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Theorem txmetcnp 13511
Description: Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)
Hypotheses
Ref Expression
metcn.2  |-  J  =  ( MetOpen `  C )
metcn.4  |-  K  =  ( MetOpen `  D )
txmetcnp.4  |-  L  =  ( MetOpen `  E )
Assertion
Ref Expression
txmetcnp  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( F  e.  ( ( ( J 
tX  K )  CnP 
L ) `  <. A ,  B >. )  <->  ( F : ( X  X.  Y ) --> Z  /\  A. z  e.  RR+  E. w  e.  RR+  A. u  e.  X  A. v  e.  Y  (
( ( A C u )  <  w  /\  ( B D v )  <  w )  ->  ( ( A F B ) E ( u F v ) )  <  z
) ) ) )
Distinct variable groups:    v, u, w, z, F    u, J, v, w, z    u, K, v, w, z    u, X, v, w, z    u, Y, v, w, z    u, Z, v, w, z    u, A, v, w, z    u, C, v, w, z    u, D, v, w, z    u, B, v, w, z    u, E, v, w, z    w, L, z
Allowed substitution hints:    L( v, u)

Proof of Theorem txmetcnp
Dummy variables  t  s  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2175 . . . 4  |-  ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y )  |->  sup ( { ( ( 1st `  r ) C ( 1st `  s ) ) ,  ( ( 2nd `  r ) D ( 2nd `  s
) ) } ,  RR* ,  <  ) )  =  ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
)
2 simp1 997 . . . . 5  |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  ->  C  e.  ( *Met `  X ) )
32adantr 276 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  C  e.  ( *Met `  X
) )
4 simp2 998 . . . . 5  |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  ->  D  e.  ( *Met `  Y ) )
54adantr 276 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  D  e.  ( *Met `  Y
) )
61, 3, 5xmetxp 13500 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
)  e.  ( *Met `  ( X  X.  Y ) ) )
7 simpl3 1002 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  E  e.  ( *Met `  Z
) )
8 simprl 529 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  A  e.  X )
9 simprr 531 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  B  e.  Y )
108, 9opelxpd 4653 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  <. A ,  B >.  e.  ( X  X.  Y ) )
11 eqid 2175 . . . 4  |-  ( MetOpen `  ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) )  =  (
MetOpen `  ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) )
12 txmetcnp.4 . . . 4  |-  L  =  ( MetOpen `  E )
1311, 12metcnp 13505 . . 3  |-  ( ( ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
)  e.  ( *Met `  ( X  X.  Y ) )  /\  E  e.  ( *Met `  Z
)  /\  <. A ,  B >.  e.  ( X  X.  Y ) )  ->  ( F  e.  ( ( ( MetOpen `  ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) )  CnP  L
) `  <. A ,  B >. )  <->  ( F : ( X  X.  Y ) --> Z  /\  A. z  e.  RR+  E. w  e.  RR+  A. t  e.  ( X  X.  Y
) ( ( <. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y )  |->  sup ( { ( ( 1st `  r ) C ( 1st `  s
) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) t )  < 
w  ->  ( ( F `  <. A ,  B >. ) E ( F `  t ) )  <  z ) ) ) )
146, 7, 10, 13syl3anc 1238 . 2  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( F  e.  ( ( ( MetOpen `  ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) )  CnP  L
) `  <. A ,  B >. )  <->  ( F : ( X  X.  Y ) --> Z  /\  A. z  e.  RR+  E. w  e.  RR+  A. t  e.  ( X  X.  Y
) ( ( <. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y )  |->  sup ( { ( ( 1st `  r ) C ( 1st `  s
) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) t )  < 
w  ->  ( ( F `  <. A ,  B >. ) E ( F `  t ) )  <  z ) ) ) )
15 metcn.2 . . . . . 6  |-  J  =  ( MetOpen `  C )
16 metcn.4 . . . . . 6  |-  K  =  ( MetOpen `  D )
171, 3, 5, 15, 16, 11xmettx 13503 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( MetOpen `  ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) )  =  ( J  tX  K ) )
1817oveq1d 5880 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( ( MetOpen
`  ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) )  CnP  L
)  =  ( ( J  tX  K )  CnP  L ) )
1918fveq1d 5509 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( (
( MetOpen `  ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) )  CnP  L
) `  <. A ,  B >. )  =  ( ( ( J  tX  K )  CnP  L
) `  <. A ,  B >. ) )
2019eleq2d 2245 . 2  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( F  e.  ( ( ( MetOpen `  ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) )  CnP  L
) `  <. A ,  B >. )  <->  F  e.  ( ( ( J 
tX  K )  CnP 
L ) `  <. A ,  B >. )
) )
21 oveq2 5873 . . . . . . . . 9  |-  ( t  =  <. u ,  v
>.  ->  ( <. A ,  B >. ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) t )  =  ( <. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) <. u ,  v
>. ) )
2221breq1d 4008 . . . . . . . 8  |-  ( t  =  <. u ,  v
>.  ->  ( ( <. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y )  |->  sup ( { ( ( 1st `  r ) C ( 1st `  s
) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) t )  < 
w  <->  ( <. A ,  B >. ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) <. u ,  v
>. )  <  w ) )
23 fveq2 5507 . . . . . . . . . 10  |-  ( t  =  <. u ,  v
>.  ->  ( F `  t )  =  ( F `  <. u ,  v >. )
)
2423oveq2d 5881 . . . . . . . . 9  |-  ( t  =  <. u ,  v
>.  ->  ( ( F `
 <. A ,  B >. ) E ( F `
 t ) )  =  ( ( F `
 <. A ,  B >. ) E ( F `
 <. u ,  v
>. ) ) )
2524breq1d 4008 . . . . . . . 8  |-  ( t  =  <. u ,  v
>.  ->  ( ( ( F `  <. A ,  B >. ) E ( F `  t ) )  <  z  <->  ( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >. )
)  <  z )
)
2622, 25imbi12d 234 . . . . . . 7  |-  ( t  =  <. u ,  v
>.  ->  ( ( (
<. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y )  |->  sup ( { ( ( 1st `  r ) C ( 1st `  s
) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) t )  < 
w  ->  ( ( F `  <. A ,  B >. ) E ( F `  t ) )  <  z )  <-> 
( ( <. A ,  B >. ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) <. u ,  v
>. )  <  w  -> 
( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >.
) )  <  z
) ) )
2726ralxp 4763 . . . . . 6  |-  ( A. t  e.  ( X  X.  Y ) ( (
<. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y )  |->  sup ( { ( ( 1st `  r ) C ( 1st `  s
) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) t )  < 
w  ->  ( ( F `  <. A ,  B >. ) E ( F `  t ) )  <  z )  <->  A. u  e.  X  A. v  e.  Y  ( ( <. A ,  B >. ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) <. u ,  v
>. )  <  w  -> 
( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >.
) )  <  z
) )
288ad4antr 494 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  A  e.  X )
299ad4antr 494 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  B  e.  Y )
3028, 29opelxpd 4653 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  <. A ,  B >.  e.  ( X  X.  Y ) )
31 simplr 528 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  u  e.  X )
32 simpr 110 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  v  e.  Y )
3331, 32opelxpd 4653 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  <. u ,  v >.  e.  ( X  X.  Y ) )
342ad5antr 496 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  C  e.  ( *Met `  X ) )
35 xmetf 13343 . . . . . . . . . . . . . . . 16  |-  ( C  e.  ( *Met `  X )  ->  C : ( X  X.  X ) --> RR* )
3634, 35syl 14 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  C : ( X  X.  X ) --> RR* )
37 op1stg 6141 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( 1st `  <. A ,  B >. )  =  A )
3828, 29, 37syl2anc 411 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( 1st `  <. A ,  B >. )  =  A )
3938, 28eqeltrd 2252 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( 1st `  <. A ,  B >. )  e.  X )
40 op1stg 6141 . . . . . . . . . . . . . . . . 17  |-  ( ( u  e.  X  /\  v  e.  Y )  ->  ( 1st `  <. u ,  v >. )  =  u )
4140adantll 476 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( 1st `  <. u ,  v
>. )  =  u
)
4241, 31eqeltrd 2252 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( 1st `  <. u ,  v
>. )  e.  X
)
4336, 39, 42fovrnd 6009 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  (
( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v >. )
)  e.  RR* )
444ad5antr 496 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  D  e.  ( *Met `  Y ) )
45 xmetf 13343 . . . . . . . . . . . . . . . 16  |-  ( D  e.  ( *Met `  Y )  ->  D : ( Y  X.  Y ) --> RR* )
4644, 45syl 14 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  D : ( Y  X.  Y ) --> RR* )
47 op2ndg 6142 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( 2nd `  <. A ,  B >. )  =  B )
4828, 29, 47syl2anc 411 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( 2nd `  <. A ,  B >. )  =  B )
4948, 29eqeltrd 2252 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( 2nd `  <. A ,  B >. )  e.  Y )
50 op2ndg 6142 . . . . . . . . . . . . . . . . 17  |-  ( ( u  e.  X  /\  v  e.  Y )  ->  ( 2nd `  <. u ,  v >. )  =  v )
5150adantll 476 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( 2nd `  <. u ,  v
>. )  =  v
)
5251, 32eqeltrd 2252 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( 2nd `  <. u ,  v
>. )  e.  Y
)
5346, 49, 52fovrnd 6009 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  (
( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
)  e.  RR* )
54 xrmaxcl 11227 . . . . . . . . . . . . . 14  |-  ( ( ( ( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v >. )
)  e.  RR*  /\  (
( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
)  e.  RR* )  ->  sup ( { ( ( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v >. )
) ,  ( ( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
) } ,  RR* ,  <  )  e.  RR* )
5543, 53, 54syl2anc 411 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  sup ( { ( ( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v
>. ) ) ,  ( ( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
) } ,  RR* ,  <  )  e.  RR* )
56 fveq2 5507 . . . . . . . . . . . . . . . . 17  |-  ( r  =  <. A ,  B >.  ->  ( 1st `  r
)  =  ( 1st `  <. A ,  B >. ) )
57 fveq2 5507 . . . . . . . . . . . . . . . . 17  |-  ( s  =  <. u ,  v
>.  ->  ( 1st `  s
)  =  ( 1st `  <. u ,  v
>. ) )
5856, 57oveqan12d 5884 . . . . . . . . . . . . . . . 16  |-  ( ( r  =  <. A ,  B >.  /\  s  =  <. u ,  v >.
)  ->  ( ( 1st `  r ) C ( 1st `  s
) )  =  ( ( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v >. )
) )
59 fveq2 5507 . . . . . . . . . . . . . . . . 17  |-  ( r  =  <. A ,  B >.  ->  ( 2nd `  r
)  =  ( 2nd `  <. A ,  B >. ) )
60 fveq2 5507 . . . . . . . . . . . . . . . . 17  |-  ( s  =  <. u ,  v
>.  ->  ( 2nd `  s
)  =  ( 2nd `  <. u ,  v
>. ) )
6159, 60oveqan12d 5884 . . . . . . . . . . . . . . . 16  |-  ( ( r  =  <. A ,  B >.  /\  s  =  <. u ,  v >.
)  ->  ( ( 2nd `  r ) D ( 2nd `  s
) )  =  ( ( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
) )
6258, 61preq12d 3674 . . . . . . . . . . . . . . 15  |-  ( ( r  =  <. A ,  B >.  /\  s  =  <. u ,  v >.
)  ->  { (
( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) }  =  { ( ( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v >. )
) ,  ( ( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
) } )
6362supeq1d 6976 . . . . . . . . . . . . . 14  |-  ( ( r  =  <. A ,  B >.  /\  s  =  <. u ,  v >.
)  ->  sup ( { ( ( 1st `  r ) C ( 1st `  s ) ) ,  ( ( 2nd `  r ) D ( 2nd `  s
) ) } ,  RR* ,  <  )  =  sup ( { ( ( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v >. )
) ,  ( ( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
) } ,  RR* ,  <  ) )
6463, 1ovmpoga 5994 . . . . . . . . . . . . 13  |-  ( (
<. A ,  B >.  e.  ( X  X.  Y
)  /\  <. u ,  v >.  e.  ( X  X.  Y )  /\  sup ( { ( ( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v >. )
) ,  ( ( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
) } ,  RR* ,  <  )  e.  RR* )  ->  ( <. A ,  B >. ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) <. u ,  v
>. )  =  sup ( { ( ( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v
>. ) ) ,  ( ( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
) } ,  RR* ,  <  ) )
6530, 33, 55, 64syl3anc 1238 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( <. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y )  |->  sup ( { ( ( 1st `  r ) C ( 1st `  s
) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) <. u ,  v
>. )  =  sup ( { ( ( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v
>. ) ) ,  ( ( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
) } ,  RR* ,  <  ) )
6638, 41oveq12d 5883 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  (
( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v >. )
)  =  ( A C u ) )
6748, 51oveq12d 5883 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  (
( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
)  =  ( B D v ) )
6866, 67preq12d 3674 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  { ( ( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v >. )
) ,  ( ( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
) }  =  {
( A C u ) ,  ( B D v ) } )
6968supeq1d 6976 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  sup ( { ( ( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v
>. ) ) ,  ( ( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
) } ,  RR* ,  <  )  =  sup ( { ( A C u ) ,  ( B D v ) } ,  RR* ,  <  ) )
7065, 69eqtrd 2208 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( <. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y )  |->  sup ( { ( ( 1st `  r ) C ( 1st `  s
) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) <. u ,  v
>. )  =  sup ( { ( A C u ) ,  ( B D v ) } ,  RR* ,  <  ) )
7170breq1d 4008 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  (
( <. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) <. u ,  v
>. )  <  w  <->  sup ( { ( A C u ) ,  ( B D v ) } ,  RR* ,  <  )  <  w ) )
72 xmetcl 13345 . . . . . . . . . . . 12  |-  ( ( C  e.  ( *Met `  X )  /\  A  e.  X  /\  u  e.  X
)  ->  ( A C u )  e. 
RR* )
7334, 28, 31, 72syl3anc 1238 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( A C u )  e. 
RR* )
74 xmetcl 13345 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  Y )  /\  B  e.  Y  /\  v  e.  Y
)  ->  ( B D v )  e. 
RR* )
7544, 29, 32, 74syl3anc 1238 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( B D v )  e. 
RR* )
76 rpxr 9630 . . . . . . . . . . . 12  |-  ( w  e.  RR+  ->  w  e. 
RR* )
7776ad3antlr 493 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  w  e.  RR* )
78 xrmaxltsup 11233 . . . . . . . . . . 11  |-  ( ( ( A C u )  e.  RR*  /\  ( B D v )  e. 
RR*  /\  w  e.  RR* )  ->  ( sup ( { ( A C u ) ,  ( B D v ) } ,  RR* ,  <  )  <  w  <->  ( ( A C u )  < 
w  /\  ( B D v )  < 
w ) ) )
7973, 75, 77, 78syl3anc 1238 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  ( sup ( { ( A C u ) ,  ( B D v ) } ,  RR* ,  <  )  <  w  <->  ( ( A C u )  <  w  /\  ( B D v )  <  w ) ) )
8071, 79bitrd 188 . . . . . . . . 9  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  (
( <. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) <. u ,  v
>. )  <  w  <->  ( ( A C u )  < 
w  /\  ( B D v )  < 
w ) ) )
81 df-ov 5868 . . . . . . . . . . . . 13  |-  ( A F B )  =  ( F `  <. A ,  B >. )
82 df-ov 5868 . . . . . . . . . . . . 13  |-  ( u F v )  =  ( F `  <. u ,  v >. )
8381, 82oveq12i 5877 . . . . . . . . . . . 12  |-  ( ( A F B ) E ( u F v ) )  =  ( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >.
) )
8483breq1i 4005 . . . . . . . . . . 11  |-  ( ( ( A F B ) E ( u F v ) )  <  z  <->  ( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >. )
)  <  z )
8584bicomi 132 . . . . . . . . . 10  |-  ( ( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >.
) )  <  z  <->  ( ( A F B ) E ( u F v ) )  <  z )
8685a1i 9 . . . . . . . . 9  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  (
( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >.
) )  <  z  <->  ( ( A F B ) E ( u F v ) )  <  z ) )
8780, 86imbi12d 234 . . . . . . . 8  |-  ( ( ( ( ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  /\  v  e.  Y )  ->  (
( ( <. A ,  B >. ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) <. u ,  v
>. )  <  w  -> 
( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >.
) )  <  z
)  <->  ( ( ( A C u )  <  w  /\  ( B D v )  < 
w )  ->  (
( A F B ) E ( u F v ) )  <  z ) ) )
8887ralbidva 2471 . . . . . . 7  |-  ( ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  z  e.  RR+ )  /\  w  e.  RR+ )  /\  u  e.  X )  ->  ( A. v  e.  Y  ( ( <. A ,  B >. ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) <. u ,  v
>. )  <  w  -> 
( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >.
) )  <  z
)  <->  A. v  e.  Y  ( ( ( A C u )  < 
w  /\  ( B D v )  < 
w )  ->  (
( A F B ) E ( u F v ) )  <  z ) ) )
8988ralbidva 2471 . . . . . 6  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  z  e.  RR+ )  /\  w  e.  RR+ )  ->  ( A. u  e.  X  A. v  e.  Y  ( ( <. A ,  B >. ( r  e.  ( X  X.  Y
) ,  s  e.  ( X  X.  Y
)  |->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) <. u ,  v
>. )  <  w  -> 
( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >.
) )  <  z
)  <->  A. u  e.  X  A. v  e.  Y  ( ( ( A C u )  < 
w  /\  ( B D v )  < 
w )  ->  (
( A F B ) E ( u F v ) )  <  z ) ) )
9027, 89bitrid 192 . . . . 5  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  z  e.  RR+ )  /\  w  e.  RR+ )  ->  ( A. t  e.  ( X  X.  Y ) ( ( <. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y ) 
|->  sup ( { ( ( 1st `  r
) C ( 1st `  s ) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) t )  < 
w  ->  ( ( F `  <. A ,  B >. ) E ( F `  t ) )  <  z )  <->  A. u  e.  X  A. v  e.  Y  ( ( ( A C u )  < 
w  /\  ( B D v )  < 
w )  ->  (
( A F B ) E ( u F v ) )  <  z ) ) )
9190rexbidva 2472 . . . 4  |-  ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  z  e.  RR+ )  ->  ( E. w  e.  RR+  A. t  e.  ( X  X.  Y
) ( ( <. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y )  |->  sup ( { ( ( 1st `  r ) C ( 1st `  s
) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) t )  < 
w  ->  ( ( F `  <. A ,  B >. ) E ( F `  t ) )  <  z )  <->  E. w  e.  RR+  A. u  e.  X  A. v  e.  Y  ( (
( A C u )  <  w  /\  ( B D v )  <  w )  -> 
( ( A F B ) E ( u F v ) )  <  z ) ) )
9291ralbidva 2471 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( A. z  e.  RR+  E. w  e.  RR+  A. t  e.  ( X  X.  Y
) ( ( <. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y )  |->  sup ( { ( ( 1st `  r ) C ( 1st `  s
) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) t )  < 
w  ->  ( ( F `  <. A ,  B >. ) E ( F `  t ) )  <  z )  <->  A. z  e.  RR+  E. w  e.  RR+  A. u  e.  X  A. v  e.  Y  ( ( ( A C u )  <  w  /\  ( B D v )  < 
w )  ->  (
( A F B ) E ( u F v ) )  <  z ) ) )
9392anbi2d 464 . 2  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( ( F : ( X  X.  Y ) --> Z  /\  A. z  e.  RR+  E. w  e.  RR+  A. t  e.  ( X  X.  Y
) ( ( <. A ,  B >. ( r  e.  ( X  X.  Y ) ,  s  e.  ( X  X.  Y )  |->  sup ( { ( ( 1st `  r ) C ( 1st `  s
) ) ,  ( ( 2nd `  r
) D ( 2nd `  s ) ) } ,  RR* ,  <  )
) t )  < 
w  ->  ( ( F `  <. A ,  B >. ) E ( F `  t ) )  <  z ) )  <->  ( F :
( X  X.  Y
) --> Z  /\  A. z  e.  RR+  E. w  e.  RR+  A. u  e.  X  A. v  e.  Y  ( ( ( A C u )  <  w  /\  ( B D v )  < 
w )  ->  (
( A F B ) E ( u F v ) )  <  z ) ) ) )
9414, 20, 933bitr3d 218 1  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( F  e.  ( ( ( J 
tX  K )  CnP 
L ) `  <. A ,  B >. )  <->  ( F : ( X  X.  Y ) --> Z  /\  A. z  e.  RR+  E. w  e.  RR+  A. u  e.  X  A. v  e.  Y  (
( ( A C u )  <  w  /\  ( B D v )  <  w )  ->  ( ( A F B ) E ( u F v ) )  <  z
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2146   A.wral 2453   E.wrex 2454   {cpr 3590   <.cop 3592   class class class wbr 3998    X. cxp 4618   -->wf 5204   ` cfv 5208  (class class class)co 5865    e. cmpo 5867   1stc1st 6129   2ndc2nd 6130   supcsup 6971   RR*cxr 7965    < clt 7966   RR+crp 9622   *Metcxmet 12973   MetOpencmopn 12978    CnP ccnp 13179    tX ctx 13245
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-map 6640  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-xneg 9741  df-xadd 9742  df-seqfrec 10414  df-exp 10488  df-cj 10818  df-re 10819  df-im 10820  df-rsqrt 10974  df-abs 10975  df-topgen 12630  df-psmet 12980  df-xmet 12981  df-bl 12983  df-mopn 12984  df-top 12989  df-topon 13002  df-bases 13034  df-cnp 13182  df-tx 13246
This theorem is referenced by:  txmetcn  13512  limccnp2cntop  13639
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