MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psmetutop Structured version   Unicode version

Theorem psmetutop 20156
Description: The topology induced by a uniform structure generated by a metric  D is generated by that metric's open balls. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
psmetutop  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (unifTop `  (metUnif `  D ) )  =  ( topGen `  ran  ( ball `  D ) ) )

Proof of Theorem psmetutop
Dummy variables  a 
b  d  e  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metuust 20145 . . . . . . . . . . . 12  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  e.  (UnifOn `  X ) )
2 utopval 19805 . . . . . . . . . . . 12  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (unifTop `  (metUnif `  D )
)  =  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  (metUnif `  D ) ( v
" { x }
)  C_  a }
)
31, 2syl 16 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (unifTop `  (metUnif `  D ) )  =  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  (metUnif `  D )
( v " {
x } )  C_  a } )
43eleq2d 2508 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( a  e.  (unifTop `  (metUnif `  D
) )  <->  a  e.  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  (metUnif `  D ) ( v
" { x }
)  C_  a }
) )
5 rabid 2895 . . . . . . . . . 10  |-  ( a  e.  { a  e. 
~P X  |  A. x  e.  a  E. v  e.  (metUnif `  D
) ( v " { x } ) 
C_  a }  <->  ( a  e.  ~P X  /\  A. x  e.  a  E. v  e.  (metUnif `  D
) ( v " { x } ) 
C_  a ) )
64, 5syl6bb 261 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( a  e.  (unifTop `  (metUnif `  D
) )  <->  ( a  e.  ~P X  /\  A. x  e.  a  E. v  e.  (metUnif `  D
) ( v " { x } ) 
C_  a ) ) )
76biimpa 484 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
( a  e.  ~P X  /\  A. x  e.  a  E. v  e.  (metUnif `  D )
( v " {
x } )  C_  a ) )
87simpld 459 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  e.  ~P X
)
98elpwid 3868 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  C_  X )
10 unirnblps 19992 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  U. ran  ( ball `  D )  =  X )
1110ad2antlr 726 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  ->  U. ran  ( ball `  D
)  =  X )
129, 11sseqtr4d 3391 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  C_  U. ran  ( ball `  D ) )
13 simpr 461 . . . . . . . 8  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnif `  D ) )  /\  ( v " {
x } )  C_  a )  ->  (
v " { x } )  C_  a
)
14 simp-5r 768 . . . . . . . . 9  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnif `  D ) )  /\  ( v " {
x } )  C_  a )  ->  D  e.  (PsMet `  X )
)
15 simplr 754 . . . . . . . . 9  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnif `  D ) )  /\  ( v " {
x } )  C_  a )  ->  v  e.  (metUnif `  D )
)
169ad3antrrr 729 . . . . . . . . . 10  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnif `  D ) )  /\  ( v " {
x } )  C_  a )  ->  a  C_  X )
17 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnif `  D ) )  /\  ( v " {
x } )  C_  a )  ->  x  e.  a )
1816, 17sseldd 3355 . . . . . . . . 9  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnif `  D ) )  /\  ( v " {
x } )  C_  a )  ->  x  e.  X )
19 metustbl 20154 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  v  e.  (metUnif `  D )  /\  x  e.  X
)  ->  E. b  e.  ran  ( ball `  D
) ( x  e.  b  /\  b  C_  ( v " {
x } ) ) )
2014, 15, 18, 19syl3anc 1218 . . . . . . . 8  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnif `  D ) )  /\  ( v " {
x } )  C_  a )  ->  E. b  e.  ran  ( ball `  D
) ( x  e.  b  /\  b  C_  ( v " {
x } ) ) )
21 sstr 3362 . . . . . . . . . . 11  |-  ( ( b  C_  ( v " { x } )  /\  ( v " { x } ) 
C_  a )  -> 
b  C_  a )
2221expcom 435 . . . . . . . . . 10  |-  ( ( v " { x } )  C_  a  ->  ( b  C_  (
v " { x } )  ->  b  C_  a ) )
2322anim2d 565 . . . . . . . . 9  |-  ( ( v " { x } )  C_  a  ->  ( ( x  e.  b  /\  b  C_  ( v " {
x } ) )  ->  ( x  e.  b  /\  b  C_  a ) ) )
2423reximdv 2825 . . . . . . . 8  |-  ( ( v " { x } )  C_  a  ->  ( E. b  e. 
ran  ( ball `  D
) ( x  e.  b  /\  b  C_  ( v " {
x } ) )  ->  E. b  e.  ran  ( ball `  D )
( x  e.  b  /\  b  C_  a
) ) )
2513, 20, 24sylc 60 . . . . . . 7  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnif `  D ) )  /\  ( v " {
x } )  C_  a )  ->  E. b  e.  ran  ( ball `  D
) ( x  e.  b  /\  b  C_  a ) )
267simprd 463 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  ->  A. x  e.  a  E. v  e.  (metUnif `  D ) ( v
" { x }
)  C_  a )
2726r19.21bi 2812 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  (unifTop `  (metUnif `  D ) ) )  /\  x  e.  a )  ->  E. v  e.  (metUnif `  D )
( v " {
x } )  C_  a )
2825, 27r19.29a 2860 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  (unifTop `  (metUnif `  D ) ) )  /\  x  e.  a )  ->  E. b  e.  ran  ( ball `  D
) ( x  e.  b  /\  b  C_  a ) )
2928ralrimiva 2797 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  ->  A. x  e.  a  E. b  e.  ran  ( ball `  D )
( x  e.  b  /\  b  C_  a
) )
3012, 29jca 532 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
( a  C_  U. ran  ( ball `  D )  /\  A. x  e.  a  E. b  e.  ran  ( ball `  D )
( x  e.  b  /\  b  C_  a
) ) )
31 fvex 5699 . . . . . . 7  |-  ( ball `  D )  e.  _V
3231rnex 6510 . . . . . 6  |-  ran  ( ball `  D )  e. 
_V
33 eltg2 18561 . . . . . 6  |-  ( ran  ( ball `  D
)  e.  _V  ->  ( a  e.  ( topGen ` 
ran  ( ball `  D
) )  <->  ( a  C_ 
U. ran  ( ball `  D )  /\  A. x  e.  a  E. b  e.  ran  ( ball `  D ) ( x  e.  b  /\  b  C_  a ) ) ) )
3432, 33ax-mp 5 . . . . 5  |-  ( a  e.  ( topGen `  ran  ( ball `  D )
)  <->  ( a  C_  U.
ran  ( ball `  D
)  /\  A. x  e.  a  E. b  e.  ran  ( ball `  D
) ( x  e.  b  /\  b  C_  a ) ) )
3534a1i 11 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
( a  e.  (
topGen `  ran  ( ball `  D ) )  <->  ( a  C_ 
U. ran  ( ball `  D )  /\  A. x  e.  a  E. b  e.  ran  ( ball `  D ) ( x  e.  b  /\  b  C_  a ) ) ) )
3630, 35mpbird 232 . . 3  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  e.  ( topGen ` 
ran  ( ball `  D
) ) )
3734a1i 11 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( a  e.  ( topGen `  ran  ( ball `  D ) )  <->  ( a  C_ 
U. ran  ( ball `  D )  /\  A. x  e.  a  E. b  e.  ran  ( ball `  D ) ( x  e.  b  /\  b  C_  a ) ) ) )
3837biimpa 484 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  ( a  C_  U.
ran  ( ball `  D
)  /\  A. x  e.  a  E. b  e.  ran  ( ball `  D
) ( x  e.  b  /\  b  C_  a ) ) )
3938simpld 459 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  C_  U. ran  ( ball `  D )
)
4010ad2antlr 726 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  U. ran  ( ball `  D )  =  X )
4139, 40sseqtrd 3390 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  C_  X
)
42 elpwg 3866 . . . . . . 7  |-  ( a  e.  ( topGen `  ran  ( ball `  D )
)  ->  ( a  e.  ~P X  <->  a  C_  X ) )
4342adantl 466 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  ( a  e. 
~P X  <->  a  C_  X ) )
4441, 43mpbird 232 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  e.  ~P X )
45 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  D  e.  (PsMet `  X ) )
4641sselda 3354 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  x  e.  X )
4738simprd 463 . . . . . . . . . . 11  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  A. x  e.  a  E. b  e.  ran  ( ball `  D )
( x  e.  b  /\  b  C_  a
) )
4847r19.21bi 2812 . . . . . . . . . 10  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  E. b  e.  ran  ( ball `  D )
( x  e.  b  /\  b  C_  a
) )
49 blssexps 19999 . . . . . . . . . . 11  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  ( E. b  e.  ran  ( ball `  D )
( x  e.  b  /\  b  C_  a
)  <->  E. d  e.  RR+  ( x ( ball `  D ) d ) 
C_  a ) )
5045, 46, 49syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  ( E. b  e. 
ran  ( ball `  D
) ( x  e.  b  /\  b  C_  a )  <->  E. d  e.  RR+  ( x (
ball `  D )
d )  C_  a
) )
5148, 50mpbid 210 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  E. d  e.  RR+  ( x ( ball `  D ) d ) 
C_  a )
52 blval2 20148 . . . . . . . . . . . . 13  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
53523expa 1187 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
5453sseq1d 3381 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
( x ( ball `  D ) d ) 
C_  a  <->  ( ( `' D " ( 0 [,) d ) )
" { x }
)  C_  a )
)
5554rexbidva 2730 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  ( E. d  e.  RR+  (
x ( ball `  D
) d )  C_  a 
<->  E. d  e.  RR+  ( ( `' D " ( 0 [,) d
) ) " {
x } )  C_  a ) )
5655biimpa 484 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  /\  E. d  e.  RR+  ( x ( ball `  D
) d )  C_  a )  ->  E. d  e.  RR+  ( ( `' D " ( 0 [,) d ) )
" { x }
)  C_  a )
5745, 46, 51, 56syl21anc 1217 . . . . . . . 8  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  E. d  e.  RR+  ( ( `' D " ( 0 [,) d
) ) " {
x } )  C_  a )
58 cnvexg 6522 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
59 imaexg 6513 . . . . . . . . . . 11  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
6058, 59syl 16 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) d ) )  e. 
_V )
6160ralrimivw 2798 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  A. d  e.  RR+  ( `' D " ( 0 [,) d
) )  e.  _V )
62 eqid 2441 . . . . . . . . . 10  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
63 imaeq1 5162 . . . . . . . . . . 11  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
v " { x } )  =  ( ( `' D "
( 0 [,) d
) ) " {
x } ) )
6463sseq1d 3381 . . . . . . . . . 10  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
( v " {
x } )  C_  a 
<->  ( ( `' D " ( 0 [,) d
) ) " {
x } )  C_  a ) )
6562, 64rexrnmpt 5851 . . . . . . . . 9  |-  ( A. d  e.  RR+  ( `' D " ( 0 [,) d ) )  e.  _V  ->  ( E. v  e.  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) 
C_  a  <->  E. d  e.  RR+  ( ( `' D " ( 0 [,) d ) )
" { x }
)  C_  a )
)
6645, 61, 653syl 20 . . . . . . . 8  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  ( E. v  e. 
ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ( v " { x } )  C_  a  <->  E. d  e.  RR+  (
( `' D "
( 0 [,) d
) ) " {
x } )  C_  a ) )
6757, 66mpbird 232 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  E. v  e.  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) 
C_  a )
68 oveq2 6097 . . . . . . . . . . . . . . 15  |-  ( d  =  e  ->  (
0 [,) d )  =  ( 0 [,) e ) )
6968imaeq2d 5167 . . . . . . . . . . . . . 14  |-  ( d  =  e  ->  ( `' D " ( 0 [,) d ) )  =  ( `' D " ( 0 [,) e
) ) )
7069cbvmptv 4381 . . . . . . . . . . . . 13  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7170rneqi 5064 . . . . . . . . . . . 12  |-  ran  (
d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ran  (
e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7271metustfbas 20139 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  e.  ( fBas `  ( X  X.  X
) ) )
73 ssfg 19443 . . . . . . . . . . 11  |-  ( ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  e.  ( fBas `  ( X  X.  X
) )  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  ( ( X  X.  X ) filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) ) ) )
7472, 73syl 16 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  ( ( X  X.  X ) filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) ) ) )
75 metuval 20123 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
7675adantl 466 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
7774, 76sseqtr4d 3391 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  (metUnif `  D
) )
78 ssrexv 3415 . . . . . . . . 9  |-  ( ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  (metUnif `  D
)  ->  ( E. v  e.  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) 
C_  a  ->  E. v  e.  (metUnif `  D )
( v " {
x } )  C_  a ) )
7977, 78syl 16 . . . . . . . 8  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( E. v  e.  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) 
C_  a  ->  E. v  e.  (metUnif `  D )
( v " {
x } )  C_  a ) )
8079ad2antrr 725 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  ( E. v  e. 
ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ( v " { x } )  C_  a  ->  E. v  e.  (metUnif `  D ) ( v
" { x }
)  C_  a )
)
8167, 80mpd 15 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  E. v  e.  (metUnif `  D ) ( v
" { x }
)  C_  a )
8281ralrimiva 2797 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  A. x  e.  a  E. v  e.  (metUnif `  D ) ( v
" { x }
)  C_  a )
8344, 82jca 532 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  ( a  e. 
~P X  /\  A. x  e.  a  E. v  e.  (metUnif `  D
) ( v " { x } ) 
C_  a ) )
846biimpar 485 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( a  e.  ~P X  /\  A. x  e.  a  E. v  e.  (metUnif `  D
) ( v " { x } ) 
C_  a ) )  ->  a  e.  (unifTop `  (metUnif `  D )
) )
8583, 84syldan 470 . . 3  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  e.  (unifTop `  (metUnif `  D )
) )
8636, 85impbida 828 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( a  e.  (unifTop `  (metUnif `  D
) )  <->  a  e.  ( topGen `  ran  ( ball `  D ) ) ) )
8786eqrdv 2439 1  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (unifTop `  (metUnif `  D ) )  =  ( topGen `  ran  ( ball `  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970    C_ wss 3326   (/)c0 3635   ~Pcpw 3858   {csn 3875   U.cuni 4089    e. cmpt 4348    X. cxp 4836   `'ccnv 4837   ran crn 4839   "cima 4841   ` cfv 5416  (class class class)co 6089   0cc0 9280   RR+crp 10989   [,)cico 11300   topGenctg 14374  PsMetcpsmet 17798   ballcbl 17801   fBascfbas 17802   filGencfg 17803  metUnifcmetu 17806  UnifOncust 19772  unifTopcutop 19803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-q 10952  df-rp 10990  df-xneg 11087  df-xadd 11088  df-xmul 11089  df-ico 11304  df-topgen 14380  df-psmet 17807  df-bl 17810  df-fbas 17812  df-fg 17813  df-metu 17815  df-fil 19417  df-ust 19773  df-utop 19804
This theorem is referenced by:  xmetutop  20157
  Copyright terms: Public domain W3C validator