MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metdseq0 Unicode version

Theorem metdseq0 18358
Description: The distance from a point to a set is zero iff the point is in the closure set. (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypotheses
Ref Expression
metdscn.f  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
metdscn.j  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
metdseq0  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( ( F `  A )  =  0  <->  A  e.  ( ( cls `  J
) `  S )
) )
Distinct variable groups:    x, y, A    x, D, y    y, J    x, S, y    x, X, y
Allowed substitution hints:    F( x, y)    J( x)

Proof of Theorem metdseq0
Dummy variables  r 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll1 994 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z ) )  ->  D  e.  ( * Met `  X ) )
2 simprl 732 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z ) )  -> 
z  e.  J )
3 simprr 733 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z ) )  ->  A  e.  z )
4 metdscn.j . . . . . . . 8  |-  J  =  ( MetOpen `  D )
54mopni2 18039 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  z  e.  J  /\  A  e.  z
)  ->  E. r  e.  RR+  ( A (
ball `  D )
r )  C_  z
)
61, 2, 3, 5syl3anc 1182 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z ) )  ->  E. r  e.  RR+  ( A ( ball `  D
) r )  C_  z )
7 simprr 733 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  -> 
( A ( ball `  D ) r ) 
C_  z )
8 ssrin 3394 . . . . . . . . . 10  |-  ( ( A ( ball `  D
) r )  C_  z  ->  ( ( A ( ball `  D
) r )  i^i 
S )  C_  (
z  i^i  S )
)
97, 8syl 15 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  -> 
( ( A (
ball `  D )
r )  i^i  S
)  C_  ( z  i^i  S ) )
10 rpgt0 10365 . . . . . . . . . . . 12  |-  ( r  e.  RR+  ->  0  < 
r )
11 0re 8838 . . . . . . . . . . . . 13  |-  0  e.  RR
12 rpre 10360 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  r  e.  RR )
13 ltnle 8902 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  r  e.  RR )  ->  ( 0  <  r  <->  -.  r  <_  0 ) )
1411, 12, 13sylancr 644 . . . . . . . . . . . 12  |-  ( r  e.  RR+  ->  ( 0  <  r  <->  -.  r  <_  0 ) )
1510, 14mpbid 201 . . . . . . . . . . 11  |-  ( r  e.  RR+  ->  -.  r  <_  0 )
1615ad2antrl 708 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  ->  -.  r  <_  0 )
17 simpllr 735 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  -> 
( F `  A
)  =  0 )
1817breq2d 4035 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  -> 
( r  <_  ( F `  A )  <->  r  <_  0 ) )
191adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  ->  D  e.  ( * Met `  X ) )
20 simpl2 959 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  S  C_  X
)
2120ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  ->  S  C_  X )
22 simpl3 960 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  A  e.  X )
2322ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  ->  A  e.  X )
24 rpxr 10361 . . . . . . . . . . . . . . 15  |-  ( r  e.  RR+  ->  r  e. 
RR* )
2524ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  -> 
r  e.  RR* )
26 metdscn.f . . . . . . . . . . . . . . 15  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
2726metdsge 18353 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  r  e.  RR* )  ->  ( r  <_  ( F `  A
)  <->  ( S  i^i  ( A ( ball `  D
) r ) )  =  (/) ) )
2819, 21, 23, 25, 27syl31anc 1185 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  -> 
( r  <_  ( F `  A )  <->  ( S  i^i  ( A ( ball `  D
) r ) )  =  (/) ) )
2918, 28bitr3d 246 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  -> 
( r  <_  0  <->  ( S  i^i  ( A ( ball `  D
) r ) )  =  (/) ) )
30 incom 3361 . . . . . . . . . . . . 13  |-  ( S  i^i  ( A (
ball `  D )
r ) )  =  ( ( A (
ball `  D )
r )  i^i  S
)
3130eqeq1i 2290 . . . . . . . . . . . 12  |-  ( ( S  i^i  ( A ( ball `  D
) r ) )  =  (/)  <->  ( ( A ( ball `  D
) r )  i^i 
S )  =  (/) )
3229, 31syl6bb 252 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  -> 
( r  <_  0  <->  ( ( A ( ball `  D ) r )  i^i  S )  =  (/) ) )
3332necon3bbid 2480 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  -> 
( -.  r  <_ 
0  <->  ( ( A ( ball `  D
) r )  i^i 
S )  =/=  (/) ) )
3416, 33mpbid 201 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  -> 
( ( A (
ball `  D )
r )  i^i  S
)  =/=  (/) )
35 ssn0 3487 . . . . . . . . 9  |-  ( ( ( ( A (
ball `  D )
r )  i^i  S
)  C_  ( z  i^i  S )  /\  (
( A ( ball `  D ) r )  i^i  S )  =/=  (/) )  ->  ( z  i^i  S )  =/=  (/) )
369, 34, 35syl2anc 642 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  (
r  e.  RR+  /\  ( A ( ball `  D
) r )  C_  z ) )  -> 
( z  i^i  S
)  =/=  (/) )
3736expr 598 . . . . . . 7  |-  ( ( ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X )  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z
) )  /\  r  e.  RR+ )  ->  (
( A ( ball `  D ) r ) 
C_  z  ->  (
z  i^i  S )  =/=  (/) ) )
3837rexlimdva 2667 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z ) )  -> 
( E. r  e.  RR+  ( A ( ball `  D ) r ) 
C_  z  ->  (
z  i^i  S )  =/=  (/) ) )
396, 38mpd 14 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z ) )  -> 
( z  i^i  S
)  =/=  (/) )
4039expr 598 . . . 4  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  /\  z  e.  J )  ->  ( A  e.  z  ->  ( z  i^i  S )  =/=  (/) ) )
4140ralrimiva 2626 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  A. z  e.  J  ( A  e.  z  ->  ( z  i^i  S )  =/=  (/) ) )
424mopntopon 17985 . . . . . . 7  |-  ( D  e.  ( * Met `  X )  ->  J  e.  (TopOn `  X )
)
43423ad2ant1 976 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  J  e.  (TopOn `  X ) )
4443adantr 451 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  J  e.  (TopOn `  X ) )
45 topontop 16664 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
4644, 45syl 15 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  J  e.  Top )
47 toponuni 16665 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
4844, 47syl 15 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  X  =  U. J )
4920, 48sseqtrd 3214 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  S  C_  U. J
)
5022, 48eleqtrd 2359 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  A  e.  U. J )
51 eqid 2283 . . . . 5  |-  U. J  =  U. J
5251elcls 16810 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  U. J  /\  A  e.  U. J )  ->  ( A  e.  ( ( cls `  J
) `  S )  <->  A. z  e.  J  ( A  e.  z  -> 
( z  i^i  S
)  =/=  (/) ) ) )
5346, 49, 50, 52syl3anc 1182 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  ( A  e.  ( ( cls `  J
) `  S )  <->  A. z  e.  J  ( A  e.  z  -> 
( z  i^i  S
)  =/=  (/) ) ) )
5441, 53mpbird 223 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  A  e.  ( ( cls `  J
) `  S )
)
55 incom 3361 . . . . . . 7  |-  ( ( A ( ball `  D
) ( F `  A ) )  i^i 
S )  =  ( S  i^i  ( A ( ball `  D
) ( F `  A ) ) )
5626metdsf 18352 . . . . . . . . . . . 12  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
57 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( F : X --> ( 0 [,]  +oo )  /\  A  e.  X )  ->  ( F `  A )  e.  ( 0 [,]  +oo ) )
5856, 57sylan 457 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  A  e.  X )  ->  ( F `  A )  e.  ( 0 [,]  +oo ) )
59583impa 1146 . . . . . . . . . 10  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( F `  A )  e.  ( 0 [,]  +oo )
)
60 elxrge0 10747 . . . . . . . . . . 11  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  A )  e.  RR*  /\  0  <_ 
( F `  A
) ) )
6160simplbi 446 . . . . . . . . . 10  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  ( F `  A )  e.  RR* )
6259, 61syl 15 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( F `  A )  e.  RR* )
63 xrleid 10484 . . . . . . . . 9  |-  ( ( F `  A )  e.  RR*  ->  ( F `
 A )  <_ 
( F `  A
) )
6462, 63syl 15 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( F `  A )  <_  ( F `  A )
)
6526metdsge 18353 . . . . . . . . 9  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  e.  RR* )  ->  ( ( F `
 A )  <_ 
( F `  A
)  <->  ( S  i^i  ( A ( ball `  D
) ( F `  A ) ) )  =  (/) ) )
6662, 65mpdan 649 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( ( F `  A )  <_  ( F `  A
)  <->  ( S  i^i  ( A ( ball `  D
) ( F `  A ) ) )  =  (/) ) )
6764, 66mpbid 201 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( S  i^i  ( A ( ball `  D ) ( F `
 A ) ) )  =  (/) )
6855, 67syl5eq 2327 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( ( A ( ball `  D
) ( F `  A ) )  i^i 
S )  =  (/) )
6968adantr 451 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  ( ( A ( ball `  D
) ( F `  A ) )  i^i 
S )  =  (/) )
7043ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  J  e.  (TopOn `  X ) )
7170, 45syl 15 . . . . . . . 8  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  J  e.  Top )
72 simpll2 995 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  S  C_  X
)
7370, 47syl 15 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  X  =  U. J )
7472, 73sseqtrd 3214 . . . . . . . 8  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  S  C_  U. J
)
75 simplr 731 . . . . . . . 8  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  A  e.  ( ( cls `  J
) `  S )
)
76 simpll1 994 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  D  e.  ( * Met `  X
) )
77 simpll3 996 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  A  e.  X )
7862ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  ( F `  A )  e.  RR* )
794blopn 18046 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  ( F `  A
)  e.  RR* )  ->  ( A ( ball `  D ) ( F `
 A ) )  e.  J )
8076, 77, 78, 79syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  ( A
( ball `  D )
( F `  A
) )  e.  J
)
81 simpr 447 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  0  <  ( F `  A ) )
82 xblcntr 17963 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  ( ( F `  A )  e.  RR*  /\  0  <  ( F `
 A ) ) )  ->  A  e.  ( A ( ball `  D
) ( F `  A ) ) )
8376, 77, 78, 81, 82syl112anc 1186 . . . . . . . 8  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  A  e.  ( A ( ball `  D
) ( F `  A ) ) )
8451clsndisj 16812 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  U. J  /\  A  e.  ( ( cls `  J ) `  S ) )  /\  ( ( A (
ball `  D )
( F `  A
) )  e.  J  /\  A  e.  ( A ( ball `  D
) ( F `  A ) ) ) )  ->  ( ( A ( ball `  D
) ( F `  A ) )  i^i 
S )  =/=  (/) )
8571, 74, 75, 80, 83, 84syl32anc 1190 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  ( ( A ( ball `  D
) ( F `  A ) )  i^i 
S )  =/=  (/) )
8685ex 423 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  ( 0  <  ( F `  A )  ->  (
( A ( ball `  D ) ( F `
 A ) )  i^i  S )  =/=  (/) ) )
8786necon2bd 2495 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  ( (
( A ( ball `  D ) ( F `
 A ) )  i^i  S )  =  (/)  ->  -.  0  <  ( F `  A ) ) )
8869, 87mpd 14 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  -.  0  <  ( F `  A
) )
8960simprbi 450 . . . . . . . 8  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  0  <_  ( F `  A
) )
9059, 89syl 15 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  0  <_  ( F `  A ) )
91 0xr 8878 . . . . . . . 8  |-  0  e.  RR*
92 xrleloe 10478 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  (
0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
9391, 62, 92sylancr 644 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( 0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
9490, 93mpbid 201 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) )
9594adantr 451 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) )
9695ord 366 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  ( -.  0  <  ( F `  A )  ->  0  =  ( F `  A ) ) )
9788, 96mpd 14 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  0  =  ( F `  A ) )
9897eqcomd 2288 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  ( F `  A )  =  0 )
9954, 98impbida 805 1  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( ( F `  A )  =  0  <->  A  e.  ( ( cls `  J
) `  S )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455   U.cuni 3827   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   RR+crp 10354   [,]cicc 10659   * Metcxmt 16369   ballcbl 16371   MetOpencmopn 16372   Topctop 16631  TopOnctopon 16632   clsccl 16755
This theorem is referenced by:  metnrmlem1a  18362  lebnumlem1  18459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-topgen 13344  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758
  Copyright terms: Public domain W3C validator