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Theorem bcthlem4 20838
Description: Lemma for bcth 20840. 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 20797 . . . . . . 7  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  D  e.  ( Met `  X ) )
4 metxmet 19909 . . . . . 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 20836 . . . . 5  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) )
15 elrp 10993 . . . . . . . . 9  |-  ( r  e.  RR+  <->  ( r  e.  RR  /\  0  < 
r ) )
16 nnrecl 10577 . . . . . . . . 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 466 . . . . . . 7  |-  ( (
ph  /\  r  e.  RR+ )  ->  E. m  e.  NN  ( 1  /  m )  <  r
)
19 peano2nn 10334 . . . . . . . . . 10  |-  ( m  e.  NN  ->  (
m  +  1 )  e.  NN )
2019adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
m  +  1 )  e.  NN )
21 oveq1 6098 . . . . . . . . . . . . . . . . 17  |-  ( k  =  m  ->  (
k  +  1 )  =  ( m  + 
1 ) )
2221fveq2d 5695 . . . . . . . . . . . . . . . 16  |-  ( k  =  m  ->  (
g `  ( k  +  1 ) )  =  ( g `  ( m  +  1
) ) )
23 id 22 . . . . . . . . . . . . . . . . 17  |-  ( k  =  m  ->  k  =  m )
24 fveq2 5691 . . . . . . . . . . . . . . . . 17  |-  ( k  =  m  ->  (
g `  k )  =  ( g `  m ) )
2523, 24oveq12d 6109 . . . . . . . . . . . . . . . 16  |-  ( k  =  m  ->  (
k F ( g `
 k ) )  =  ( m F ( g `  m
) ) )
2622, 25eleq12d 2511 . . . . . . . . . . . . . . 15  |-  ( k  =  m  ->  (
( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  <->  ( g `  ( m  +  1 ) )  e.  ( m F ( g `
 m ) ) ) )
2726rspccva 3072 . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 ( m  + 
1 ) )  e.  ( m F ( g `  m ) ) )
296ffvelrnda 5843 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 m )  e.  ( X  X.  RR+ ) )
307, 1, 8bcthlem1 20835 . . . . . . . . . . . . . . 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 615 . . . . . . . . . . . . . 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 1001 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  <  (
1  /  m ) )
3534adantlr 714 . . . . . . . . . 10  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  ( 2nd `  ( g `  ( m  +  1
) ) )  < 
( 1  /  m
) )
3633simp1d 1000 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 ( m  + 
1 ) )  e.  ( X  X.  RR+ ) )
37 xp2nd 6607 . . . . . . . . . . . . . 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 11027 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  e.  RR )
4039adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  ( 2nd `  ( g `  ( m  +  1
) ) )  e.  RR )
41 nnrecre 10358 . . . . . . . . . . . 12  |-  ( m  e.  NN  ->  (
1  /  m )  e.  RR )
4241adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
1  /  m )  e.  RR )
43 rpre 10997 . . . . . . . . . . . 12  |-  ( r  e.  RR+  ->  r  e.  RR )
4443ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  r  e.  RR )
45 lttr 9451 . . . . . . . . . . 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 1218 . . . . . . . . . 10  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
( ( 2nd `  (
g `  ( m  +  1 ) ) )  <  ( 1  /  m )  /\  ( 1  /  m
)  <  r )  ->  ( 2nd `  (
g `  ( m  +  1 ) ) )  <  r ) )
4735, 46mpand 675 . . . . . . . . 9  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
( 1  /  m
)  <  r  ->  ( 2nd `  ( g `
 ( m  + 
1 ) ) )  <  r ) )
48 fveq2 5691 . . . . . . . . . . . 12  |-  ( n  =  ( m  + 
1 )  ->  (
g `  n )  =  ( g `  ( m  +  1
) ) )
4948fveq2d 5695 . . . . . . . . . . 11  |-  ( n  =  ( m  + 
1 )  ->  ( 2nd `  ( g `  n ) )  =  ( 2nd `  (
g `  ( m  +  1 ) ) ) )
5049breq1d 4302 . . . . . . . . . 10  |-  ( n  =  ( m  + 
1 )  ->  (
( 2nd `  (
g `  n )
)  <  r  <->  ( 2nd `  ( g `  (
m  +  1 ) ) )  <  r
) )
5150rspcev 3073 . . . . . . . . 9  |-  ( ( ( m  +  1 )  e.  NN  /\  ( 2nd `  ( g `
 ( m  + 
1 ) ) )  <  r )  ->  E. n  e.  NN  ( 2nd `  ( g `
 n ) )  <  r )
5220, 47, 51syl6an 545 . . . . . . . 8  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
( 1  /  m
)  <  r  ->  E. n  e.  NN  ( 2nd `  ( g `  n ) )  < 
r ) )
5352rexlimdva 2841 . . . . . . 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 2799 . . . . 5  |-  ( ph  ->  A. r  e.  RR+  E. n  e.  NN  ( 2nd `  ( g `  n ) )  < 
r )
565, 6, 14, 55caubl 20818 . . . 4  |-  ( ph  ->  ( 1st  o.  g
)  e.  ( Cau `  D ) )
577cmetcau 20800 . . . 4  |-  ( ( D  e.  ( CMet `  X )  /\  ( 1st  o.  g )  e.  ( Cau `  D
) )  ->  ( 1st  o.  g )  e. 
dom  ( ~~> t `  J ) )
581, 56, 57syl2anc 661 . . 3  |-  ( ph  ->  ( 1st  o.  g
)  e.  dom  ( ~~> t `  J )
)
59 fo1st 6596 . . . . . 6  |-  1st : _V -onto-> _V
60 fofun 5621 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
6159, 60ax-mp 5 . . . . 5  |-  Fun  1st
62 vex 2975 . . . . 5  |-  g  e. 
_V
63 cofunexg 6541 . . . . 5  |-  ( ( Fun  1st  /\  g  e.  _V )  ->  ( 1st  o.  g )  e. 
_V )
6461, 62, 63mp2an 672 . . . 4  |-  ( 1st 
o.  g )  e. 
_V
6564eldm 5037 . . 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 10333 . . . . . 6  |-  1  e.  NN
687, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 20837 . . . . . 6  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  1  e.  NN )  ->  x  e.  ( (
ball `  D ) `  ( g `  1
) ) )
6967, 68mp3an3 1303 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  x  e.  ( ( ball `  D
) `  ( g `  1 ) ) )
7012fveq2d 5695 . . . . . . 7  |-  ( ph  ->  ( ( ball `  D
) `  ( g `  1 ) )  =  ( ( ball `  D ) `  <. C ,  R >. )
)
71 df-ov 6094 . . . . . . 7  |-  ( C ( ball `  D
) R )  =  ( ( ball `  D
) `  <. C ,  R >. )
7270, 71syl6eqr 2493 . . . . . 6  |-  ( ph  ->  ( ( ball `  D
) `  ( g `  1 ) )  =  ( C (
ball `  D ) R ) )
7372adantr 465 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  ( ( ball `  D ) `  (
g `  1 )
)  =  ( C ( ball `  D
) R ) )
7469, 73eleqtrd 2519 . . . 4  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  x  e.  ( C ( ball `  D
) R ) )
757mopntop 20015 . . . . . . . . . . . . . 14  |-  ( D  e.  ( *Met `  X )  ->  J  e.  Top )
765, 75syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  Top )
7776adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  J  e. 
Top )
785adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  D  e.  ( *Met `  X ) )
79 xp1st 6606 . . . . . . . . . . . . . . 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 11028 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  e.  RR* )
82 blssm 19993 . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( 1st `  ( g `
 ( m  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  C_  X
)
84 1st2nd2 6613 . . . . . . . . . . . . . . . 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 5695 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
m  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( m  + 
1 ) ) )
>. ) )
87 df-ov 6094 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( g `
 ( m  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
m  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( m  + 
1 ) ) )
>. )
8886, 87syl6reqr 2494 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( 1st `  ( g `
 ( m  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  =  ( ( ball `  D
) `  ( g `  ( m  +  1 ) ) ) )
897mopnuni 20016 . . . . . . . . . . . . . . 15  |-  ( D  e.  ( *Met `  X )  ->  X  =  U. J )
905, 89syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  X  =  U. J
)
9190adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  X  = 
U. J )
9283, 88, 913sstr3d 3398 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  U. J
)
93 eqid 2443 . . . . . . . . . . . . 13  |-  U. J  =  U. J
9493sscls 18660 . . . . . . . . . . . 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 661 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
m  +  1 ) ) ) ) )
9633simp3d 1002 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( cls `  J ) `
 ( ( ball `  D ) `  (
g `  ( m  +  1 ) ) ) )  C_  (
( ( ball `  D
) `  ( g `  m ) )  \ 
( M `  m
) ) )
9795, 96sstrd 3366 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  (
( ( ball `  D
) `  ( g `  m ) )  \ 
( M `  m
) ) )
98973adant2 1007 . . . . . . . . 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 20837 . . . . . . . . . 10  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  ( m  +  1
)  e.  NN )  ->  x  e.  ( ( ball `  D
) `  ( g `  ( m  +  1 ) ) ) )
10019, 99syl3an3 1253 . . . . . . . . 9  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  x  e.  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) ) )
10198, 100sseldd 3357 . . . . . . . 8  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  x  e.  ( ( ( ball `  D
) `  ( g `  m ) )  \ 
( M `  m
) ) )
102101eldifbd 3341 . . . . . . 7  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  -.  x  e.  ( M `  m ) )
1031023expa 1187 . . . . . 6  |-  ( ( ( ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  /\  m  e.  NN )  ->  -.  x  e.  ( M `  m ) )
104103ralrimiva 2799 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  A. m  e.  NN  -.  x  e.  ( M `  m )
)
105 eluni2 4095 . . . . . . . . 9  |-  ( x  e.  U. ran  M  <->  E. y  e.  ran  M  x  e.  y )
106 ffn 5559 . . . . . . . . . . 11  |-  ( M : NN --> ( Clsd `  J )  ->  M  Fn  NN )
1079, 106syl 16 . . . . . . . . . 10  |-  ( ph  ->  M  Fn  NN )
108 eleq2 2504 . . . . . . . . . . 11  |-  ( y  =  ( M `  m )  ->  (
x  e.  y  <->  x  e.  ( M `  m ) ) )
109108rexrn 5845 . . . . . . . . . 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 2725 . . . . . . 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 485 . . . . 5  |-  ( (
ph  /\  A. m  e.  NN  -.  x  e.  ( M `  m
) )  ->  -.  x  e.  U. ran  M
)
116104, 115syldan 470 . . . 4  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  -.  x  e.  U.
ran  M )
11774, 116eldifd 3339 . . 3  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  x  e.  ( ( C ( ball `  D ) R ) 
\  U. ran  M ) )
118 ne0i 3643 . . 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 1692 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 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   _Vcvv 2972    \ cdif 3325    C_ wss 3328   (/)c0 3637   <.cop 3883   U.cuni 4091   class class class wbr 4292   {copab 4349    X. cxp 4838   dom cdm 4840   ran crn 4841    o. ccom 4844   Fun wfun 5412    Fn wfn 5413   -->wf 5414   -onto->wfo 5416   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1stc1st 6575   2ndc2nd 6576   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285   RR*cxr 9417    < clt 9418    / cdiv 9993   NNcn 10322   RR+crp 10991   *Metcxmt 17801   Metcme 17802   ballcbl 17803   MetOpencmopn 17806   Topctop 18498   Clsdccld 18620   clsccl 18622   ~~> tclm 18830   Caucca 20764   CMetcms 20765
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ico 11306  df-rest 14361  df-topgen 14382  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-top 18503  df-bases 18505  df-topon 18506  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lm 18833  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-cfil 20766  df-cau 20767  df-cmet 20768
This theorem is referenced by:  bcthlem5  20839
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