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Theorem txmetcnp 18109
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 2296 . . . 4  |-  ( dist `  ( (toMetSp `  C
)  X.s  (toMetSp `  D )
) )  =  (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) )
2 simpl1 958 . . . 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 959 . . . 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 18098 . . 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 960 . . 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 4737 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
76adantl 452 . . 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 2296 . . . 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 18103 . . 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 1182 . 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 18099 . . . . 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 5889 . . . 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 5543 . . 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 2363 . 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 5882 . . . . . . . . 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 4049 . . . . . . . 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 5877 . . . . . . . . . . 11  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2120oveq1i 5884 . . . . . . . . . 10  |-  ( ( A F B ) E ( F `  x ) )  =  ( ( F `  <. A ,  B >. ) E ( F `  x ) )
22 fveq2 5541 . . . . . . . . . . . 12  |-  ( x  =  <. u ,  v
>.  ->  ( F `  x )  =  ( F `  <. u ,  v >. )
)
23 df-ov 5877 . . . . . . . . . . . 12  |-  ( u F v )  =  ( F `  <. u ,  v >. )
2422, 23syl6eqr 2346 . . . . . . . . . . 11  |-  ( x  =  <. u ,  v
>.  ->  ( F `  x )  =  ( u F v ) )
2524oveq2d 5890 . . . . . . . . . 10  |-  ( x  =  <. u ,  v
>.  ->  ( ( A F B ) E ( F `  x
) )  =  ( ( A F B ) E ( u F v ) ) )
2621, 25syl5eqr 2342 . . . . . . . . 9  |-  ( x  =  <. u ,  v
>.  ->  ( ( F `
 <. A ,  B >. ) E ( F `
 x ) )  =  ( ( A F B ) E ( u F v ) ) )
2726breq1d 4049 . . . . . . . 8  |-  ( x  =  <. u ,  v
>.  ->  ( ( ( F `  <. A ,  B >. ) E ( F `  x ) )  <  z  <->  ( ( A F B ) E ( u F v ) )  <  z
) )
2819, 27imbi12d 311 . . . . . . 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 4843 . . . . . 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 706 . . . . . . . . . . . 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 706 . . . . . . . . . . . 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 735 . . . . . . . . . . . . 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 445 . . . . . . . . . . . 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 449 . . . . . . . . . . . 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 740 . . . . . . . . . . . 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 741 . . . . . . . . . . . 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 18101 . . . . . . . . . . 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 4049 . . . . . . . . . 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 17912 . . . . . . . . . . . 12  |-  ( ( C  e.  ( * Met `  X )  /\  A  e.  X  /\  u  e.  X
)  ->  ( A C u )  e. 
RR* )
4030, 33, 35, 39syl3anc 1182 . . . . . . . . . . 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 17912 . . . . . . . . . . . 12  |-  ( ( D  e.  ( * Met `  Y )  /\  B  e.  Y  /\  v  e.  Y
)  ->  ( B D v )  e. 
RR* )
4231, 34, 36, 41syl3anc 1182 . . . . . . . . . . 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 10377 . . . . . . . . . . . 12  |-  ( w  e.  RR+  ->  w  e. 
RR* )
4443ad2antrl 708 . . . . . . . . . . 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 10526 . . . . . . . . . . 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 1182 . . . . . . . . . 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 244 . . . . . . . . 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 308 . . . . . . . 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 629 . . . . . . 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 2596 . . . . . 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 248 . . . . 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 2573 . . . 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 2576 . . 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 622 . 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 274 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   ifcif 3578   <.cop 3656   class class class wbr 4039    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   RR*cxr 8882    < clt 8883    <_ cle 8884   RR+crp 10370   distcds 13233    X.s cxps 13425   * Metcxmt 16385   MetOpencmopn 16388    CnP ccnp 16971    tX ctx 17271  toMetSpctmt 17900
This theorem is referenced by:  txmetcn  18110  cxpcn3  20104
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-cnp 16974  df-tx 17273  df-hmeo 17462  df-xms 17901  df-tms 17903
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