( *Met `  X )  /\  ( 1st `  r
)  e.  X  /\  ( 1st `  s )  e.  X )  -> 
( ( 1st `  r
) M ( 1st `  s ) )  e. 
RR* )
1711, 13, 15, 16syl3anc 1238 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( 1st `  r
) M ( 1st `  s ) )  e. 
RR* )
185adantr 276 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  ->  N  e.  ( *Met `  Y ) )
19 xp2nd 6167 . . . . . . 7  |-  ( r  e.  ( X  X.  Y )  ->  ( 2nd `  r )  e.  Y )
2019ad2antrl 490 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( 2nd `  r
)  e.  Y )
21 xp2nd 6167 . . . . . . 7  |-  ( s  e.  ( X  X.  Y )  ->  ( 2nd `  s )  e.  Y )
2221ad2antll 491 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( 2nd `  s
)  e.  Y )
23 xmetcl 13855 . . . . . 6  |-  ( ( N  e.  ( *Met `  Y )  /\  ( 2nd `  r
)  e.  Y  /\  ( 2nd `  s )  e.  Y )  -> 
( ( 2nd `  r
) N ( 2nd `  s ) )  e. 
RR* )
2418, 20, 22, 23syl3anc 1238 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( 2nd `  r
) N ( 2nd `  s ) )  e. 
RR* )
25 xrmaxcl 11260 . . . . 5  |-  ( ( ( ( 1st `  r
) M ( 1st `  s ) )  e. 
RR*  /\  ( ( 2nd `  r ) N ( 2nd `  s
) )  e.  RR* )  ->  sup ( { ( ( 1st `  r
) M ( 1st `  s ) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )  e.  RR* )
2617, 24, 25syl2anc 411 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  ->  sup ( { ( ( 1st `  r ) M ( 1st `  s
) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )  e.  RR* )
2726ralrimivva 2559 . . 3  |-  ( ph  ->  A. r  e.  ( X  X.  Y ) A. s  e.  ( X  X.  Y ) sup ( { ( ( 1st `  r
) M ( 1st `  s ) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )  e.  RR* )
28 xmetxp.p . . . . 5  |-  P  =  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  u
) M ( 1st `  v ) ) ,  ( ( 2nd `  u
) N ( 2nd `  v ) ) } ,  RR* ,  <  )
)
29 fveq2 5516 . . . . . . . . 9  |-  ( u  =  r  ->  ( 1st `  u )  =  ( 1st `  r
) )
3029oveq1d 5890 . . . . . . . 8  |-  ( u  =  r  ->  (
( 1st `  u
) M ( 1st `  v ) )  =  ( ( 1st `  r
) M ( 1st `  v ) ) )
31 fveq2 5516 . . . . . . . . 9  |-  ( u  =  r  ->  ( 2nd `  u )  =  ( 2nd `  r
) )
3231oveq1d 5890 . . . . . . . 8  |-  ( u  =  r  ->  (
( 2nd `  u
) N ( 2nd `  v ) )  =  ( ( 2nd `  r
) N ( 2nd `  v ) ) )
3330, 32preq12d 3678 . . . . . . 7  |-  ( u  =  r  ->  { ( ( 1st `  u
) M ( 1st `  v ) ) ,  ( ( 2nd `  u
) N ( 2nd `  v ) ) }  =  { ( ( 1st `  r ) M ( 1st `  v
) ) ,  ( ( 2nd `  r
) N ( 2nd `  v ) ) } )
3433supeq1d 6986 . . . . . 6  |-  ( u  =  r  ->  sup ( { ( ( 1st `  u ) M ( 1st `  v ) ) ,  ( ( 2nd `  u ) N ( 2nd `  v
) ) } ,  RR* ,  <  )  =  sup ( { ( ( 1st `  r
) M ( 1st `  v ) ) ,  ( ( 2nd `  r
) N ( 2nd `  v ) ) } ,  RR* ,  <  )
)
35 fveq2 5516 . . . . . . . . 9  |-  ( v  =  s  ->  ( 1st `  v )  =  ( 1st `  s
) )
3635oveq2d 5891 . . . . . . . 8  |-  ( v  =  s  ->  (
( 1st `  r
) M ( 1st `  v ) )  =  ( ( 1st `  r
) M ( 1st `  s ) ) )
37 fveq2 5516 . . . . . . . . 9  |-  ( v  =  s  ->  ( 2nd `  v )  =  ( 2nd `  s
) )
3837oveq2d 5891 . . . . . . . 8  |-  ( v  =  s  ->  (
( 2nd `  r
) N ( 2nd `  v ) )  =  ( ( 2nd `  r
) N ( 2nd `  s ) ) )
3936, 38preq12d 3678 . . . . . . 7  |-  ( v  =  s  ->  { ( ( 1st `  r
) M ( 1st `  v ) ) ,  ( ( 2nd `  r
) N ( 2nd `  v ) ) }  =  { ( ( 1st `  r ) M ( 1st `  s
) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } )
4039supeq1d 6986 . . . . . 6  |-  ( v  =  s  ->  sup ( { ( ( 1st `  r ) M ( 1st `  v ) ) ,  ( ( 2nd `  r ) N ( 2nd `  v
) ) } ,  RR* ,  <  )  =  sup ( { ( ( 1st `  r
) M ( 1st `  s ) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )
)
4134, 40cbvmpov 5955 . . . . 5  |-  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  sup ( { ( ( 1st `  u ) M ( 1st `  v ) ) ,  ( ( 2nd `  u ) N ( 2nd `  v
) ) } ,  RR* ,  <  ) )  =  ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) M ( 1st `  s ) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )
)
4228, 41eqtri 2198 . . . 4  |-  P  =  ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  r
) M ( 1st `  s ) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )
)
4342fmpo 6202 . . 3  |-  ( A. r  e.  ( X  X.  Y ) A. s  e.  ( X  X.  Y
) sup ( { ( ( 1st `  r
) M ( 1st `  s ) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )  e.  RR*  <->  P : ( ( X  X.  Y )  X.  ( X  X.  Y ) ) --> RR* )
4427, 43sylib 122 . 2  |-  ( ph  ->  P : ( ( X  X.  Y )  X.  ( X  X.  Y ) ) --> RR* )
45 simprl 529 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
r  e.  ( X  X.  Y ) )
46 simprr 531 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
s  e.  ( X  X.  Y ) )
4734, 40, 28ovmpog 6009 . . . . . . . 8  |-  ( ( r  e.  ( X  X.  Y )  /\  s  e.  ( X  X.  Y )  /\  sup ( { ( ( 1st `  r ) M ( 1st `  s ) ) ,  ( ( 2nd `  r ) N ( 2nd `  s
) ) } ,  RR* ,  <  )  e. 
RR* )  ->  (
r P s )  =  sup ( { ( ( 1st `  r
) M ( 1st `  s ) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )
)
4845, 46, 26, 47syl3anc 1238 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( r P s )  =  sup ( { ( ( 1st `  r ) M ( 1st `  s ) ) ,  ( ( 2nd `  r ) N ( 2nd `  s
) ) } ,  RR* ,  <  ) )
4948, 26eqeltrd 2254 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( r P s )  e.  RR* )
50 0xr 8004 . . . . . . 7  |-  0  e.  RR*
5150a1i 9 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
0  e.  RR* )
52 xrletri3 9804 . . . . . 6  |-  ( ( ( r P s )  e.  RR*  /\  0  e.  RR* )  ->  (
( r P s )  =  0  <->  (
( r P s )  <_  0  /\  0  <_  ( r P s ) ) ) )
5349, 51, 52syl2anc 411 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( r P s )  =  0  <-> 
( ( r P s )  <_  0  /\  0  <_  ( r P s ) ) ) )
54 xmetge0 13868 . . . . . . . . 9  |-  ( ( M  e.  ( *Met `  X )  /\  ( 1st `  r
)  e.  X  /\  ( 1st `  s )  e.  X )  -> 
0  <_  ( ( 1st `  r ) M ( 1st `  s
) ) )
5511, 13, 15, 54syl3anc 1238 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
0  <_  ( ( 1st `  r ) M ( 1st `  s
) ) )
56 xrmax1sup 11261 . . . . . . . . 9  |-  ( ( ( ( 1st `  r
) M ( 1st `  s ) )  e. 
RR*  /\  ( ( 2nd `  r ) N ( 2nd `  s
) )  e.  RR* )  ->  ( ( 1st `  r ) M ( 1st `  s ) )  <_  sup ( { ( ( 1st `  r ) M ( 1st `  s ) ) ,  ( ( 2nd `  r ) N ( 2nd `  s
) ) } ,  RR* ,  <  ) )
5717, 24, 56syl2anc 411 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( 1st `  r
) M ( 1st `  s ) )  <_  sup ( { ( ( 1st `  r ) M ( 1st `  s
) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )
)
5851, 17, 26, 55, 57xrletrd 9812 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
0  <_  sup ( { ( ( 1st `  r ) M ( 1st `  s ) ) ,  ( ( 2nd `  r ) N ( 2nd `  s
) ) } ,  RR* ,  <  ) )
5958, 48breqtrrd 4032 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
0  <_  ( r P s ) )
6059biantrud 304 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( r P s )  <_  0  <->  ( ( r P s )  <_  0  /\  0  <_  ( r P s ) ) ) )
6153, 60bitr4d 191 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( r P s )  =  0  <-> 
( r P s )  <_  0 ) )
6248breq1d 4014 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( r P s )  <_  0  <->  sup ( { ( ( 1st `  r ) M ( 1st `  s
) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )  <_  0 ) )
63 xrmaxlesup 11267 . . . . 5  |-  ( ( ( ( 1st `  r
) M ( 1st `  s ) )  e. 
RR*  /\  ( ( 2nd `  r ) N ( 2nd `  s
) )  e.  RR*  /\  0  e.  RR* )  ->  ( sup ( { ( ( 1st `  r
) M ( 1st `  s ) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )  <_  0  <->  ( ( ( 1st `  r ) M ( 1st `  s
) )  <_  0  /\  ( ( 2nd `  r
) N ( 2nd `  s ) )  <_ 
0 ) ) )
6417, 24, 51, 63syl3anc 1238 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( sup ( { ( ( 1st `  r
) M ( 1st `  s ) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )  <_  0  <->  ( ( ( 1st `  r ) M ( 1st `  s
) )  <_  0  /\  ( ( 2nd `  r
) N ( 2nd `  s ) )  <_ 
0 ) ) )
6561, 62, 643bitrd 214 . . 3  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( r P s )  =  0  <-> 
( ( ( 1st `  r ) M ( 1st `  s ) )  <_  0  /\  ( ( 2nd `  r
) N ( 2nd `  s ) )  <_ 
0 ) ) )
6655biantrud 304 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( 1st `  r ) M ( 1st `  s ) )  <_  0  <->  ( (
( 1st `  r
) M ( 1st `  s ) )  <_ 
0  /\  0  <_  ( ( 1st `  r
) M ( 1st `  s ) ) ) ) )
67 xrletri3 9804 . . . . . 6  |-  ( ( ( ( 1st `  r
) M ( 1st `  s ) )  e. 
RR*  /\  0  e.  RR* )  ->  ( (
( 1st `  r
) M ( 1st `  s ) )  =  0  <->  ( ( ( 1st `  r ) M ( 1st `  s
) )  <_  0  /\  0  <_  ( ( 1st `  r ) M ( 1st `  s
) ) ) ) )
6817, 51, 67syl2anc 411 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( 1st `  r ) M ( 1st `  s ) )  =  0  <->  (
( ( 1st `  r
) M ( 1st `  s ) )  <_ 
0  /\  0  <_  ( ( 1st `  r
) M ( 1st `  s ) ) ) ) )
6966, 68bitr4d 191 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( 1st `  r ) M ( 1st `  s ) )  <_  0  <->  ( ( 1st `  r ) M ( 1st `  s
) )  =  0 ) )
70 xmetge0 13868 . . . . . . 7  |-  ( ( N  e.  ( *Met `  Y )  /\  ( 2nd `  r
)  e.  Y  /\  ( 2nd `  s )  e.  Y )  -> 
0  <_  ( ( 2nd `  r ) N ( 2nd `  s
) ) )
7118, 20, 22, 70syl3anc 1238 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
0  <_  ( ( 2nd `  r ) N ( 2nd `  s
) ) )
7271biantrud 304 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( 2nd `  r ) N ( 2nd `  s ) )  <_  0  <->  ( (
( 2nd `  r
) N ( 2nd `  s ) )  <_ 
0  /\  0  <_  ( ( 2nd `  r
) N ( 2nd `  s ) ) ) ) )
73 xrletri3 9804 . . . . . 6  |-  ( ( ( ( 2nd `  r
) N ( 2nd `  s ) )  e. 
RR*  /\  0  e.  RR* )  ->  ( (
( 2nd `  r
) N ( 2nd `  s ) )  =  0  <->  ( ( ( 2nd `  r ) N ( 2nd `  s
) )  <_  0  /\  0  <_  ( ( 2nd `  r ) N ( 2nd `  s
) ) ) ) )
7424, 51, 73syl2anc 411 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( 2nd `  r ) N ( 2nd `  s ) )  =  0  <->  (
( ( 2nd `  r
) N ( 2nd `  s ) )  <_ 
0  /\  0  <_  ( ( 2nd `  r
) N ( 2nd `  s ) ) ) ) )
7572, 74bitr4d 191 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( 2nd `  r ) N ( 2nd `  s ) )  <_  0  <->  ( ( 2nd `  r ) N ( 2nd `  s
) )  =  0 ) )
7669, 75anbi12d 473 . . 3  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( ( 1st `  r ) M ( 1st `  s
) )  <_  0  /\  ( ( 2nd `  r
) N ( 2nd `  s ) )  <_ 
0 )  <->  ( (
( 1st `  r
) M ( 1st `  s ) )  =  0  /\  ( ( 2nd `  r ) N ( 2nd `  s
) )  =  0 ) ) )
77 xmeteq0 13862 . . . . . 6  |-  ( ( M  e.  ( *Met `  X )  /\  ( 1st `  r
)  e.  X  /\  ( 1st `  s )  e.  X )  -> 
( ( ( 1st `  r ) M ( 1st `  s ) )  =  0  <->  ( 1st `  r )  =  ( 1st `  s
) ) )
7811, 13, 15, 77syl3anc 1238 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( 1st `  r ) M ( 1st `  s ) )  =  0  <->  ( 1st `  r )  =  ( 1st `  s
) ) )
79 xmeteq0 13862 . . . . . 6  |-  ( ( N  e.  ( *Met `  Y )  /\  ( 2nd `  r
)  e.  Y  /\  ( 2nd `  s )  e.  Y )  -> 
( ( ( 2nd `  r ) N ( 2nd `  s ) )  =  0  <->  ( 2nd `  r )  =  ( 2nd `  s
) ) )
8018, 20, 22, 79syl3anc 1238 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( 2nd `  r ) N ( 2nd `  s ) )  =  0  <->  ( 2nd `  r )  =  ( 2nd `  s
) ) )
8178, 80anbi12d 473 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( ( 1st `  r ) M ( 1st `  s
) )  =  0  /\  ( ( 2nd `  r ) N ( 2nd `  s ) )  =  0 )  <-> 
( ( 1st `  r
)  =  ( 1st `  s )  /\  ( 2nd `  r )  =  ( 2nd `  s
) ) ) )
82 xpopth 6177 . . . . 5  |-  ( ( r  e.  ( X  X.  Y )  /\  s  e.  ( X  X.  Y ) )  -> 
( ( ( 1st `  r )  =  ( 1st `  s )  /\  ( 2nd `  r
)  =  ( 2nd `  s ) )  <->  r  =  s ) )
8382adantl 277 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( 1st `  r )  =  ( 1st `  s )  /\  ( 2nd `  r
)  =  ( 2nd `  s ) )  <->  r  =  s ) )
8481, 83bitrd 188 . . 3  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( ( 1st `  r ) M ( 1st `  s
) )  =  0  /\  ( ( 2nd `  r ) N ( 2nd `  s ) )  =  0 )  <-> 
r  =  s ) )
8565, 76, 843bitrd 214 . 2  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( r P s )  =  0  <-> 
r  =  s ) )
86483adantr3 1158 . . 3  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( r P s )  =  sup ( { ( ( 1st `  r ) M ( 1st `  s ) ) ,  ( ( 2nd `  r ) N ( 2nd `  s
) ) } ,  RR* ,  <  ) )
87173adantr3 1158 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 1st `  r
) M ( 1st `  s ) )  e. 
RR* )
881adantr 276 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  ->  M  e.  ( *Met `  X ) )
89 simpr3 1005 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
t  e.  ( X  X.  Y ) )
90 xp1st 6166 . . . . . . . 8  |-  ( t  e.  ( X  X.  Y )  ->  ( 1st `  t )  e.  X )
9189, 90syl 14 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( 1st `  t
)  e.  X )
92 simpr1 1003 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
r  e.  ( X  X.  Y ) )
9392, 12syl 14 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( 1st `  r
)  e.  X )
94 xmetcl 13855 . . . . . . 7  |-  ( ( M  e.  ( *Met `  X )  /\  ( 1st `  t
)  e.  X  /\  ( 1st `  r )  e.  X )  -> 
( ( 1st `  t
) M ( 1st `  r ) )  e. 
RR* )
9588, 91, 93, 94syl3anc 1238 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 1st `  t
) M ( 1st `  r ) )  e. 
RR* )
96153adantr3 1158 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( 1st `  s
)  e.  X )
97 xmetcl 13855 . . . . . . 7  |-  ( ( M  e.  ( *Met `  X )  /\  ( 1st `  t
)  e.  X  /\  ( 1st `  s )  e.  X )  -> 
( ( 1st `  t
) M ( 1st `  s ) )  e. 
RR* )
9888, 91, 96, 97syl3anc 1238 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 1st `  t
) M ( 1st `  s ) )  e. 
RR* )
9995, 98xaddcld 9884 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( ( 1st `  t ) M ( 1st `  r ) ) +e ( ( 1st `  t
) M ( 1st `  s ) ) )  e.  RR* )
1005adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  ->  N  e.  ( *Met `  Y ) )
101 xp2nd 6167 . . . . . . . . . . 11  |-  ( t  e.  ( X  X.  Y )  ->  ( 2nd `  t )  e.  Y )
10289, 101syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( 2nd `  t
)  e.  Y )
10392, 19syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( 2nd `  r
)  e.  Y )
104 xmetcl 13855 . . . . . . . . . 10  |-  ( ( N  e.  ( *Met `  Y )  /\  ( 2nd `  t
)  e.  Y  /\  ( 2nd `  r )  e.  Y )  -> 
( ( 2nd `  t
) N ( 2nd `  r ) )  e. 
RR* )
105100, 102, 103, 104syl3anc 1238 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 2nd `  t
) N ( 2nd `  r ) )  e. 
RR* )
106 xrmaxcl 11260 . . . . . . . . 9  |-  ( ( ( ( 1st `  t
) M ( 1st `  r ) )  e. 
RR*  /\  ( ( 2nd `  t ) N ( 2nd `  r
) )  e.  RR* )  ->  sup ( { ( ( 1st `  t
) M ( 1st `  r ) ) ,  ( ( 2nd `  t
) N ( 2nd `  r ) ) } ,  RR* ,  <  )  e.  RR* )
10795, 105, 106syl2anc 411 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  ->  sup ( { ( ( 1st `  t ) M ( 1st `  r
) ) ,  ( ( 2nd `  t
) N ( 2nd `  r ) ) } ,  RR* ,  <  )  e.  RR* )
108 fveq2 5516 . . . . . . . . . . . 12  |-  ( u  =  t  ->  ( 1st `  u )  =  ( 1st `  t
) )
109 fveq2 5516 . . . . . . . . . . . 12  |-  ( v  =  r  ->  ( 1st `  v )  =  ( 1st `  r
) )
110108, 109oveqan12d 5894 . . . . . . . . . . 11  |-  ( ( u  =  t  /\  v  =  r )  ->  ( ( 1st `  u
) M ( 1st `  v ) )  =  ( ( 1st `  t
) M ( 1st `  r ) ) )
111 fveq2 5516 . . . . . . . . . . . 12  |-  ( u  =  t  ->  ( 2nd `  u )  =  ( 2nd `  t
) )
112 fveq2 5516 . . . . . . . . . . . 12  |-  ( v  =  r  ->  ( 2nd `  v )  =  ( 2nd `  r
) )
113111, 112oveqan12d 5894 . . . . . . . . . . 11  |-  ( ( u  =  t  /\  v  =  r )  ->  ( ( 2nd `  u
) N ( 2nd `  v ) )  =  ( ( 2nd `  t
) N ( 2nd `  r ) ) )
114110, 113preq12d 3678 . . . . . . . . . 10  |-  ( ( u  =  t  /\  v  =  r )  ->  { ( ( 1st `  u ) M ( 1st `  v ) ) ,  ( ( 2nd `  u ) N ( 2nd `  v
) ) }  =  { ( ( 1st `  t ) M ( 1st `  r ) ) ,  ( ( 2nd `  t ) N ( 2nd `  r
) ) } )
115114supeq1d 6986 . . . . . . . . 9  |-  ( ( u  =  t  /\  v  =  r )  ->  sup ( { ( ( 1st `  u
) M ( 1st `  v ) ) ,  ( ( 2nd `  u
) N ( 2nd `  v ) ) } ,  RR* ,  <  )  =  sup ( { ( ( 1st `  t
) M ( 1st `  r ) ) ,  ( ( 2nd `  t
) N ( 2nd `  r ) ) } ,  RR* ,  <  )
)
116115, 28ovmpoga 6004 . . . . . . . 8  |-  ( ( t  e.  ( X  X.  Y )  /\  r  e.  ( X  X.  Y )  /\  sup ( { ( ( 1st `  t ) M ( 1st `  r ) ) ,  ( ( 2nd `  t ) N ( 2nd `  r
) ) } ,  RR* ,  <  )  e. 
RR* )  ->  (
t P r )  =  sup ( { ( ( 1st `  t
) M ( 1st `  r ) ) ,  ( ( 2nd `  t
) N ( 2nd `  r ) ) } ,  RR* ,  <  )
)
11789, 92, 107, 116syl3anc 1238 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( t P r )  =  sup ( { ( ( 1st `  t ) M ( 1st `  r ) ) ,  ( ( 2nd `  t ) N ( 2nd `  r
) ) } ,  RR* ,  <  ) )
118117, 107eqeltrd 2254 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( t P r )  e.  RR* )
119 simpr2 1004 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
s  e.  ( X  X.  Y ) )
120223adantr3 1158 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( 2nd `  s
)  e.  Y )
121 xmetcl 13855 . . . . . . . . . 10  |-  ( ( N  e.  ( *Met `  Y )  /\  ( 2nd `  t
)  e.  Y  /\  ( 2nd `  s )  e.  Y )  -> 
( ( 2nd `  t
) N ( 2nd `  s ) )  e. 
RR* )
122100, 102, 120, 121syl3anc 1238 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 2nd `  t
) N ( 2nd `  s ) )  e. 
RR* )
123 xrmaxcl 11260 . . . . . . . . 9  |-  ( ( ( ( 1st `  t
) M ( 1st `  s ) )  e. 
RR*  /\  ( ( 2nd `  t ) N ( 2nd `  s
) )  e.  RR* )  ->  sup ( { ( ( 1st `  t
) M ( 1st `  s ) ) ,  ( ( 2nd `  t
) N ( 2nd `  s ) ) } ,  RR* ,  <  )  e.  RR* )
12498, 122, 123syl2anc 411 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  ->  sup ( { ( ( 1st `  t ) M ( 1st `  s
) ) ,  ( ( 2nd `  t
) N ( 2nd `  s ) ) } ,  RR* ,  <  )  e.  RR* )
125108, 35oveqan12d 5894 . . . . . . . . . . 11  |-  ( ( u  =  t  /\  v  =  s )  ->  ( ( 1st `  u
) M ( 1st `  v ) )  =  ( ( 1st `  t
) M ( 1st `  s ) ) )
126111, 37oveqan12d 5894 . . . . . . . . . . 11  |-  ( ( u  =  t  /\  v  =  s )  ->  ( ( 2nd `  u
) N ( 2nd `  v ) )  =  ( ( 2nd `  t
) N ( 2nd `  s ) ) )
127125, 126preq12d 3678 . . . . . . . . . 10  |-  ( ( u  =  t  /\  v  =  s )  ->  { ( ( 1st `  u ) M ( 1st `  v ) ) ,  ( ( 2nd `  u ) N ( 2nd `  v
) ) }  =  { ( ( 1st `  t ) M ( 1st `  s ) ) ,  ( ( 2nd `  t ) N ( 2nd `  s
) ) } )
128127supeq1d 6986 . . . . . . . . 9  |-  ( ( u  =  t  /\  v  =  s )  ->  sup ( { ( ( 1st `  u
) M ( 1st `  v ) ) ,  ( ( 2nd `  u
) N ( 2nd `  v ) ) } ,  RR* ,  <  )  =  sup ( { ( ( 1st `  t
) M ( 1st `  s ) ) ,  ( ( 2nd `  t
) N ( 2nd `  s ) ) } ,  RR* ,  <  )
)
129128, 28ovmpoga 6004 . . . . . . . 8  |-  ( ( t  e.  ( X  X.  Y )  /\  s  e.  ( X  X.  Y )  /\  sup ( { ( ( 1st `  t ) M ( 1st `  s ) ) ,  ( ( 2nd `  t ) N ( 2nd `  s
) ) } ,  RR* ,  <  )  e. 
RR* )  ->  (
t P s )  =  sup ( { ( ( 1st `  t
) M ( 1st `  s ) ) ,  ( ( 2nd `  t
) N ( 2nd `  s ) ) } ,  RR* ,  <  )
)
13089, 119, 124, 129syl3anc 1238 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( t P s )  =  sup ( { ( ( 1st `  t ) M ( 1st `  s ) ) ,  ( ( 2nd `  t ) N ( 2nd `  s
) ) } ,  RR* ,  <  ) )
131130, 124eqeltrd 2254 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( t P s )  e.  RR* )
132118, 131xaddcld 9884 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( t P r ) +e
( t P s ) )  e.  RR* )
133 xmettri2 13864 . . . . . 6  |-  ( ( M  e.  ( *Met `  X )  /\  ( ( 1st `  t )  e.  X  /\  ( 1st `  r
)  e.  X  /\  ( 1st `  s )  e.  X ) )  ->  ( ( 1st `  r ) M ( 1st `  s ) )  <_  ( (
( 1st `  t
) M ( 1st `  r ) ) +e ( ( 1st `  t ) M ( 1st `  s ) ) ) )
13488, 91, 93, 96, 133syl13anc 1240 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 1st `  r
) M ( 1st `  s ) )  <_ 
( ( ( 1st `  t ) M ( 1st `  r ) ) +e ( ( 1st `  t
) M ( 1st `  s ) ) ) )
135 xrmax1sup 11261 . . . . . . . 8  |-  ( ( ( ( 1st `  t
) M ( 1st `  r ) )  e. 
RR*  /\  ( ( 2nd `  t ) N ( 2nd `  r
) )  e.  RR* )  ->  ( ( 1st `  t ) M ( 1st `  r ) )  <_  sup ( { ( ( 1st `  t ) M ( 1st `  r ) ) ,  ( ( 2nd `  t ) N ( 2nd `  r
) ) } ,  RR* ,  <  ) )
13695, 105, 135syl2anc 411 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 1st `  t
) M ( 1st `  r ) )  <_  sup ( { ( ( 1st `  t ) M ( 1st `  r
) ) ,  ( ( 2nd `  t
) N ( 2nd `  r ) ) } ,  RR* ,  <  )
)
137136, 117breqtrrd 4032 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 1st `  t
) M ( 1st `  r ) )  <_ 
( t P r ) )
138 xrmax1sup 11261 . . . . . . . 8  |-  ( ( ( ( 1st `  t
) M ( 1st `  s ) )  e. 
RR*  /\  ( ( 2nd `  t ) N ( 2nd `  s
) )  e.  RR* )  ->  ( ( 1st `  t ) M ( 1st `  s ) )  <_  sup ( { ( ( 1st `  t ) M ( 1st `  s ) ) ,  ( ( 2nd `  t ) N ( 2nd `  s
) ) } ,  RR* ,  <  ) )
13998, 122, 138syl2anc 411 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 1st `  t
) M ( 1st `  s ) )  <_  sup ( { ( ( 1st `  t ) M ( 1st `  s
) ) ,  ( ( 2nd `  t
) N ( 2nd `  s ) ) } ,  RR* ,  <  )
)
140139, 130breqtrrd 4032 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 1st `  t
) M ( 1st `  s ) )  <_ 
( t P s ) )
141 xle2add 9879 . . . . . . 7  |-  ( ( ( ( ( 1st `  t ) M ( 1st `  r ) )  e.  RR*  /\  (
( 1st `  t
) M ( 1st `  s ) )  e. 
RR* )  /\  (
( t P r )  e.  RR*  /\  (
t P s )  e.  RR* ) )  -> 
( ( ( ( 1st `  t ) M ( 1st `  r
) )  <_  (
t P r )  /\  ( ( 1st `  t ) M ( 1st `  s ) )  <_  ( t P s ) )  ->  ( ( ( 1st `  t ) M ( 1st `  r
) ) +e
( ( 1st `  t
) M ( 1st `  s ) ) )  <_  ( ( t P r ) +e ( t P s ) ) ) )
14295, 98, 118, 131, 141syl22anc 1239 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( ( ( 1st `  t ) M ( 1st `  r
) )  <_  (
t P r )  /\  ( ( 1st `  t ) M ( 1st `  s ) )  <_  ( t P s ) )  ->  ( ( ( 1st `  t ) M ( 1st `  r
) ) +e
( ( 1st `  t
) M ( 1st `  s ) ) )  <_  ( ( t P r ) +e ( t P s ) ) ) )
143137, 140, 142mp2and 433 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( ( 1st `  t ) M ( 1st `  r ) ) +e ( ( 1st `  t
) M ( 1st `  s ) ) )  <_  ( ( t P r ) +e ( t P s ) ) )
14487, 99, 132, 134, 143xrletrd 9812 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 1st `  r
) M ( 1st `  s ) )  <_ 
( ( t P r ) +e
( t P s ) ) )
145243adantr3 1158 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 2nd `  r
) N ( 2nd `  s ) )  e. 
RR* )
146105, 122xaddcld 9884 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( ( 2nd `  t ) N ( 2nd `  r ) ) +e ( ( 2nd `  t
) N ( 2nd `  s ) ) )  e.  RR* )
147 xmettri2 13864 . . . . . 6  |-  ( ( N  e.  ( *Met `  Y )  /\  ( ( 2nd `  t )  e.  Y  /\  ( 2nd `  r
)  e.  Y  /\  ( 2nd `  s )  e.  Y ) )  ->  ( ( 2nd `  r ) N ( 2nd `  s ) )  <_  ( (
( 2nd `  t
) N ( 2nd `  r ) ) +e ( ( 2nd `  t ) N ( 2nd `  s ) ) ) )
148100, 102, 103, 120, 147syl13anc 1240 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 2nd `  r
) N ( 2nd `  s ) )  <_ 
( ( ( 2nd `  t ) N ( 2nd `  r ) ) +e ( ( 2nd `  t
) N ( 2nd `  s ) ) ) )
149 xrmax2sup 11262 . . . . . . . 8  |-  ( ( ( ( 1st `  t
) M ( 1st `  r ) )  e. 
RR*  /\  ( ( 2nd `  t ) N ( 2nd `  r
) )  e.  RR* )  ->  ( ( 2nd `  t ) N ( 2nd `  r ) )  <_  sup ( { ( ( 1st `  t ) M ( 1st `  r ) ) ,  ( ( 2nd `  t ) N ( 2nd `  r
) ) } ,  RR* ,  <  ) )
15095, 105, 149syl2anc 411 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 2nd `  t
) N ( 2nd `  r ) )  <_  sup ( { ( ( 1st `  t ) M ( 1st `  r
) ) ,  ( ( 2nd `  t
) N ( 2nd `  r ) ) } ,  RR* ,  <  )
)
151150, 117breqtrrd 4032 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 2nd `  t
) N ( 2nd `  r ) )  <_ 
( t P r ) )
152 xrmax2sup 11262 . . . . . . . 8  |-  ( ( ( ( 1st `  t
) M ( 1st `  s ) )  e. 
RR*  /\  ( ( 2nd `  t ) N ( 2nd `  s
) )  e.  RR* )  ->  ( ( 2nd `  t ) N ( 2nd `  s ) )  <_  sup ( { ( ( 1st `  t ) M ( 1st `  s ) ) ,  ( ( 2nd `  t ) N ( 2nd `  s
) ) } ,  RR* ,  <  ) )
15398, 122, 152syl2anc 411 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 2nd `  t
) N ( 2nd `  s ) )  <_  sup ( { ( ( 1st `  t ) M ( 1st `  s
) ) ,  ( ( 2nd `  t
) N ( 2nd `  s ) ) } ,  RR* ,  <  )
)
154153, 130breqtrrd 4032 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 2nd `  t
) N ( 2nd `  s ) )  <_ 
( t P s ) )
155 xle2add 9879 . . . . . . 7  |-  ( ( ( ( ( 2nd `  t ) N ( 2nd `  r ) )  e.  RR*  /\  (
( 2nd `  t
) N ( 2nd `  s ) )  e. 
RR* )  /\  (
( t P r )  e.  RR*  /\  (
t P s )  e.  RR* ) )  -> 
( ( ( ( 2nd `  t ) N ( 2nd `  r
) )  <_  (
t P r )  /\  ( ( 2nd `  t ) N ( 2nd `  s ) )  <_  ( t P s ) )  ->  ( ( ( 2nd `  t ) N ( 2nd `  r
) ) +e
( ( 2nd `  t
) N ( 2nd `  s ) ) )  <_  ( ( t P r ) +e ( t P s ) ) ) )
156105, 122, 118, 131, 155syl22anc 1239 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( ( ( 2nd `  t ) N ( 2nd `  r
) )  <_  (
t P r )  /\  ( ( 2nd `  t ) N ( 2nd `  s ) )  <_  ( t P s ) )  ->  ( ( ( 2nd `  t ) N ( 2nd `  r
) ) +e
( ( 2nd `  t
) N ( 2nd `  s ) ) )  <_  ( ( t P r ) +e ( t P s ) ) ) )
157151, 154, 156mp2and 433 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( ( 2nd `  t ) N ( 2nd `  r ) ) +e ( ( 2nd `  t
) N ( 2nd `  s ) ) )  <_  ( ( t P r ) +e ( t P s ) ) )
158145, 146, 132, 148, 157xrletrd 9812 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( ( 2nd `  r
) N ( 2nd `  s ) )  <_ 
( ( t P r ) +e
( t P s ) ) )
159 xrmaxlesup 11267 . . . . 5  |-  ( ( ( ( 1st `  r
) M ( 1st `  s ) )  e. 
RR*  /\  ( ( 2nd `  r ) N ( 2nd `  s
) )  e.  RR*  /\  ( ( t P r ) +e
( t P s ) )  e.  RR* )  ->  ( sup ( { ( ( 1st `  r ) M ( 1st `  s ) ) ,  ( ( 2nd `  r ) N ( 2nd `  s
) ) } ,  RR* ,  <  )  <_ 
( ( t P r ) +e
( t P s ) )  <->  ( (
( 1st `  r
) M ( 1st `  s ) )  <_ 
( ( t P r ) +e
( t P s ) )  /\  (
( 2nd `  r
) N ( 2nd `  s ) )  <_ 
( ( t P r ) +e
( t P s ) ) ) ) )
16087, 145, 132, 159syl3anc 1238 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( sup ( { ( ( 1st `  r
) M ( 1st `  s ) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )  <_  ( ( t P r ) +e
( t P s ) )  <->  ( (
( 1st `  r
) M ( 1st `  s ) )  <_ 
( ( t P r ) +e
( t P s ) )  /\  (
( 2nd `  r
) N ( 2nd `  s ) )  <_ 
( ( t P r ) +e
( t P s ) ) ) ) )
161144, 158, 160mpbir2and 944 . . 3  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  ->  sup ( { ( ( 1st `  r ) M ( 1st `  s
) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } ,  RR* ,  <  )  <_  ( ( t P r ) +e
( t P s ) ) )
16286, 161eqbrtrd 4026 . 2  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( r P s )  <_  ( (
t P r ) +e ( t P s ) ) )
16310, 44, 85, 162isxmetd 13850 1  |-  ( ph  ->  P  e.  ( *Met `  ( X  X.  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   _Vcvv 2738   {cpr 3594   class class class wbr 4004    X. cxp 4625   -->wf 5213   ` cfv 5217  (class class class)co 5875    e. cmpo 5877   1stc1st 6139   2ndc2nd 6140   supcsup 6981   0cc0 7811   RR*cxr 7991    < clt 7992    <_ cle 7993   +ecxad 9770   *Metcxmet 13443   MetOpencmopn 13448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-map 6650  df-sup 6983  df-inf 6984  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-xneg 9772  df-xadd 9773  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-topgen 12709  df-psmet 13450  df-xmet 13451  df-bl 13453  df-mopn 13454  df-top 13501  df-topon 13514  df-bases 13546
This theorem is referenced by:  xmetxpbl  14011  xmettxlem  14012  xmettx  14013  txmetcnp  14021
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