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Theorem txmetcnp 21549
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 2422 . . . 4  |-  ( dist `  ( (toMetSp `  C
)  X.s  (toMetSp `  D )
) )  =  (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) )
2 simpl1 1008 . . . 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 1009 . . . 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 21538 . . 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 1010 . . 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 4882 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
76adantl 467 . . 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 2422 . . . 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 21543 . . 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 1264 . 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 21539 . . . . 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 6317 . . . 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 5880 . . 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 2492 . 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 6310 . . . . . . . . 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 4430 . . . . . . . 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 6305 . . . . . . . . . . 11  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2120oveq1i 6312 . . . . . . . . . 10  |-  ( ( A F B ) E ( F `  x ) )  =  ( ( F `  <. A ,  B >. ) E ( F `  x ) )
22 fveq2 5878 . . . . . . . . . . . 12  |-  ( x  =  <. u ,  v
>.  ->  ( F `  x )  =  ( F `  <. u ,  v >. )
)
23 df-ov 6305 . . . . . . . . . . . 12  |-  ( u F v )  =  ( F `  <. u ,  v >. )
2422, 23syl6eqr 2481 . . . . . . . . . . 11  |-  ( x  =  <. u ,  v
>.  ->  ( F `  x )  =  ( u F v ) )
2524oveq2d 6318 . . . . . . . . . 10  |-  ( x  =  <. u ,  v
>.  ->  ( ( A F B ) E ( F `  x
) )  =  ( ( A F B ) E ( u F v ) ) )
2621, 25syl5eqr 2477 . . . . . . . . 9  |-  ( x  =  <. u ,  v
>.  ->  ( ( F `
 <. A ,  B >. ) E ( F `
 x ) )  =  ( ( A F B ) E ( u F v ) ) )
2726breq1d 4430 . . . . . . . 8  |-  ( x  =  <. u ,  v
>.  ->  ( ( ( F `  <. A ,  B >. ) E ( F `  x ) )  <  z  <->  ( ( A F B ) E ( u F v ) )  <  z
) )
2819, 27imbi12d 321 . . . . . . 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 4992 . . . . . 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 730 . . . . . . . . . . . 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 730 . . . . . . . . . . . 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 767 . . . . . . . . . . . . 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 460 . . . . . . . . . . . 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 464 . . . . . . . . . . . 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 772 . . . . . . . . . . . 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 773 . . . . . . . . . . . 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 21541 . . . . . . . . . . 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 4430 . . . . . . . . . 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 21333 . . . . . . . . . . . 12  |-  ( ( C  e.  ( *Met `  X )  /\  A  e.  X  /\  u  e.  X
)  ->  ( A C u )  e. 
RR* )
4030, 33, 35, 39syl3anc 1264 . . . . . . . . . . 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 21333 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  Y )  /\  B  e.  Y  /\  v  e.  Y
)  ->  ( B D v )  e. 
RR* )
4231, 34, 36, 41syl3anc 1264 . . . . . . . . . . 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 11310 . . . . . . . . . . . 12  |-  ( w  e.  RR+  ->  w  e. 
RR* )
4443ad2antrl 732 . . . . . . . . . . 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 11477 . . . . . . . . . . 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 1264 . . . . . . . . . 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 256 . . . . . . . . 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 318 . . . . . . . 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 652 . . . . . . 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 2867 . . . . . 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 260 . . . . 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 2936 . . . 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 2864 . . 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 645 . 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 286 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776   ifcif 3909   <.cop 4002   class class class wbr 4420    X. cxp 4848   -->wf 5594   ` cfv 5598  (class class class)co 6302   RR*cxr 9675    < clt 9676    <_ cle 9677   RR+crp 11303   distcds 15187    X.s cxps 15393   *Metcxmt 18943   MetOpencmopn 18948    CnP ccnp 20228    tX ctx 20562  toMetSpctmt 21321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-icc 11643  df-fz 11786  df-fzo 11917  df-seq 12214  df-hash 12516  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-pt 15331  df-prds 15334  df-xrs 15388  df-qtop 15394  df-imas 15395  df-xps 15398  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-mulg 16664  df-cntz 16959  df-cmn 17420  df-psmet 18950  df-xmet 18951  df-bl 18953  df-mopn 18954  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-cn 20230  df-cnp 20231  df-tx 20564  df-hmeo 20757  df-xms 21322  df-tms 21324
This theorem is referenced by:  txmetcn  21550  cxpcn3  23675
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