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Theorem xmetxpbl 12572
Description: The maximum metric (Chebyshev distance) on the product of two sets, expressed in terms of balls centered on a point  C with radius  R. (Contributed by Jim Kingdon, 22-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 ) )
xmetxpbl.r  |-  ( ph  ->  R  e.  RR* )
xmetxpbl.c  |-  ( ph  ->  C  e.  ( X  X.  Y ) )
Assertion
Ref Expression
xmetxpbl  |-  ( ph  ->  ( C ( ball `  P ) R )  =  ( ( ( 1st `  C ) ( ball `  M
) R )  X.  ( ( 2nd `  C
) ( ball `  N
) R ) ) )
Distinct variable groups:    u, C, v   
u, M, v    u, N, v    u, X, v   
u, Y, v
Allowed substitution hints:    ph( v, u)    P( v, u)    R( v, u)

Proof of Theorem xmetxpbl
Dummy variables  n  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xmetxp.p . . . 4  |-  P  =  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  u
) M ( 1st `  v ) ) ,  ( ( 2nd `  u
) N ( 2nd `  v ) ) } ,  RR* ,  <  )
)
2 xmetxp.1 . . . 4  |-  ( ph  ->  M  e.  ( *Met `  X ) )
3 xmetxp.2 . . . 4  |-  ( ph  ->  N  e.  ( *Met `  Y ) )
41, 2, 3xmetxp 12571 . . 3  |-  ( ph  ->  P  e.  ( *Met `  ( X  X.  Y ) ) )
5 xmetxpbl.c . . 3  |-  ( ph  ->  C  e.  ( X  X.  Y ) )
6 xmetxpbl.r . . 3  |-  ( ph  ->  R  e.  RR* )
7 blval 12453 . . 3  |-  ( ( P  e.  ( *Met `  ( X  X.  Y ) )  /\  C  e.  ( X  X.  Y )  /\  R  e.  RR* )  ->  ( C (
ball `  P ) R )  =  {
t  e.  ( X  X.  Y )  |  ( C P t )  <  R }
)
84, 5, 6, 7syl3anc 1199 . 2  |-  ( ph  ->  ( C ( ball `  P ) R )  =  { t  e.  ( X  X.  Y
)  |  ( C P t )  < 
R } )
95adantr 272 . . . . . 6  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  C  e.  ( X  X.  Y
) )
10 simpr 109 . . . . . 6  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  t  e.  ( X  X.  Y
) )
112adantr 272 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  M  e.  ( *Met `  X ) )
12 xp1st 6029 . . . . . . . . 9  |-  ( C  e.  ( X  X.  Y )  ->  ( 1st `  C )  e.  X )
139, 12syl 14 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  ( 1st `  C )  e.  X )
14 xp1st 6029 . . . . . . . . 9  |-  ( t  e.  ( X  X.  Y )  ->  ( 1st `  t )  e.  X )
1514adantl 273 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  ( 1st `  t )  e.  X )
16 xmetcl 12416 . . . . . . . 8  |-  ( ( M  e.  ( *Met `  X )  /\  ( 1st `  C
)  e.  X  /\  ( 1st `  t )  e.  X )  -> 
( ( 1st `  C
) M ( 1st `  t ) )  e. 
RR* )
1711, 13, 15, 16syl3anc 1199 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  (
( 1st `  C
) M ( 1st `  t ) )  e. 
RR* )
183adantr 272 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  N  e.  ( *Met `  Y ) )
19 xp2nd 6030 . . . . . . . . 9  |-  ( C  e.  ( X  X.  Y )  ->  ( 2nd `  C )  e.  Y )
209, 19syl 14 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  ( 2nd `  C )  e.  Y )
21 xp2nd 6030 . . . . . . . . 9  |-  ( t  e.  ( X  X.  Y )  ->  ( 2nd `  t )  e.  Y )
2221adantl 273 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  ( 2nd `  t )  e.  Y )
23 xmetcl 12416 . . . . . . . 8  |-  ( ( N  e.  ( *Met `  Y )  /\  ( 2nd `  C
)  e.  Y  /\  ( 2nd `  t )  e.  Y )  -> 
( ( 2nd `  C
) N ( 2nd `  t ) )  e. 
RR* )
2418, 20, 22, 23syl3anc 1199 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  (
( 2nd `  C
) N ( 2nd `  t ) )  e. 
RR* )
25 xrmaxcl 10961 . . . . . . 7  |-  ( ( ( ( 1st `  C
) M ( 1st `  t ) )  e. 
RR*  /\  ( ( 2nd `  C ) N ( 2nd `  t
) )  e.  RR* )  ->  sup ( { ( ( 1st `  C
) M ( 1st `  t ) ) ,  ( ( 2nd `  C
) N ( 2nd `  t ) ) } ,  RR* ,  <  )  e.  RR* )
2617, 24, 25syl2anc 406 . . . . . 6  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  sup ( { ( ( 1st `  C ) M ( 1st `  t ) ) ,  ( ( 2nd `  C ) N ( 2nd `  t
) ) } ,  RR* ,  <  )  e. 
RR* )
27 fveq2 5387 . . . . . . . . . 10  |-  ( u  =  C  ->  ( 1st `  u )  =  ( 1st `  C
) )
28 fveq2 5387 . . . . . . . . . 10  |-  ( v  =  t  ->  ( 1st `  v )  =  ( 1st `  t
) )
2927, 28oveqan12d 5759 . . . . . . . . 9  |-  ( ( u  =  C  /\  v  =  t )  ->  ( ( 1st `  u
) M ( 1st `  v ) )  =  ( ( 1st `  C
) M ( 1st `  t ) ) )
30 fveq2 5387 . . . . . . . . . 10  |-  ( u  =  C  ->  ( 2nd `  u )  =  ( 2nd `  C
) )
31 fveq2 5387 . . . . . . . . . 10  |-  ( v  =  t  ->  ( 2nd `  v )  =  ( 2nd `  t
) )
3230, 31oveqan12d 5759 . . . . . . . . 9  |-  ( ( u  =  C  /\  v  =  t )  ->  ( ( 2nd `  u
) N ( 2nd `  v ) )  =  ( ( 2nd `  C
) N ( 2nd `  t ) ) )
3329, 32preq12d 3576 . . . . . . . 8  |-  ( ( u  =  C  /\  v  =  t )  ->  { ( ( 1st `  u ) M ( 1st `  v ) ) ,  ( ( 2nd `  u ) N ( 2nd `  v
) ) }  =  { ( ( 1st `  C ) M ( 1st `  t ) ) ,  ( ( 2nd `  C ) N ( 2nd `  t
) ) } )
3433supeq1d 6840 . . . . . . 7  |-  ( ( u  =  C  /\  v  =  t )  ->  sup ( { ( ( 1st `  u
) M ( 1st `  v ) ) ,  ( ( 2nd `  u
) N ( 2nd `  v ) ) } ,  RR* ,  <  )  =  sup ( { ( ( 1st `  C
) M ( 1st `  t ) ) ,  ( ( 2nd `  C
) N ( 2nd `  t ) ) } ,  RR* ,  <  )
)
3534, 1ovmpoga 5866 . . . . . 6  |-  ( ( C  e.  ( X  X.  Y )  /\  t  e.  ( X  X.  Y )  /\  sup ( { ( ( 1st `  C ) M ( 1st `  t ) ) ,  ( ( 2nd `  C ) N ( 2nd `  t
) ) } ,  RR* ,  <  )  e. 
RR* )  ->  ( C P t )  =  sup ( { ( ( 1st `  C
) M ( 1st `  t ) ) ,  ( ( 2nd `  C
) N ( 2nd `  t ) ) } ,  RR* ,  <  )
)
369, 10, 26, 35syl3anc 1199 . . . . 5  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  ( C P t )  =  sup ( { ( ( 1st `  C
) M ( 1st `  t ) ) ,  ( ( 2nd `  C
) N ( 2nd `  t ) ) } ,  RR* ,  <  )
)
3736breq1d 3907 . . . 4  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  (
( C P t )  <  R  <->  sup ( { ( ( 1st `  C ) M ( 1st `  t ) ) ,  ( ( 2nd `  C ) N ( 2nd `  t
) ) } ,  RR* ,  <  )  < 
R ) )
386adantr 272 . . . . 5  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  R  e.  RR* )
39 xrmaxltsup 10967 . . . . 5  |-  ( ( ( ( 1st `  C
) M ( 1st `  t ) )  e. 
RR*  /\  ( ( 2nd `  C ) N ( 2nd `  t
) )  e.  RR*  /\  R  e.  RR* )  ->  ( sup ( { ( ( 1st `  C
) M ( 1st `  t ) ) ,  ( ( 2nd `  C
) N ( 2nd `  t ) ) } ,  RR* ,  <  )  <  R  <->  ( ( ( 1st `  C ) M ( 1st `  t
) )  <  R  /\  ( ( 2nd `  C
) N ( 2nd `  t ) )  < 
R ) ) )
4017, 24, 38, 39syl3anc 1199 . . . 4  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  ( sup ( { ( ( 1st `  C ) M ( 1st `  t
) ) ,  ( ( 2nd `  C
) N ( 2nd `  t ) ) } ,  RR* ,  <  )  <  R  <->  ( ( ( 1st `  C ) M ( 1st `  t
) )  <  R  /\  ( ( 2nd `  C
) N ( 2nd `  t ) )  < 
R ) ) )
4137, 40bitrd 187 . . 3  |-  ( (
ph  /\  t  e.  ( X  X.  Y
) )  ->  (
( C P t )  <  R  <->  ( (
( 1st `  C
) M ( 1st `  t ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  t
) )  <  R
) ) )
4241rabbidva 2646 . 2  |-  ( ph  ->  { t  e.  ( X  X.  Y )  |  ( C P t )  <  R }  =  { t  e.  ( X  X.  Y
)  |  ( ( ( 1st `  C
) M ( 1st `  t ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  t
) )  <  R
) } )
43 1st2nd2 6039 . . . . . . 7  |-  ( n  e.  ( X  X.  Y )  ->  n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >. )
4443ad2antrl 479 . . . . . 6  |-  ( (
ph  /\  ( n  e.  ( X  X.  Y
)  /\  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )  ->  n  =  <. ( 1st `  n ) ,  ( 2nd `  n )
>. )
45 xp1st 6029 . . . . . . . 8  |-  ( n  e.  ( X  X.  Y )  ->  ( 1st `  n )  e.  X )
4645ad2antrl 479 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( X  X.  Y
)  /\  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )  -> 
( 1st `  n
)  e.  X )
47 simprrl 511 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( X  X.  Y
)  /\  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )  -> 
( ( 1st `  C
) M ( 1st `  n ) )  < 
R )
485, 12syl 14 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  C
)  e.  X )
49 elbl 12455 . . . . . . . . 9  |-  ( ( M  e.  ( *Met `  X )  /\  ( 1st `  C
)  e.  X  /\  R  e.  RR* )  -> 
( ( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  <->  ( ( 1st `  n )  e.  X  /\  ( ( 1st `  C ) M ( 1st `  n
) )  <  R
) ) )
502, 48, 6, 49syl3anc 1199 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  <->  ( ( 1st `  n )  e.  X  /\  ( ( 1st `  C ) M ( 1st `  n
) )  <  R
) ) )
5150adantr 272 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( X  X.  Y
)  /\  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )  -> 
( ( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  <->  ( ( 1st `  n )  e.  X  /\  ( ( 1st `  C ) M ( 1st `  n
) )  <  R
) ) )
5246, 47, 51mpbir2and 911 . . . . . 6  |-  ( (
ph  /\  ( n  e.  ( X  X.  Y
)  /\  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )  -> 
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R ) )
53 xp2nd 6030 . . . . . . . 8  |-  ( n  e.  ( X  X.  Y )  ->  ( 2nd `  n )  e.  Y )
5453ad2antrl 479 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( X  X.  Y
)  /\  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )  -> 
( 2nd `  n
)  e.  Y )
55 simprrr 512 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( X  X.  Y
)  /\  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )  -> 
( ( 2nd `  C
) N ( 2nd `  n ) )  < 
R )
565, 19syl 14 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  C
)  e.  Y )
57 elbl 12455 . . . . . . . . 9  |-  ( ( N  e.  ( *Met `  Y )  /\  ( 2nd `  C
)  e.  Y  /\  R  e.  RR* )  -> 
( ( 2nd `  n
)  e.  ( ( 2nd `  C ) ( ball `  N
) R )  <->  ( ( 2nd `  n )  e.  Y  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )
583, 56, 6, 57syl3anc 1199 . . . . . . . 8  |-  ( ph  ->  ( ( 2nd `  n
)  e.  ( ( 2nd `  C ) ( ball `  N
) R )  <->  ( ( 2nd `  n )  e.  Y  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )
5958adantr 272 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( X  X.  Y
)  /\  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )  -> 
( ( 2nd `  n
)  e.  ( ( 2nd `  C ) ( ball `  N
) R )  <->  ( ( 2nd `  n )  e.  Y  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )
6054, 55, 59mpbir2and 911 . . . . . 6  |-  ( (
ph  /\  ( n  e.  ( X  X.  Y
)  /\  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )  -> 
( 2nd `  n
)  e.  ( ( 2nd `  C ) ( ball `  N
) R ) )
6144, 52, 60jca32 306 . . . . 5  |-  ( (
ph  /\  ( n  e.  ( X  X.  Y
)  /\  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )  -> 
( n  =  <. ( 1st `  n ) ,  ( 2nd `  n
) >.  /\  ( ( 1st `  n )  e.  ( ( 1st `  C
) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )
62 simprl 503 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  ->  n  =  <. ( 1st `  n ) ,  ( 2nd `  n )
>. )
63 simprrl 511 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  -> 
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R ) )
6450adantr 272 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  -> 
( ( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  <->  ( ( 1st `  n )  e.  X  /\  ( ( 1st `  C ) M ( 1st `  n
) )  <  R
) ) )
6563, 64mpbid 146 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  -> 
( ( 1st `  n
)  e.  X  /\  ( ( 1st `  C
) M ( 1st `  n ) )  < 
R ) )
6665simpld 111 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  -> 
( 1st `  n
)  e.  X )
67 simprrr 512 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  -> 
( 2nd `  n
)  e.  ( ( 2nd `  C ) ( ball `  N
) R ) )
6858adantr 272 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  -> 
( ( 2nd `  n
)  e.  ( ( 2nd `  C ) ( ball `  N
) R )  <->  ( ( 2nd `  n )  e.  Y  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )
6967, 68mpbid 146 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  -> 
( ( 2nd `  n
)  e.  Y  /\  ( ( 2nd `  C
) N ( 2nd `  n ) )  < 
R ) )
7069simpld 111 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  -> 
( 2nd `  n
)  e.  Y )
7162, 66, 70jca32 306 . . . . . . 7  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  -> 
( n  =  <. ( 1st `  n ) ,  ( 2nd `  n
) >.  /\  ( ( 1st `  n )  e.  X  /\  ( 2nd `  n )  e.  Y
) ) )
72 elxp6 6033 . . . . . . 7  |-  ( n  e.  ( X  X.  Y )  <->  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  X  /\  ( 2nd `  n )  e.  Y ) ) )
7371, 72sylibr 133 . . . . . 6  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  ->  n  e.  ( X  X.  Y ) )
7465simprd 113 . . . . . 6  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  -> 
( ( 1st `  C
) M ( 1st `  n ) )  < 
R )
7569simprd 113 . . . . . 6  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  -> 
( ( 2nd `  C
) N ( 2nd `  n ) )  < 
R )
7673, 74, 75jca32 306 . . . . 5  |-  ( (
ph  /\  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )  -> 
( n  e.  ( X  X.  Y )  /\  ( ( ( 1st `  C ) M ( 1st `  n
) )  <  R  /\  ( ( 2nd `  C
) N ( 2nd `  n ) )  < 
R ) ) )
7761, 76impbida 568 . . . 4  |-  ( ph  ->  ( ( n  e.  ( X  X.  Y
)  /\  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) )  <->  ( n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >.  /\  (
( 1st `  n
)  e.  ( ( 1st `  C ) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) ) )
78 fveq2 5387 . . . . . . . 8  |-  ( t  =  n  ->  ( 1st `  t )  =  ( 1st `  n
) )
7978oveq2d 5756 . . . . . . 7  |-  ( t  =  n  ->  (
( 1st `  C
) M ( 1st `  t ) )  =  ( ( 1st `  C
) M ( 1st `  n ) ) )
8079breq1d 3907 . . . . . 6  |-  ( t  =  n  ->  (
( ( 1st `  C
) M ( 1st `  t ) )  < 
R  <->  ( ( 1st `  C ) M ( 1st `  n ) )  <  R ) )
81 fveq2 5387 . . . . . . . 8  |-  ( t  =  n  ->  ( 2nd `  t )  =  ( 2nd `  n
) )
8281oveq2d 5756 . . . . . . 7  |-  ( t  =  n  ->  (
( 2nd `  C
) N ( 2nd `  t ) )  =  ( ( 2nd `  C
) N ( 2nd `  n ) ) )
8382breq1d 3907 . . . . . 6  |-  ( t  =  n  ->  (
( ( 2nd `  C
) N ( 2nd `  t ) )  < 
R  <->  ( ( 2nd `  C ) N ( 2nd `  n ) )  <  R ) )
8480, 83anbi12d 462 . . . . 5  |-  ( t  =  n  ->  (
( ( ( 1st `  C ) M ( 1st `  t ) )  <  R  /\  ( ( 2nd `  C
) N ( 2nd `  t ) )  < 
R )  <->  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )
8584elrab 2811 . . . 4  |-  ( n  e.  { t  e.  ( X  X.  Y
)  |  ( ( ( 1st `  C
) M ( 1st `  t ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  t
) )  <  R
) }  <->  ( n  e.  ( X  X.  Y
)  /\  ( (
( 1st `  C
) M ( 1st `  n ) )  < 
R  /\  ( ( 2nd `  C ) N ( 2nd `  n
) )  <  R
) ) )
86 elxp6 6033 . . . 4  |-  ( n  e.  ( ( ( 1st `  C ) ( ball `  M
) R )  X.  ( ( 2nd `  C
) ( ball `  N
) R ) )  <-> 
( n  =  <. ( 1st `  n ) ,  ( 2nd `  n
) >.  /\  ( ( 1st `  n )  e.  ( ( 1st `  C
) ( ball `  M
) R )  /\  ( 2nd `  n )  e.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )
8777, 85, 863bitr4g 222 . . 3  |-  ( ph  ->  ( n  e.  {
t  e.  ( X  X.  Y )  |  ( ( ( 1st `  C ) M ( 1st `  t ) )  <  R  /\  ( ( 2nd `  C
) N ( 2nd `  t ) )  < 
R ) }  <->  n  e.  ( ( ( 1st `  C ) ( ball `  M ) R )  X.  ( ( 2nd `  C ) ( ball `  N ) R ) ) ) )
8887eqrdv 2113 . 2  |-  ( ph  ->  { t  e.  ( X  X.  Y )  |  ( ( ( 1st `  C ) M ( 1st `  t
) )  <  R  /\  ( ( 2nd `  C
) N ( 2nd `  t ) )  < 
R ) }  =  ( ( ( 1st `  C ) ( ball `  M ) R )  X.  ( ( 2nd `  C ) ( ball `  N ) R ) ) )
898, 42, 883eqtrd 2152 1  |-  ( ph  ->  ( C ( ball `  P ) R )  =  ( ( ( 1st `  C ) ( ball `  M
) R )  X.  ( ( 2nd `  C
) ( ball `  N
) R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   {crab 2395   {cpr 3496   <.cop 3498   class class class wbr 3897    X. cxp 4505   ` cfv 5091  (class class class)co 5740    e. cmpo 5742   1stc1st 6002   2ndc2nd 6003   supcsup 6835   RR*cxr 7763    < clt 7764   *Metcxmet 12044   ballcbl 12046
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:  xmettxlem  12573  xmettx  12574
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