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Theorem txmetcnp 21028
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 2443 . . . 4  |-  ( dist `  ( (toMetSp `  C
)  X.s  (toMetSp `  D )
) )  =  (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) )
2 simpl1 1000 . . . 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 1001 . . . 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 21017 . . 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 1002 . . 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 5021 . . . 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 2443 . . . 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 21022 . . 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 1229 . 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 21018 . . . . 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 6296 . . . 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 5858 . . 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 2513 . 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 6289 . . . . . . . . 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 4447 . . . . . . . 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 6284 . . . . . . . . . . 11  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2120oveq1i 6291 . . . . . . . . . 10  |-  ( ( A F B ) E ( F `  x ) )  =  ( ( F `  <. A ,  B >. ) E ( F `  x ) )
22 fveq2 5856 . . . . . . . . . . . 12  |-  ( x  =  <. u ,  v
>.  ->  ( F `  x )  =  ( F `  <. u ,  v >. )
)
23 df-ov 6284 . . . . . . . . . . . 12  |-  ( u F v )  =  ( F `  <. u ,  v >. )
2422, 23syl6eqr 2502 . . . . . . . . . . 11  |-  ( x  =  <. u ,  v
>.  ->  ( F `  x )  =  ( u F v ) )
2524oveq2d 6297 . . . . . . . . . 10  |-  ( x  =  <. u ,  v
>.  ->  ( ( A F B ) E ( F `  x
) )  =  ( ( A F B ) E ( u F v ) ) )
2621, 25syl5eqr 2498 . . . . . . . . 9  |-  ( x  =  <. u ,  v
>.  ->  ( ( F `
 <. A ,  B >. ) E ( F `
 x ) )  =  ( ( A F B ) E ( u F v ) ) )
2726breq1d 4447 . . . . . . . 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 5134 . . . . . 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 760 . . . . . . . . . . . . 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 765 . . . . . . . . . . . 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 766 . . . . . . . . . . . 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 21020 . . . . . . . . . . 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 4447 . . . . . . . . . 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 20812 . . . . . . . . . . . 12  |-  ( ( C  e.  ( *Met `  X )  /\  A  e.  X  /\  u  e.  X
)  ->  ( A C u )  e. 
RR* )
4030, 33, 35, 39syl3anc 1229 . . . . . . . . . . 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 20812 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  Y )  /\  B  e.  Y  /\  v  e.  Y
)  ->  ( B D v )  e. 
RR* )
4231, 34, 36, 41syl3anc 1229 . . . . . . . . . . 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 11238 . . . . . . . . . . . 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 11393 . . . . . . . . . . 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 1229 . . . . . . . . . 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 2885 . . . . . 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 2951 . . . 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 2882 . . 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 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   ifcif 3926   <.cop 4020   class class class wbr 4437    X. cxp 4987   -->wf 5574   ` cfv 5578  (class class class)co 6281   RR*cxr 9630    < clt 9631    <_ cle 9632   RR+crp 11231   distcds 14688    X.s cxps 14885   *Metcxmt 18382   MetOpencmopn 18387    CnP ccnp 19704    tX ctx 20039  toMetSpctmt 20800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-icc 11547  df-fz 11684  df-fzo 11807  df-seq 12090  df-hash 12388  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-hom 14703  df-cco 14704  df-rest 14802  df-topn 14803  df-0g 14821  df-gsum 14822  df-topgen 14823  df-pt 14824  df-prds 14827  df-xrs 14881  df-qtop 14886  df-imas 14887  df-xps 14889  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-submnd 15946  df-mulg 16039  df-cntz 16334  df-cmn 16779  df-psmet 18390  df-xmet 18391  df-bl 18393  df-mopn 18394  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cn 19706  df-cnp 19707  df-tx 20041  df-hmeo 20234  df-xms 20801  df-tms 20803
This theorem is referenced by:  txmetcn  21029  cxpcn3  23100
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