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Theorem bcthlem4 20797
Description: Lemma for bcth 20799. Given any open ball  ( C ( ball `  D
) R ) as starting point (and in particular, a ball in  int ( U. ran  M )), the limit point  x of the centers of the induced sequence of balls  g is outside  U. ran  M. Note that a set  A has empty interior iff every nonempty open set  U contains points outside  A, i.e.  ( U  \  A )  =/=  (/). (Contributed by Mario Carneiro, 7-Jan-2014.)
Hypotheses
Ref Expression
bcth.2  |-  J  =  ( MetOpen `  D )
bcthlem.4  |-  ( ph  ->  D  e.  ( CMet `  X ) )
bcthlem.5  |-  F  =  ( k  e.  NN ,  z  e.  ( X  X.  RR+ )  |->  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  k )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) ) ) } )
bcthlem.6  |-  ( ph  ->  M : NN --> ( Clsd `  J ) )
bcthlem.7  |-  ( ph  ->  R  e.  RR+ )
bcthlem.8  |-  ( ph  ->  C  e.  X )
bcthlem.9  |-  ( ph  ->  g : NN --> ( X  X.  RR+ ) )
bcthlem.10  |-  ( ph  ->  ( g `  1
)  =  <. C ,  R >. )
bcthlem.11  |-  ( ph  ->  A. k  e.  NN  ( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) ) )
Assertion
Ref Expression
bcthlem4  |-  ( ph  ->  ( ( C (
ball `  D ) R )  \  U. ran  M )  =/=  (/) )
Distinct variable groups:    k, r, x, z    C, r, x   
g, k, r, x, z, D    g, F, k, r, x, z    g, J, k, r, x, z   
g, M, k, r, x, z    ph, k,
r, x, z    x, R    g, X, k, r, x, z
Allowed substitution hints:    ph( g)    C( z, g, k)    R( z, g, k, r)

Proof of Theorem bcthlem4
Dummy variables  n  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bcthlem.4 . . . 4  |-  ( ph  ->  D  e.  ( CMet `  X ) )
2 cmetmet 20756 . . . . . . 7  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  D  e.  ( Met `  X ) )
4 metxmet 19868 . . . . . 6  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
53, 4syl 16 . . . . 5  |-  ( ph  ->  D  e.  ( *Met `  X ) )
6 bcthlem.9 . . . . 5  |-  ( ph  ->  g : NN --> ( X  X.  RR+ ) )
7 bcth.2 . . . . . 6  |-  J  =  ( MetOpen `  D )
8 bcthlem.5 . . . . . 6  |-  F  =  ( k  e.  NN ,  z  e.  ( X  X.  RR+ )  |->  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  k )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) ) ) } )
9 bcthlem.6 . . . . . 6  |-  ( ph  ->  M : NN --> ( Clsd `  J ) )
10 bcthlem.7 . . . . . 6  |-  ( ph  ->  R  e.  RR+ )
11 bcthlem.8 . . . . . 6  |-  ( ph  ->  C  e.  X )
12 bcthlem.10 . . . . . 6  |-  ( ph  ->  ( g `  1
)  =  <. C ,  R >. )
13 bcthlem.11 . . . . . 6  |-  ( ph  ->  A. k  e.  NN  ( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) ) )
147, 1, 8, 9, 10, 11, 6, 12, 13bcthlem2 20795 . . . . 5  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) )
15 elrp 10989 . . . . . . . . 9  |-  ( r  e.  RR+  <->  ( r  e.  RR  /\  0  < 
r ) )
16 nnrecl 10573 . . . . . . . . 9  |-  ( ( r  e.  RR  /\  0  <  r )  ->  E. m  e.  NN  ( 1  /  m
)  <  r )
1715, 16sylbi 195 . . . . . . . 8  |-  ( r  e.  RR+  ->  E. m  e.  NN  ( 1  /  m )  <  r
)
1817adantl 463 . . . . . . 7  |-  ( (
ph  /\  r  e.  RR+ )  ->  E. m  e.  NN  ( 1  /  m )  <  r
)
19 peano2nn 10330 . . . . . . . . . 10  |-  ( m  e.  NN  ->  (
m  +  1 )  e.  NN )
2019adantl 463 . . . . . . . . 9  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
m  +  1 )  e.  NN )
21 oveq1 6097 . . . . . . . . . . . . . . . . 17  |-  ( k  =  m  ->  (
k  +  1 )  =  ( m  + 
1 ) )
2221fveq2d 5692 . . . . . . . . . . . . . . . 16  |-  ( k  =  m  ->  (
g `  ( k  +  1 ) )  =  ( g `  ( m  +  1
) ) )
23 id 22 . . . . . . . . . . . . . . . . 17  |-  ( k  =  m  ->  k  =  m )
24 fveq2 5688 . . . . . . . . . . . . . . . . 17  |-  ( k  =  m  ->  (
g `  k )  =  ( g `  m ) )
2523, 24oveq12d 6108 . . . . . . . . . . . . . . . 16  |-  ( k  =  m  ->  (
k F ( g `
 k ) )  =  ( m F ( g `  m
) ) )
2622, 25eleq12d 2509 . . . . . . . . . . . . . . 15  |-  ( k  =  m  ->  (
( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  <->  ( g `  ( m  +  1 ) )  e.  ( m F ( g `
 m ) ) ) )
2726rspccva 3069 . . . . . . . . . . . . . 14  |-  ( ( A. k  e.  NN  ( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  /\  m  e.  NN )  ->  ( g `  (
m  +  1 ) )  e.  ( m F ( g `  m ) ) )
2813, 27sylan 468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 ( m  + 
1 ) )  e.  ( m F ( g `  m ) ) )
296ffvelrnda 5840 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 m )  e.  ( X  X.  RR+ ) )
307, 1, 8bcthlem1 20794 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( m  e.  NN  /\  ( g `
 m )  e.  ( X  X.  RR+ ) ) )  -> 
( ( g `  ( m  +  1
) )  e.  ( m F ( g `
 m ) )  <-> 
( ( g `  ( m  +  1
) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  (
g `  ( m  +  1 ) ) )  <  ( 1  /  m )  /\  ( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
m  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  m )
)  \  ( M `  m ) ) ) ) )
3130expr 612 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( g `  m )  e.  ( X  X.  RR+ )  ->  ( (
g `  ( m  +  1 ) )  e.  ( m F ( g `  m
) )  <->  ( (
g `  ( m  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  (
m  +  1 ) ) )  <  (
1  /  m )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  (
m  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  m )
)  \  ( M `  m ) ) ) ) ) )
3229, 31mpd 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( g `  ( m  +  1 ) )  e.  ( m F ( g `  m
) )  <->  ( (
g `  ( m  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  (
m  +  1 ) ) )  <  (
1  /  m )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  (
m  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  m )
)  \  ( M `  m ) ) ) ) )
3328, 32mpbid 210 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( g `  ( m  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  (
m  +  1 ) ) )  <  (
1  /  m )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  (
m  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  m )
)  \  ( M `  m ) ) ) )
3433simp2d 996 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  <  (
1  /  m ) )
3534adantlr 709 . . . . . . . . . 10  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  ( 2nd `  ( g `  ( m  +  1
) ) )  < 
( 1  /  m
) )
3633simp1d 995 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 ( m  + 
1 ) )  e.  ( X  X.  RR+ ) )
37 xp2nd 6606 . . . . . . . . . . . . . 14  |-  ( ( g `  ( m  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  e.  RR+ )
3836, 37syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  e.  RR+ )
3938rpred 11023 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  e.  RR )
4039adantlr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  ( 2nd `  ( g `  ( m  +  1
) ) )  e.  RR )
41 nnrecre 10354 . . . . . . . . . . . 12  |-  ( m  e.  NN  ->  (
1  /  m )  e.  RR )
4241adantl 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
1  /  m )  e.  RR )
43 rpre 10993 . . . . . . . . . . . 12  |-  ( r  e.  RR+  ->  r  e.  RR )
4443ad2antlr 721 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  r  e.  RR )
45 lttr 9447 . . . . . . . . . . 11  |-  ( ( ( 2nd `  (
g `  ( m  +  1 ) ) )  e.  RR  /\  ( 1  /  m
)  e.  RR  /\  r  e.  RR )  ->  ( ( ( 2nd `  ( g `  (
m  +  1 ) ) )  <  (
1  /  m )  /\  ( 1  /  m )  <  r
)  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  <  r
) )
4640, 42, 44, 45syl3anc 1213 . . . . . . . . . 10  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
( ( 2nd `  (
g `  ( m  +  1 ) ) )  <  ( 1  /  m )  /\  ( 1  /  m
)  <  r )  ->  ( 2nd `  (
g `  ( m  +  1 ) ) )  <  r ) )
4735, 46mpand 670 . . . . . . . . 9  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
( 1  /  m
)  <  r  ->  ( 2nd `  ( g `
 ( m  + 
1 ) ) )  <  r ) )
48 fveq2 5688 . . . . . . . . . . . 12  |-  ( n  =  ( m  + 
1 )  ->  (
g `  n )  =  ( g `  ( m  +  1
) ) )
4948fveq2d 5692 . . . . . . . . . . 11  |-  ( n  =  ( m  + 
1 )  ->  ( 2nd `  ( g `  n ) )  =  ( 2nd `  (
g `  ( m  +  1 ) ) ) )
5049breq1d 4299 . . . . . . . . . 10  |-  ( n  =  ( m  + 
1 )  ->  (
( 2nd `  (
g `  n )
)  <  r  <->  ( 2nd `  ( g `  (
m  +  1 ) ) )  <  r
) )
5150rspcev 3070 . . . . . . . . 9  |-  ( ( ( m  +  1 )  e.  NN  /\  ( 2nd `  ( g `
 ( m  + 
1 ) ) )  <  r )  ->  E. n  e.  NN  ( 2nd `  ( g `
 n ) )  <  r )
5220, 47, 51syl6an 542 . . . . . . . 8  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
( 1  /  m
)  <  r  ->  E. n  e.  NN  ( 2nd `  ( g `  n ) )  < 
r ) )
5352rexlimdva 2839 . . . . . . 7  |-  ( (
ph  /\  r  e.  RR+ )  ->  ( E. m  e.  NN  (
1  /  m )  <  r  ->  E. n  e.  NN  ( 2nd `  (
g `  n )
)  <  r )
)
5418, 53mpd 15 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  E. n  e.  NN  ( 2nd `  (
g `  n )
)  <  r )
5554ralrimiva 2797 . . . . 5  |-  ( ph  ->  A. r  e.  RR+  E. n  e.  NN  ( 2nd `  ( g `  n ) )  < 
r )
565, 6, 14, 55caubl 20777 . . . 4  |-  ( ph  ->  ( 1st  o.  g
)  e.  ( Cau `  D ) )
577cmetcau 20759 . . . 4  |-  ( ( D  e.  ( CMet `  X )  /\  ( 1st  o.  g )  e.  ( Cau `  D
) )  ->  ( 1st  o.  g )  e. 
dom  ( ~~> t `  J ) )
581, 56, 57syl2anc 656 . . 3  |-  ( ph  ->  ( 1st  o.  g
)  e.  dom  ( ~~> t `  J )
)
59 fo1st 6595 . . . . . 6  |-  1st : _V -onto-> _V
60 fofun 5618 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
6159, 60ax-mp 5 . . . . 5  |-  Fun  1st
62 vex 2973 . . . . 5  |-  g  e. 
_V
63 cofunexg 6540 . . . . 5  |-  ( ( Fun  1st  /\  g  e.  _V )  ->  ( 1st  o.  g )  e. 
_V )
6461, 62, 63mp2an 667 . . . 4  |-  ( 1st 
o.  g )  e. 
_V
6564eldm 5033 . . 3  |-  ( ( 1st  o.  g )  e.  dom  ( ~~> t `  J )  <->  E. x
( 1st  o.  g
) ( ~~> t `  J ) x )
6658, 65sylib 196 . 2  |-  ( ph  ->  E. x ( 1st 
o.  g ) ( ~~> t `  J ) x )
67 1nn 10329 . . . . . 6  |-  1  e.  NN
687, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 20796 . . . . . 6  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  1  e.  NN )  ->  x  e.  ( (
ball `  D ) `  ( g `  1
) ) )
6967, 68mp3an3 1298 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  x  e.  ( ( ball `  D
) `  ( g `  1 ) ) )
7012fveq2d 5692 . . . . . . 7  |-  ( ph  ->  ( ( ball `  D
) `  ( g `  1 ) )  =  ( ( ball `  D ) `  <. C ,  R >. )
)
71 df-ov 6093 . . . . . . 7  |-  ( C ( ball `  D
) R )  =  ( ( ball `  D
) `  <. C ,  R >. )
7270, 71syl6eqr 2491 . . . . . 6  |-  ( ph  ->  ( ( ball `  D
) `  ( g `  1 ) )  =  ( C (
ball `  D ) R ) )
7372adantr 462 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  ( ( ball `  D ) `  (
g `  1 )
)  =  ( C ( ball `  D
) R ) )
7469, 73eleqtrd 2517 . . . 4  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  x  e.  ( C ( ball `  D
) R ) )
757mopntop 19974 . . . . . . . . . . . . . 14  |-  ( D  e.  ( *Met `  X )  ->  J  e.  Top )
765, 75syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  Top )
7776adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  J  e. 
Top )
785adantr 462 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  D  e.  ( *Met `  X ) )
79 xp1st 6605 . . . . . . . . . . . . . . 15  |-  ( ( g `  ( m  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( g `  (
m  +  1 ) ) )  e.  X
)
8036, 79syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( 1st `  ( g `  (
m  +  1 ) ) )  e.  X
)
8138rpxrd 11024 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  e.  RR* )
82 blssm 19952 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( *Met `  X )  /\  ( 1st `  (
g `  ( m  +  1 ) ) )  e.  X  /\  ( 2nd `  ( g `
 ( m  + 
1 ) ) )  e.  RR* )  ->  (
( 1st `  (
g `  ( m  +  1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  C_  X
)
8378, 80, 81, 82syl3anc 1213 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( 1st `  ( g `
 ( m  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  C_  X
)
84 1st2nd2 6612 . . . . . . . . . . . . . . . 16  |-  ( ( g `  ( m  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( g `  ( m  +  1 ) )  =  <. ( 1st `  ( g `
 ( m  + 
1 ) ) ) ,  ( 2nd `  (
g `  ( m  +  1 ) ) ) >. )
8536, 84syl 16 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 ( m  + 
1 ) )  = 
<. ( 1st `  (
g `  ( m  +  1 ) ) ) ,  ( 2nd `  ( g `  (
m  +  1 ) ) ) >. )
8685fveq2d 5692 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
m  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( m  + 
1 ) ) )
>. ) )
87 df-ov 6093 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( g `
 ( m  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
m  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( m  + 
1 ) ) )
>. )
8886, 87syl6reqr 2492 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( 1st `  ( g `
 ( m  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  =  ( ( ball `  D
) `  ( g `  ( m  +  1 ) ) ) )
897mopnuni 19975 . . . . . . . . . . . . . . 15  |-  ( D  e.  ( *Met `  X )  ->  X  =  U. J )
905, 89syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  X  =  U. J
)
9190adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  X  = 
U. J )
9283, 88, 913sstr3d 3395 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  U. J
)
93 eqid 2441 . . . . . . . . . . . . 13  |-  U. J  =  U. J
9493sscls 18619 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  ( ( ball `  D
) `  ( g `  ( m  +  1 ) ) )  C_  U. J )  ->  (
( ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
m  +  1 ) ) ) ) )
9577, 92, 94syl2anc 656 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
m  +  1 ) ) ) ) )
9633simp3d 997 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( cls `  J ) `
 ( ( ball `  D ) `  (
g `  ( m  +  1 ) ) ) )  C_  (
( ( ball `  D
) `  ( g `  m ) )  \ 
( M `  m
) ) )
9795, 96sstrd 3363 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  (
( ( ball `  D
) `  ( g `  m ) )  \ 
( M `  m
) ) )
98973adant2 1002 . . . . . . . . 9  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  ( ( ball `  D
) `  ( g `  ( m  +  1 ) ) )  C_  ( ( ( ball `  D ) `  (
g `  m )
)  \  ( M `  m ) ) )
997, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 20796 . . . . . . . . . 10  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  ( m  +  1
)  e.  NN )  ->  x  e.  ( ( ball `  D
) `  ( g `  ( m  +  1 ) ) ) )
10019, 99syl3an3 1248 . . . . . . . . 9  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  x  e.  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) ) )
10198, 100sseldd 3354 . . . . . . . 8  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  x  e.  ( ( ( ball `  D
) `  ( g `  m ) )  \ 
( M `  m
) ) )
102101eldifbd 3338 . . . . . . 7  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  -.  x  e.  ( M `  m ) )
1031023expa 1182 . . . . . 6  |-  ( ( ( ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  /\  m  e.  NN )  ->  -.  x  e.  ( M `  m ) )
104103ralrimiva 2797 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  A. m  e.  NN  -.  x  e.  ( M `  m )
)
105 eluni2 4092 . . . . . . . . 9  |-  ( x  e.  U. ran  M  <->  E. y  e.  ran  M  x  e.  y )
106 ffn 5556 . . . . . . . . . . 11  |-  ( M : NN --> ( Clsd `  J )  ->  M  Fn  NN )
1079, 106syl 16 . . . . . . . . . 10  |-  ( ph  ->  M  Fn  NN )
108 eleq2 2502 . . . . . . . . . . 11  |-  ( y  =  ( M `  m )  ->  (
x  e.  y  <->  x  e.  ( M `  m ) ) )
109108rexrn 5842 . . . . . . . . . 10  |-  ( M  Fn  NN  ->  ( E. y  e.  ran  M  x  e.  y  <->  E. m  e.  NN  x  e.  ( M `  m ) ) )
110107, 109syl 16 . . . . . . . . 9  |-  ( ph  ->  ( E. y  e. 
ran  M  x  e.  y 
<->  E. m  e.  NN  x  e.  ( M `  m ) ) )
111105, 110syl5bb 257 . . . . . . . 8  |-  ( ph  ->  ( x  e.  U. ran  M  <->  E. m  e.  NN  x  e.  ( M `  m ) ) )
112111notbid 294 . . . . . . 7  |-  ( ph  ->  ( -.  x  e. 
U. ran  M  <->  -.  E. m  e.  NN  x  e.  ( M `  m ) ) )
113 ralnex 2723 . . . . . . 7  |-  ( A. m  e.  NN  -.  x  e.  ( M `  m )  <->  -.  E. m  e.  NN  x  e.  ( M `  m ) )
114112, 113syl6bbr 263 . . . . . 6  |-  ( ph  ->  ( -.  x  e. 
U. ran  M  <->  A. m  e.  NN  -.  x  e.  ( M `  m
) ) )
115114biimpar 482 . . . . 5  |-  ( (
ph  /\  A. m  e.  NN  -.  x  e.  ( M `  m
) )  ->  -.  x  e.  U. ran  M
)
116104, 115syldan 467 . . . 4  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  -.  x  e.  U.
ran  M )
11774, 116eldifd 3336 . . 3  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  x  e.  ( ( C ( ball `  D ) R ) 
\  U. ran  M ) )
118 ne0i 3640 . . 3  |-  ( x  e.  ( ( C ( ball `  D
) R )  \  U. ran  M )  -> 
( ( C (
ball `  D ) R )  \  U. ran  M )  =/=  (/) )
119117, 118syl 16 . 2  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  ( ( C ( ball `  D
) R )  \  U. ran  M )  =/=  (/) )
12066, 119exlimddv 1697 1  |-  ( ph  ->  ( ( C (
ball `  D ) R )  \  U. ran  M )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364   E.wex 1591    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   _Vcvv 2970    \ cdif 3322    C_ wss 3325   (/)c0 3634   <.cop 3880   U.cuni 4088   class class class wbr 4289   {copab 4346    X. cxp 4834   dom cdm 4836   ran crn 4837    o. ccom 4840   Fun wfun 5409    Fn wfn 5410   -->wf 5411   -onto->wfo 5413   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281   RR*cxr 9413    < clt 9414    / cdiv 9989   NNcn 10318   RR+crp 10987   *Metcxmt 17760   Metcme 17761   ballcbl 17762   MetOpencmopn 17765   Topctop 18457   Clsdccld 18579   clsccl 18581   ~~> tclm 18789   Caucca 20723   CMetcms 20724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  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 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ico 11302  df-rest 14357  df-topgen 14378  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-top 18462  df-bases 18464  df-topon 18465  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lm 18792  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-cfil 20725  df-cau 20726  df-cmet 20727
This theorem is referenced by:  bcthlem5  20798
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