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Theorem xmetxp 12571
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 2115 . . . . 5  |-  ( MetOpen `  M )  =  (
MetOpen `  M )
32mopnm 12512 . . . 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 2115 . . . . 5  |-  ( MetOpen `  N )  =  (
MetOpen `  N )
76mopnm 12512 . . . 4  |-  ( N  e.  ( *Met `  Y )  ->  Y  e.  ( MetOpen `  N )
)
85, 7syl 14 . . 3  |-  ( ph  ->  Y  e.  ( MetOpen `  N ) )
9 xpexg 4621 . . 3  |-  ( ( X  e.  ( MetOpen `  M )  /\  Y  e.  ( MetOpen `  N )
)  ->  ( X  X.  Y )  e.  _V )
104, 8, 9syl2anc 406 . 2  |-  ( ph  ->  ( X  X.  Y
)  e.  _V )
111adantr 272 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  ->  M  e.  ( *Met `  X ) )
12 xp1st 6029 . . . . . . 7  |-  ( r  e.  ( X  X.  Y )  ->  ( 1st `  r )  e.  X )
1312ad2antrl 479 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( 1st `  r
)  e.  X )
14 xp1st 6029 . . . . . . 7  |-  ( s  e.  ( X  X.  Y )  ->  ( 1st `  s )  e.  X )
1514ad2antll 480 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( 1st `  s
)  e.  X )
16 xmetcl 12416 . . . . . 6  |-  ( ( M  e.  ( *Met `  X )  /\  ( 1st `  r
)  e.  X  /\  ( 1st `  s )  e.  X )  -> 
( ( 1st `  r
) M ( 1st `  s ) )  e. 
RR* )
1711, 13, 15, 16syl3anc 1199 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( 1st `  r
) M ( 1st `  s ) )  e. 
RR* )
185adantr 272 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  ->  N  e.  ( *Met `  Y ) )
19 xp2nd 6030 . . . . . . 7  |-  ( r  e.  ( X  X.  Y )  ->  ( 2nd `  r )  e.  Y )
2019ad2antrl 479 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( 2nd `  r
)  e.  Y )
21 xp2nd 6030 . . . . . . 7  |-  ( s  e.  ( X  X.  Y )  ->  ( 2nd `  s )  e.  Y )
2221ad2antll 480 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( 2nd `  s
)  e.  Y )
23 xmetcl 12416 . . . . . 6  |-  ( ( N  e.  ( *Met `  Y )  /\  ( 2nd `  r
)  e.  Y  /\  ( 2nd `  s )  e.  Y )  -> 
( ( 2nd `  r
) N ( 2nd `  s ) )  e. 
RR* )
2418, 20, 22, 23syl3anc 1199 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( 2nd `  r
) N ( 2nd `  s ) )  e. 
RR* )
25 xrmaxcl 10961 . . . . 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 406 . . . 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 2489 . . 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 5387 . . . . . . . . 9  |-  ( u  =  r  ->  ( 1st `  u )  =  ( 1st `  r
) )
3029oveq1d 5755 . . . . . . . 8  |-  ( u  =  r  ->  (
( 1st `  u
) M ( 1st `  v ) )  =  ( ( 1st `  r
) M ( 1st `  v ) ) )
31 fveq2 5387 . . . . . . . . 9  |-  ( u  =  r  ->  ( 2nd `  u )  =  ( 2nd `  r
) )
3231oveq1d 5755 . . . . . . . 8  |-  ( u  =  r  ->  (
( 2nd `  u
) N ( 2nd `  v ) )  =  ( ( 2nd `  r
) N ( 2nd `  v ) ) )
3330, 32preq12d 3576 . . . . . . 7  |-  ( u  =  r  ->  { ( ( 1st `  u
) M ( 1st `  v ) ) ,  ( ( 2nd `  u
) N ( 2nd `  v ) ) }  =  { ( ( 1st `  r ) M ( 1st `  v
) ) ,  ( ( 2nd `  r
) N ( 2nd `  v ) ) } )
3433supeq1d 6840 . . . . . 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 5387 . . . . . . . . 9  |-  ( v  =  s  ->  ( 1st `  v )  =  ( 1st `  s
) )
3635oveq2d 5756 . . . . . . . 8  |-  ( v  =  s  ->  (
( 1st `  r
) M ( 1st `  v ) )  =  ( ( 1st `  r
) M ( 1st `  s ) ) )
37 fveq2 5387 . . . . . . . . 9  |-  ( v  =  s  ->  ( 2nd `  v )  =  ( 2nd `  s
) )
3837oveq2d 5756 . . . . . . . 8  |-  ( v  =  s  ->  (
( 2nd `  r
) N ( 2nd `  v ) )  =  ( ( 2nd `  r
) N ( 2nd `  s ) ) )
3936, 38preq12d 3576 . . . . . . 7  |-  ( v  =  s  ->  { ( ( 1st `  r
) M ( 1st `  v ) ) ,  ( ( 2nd `  r
) N ( 2nd `  v ) ) }  =  { ( ( 1st `  r ) M ( 1st `  s
) ) ,  ( ( 2nd `  r
) N ( 2nd `  s ) ) } )
4039supeq1d 6840 . . . . . 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 5817 . . . . 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 2136 . . . 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 6065 . . 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 121 . 2  |-  ( ph  ->  P : ( ( X  X.  Y )  X.  ( X  X.  Y ) ) --> RR* )
45 simprl 503 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
r  e.  ( X  X.  Y ) )
46 simprr 504 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
s  e.  ( X  X.  Y ) )
4734, 40, 28ovmpog 5871 . . . . . . . 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 1199 . . . . . . 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 2192 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( r P s )  e.  RR* )
50 0xr 7776 . . . . . . 7  |-  0  e.  RR*
5150a1i 9 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
0  e.  RR* )
52 xrletri3 9528 . . . . . 6  |-  ( ( ( r P s )  e.  RR*  /\  0  e.  RR* )  ->  (
( r P s )  =  0  <->  (
( r P s )  <_  0  /\  0  <_  ( r P s ) ) ) )
5349, 51, 52syl2anc 406 . . . . 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 12429 . . . . . . . . 9  |-  ( ( M  e.  ( *Met `  X )  /\  ( 1st `  r
)  e.  X  /\  ( 1st `  s )  e.  X )  -> 
0  <_  ( ( 1st `  r ) M ( 1st `  s
) ) )
5511, 13, 15, 54syl3anc 1199 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
0  <_  ( ( 1st `  r ) M ( 1st `  s
) ) )
56 xrmax1sup 10962 . . . . . . . . 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 406 . . . . . . . 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 9535 . . . . . . 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 3924 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
0  <_  ( r P s ) )
6059biantrud 300 . . . . 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 190 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( r P s )  =  0  <-> 
( r P s )  <_  0 ) )
6248breq1d 3907 . . . 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 10968 . . . . 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 1199 . . . 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 213 . . 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 300 . . . . 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 9528 . . . . . 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 406 . . . . 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 190 . . . 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 12429 . . . . . . 7  |-  ( ( N  e.  ( *Met `  Y )  /\  ( 2nd `  r
)  e.  Y  /\  ( 2nd `  s )  e.  Y )  -> 
0  <_  ( ( 2nd `  r ) N ( 2nd `  s
) ) )
7118, 20, 22, 70syl3anc 1199 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
0  <_  ( ( 2nd `  r ) N ( 2nd `  s
) ) )
7271biantrud 300 . . . . 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 9528 . . . . . 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 406 . . . . 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 190 . . . 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 462 . . 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 12423 . . . . . 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 1199 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( 1st `  r ) M ( 1st `  s ) )  =  0  <->  ( 1st `  r )  =  ( 1st `  s
) ) )
79 xmeteq0 12423 . . . . . 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 1199 . . . . 5  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( 2nd `  r ) N ( 2nd `  s ) )  =  0  <->  ( 2nd `  r )  =  ( 2nd `  s
) ) )
8178, 80anbi12d 462 . . . 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 6040 . . . . 5  |-  ( ( r  e.  ( X  X.  Y )  /\  s  e.  ( X  X.  Y ) )  -> 
( ( ( 1st `  r )  =  ( 1st `  s )  /\  ( 2nd `  r
)  =  ( 2nd `  s ) )  <->  r  =  s ) )
8382adantl 273 . . . 4  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( ( 1st `  r )  =  ( 1st `  s )  /\  ( 2nd `  r
)  =  ( 2nd `  s ) )  <->  r  =  s ) )
8481, 83bitrd 187 . . 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 213 . 2  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
) ) )  -> 
( ( r P s )  =  0  <-> 
r  =  s ) )
86483adantr3 1125 . . 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 1125 . . . . 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 272 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  ->  M  e.  ( *Met `  X ) )
89 simpr3 972 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
t  e.  ( X  X.  Y ) )
90 xp1st 6029 . . . . . . . 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 970 . . . . . . . 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 12416 . . . . . . 7  |-  ( ( M  e.  ( *Met `  X )  /\  ( 1st `  t
)  e.  X  /\  ( 1st `  r )  e.  X )  -> 
( ( 1st `  t
) M ( 1st `  r ) )  e. 
RR* )
9588, 91, 93, 94syl3anc 1199 . . . . . 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 1125 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( 1st `  s
)  e.  X )
97 xmetcl 12416 . . . . . . 7  |-  ( ( M  e.  ( *Met `  X )  /\  ( 1st `  t
)  e.  X  /\  ( 1st `  s )  e.  X )  -> 
( ( 1st `  t
) M ( 1st `  s ) )  e. 
RR* )
9888, 91, 96, 97syl3anc 1199 . . . . . 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 9607 . . . . 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 272 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  ->  N  e.  ( *Met `  Y ) )
101 xp2nd 6030 . . . . . . . . . . 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 12416 . . . . . . . . . 10  |-  ( ( N  e.  ( *Met `  Y )  /\  ( 2nd `  t
)  e.  Y  /\  ( 2nd `  r )  e.  Y )  -> 
( ( 2nd `  t
) N ( 2nd `  r ) )  e. 
RR* )
105100, 102, 103, 104syl3anc 1199 . . . . . . . . 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 10961 . . . . . . . . 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 406 . . . . . . . 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 5387 . . . . . . . . . . . 12  |-  ( u  =  t  ->  ( 1st `  u )  =  ( 1st `  t
) )
109 fveq2 5387 . . . . . . . . . . . 12  |-  ( v  =  r  ->  ( 1st `  v )  =  ( 1st `  r
) )
110108, 109oveqan12d 5759 . . . . . . . . . . 11  |-  ( ( u  =  t  /\  v  =  r )  ->  ( ( 1st `  u
) M ( 1st `  v ) )  =  ( ( 1st `  t
) M ( 1st `  r ) ) )
111 fveq2 5387 . . . . . . . . . . . 12  |-  ( u  =  t  ->  ( 2nd `  u )  =  ( 2nd `  t
) )
112 fveq2 5387 . . . . . . . . . . . 12  |-  ( v  =  r  ->  ( 2nd `  v )  =  ( 2nd `  r
) )
113111, 112oveqan12d 5759 . . . . . . . . . . 11  |-  ( ( u  =  t  /\  v  =  r )  ->  ( ( 2nd `  u
) N ( 2nd `  v ) )  =  ( ( 2nd `  t
) N ( 2nd `  r ) ) )
114110, 113preq12d 3576 . . . . . . . . . 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 6840 . . . . . . . . 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 5866 . . . . . . . 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 1199 . . . . . . 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 2192 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( t P r )  e.  RR* )
119 simpr2 971 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
s  e.  ( X  X.  Y ) )
120223adantr3 1125 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( 2nd `  s
)  e.  Y )
121 xmetcl 12416 . . . . . . . . . 10  |-  ( ( N  e.  ( *Met `  Y )  /\  ( 2nd `  t
)  e.  Y  /\  ( 2nd `  s )  e.  Y )  -> 
( ( 2nd `  t
) N ( 2nd `  s ) )  e. 
RR* )
122100, 102, 120, 121syl3anc 1199 . . . . . . . . 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 10961 . . . . . . . . 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 406 . . . . . . . 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 5759 . . . . . . . . . . 11  |-  ( ( u  =  t  /\  v  =  s )  ->  ( ( 1st `  u
) M ( 1st `  v ) )  =  ( ( 1st `  t
) M ( 1st `  s ) ) )
126111, 37oveqan12d 5759 . . . . . . . . . . 11  |-  ( ( u  =  t  /\  v  =  s )  ->  ( ( 2nd `  u
) N ( 2nd `  v ) )  =  ( ( 2nd `  t
) N ( 2nd `  s ) ) )
127125, 126preq12d 3576 . . . . . . . . . 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 6840 . . . . . . . . 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 5866 . . . . . . . 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 1199 . . . . . . 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 2192 . . . . . 6  |-  ( (
ph  /\  ( r  e.  ( X  X.  Y
)  /\  s  e.  ( X  X.  Y
)  /\  t  e.  ( X  X.  Y
) ) )  -> 
( t P s )  e.  RR* )
132118, 131xaddcld 9607 . . . . 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 12425 . . . . . 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 1201 . . . . 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 10962 . . . . . . . 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 406 . . . . . . 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 3924 . . . . . 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 10962 . . . . . . . 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 406 . . . . . . 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 3924 . . . . . 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 9602 . . . . . . 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 1200 . . . . . 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 427 . . . . 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 9535 . . . 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 1125 . . . . 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 9607 . . . . 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 12425 . . . . . 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 1201 . . . . 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 10963 . . . . . . . 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 406 . . . . . . 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 3924 . . . . . 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 10963 . . . . . . . 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 406 . . . . . . 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 3924 . . . . . 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 9602 . . . . . . 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 1200 . . . . . 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 427 . . . . 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 9535 . . . 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 10968 . . . . 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 1199 . . . 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 911 . . 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 3918 . 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 12411 1  |-  ( ph  ->  P  e.  ( *Met `  ( X  X.  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   A.wral 2391   _Vcvv 2658   {cpr 3496   class class class wbr 3897    X. cxp 4505   -->wf 5087   ` cfv 5091  (class class class)co 5740    e. cmpo 5742   1stc1st 6002   2ndc2nd 6003   supcsup 6835   0cc0 7584   RR*cxr 7763    < clt 7764    <_ cle 7765   +ecxad 9497   *Metcxmet 12044   MetOpencmopn 12049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-map 6510  df-sup 6837  df-inf 6838  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-q 9361  df-rp 9391  df-xneg 9499  df-xadd 9500  df-seqfrec 10159  df-exp 10233  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711  df-topgen 12036  df-psmet 12051  df-xmet 12052  df-bl 12054  df-mopn 12055  df-top 12060  df-topon 12073  df-bases 12105
This theorem is referenced by:  xmetxpbl  12572  xmettxlem  12573  xmettx  12574  txmetcnp  12582
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