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Theorem txmetcnp 20785
Description: Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
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 variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . 4  |-  ( dist `  ( (toMetSp `  C
)  X.s  (toMetSp `  D )
) )  =  (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) )
2 simpl1 999 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  C  e.  ( *Met `  X
) )
3 simpl2 1000 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  D  e.  ( *Met `  Y
) )
41, 2, 3tmsxps 20774 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( dist `  ( (toMetSp `  C
)  X.s  (toMetSp `  D )
) )  e.  ( *Met `  ( X  X.  Y ) ) )
5 simpl3 1001 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  E  e.  ( *Met `  Z
) )
6 opelxpi 5030 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
76adantl 466 . . 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 ) )
8 eqid 2467 . . . 4  |-  ( MetOpen `  ( dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) )  =  ( MetOpen `  ( dist `  ( (toMetSp `  C
)  X.s  (toMetSp `  D )
) ) )
9 txmetcnp.4 . . . 4  |-  L  =  ( MetOpen `  E )
108, 9metcnp 20779 . . 3  |-  ( ( ( dist `  (
(toMetSp `  C )  X.s  (toMetSp `  D ) ) )  e.  ( *Met `  ( X  X.  Y
) )  /\  E  e.  ( *Met `  Z )  /\  <. A ,  B >.  e.  ( X  X.  Y ) )  ->  ( F  e.  ( ( ( MetOpen `  ( dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) )  CnP  L ) `  <. A ,  B >. )  <-> 
( F : ( X  X.  Y ) --> Z  /\  A. z  e.  RR+  E. w  e.  RR+  A. x  e.  ( X  X.  Y ) ( ( <. A ,  B >. ( dist `  (
(toMetSp `  C )  X.s  (toMetSp `  D ) ) ) x )  <  w  ->  ( ( F `  <. A ,  B >. ) E ( F `  x ) )  < 
z ) ) ) )
114, 5, 7, 10syl3anc 1228 . 2  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( F  e.  ( ( ( MetOpen `  ( dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) )  CnP  L ) `  <. A ,  B >. )  <-> 
( F : ( X  X.  Y ) --> Z  /\  A. z  e.  RR+  E. w  e.  RR+  A. x  e.  ( X  X.  Y ) ( ( <. A ,  B >. ( dist `  (
(toMetSp `  C )  X.s  (toMetSp `  D ) ) ) x )  <  w  ->  ( ( F `  <. A ,  B >. ) E ( F `  x ) )  < 
z ) ) ) )
12 metcn.2 . . . . . 6  |-  J  =  ( MetOpen `  C )
13 metcn.4 . . . . . 6  |-  K  =  ( MetOpen `  D )
141, 2, 3, 12, 13, 8tmsxpsmopn 20775 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( MetOpen `  ( dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) )  =  ( J  tX  K ) )
1514oveq1d 6297 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( ( MetOpen
`  ( dist `  (
(toMetSp `  C )  X.s  (toMetSp `  D ) ) ) )  CnP  L )  =  ( ( J 
tX  K )  CnP 
L ) )
1615fveq1d 5866 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( (
( MetOpen `  ( dist `  ( (toMetSp `  C
)  X.s  (toMetSp `  D )
) ) )  CnP 
L ) `  <. A ,  B >. )  =  ( ( ( J  tX  K )  CnP  L ) `  <. A ,  B >. ) )
1716eleq2d 2537 . 2  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  ->  ( F  e.  ( ( ( MetOpen `  ( dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) )  CnP  L ) `  <. A ,  B >. )  <-> 
F  e.  ( ( ( J  tX  K
)  CnP  L ) `  <. A ,  B >. ) ) )
18 oveq2 6290 . . . . . . . . 9  |-  ( x  =  <. u ,  v
>.  ->  ( <. A ,  B >. ( dist `  (
(toMetSp `  C )  X.s  (toMetSp `  D ) ) ) x )  =  (
<. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) <.
u ,  v >.
) )
1918breq1d 4457 . . . . . . . 8  |-  ( x  =  <. u ,  v
>.  ->  ( ( <. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) x )  <  w  <->  ( <. A ,  B >. ( dist `  ( (toMetSp `  C
)  X.s  (toMetSp `  D )
) ) <. u ,  v >. )  <  w ) )
20 df-ov 6285 . . . . . . . . . . 11  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2120oveq1i 6292 . . . . . . . . . 10  |-  ( ( A F B ) E ( F `  x ) )  =  ( ( F `  <. A ,  B >. ) E ( F `  x ) )
22 fveq2 5864 . . . . . . . . . . . 12  |-  ( x  =  <. u ,  v
>.  ->  ( F `  x )  =  ( F `  <. u ,  v >. )
)
23 df-ov 6285 . . . . . . . . . . . 12  |-  ( u F v )  =  ( F `  <. u ,  v >. )
2422, 23syl6eqr 2526 . . . . . . . . . . 11  |-  ( x  =  <. u ,  v
>.  ->  ( F `  x )  =  ( u F v ) )
2524oveq2d 6298 . . . . . . . . . 10  |-  ( x  =  <. u ,  v
>.  ->  ( ( A F B ) E ( F `  x
) )  =  ( ( A F B ) E ( u F v ) ) )
2621, 25syl5eqr 2522 . . . . . . . . 9  |-  ( x  =  <. u ,  v
>.  ->  ( ( F `
 <. A ,  B >. ) E ( F `
 x ) )  =  ( ( A F B ) E ( u F v ) ) )
2726breq1d 4457 . . . . . . . 8  |-  ( x  =  <. u ,  v
>.  ->  ( ( ( F `  <. A ,  B >. ) E ( F `  x ) )  <  z  <->  ( ( A F B ) E ( u F v ) )  <  z
) )
2819, 27imbi12d 320 . . . . . . 7  |-  ( x  =  <. u ,  v
>.  ->  ( ( (
<. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) x )  <  w  -> 
( ( F `  <. A ,  B >. ) E ( F `  x ) )  < 
z )  <->  ( ( <. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) <.
u ,  v >.
)  <  w  ->  ( ( A F B ) E ( u F v ) )  <  z ) ) )
2928ralxp 5142 . . . . . 6  |-  ( A. x  e.  ( X  X.  Y ) ( (
<. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) x )  <  w  -> 
( ( F `  <. A ,  B >. ) E ( F `  x ) )  < 
z )  <->  A. u  e.  X  A. v  e.  Y  ( ( <. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) <.
u ,  v >.
)  <  w  ->  ( ( A F B ) E ( u F v ) )  <  z ) )
302ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  C  e.  ( *Met `  X
) )
313ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  D  e.  ( *Met `  Y
) )
32 simpllr 758 . . . . . . . . . . . . 13  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  ( A  e.  X  /\  B  e.  Y ) )
3332simpld 459 . . . . . . . . . . . 12  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  A  e.  X
)
3432simprd 463 . . . . . . . . . . . 12  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  B  e.  Y
)
35 simprrl 763 . . . . . . . . . . . 12  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  u  e.  X
)
36 simprrr 764 . . . . . . . . . . . 12  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  v  e.  Y
)
371, 30, 31, 33, 34, 35, 36tmsxpsval2 20777 . . . . . . . . . . 11  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  ( <. A ,  B >. ( dist `  (
(toMetSp `  C )  X.s  (toMetSp `  D ) ) )
<. u ,  v >.
)  =  if ( ( A C u )  <_  ( B D v ) ,  ( B D v ) ,  ( A C u ) ) )
3837breq1d 4457 . . . . . . . . . 10  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  ( ( <. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) <.
u ,  v >.
)  <  w  <->  if (
( A C u )  <_  ( B D v ) ,  ( B D v ) ,  ( A C u ) )  <  w ) )
39 xmetcl 20569 . . . . . . . . . . . 12  |-  ( ( C  e.  ( *Met `  X )  /\  A  e.  X  /\  u  e.  X
)  ->  ( A C u )  e. 
RR* )
4030, 33, 35, 39syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  ( A C u )  e.  RR* )
41 xmetcl 20569 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  Y )  /\  B  e.  Y  /\  v  e.  Y
)  ->  ( B D v )  e. 
RR* )
4231, 34, 36, 41syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  ( B D v )  e.  RR* )
43 rpxr 11223 . . . . . . . . . . . 12  |-  ( w  e.  RR+  ->  w  e. 
RR* )
4443ad2antrl 727 . . . . . . . . . . 11  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  w  e.  RR* )
45 xrmaxlt 11378 . . . . . . . . . . 11  |-  ( ( ( A C u )  e.  RR*  /\  ( B D v )  e. 
RR*  /\  w  e.  RR* )  ->  ( if ( ( A C u )  <_  ( B D v ) ,  ( B D v ) ,  ( A C u ) )  <  w  <->  ( ( A C u )  < 
w  /\  ( B D v )  < 
w ) ) )
4640, 42, 44, 45syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  ( if ( ( A C u )  <_  ( B D v ) ,  ( B D v ) ,  ( A C u ) )  <  w  <->  ( ( A C u )  < 
w  /\  ( B D v )  < 
w ) ) )
4738, 46bitrd 253 . . . . . . . . 9  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  ( ( <. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) <.
u ,  v >.
)  <  w  <->  ( ( A C u )  < 
w  /\  ( B D v )  < 
w ) ) )
4847imbi1d 317 . . . . . . . 8  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  ( w  e.  RR+  /\  ( u  e.  X  /\  v  e.  Y ) ) )  ->  ( ( (
<. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) <.
u ,  v >.
)  <  w  ->  ( ( A F B ) E ( u F v ) )  <  z )  <->  ( (
( A C u )  <  w  /\  ( B D v )  <  w )  -> 
( ( A F B ) E ( u F v ) )  <  z ) ) )
4948anassrs 648 . . . . . . 7  |-  ( ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y
)  /\  E  e.  ( *Met `  Z
) )  /\  ( A  e.  X  /\  B  e.  Y )
)  /\  F :
( X  X.  Y
) --> Z )  /\  w  e.  RR+ )  /\  ( u  e.  X  /\  v  e.  Y
) )  ->  (
( ( <. A ,  B >. ( dist `  (
(toMetSp `  C )  X.s  (toMetSp `  D ) ) )
<. u ,  v >.
)  <  w  ->  ( ( A F B ) E ( u F v ) )  <  z )  <->  ( (
( A C u )  <  w  /\  ( B D v )  <  w )  -> 
( ( A F B ) E ( u F v ) )  <  z ) ) )
50492ralbidva 2906 . . . . . 6  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  w  e.  RR+ )  ->  ( A. u  e.  X  A. v  e.  Y  ( ( <. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) <.
u ,  v >.
)  <  w  ->  ( ( A F B ) E ( u F 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 ) ) )
5129, 50syl5bb 257 . . . . 5  |-  ( ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  /\  w  e.  RR+ )  ->  ( A. x  e.  ( X  X.  Y
) ( ( <. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) x )  <  w  -> 
( ( F `  <. A ,  B >. ) E ( F `  x ) )  < 
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 ) ) )
5251rexbidva 2970 . . . 4  |-  ( ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) )  /\  ( A  e.  X  /\  B  e.  Y
) )  /\  F : ( X  X.  Y ) --> Z )  ->  ( E. w  e.  RR+  A. x  e.  ( X  X.  Y
) ( ( <. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) x )  <  w  -> 
( ( F `  <. A ,  B >. ) E ( F `  x ) )  < 
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 ) ) )
5352ralbidv 2903 . . 3  |-  ( ( ( ( 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. x  e.  ( X  X.  Y ) ( ( <. A ,  B >. ( dist `  (
(toMetSp `  C )  X.s  (toMetSp `  D ) ) ) x )  <  w  ->  ( ( F `  <. A ,  B >. ) E ( F `  x ) )  < 
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 ) ) )
5453pm5.32da 641 . 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. x  e.  ( X  X.  Y
) ( ( <. A ,  B >. (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) ) x )  <  w  -> 
( ( F `  <. A ,  B >. ) E ( F `  x ) )  < 
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 ) ) ) )
5511, 17, 543bitr3d 283 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   ifcif 3939   <.cop 4033   class class class wbr 4447    X. cxp 4997   -->wf 5582   ` cfv 5586  (class class class)co 6282   RR*cxr 9623    < clt 9624    <_ cle 9625   RR+crp 11216   distcds 14560    X.s cxps 14757   *Metcxmt 18174   MetOpencmopn 18179    CnP ccnp 19492    tX ctx 19796  toMetSpctmt 20557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-bl 18185  df-mopn 18186  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cn 19494  df-cnp 19495  df-tx 19798  df-hmeo 19991  df-xms 20558  df-tms 20560
This theorem is referenced by:  txmetcn  20786  cxpcn3  22850
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