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Theorem metdseq0 18837
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 996 . . . . . . 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 733 . . . . . . 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 734 . . . . . . 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 18476 . . . . . . 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 1184 . . . . . 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 734 . . . . . . . 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 ) )  -> 
( A ( ball `  D ) r ) 
C_  z )
8 ssrin 3526 . . . . . . . 8  |-  ( ( A ( ball `  D
) r )  C_  z  ->  ( ( A ( ball `  D
) r )  i^i 
S )  C_  (
z  i^i  S )
)
97, 8syl 16 . . . . . . 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 ) )  -> 
( ( A (
ball `  D )
r )  i^i  S
)  C_  ( z  i^i  S ) )
10 rpgt0 10579 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  0  < 
r )
11 0re 9047 . . . . . . . . . . 11  |-  0  e.  RR
12 rpre 10574 . . . . . . . . . . 11  |-  ( r  e.  RR+  ->  r  e.  RR )
13 ltnle 9111 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  r  e.  RR )  ->  ( 0  <  r  <->  -.  r  <_  0 ) )
1411, 12, 13sylancr 645 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  ( 0  <  r  <->  -.  r  <_  0 ) )
1510, 14mpbid 202 . . . . . . . . 9  |-  ( r  e.  RR+  ->  -.  r  <_  0 )
1615ad2antrl 709 . . . . . . . 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 ) )  ->  -.  r  <_  0 )
17 simpllr 736 . . . . . . . . . . . 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 ) )  -> 
( F `  A
)  =  0 )
1817breq2d 4184 . . . . . . . . . . 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  <_  ( F `  A )  <->  r  <_  0 ) )
191adantr 452 . . . . . . . . . . . 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 ) )  ->  D  e.  ( * Met `  X ) )
20 simpl2 961 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  S  C_  X
)
2120ad2antrr 707 . . . . . . . . . . . 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 ) )  ->  S  C_  X )
22 simpl3 962 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  A  e.  X )
2322ad2antrr 707 . . . . . . . . . . . 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 ) )  ->  A  e.  X )
24 rpxr 10575 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  r  e. 
RR* )
2524ad2antrl 709 . . . . . . . . . . . 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  e.  RR* )
26 metdscn.f . . . . . . . . . . . . 13  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
2726metdsge 18832 . . . . . . . . . . . 12  |-  ( ( ( 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 1187 . . . . . . . . . . 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  <_  ( F `  A )  <->  ( S  i^i  ( A ( ball `  D
) r ) )  =  (/) ) )
2918, 28bitr3d 247 . . . . . . . . . 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  <->  ( S  i^i  ( A ( ball `  D
) r ) )  =  (/) ) )
30 incom 3493 . . . . . . . . . . 11  |-  ( S  i^i  ( A (
ball `  D )
r ) )  =  ( ( A (
ball `  D )
r )  i^i  S
)
3130eqeq1i 2411 . . . . . . . . . 10  |-  ( ( S  i^i  ( A ( ball `  D
) r ) )  =  (/)  <->  ( ( A ( ball `  D
) r )  i^i 
S )  =  (/) )
3229, 31syl6bb 253 . . . . . . . . 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 ) )  -> 
( r  <_  0  <->  ( ( A ( ball `  D ) r )  i^i  S )  =  (/) ) )
3332necon3bbid 2601 . . . . . . . 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 ) )  -> 
( -.  r  <_ 
0  <->  ( ( A ( ball `  D
) r )  i^i 
S )  =/=  (/) ) )
3416, 33mpbid 202 . . . . . . 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 ) )  -> 
( ( A (
ball `  D )
r )  i^i  S
)  =/=  (/) )
35 ssn0 3620 . . . . . . 7  |-  ( ( ( ( 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 643 . . . . . 6  |-  ( ( ( ( ( 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
)  =/=  (/) )
376, 36rexlimddv 2794 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  /\  ( z  e.  J  /\  A  e.  z ) )  -> 
( z  i^i  S
)  =/=  (/) )
3837expr 599 . . . 4  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  /\  z  e.  J )  ->  ( A  e.  z  ->  ( z  i^i  S )  =/=  (/) ) )
3938ralrimiva 2749 . . 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 )  =/=  (/) ) )
404mopntopon 18422 . . . . . . 7  |-  ( D  e.  ( * Met `  X )  ->  J  e.  (TopOn `  X )
)
41403ad2ant1 978 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  J  e.  (TopOn `  X ) )
4241adantr 452 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  J  e.  (TopOn `  X ) )
43 topontop 16946 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
4442, 43syl 16 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  J  e.  Top )
45 toponuni 16947 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
4642, 45syl 16 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  X  =  U. J )
4720, 46sseqtrd 3344 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  S  C_  U. J
)
4822, 46eleqtrd 2480 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  A  e.  U. J )
49 eqid 2404 . . . . 5  |-  U. J  =  U. J
5049elcls 17092 . . . 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
)  =/=  (/) ) ) )
5144, 47, 48, 50syl3anc 1184 . . 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
)  =/=  (/) ) ) )
5239, 51mpbird 224 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  ( F `  A )  =  0 )  ->  A  e.  ( ( cls `  J
) `  S )
)
53 incom 3493 . . . . . . 7  |-  ( ( A ( ball `  D
) ( F `  A ) )  i^i 
S )  =  ( S  i^i  ( A ( ball `  D
) ( F `  A ) ) )
5426metdsf 18831 . . . . . . . . . . . 12  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  F : X
--> ( 0 [,]  +oo ) )
5554ffvelrnda 5829 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  A  e.  X )  ->  ( F `  A )  e.  ( 0 [,]  +oo ) )
56553impa 1148 . . . . . . . . . 10  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( F `  A )  e.  ( 0 [,]  +oo )
)
57 elxrge0 10964 . . . . . . . . . . 11  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  <->  ( ( F `  A )  e.  RR*  /\  0  <_ 
( F `  A
) ) )
5857simplbi 447 . . . . . . . . . 10  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  ( F `  A )  e.  RR* )
5956, 58syl 16 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( F `  A )  e.  RR* )
60 xrleid 10699 . . . . . . . . 9  |-  ( ( F `  A )  e.  RR*  ->  ( F `
 A )  <_ 
( F `  A
) )
6159, 60syl 16 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( F `  A )  <_  ( F `  A )
)
6226metdsge 18832 . . . . . . . . 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 ) ) )  =  (/) ) )
6359, 62mpdan 650 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( ( F `  A )  <_  ( F `  A
)  <->  ( S  i^i  ( A ( ball `  D
) ( F `  A ) ) )  =  (/) ) )
6461, 63mpbid 202 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( S  i^i  ( A ( ball `  D ) ( F `
 A ) ) )  =  (/) )
6553, 64syl5eq 2448 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( ( A ( ball `  D
) ( F `  A ) )  i^i 
S )  =  (/) )
6665adantr 452 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  ( ( A ( ball `  D
) ( F `  A ) )  i^i 
S )  =  (/) )
6741ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  J  e.  (TopOn `  X ) )
6867, 43syl 16 . . . . . . . 8  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  J  e.  Top )
69 simpll2 997 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  S  C_  X
)
7067, 45syl 16 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  X  =  U. J )
7169, 70sseqtrd 3344 . . . . . . . 8  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  S  C_  U. J
)
72 simplr 732 . . . . . . . 8  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  A  e.  ( ( cls `  J
) `  S )
)
73 simpll1 996 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  D  e.  ( * Met `  X
) )
74 simpll3 998 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  A  e.  X )
7559ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  ( F `  A )  e.  RR* )
764blopn 18483 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  ( F `  A
)  e.  RR* )  ->  ( A ( ball `  D ) ( F `
 A ) )  e.  J )
7773, 74, 75, 76syl3anc 1184 . . . . . . . 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
)
78 simpr 448 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  /\  0  <  ( F `  A ) )  ->  0  <  ( F `  A ) )
79 xblcntr 18394 . . . . . . . . 9  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  ( ( F `  A )  e.  RR*  /\  0  <  ( F `
 A ) ) )  ->  A  e.  ( A ( ball `  D
) ( F `  A ) ) )
8073, 74, 75, 78, 79syl112anc 1188 . . . . . . . 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 ) ) )
8149clsndisj 17094 . . . . . . . 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 )  =/=  (/) )
8268, 71, 72, 77, 80, 81syl32anc 1192 . . . . . . 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 )  =/=  (/) )
8382ex 424 . . . . . 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 )  =/=  (/) ) )
8483necon2bd 2616 . . . . 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 ) ) )
8566, 84mpd 15 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  -.  0  <  ( F `  A
) )
8657simprbi 451 . . . . . . . 8  |-  ( ( F `  A )  e.  ( 0 [,] 
+oo )  ->  0  <_  ( F `  A
) )
8756, 86syl 16 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  0  <_  ( F `  A ) )
88 0xr 9087 . . . . . . . 8  |-  0  e.  RR*
89 xrleloe 10693 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  ( F `  A )  e.  RR* )  ->  (
0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
9088, 59, 89sylancr 645 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( 0  <_  ( F `  A )  <->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) ) )
9187, 90mpbid 202 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  X
)  ->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) )
9291adantr 452 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  ( 0  <  ( F `  A )  \/  0  =  ( F `  A ) ) )
9392ord 367 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  ( -.  0  <  ( F `  A )  ->  0  =  ( F `  A ) ) )
9485, 93mpd 15 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  0  =  ( F `  A ) )
9594eqcomd 2409 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  A  e.  ( ( cls `  J
) `  S )
)  ->  ( F `  A )  =  0 )
9652, 95impbida 806 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    i^i cin 3279    C_ wss 3280   (/)c0 3588   U.cuni 3975   class class class wbr 4172    e. cmpt 4226   `'ccnv 4836   ran crn 4838   ` cfv 5413  (class class class)co 6040   supcsup 7403   RRcr 8945   0cc0 8946    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077   RR+crp 10568   [,]cicc 10875   * Metcxmt 16641   ballcbl 16643   MetOpencmopn 16646   Topctop 16913  TopOnctopon 16914   clsccl 17037
This theorem is referenced by:  metnrmlem1a  18841  lebnumlem1  18939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-icc 10879  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-cld 17038  df-ntr 17039  df-cls 17040
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