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Theorem txmetcnp 12593
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 2115 . . . 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 964 . . . . 5  |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  ->  C  e.  ( *Met `  X ) )
32adantr 272 . . . 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 965 . . . . 5  |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  ->  D  e.  ( *Met `  Y ) )
54adantr 272 . . . 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 12582 . . 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 969 . . 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 503 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  A  e.  X )
9 simprr 504 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  B  e.  Y )
108, 9opelxpd 4540 . . 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 2115 . . . 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 12587 . . 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 1199 . 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 12585 . . . . 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 5755 . . . 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 5389 . . 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 2185 . 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 5748 . . . . . . . . 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 3907 . . . . . . . 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 5387 . . . . . . . . . 10  |-  ( t  =  <. u ,  v
>.  ->  ( F `  t )  =  ( F `  <. u ,  v >. )
)
2423oveq2d 5756 . . . . . . . . 9  |-  ( t  =  <. u ,  v
>.  ->  ( ( F `
 <. A ,  B >. ) E ( F `
 t ) )  =  ( ( F `
 <. A ,  B >. ) E ( F `
 <. u ,  v
>. ) ) )
2524breq1d 3907 . . . . . . . 8  |-  ( t  =  <. u ,  v
>.  ->  ( ( ( F `  <. A ,  B >. ) E ( F `  t ) )  <  z  <->  ( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >. )
)  <  z )
)
2622, 25imbi12d 233 . . . . . . 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 4650 . . . . . 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 483 . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 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 4540 . . . . . . . . . . . . 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 502 . . . . . . . . . . . . . 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 109 . . . . . . . . . . . . . 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 4540 . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . 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 12425 . . . . . . . . . . . . . . . 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 6014 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( 1st `  <. A ,  B >. )  =  A )
3828, 29, 37syl2anc 406 . . . . . . . . . . . . . . . 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 2192 . . . . . . . . . . . . . . 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 6014 . . . . . . . . . . . . . . . . 17  |-  ( ( u  e.  X  /\  v  e.  Y )  ->  ( 1st `  <. u ,  v >. )  =  u )
4140adantll 465 . . . . . . . . . . . . . . . 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 2192 . . . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . 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 12425 . . . . . . . . . . . . . . . 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 6015 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( 2nd `  <. A ,  B >. )  =  B )
4828, 29, 47syl2anc 406 . . . . . . . . . . . . . . . 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 2192 . . . . . . . . . . . . . . 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 6015 . . . . . . . . . . . . . . . . 17  |-  ( ( u  e.  X  /\  v  e.  Y )  ->  ( 2nd `  <. u ,  v >. )  =  v )
5150adantll 465 . . . . . . . . . . . . . . . 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 2192 . . . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . . 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 10972 . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . 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 5387 . . . . . . . . . . . . . . . . 17  |-  ( r  =  <. A ,  B >.  ->  ( 1st `  r
)  =  ( 1st `  <. A ,  B >. ) )
57 fveq2 5387 . . . . . . . . . . . . . . . . 17  |-  ( s  =  <. u ,  v
>.  ->  ( 1st `  s
)  =  ( 1st `  <. u ,  v
>. ) )
5856, 57oveqan12d 5759 . . . . . . . . . . . . . . . 16  |-  ( ( r  =  <. A ,  B >.  /\  s  =  <. u ,  v >.
)  ->  ( ( 1st `  r ) C ( 1st `  s
) )  =  ( ( 1st `  <. A ,  B >. ) C ( 1st `  <. u ,  v >. )
) )
59 fveq2 5387 . . . . . . . . . . . . . . . . 17  |-  ( r  =  <. A ,  B >.  ->  ( 2nd `  r
)  =  ( 2nd `  <. A ,  B >. ) )
60 fveq2 5387 . . . . . . . . . . . . . . . . 17  |-  ( s  =  <. u ,  v
>.  ->  ( 2nd `  s
)  =  ( 2nd `  <. u ,  v
>. ) )
6159, 60oveqan12d 5759 . . . . . . . . . . . . . . . 16  |-  ( ( r  =  <. A ,  B >.  /\  s  =  <. u ,  v >.
)  ->  ( ( 2nd `  r ) D ( 2nd `  s
) )  =  ( ( 2nd `  <. A ,  B >. ) D ( 2nd `  <. u ,  v >. )
) )
6258, 61preq12d 3576 . . . . . . . . . . . . . . 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 6840 . . . . . . . . . . . . . 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 5866 . . . . . . . . . . . . 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 1199 . . . . . . . . . . . 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 5758 . . . . . . . . . . . . . 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 5758 . . . . . . . . . . . . . 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 3576 . . . . . . . . . . . . 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 6840 . . . . . . . . . . . 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 2148 . . . . . . . . . . 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 3907 . . . . . . . . . 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 12427 . . . . . . . . . . . 12  |-  ( ( C  e.  ( *Met `  X )  /\  A  e.  X  /\  u  e.  X
)  ->  ( A C u )  e. 
RR* )
7334, 28, 31, 72syl3anc 1199 . . . . . . . . . . 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 12427 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  Y )  /\  B  e.  Y  /\  v  e.  Y
)  ->  ( B D v )  e. 
RR* )
7544, 29, 32, 74syl3anc 1199 . . . . . . . . . . 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 9400 . . . . . . . . . . . 12  |-  ( w  e.  RR+  ->  w  e. 
RR* )
7776ad3antlr 482 . . . . . . . . . . 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 10978 . . . . . . . . . . 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 1199 . . . . . . . . . 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 187 . . . . . . . . 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 5743 . . . . . . . . . . . . 13  |-  ( A F B )  =  ( F `  <. A ,  B >. )
82 df-ov 5743 . . . . . . . . . . . . 13  |-  ( u F v )  =  ( F `  <. u ,  v >. )
8381, 82oveq12i 5752 . . . . . . . . . . . 12  |-  ( ( A F B ) E ( u F v ) )  =  ( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >.
) )
8483breq1i 3904 . . . . . . . . . . 11  |-  ( ( ( A F B ) E ( u F v ) )  <  z  <->  ( ( F `  <. A ,  B >. ) E ( F `  <. u ,  v >. )
)  <  z )
8584bicomi 131 . . . . . . . . . 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 233 . . . . . . . 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 2408 . . . . . . 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 2408 . . . . . 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, 89syl5bb 191 . . . . 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 2409 . . . 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 2408 . . 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 457 . 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 217 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 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   A.wral 2391   E.wrex 2392   {cpr 3496   <.cop 3498   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   RR*cxr 7763    < clt 7764   RR+crp 9393   *Metcxmet 12055   MetOpencmopn 12060    CnP ccnp 12261    tX ctx 12327
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 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-xneg 9510  df-xadd 9511  df-seqfrec 10170  df-exp 10244  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-topgen 12047  df-psmet 12062  df-xmet 12063  df-bl 12065  df-mopn 12066  df-top 12071  df-topon 12084  df-bases 12116  df-cnp 12264  df-tx 12328
This theorem is referenced by:  txmetcn  12594  limccnp2cntop  12721
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