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Theorem txmetcnp 18578
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 2437 . . . 4  |-  ( dist `  ( (toMetSp `  C
)  X.s  (toMetSp `  D )
) )  =  (
dist `  ( (toMetSp `  C )  X.s  (toMetSp `  D
) ) )
2 simpl1 961 . . . 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 962 . . . 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 18567 . . 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 963 . . 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 4911 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
76adantl 454 . . 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 2437 . . . 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 18572 . . 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 1185 . 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 18568 . . . . 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 6097 . . . 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 5731 . . 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 2504 . 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 6090 . . . . . . . . 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 4223 . . . . . . . 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 6085 . . . . . . . . . . 11  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2120oveq1i 6092 . . . . . . . . . 10  |-  ( ( A F B ) E ( F `  x ) )  =  ( ( F `  <. A ,  B >. ) E ( F `  x ) )
22 fveq2 5729 . . . . . . . . . . . 12  |-  ( x  =  <. u ,  v
>.  ->  ( F `  x )  =  ( F `  <. u ,  v >. )
)
23 df-ov 6085 . . . . . . . . . . . 12  |-  ( u F v )  =  ( F `  <. u ,  v >. )
2422, 23syl6eqr 2487 . . . . . . . . . . 11  |-  ( x  =  <. u ,  v
>.  ->  ( F `  x )  =  ( u F v ) )
2524oveq2d 6098 . . . . . . . . . 10  |-  ( x  =  <. u ,  v
>.  ->  ( ( A F B ) E ( F `  x
) )  =  ( ( A F B ) E ( u F v ) ) )
2621, 25syl5eqr 2483 . . . . . . . . 9  |-  ( x  =  <. u ,  v
>.  ->  ( ( F `
 <. A ,  B >. ) E ( F `
 x ) )  =  ( ( A F B ) E ( u F v ) ) )
2726breq1d 4223 . . . . . . . 8  |-  ( x  =  <. u ,  v
>.  ->  ( ( ( F `  <. A ,  B >. ) E ( F `  x ) )  <  z  <->  ( ( A F B ) E ( u F v ) )  <  z
) )
2819, 27imbi12d 313 . . . . . . 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 5017 . . . . . 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 708 . . . . . . . . . . . 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 708 . . . . . . . . . . . 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 737 . . . . . . . . . . . . 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 447 . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 742 . . . . . . . . . . . 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 743 . . . . . . . . . . . 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 18570 . . . . . . . . . . 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 4223 . . . . . . . . . 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 18362 . . . . . . . . . . . 12  |-  ( ( C  e.  ( * Met `  X )  /\  A  e.  X  /\  u  e.  X
)  ->  ( A C u )  e. 
RR* )
4030, 33, 35, 39syl3anc 1185 . . . . . . . . . . 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 18362 . . . . . . . . . . . 12  |-  ( ( D  e.  ( * Met `  Y )  /\  B  e.  Y  /\  v  e.  Y
)  ->  ( B D v )  e. 
RR* )
4231, 34, 36, 41syl3anc 1185 . . . . . . . . . . 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 10620 . . . . . . . . . . . 12  |-  ( w  e.  RR+  ->  w  e. 
RR* )
4443ad2antrl 710 . . . . . . . . . . 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 10770 . . . . . . . . . . 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 1185 . . . . . . . . . 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 246 . . . . . . . . 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 310 . . . . . . . 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 631 . . . . . . 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 2746 . . . . . 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 250 . . . . 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 2723 . . . 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 2726 . . 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 624 . 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 276 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707   ifcif 3740   <.cop 3818   class class class wbr 4213    X. cxp 4877   -->wf 5451   ` cfv 5455  (class class class)co 6082   RR*cxr 9120    < clt 9121    <_ cle 9122   RR+crp 10613   distcds 13539    X.s cxps 13733   * Metcxmt 16687   MetOpencmopn 16692    CnP ccnp 17290    tX ctx 17593  toMetSpctmt 18350
This theorem is referenced by:  txmetcn  18579  cxpcn3  20633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-map 7021  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-fi 7417  df-sup 7447  df-oi 7480  df-card 7827  df-cda 8049  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-icc 10924  df-fz 11045  df-fzo 11137  df-seq 11325  df-hash 11620  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-sca 13546  df-vsca 13547  df-tset 13549  df-ple 13550  df-ds 13552  df-hom 13554  df-cco 13555  df-rest 13651  df-topn 13652  df-topgen 13668  df-pt 13669  df-prds 13672  df-xrs 13727  df-0g 13728  df-gsum 13729  df-qtop 13734  df-imas 13735  df-xps 13737  df-mre 13812  df-mrc 13813  df-acs 13815  df-mnd 14691  df-submnd 14740  df-mulg 14816  df-cntz 15117  df-cmn 15415  df-psmet 16695  df-xmet 16696  df-bl 16698  df-mopn 16699  df-top 16964  df-bases 16966  df-topon 16967  df-topsp 16968  df-cn 17292  df-cnp 17293  df-tx 17595  df-hmeo 17788  df-xms 18351  df-tms 18353
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