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Theorem bcthlem4 19285
Description: Lemma for bcth 19287. 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 19244 . . . . . . 7  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  D  e.  ( Met `  X ) )
4 metxmet 18369 . . . . . 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 19283 . . . . 5  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) )
15 elrp 10619 . . . . . . . . 9  |-  ( r  e.  RR+  <->  ( r  e.  RR  /\  0  < 
r ) )
16 nnrecl 10224 . . . . . . . . 9  |-  ( ( r  e.  RR  /\  0  <  r )  ->  E. m  e.  NN  ( 1  /  m
)  <  r )
1715, 16sylbi 189 . . . . . . . 8  |-  ( r  e.  RR+  ->  E. m  e.  NN  ( 1  /  m )  <  r
)
1817adantl 454 . . . . . . 7  |-  ( (
ph  /\  r  e.  RR+ )  ->  E. m  e.  NN  ( 1  /  m )  <  r
)
19 peano2nn 10017 . . . . . . . . . 10  |-  ( m  e.  NN  ->  (
m  +  1 )  e.  NN )
2019adantl 454 . . . . . . . . 9  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
m  +  1 )  e.  NN )
21 oveq1 6091 . . . . . . . . . . . . . . . . 17  |-  ( k  =  m  ->  (
k  +  1 )  =  ( m  + 
1 ) )
2221fveq2d 5735 . . . . . . . . . . . . . . . 16  |-  ( k  =  m  ->  (
g `  ( k  +  1 ) )  =  ( g `  ( m  +  1
) ) )
23 id 21 . . . . . . . . . . . . . . . . 17  |-  ( k  =  m  ->  k  =  m )
24 fveq2 5731 . . . . . . . . . . . . . . . . 17  |-  ( k  =  m  ->  (
g `  k )  =  ( g `  m ) )
2523, 24oveq12d 6102 . . . . . . . . . . . . . . . 16  |-  ( k  =  m  ->  (
k F ( g `
 k ) )  =  ( m F ( g `  m
) ) )
2622, 25eleq12d 2506 . . . . . . . . . . . . . . 15  |-  ( k  =  m  ->  (
( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  <->  ( g `  ( m  +  1 ) )  e.  ( m F ( g `
 m ) ) ) )
2726rspccva 3053 . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 ( m  + 
1 ) )  e.  ( m F ( g `  m ) ) )
296ffvelrnda 5873 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 m )  e.  ( X  X.  RR+ ) )
307, 1, 8bcthlem1 19282 . . . . . . . . . . . . . . 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 600 . . . . . . . . . . . . . 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 203 . . . . . . . . . . . 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 971 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  <  (
1  /  m ) )
3534adantlr 697 . . . . . . . . . 10  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  ( 2nd `  ( g `  ( m  +  1
) ) )  < 
( 1  /  m
) )
3633simp1d 970 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 ( m  + 
1 ) )  e.  ( X  X.  RR+ ) )
37 xp2nd 6380 . . . . . . . . . . . . . 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 10653 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  e.  RR )
4039adantlr 697 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  ( 2nd `  ( g `  ( m  +  1
) ) )  e.  RR )
41 nnrecre 10041 . . . . . . . . . . . 12  |-  ( m  e.  NN  ->  (
1  /  m )  e.  RR )
4241adantl 454 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
1  /  m )  e.  RR )
43 rpre 10623 . . . . . . . . . . . 12  |-  ( r  e.  RR+  ->  r  e.  RR )
4443ad2antlr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  r  e.  RR )
45 lttr 9157 . . . . . . . . . . 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 1185 . . . . . . . . . 10  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
( ( 2nd `  (
g `  ( m  +  1 ) ) )  <  ( 1  /  m )  /\  ( 1  /  m
)  <  r )  ->  ( 2nd `  (
g `  ( m  +  1 ) ) )  <  r ) )
4735, 46mpand 658 . . . . . . . . 9  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
( 1  /  m
)  <  r  ->  ( 2nd `  ( g `
 ( m  + 
1 ) ) )  <  r ) )
48 fveq2 5731 . . . . . . . . . . . 12  |-  ( n  =  ( m  + 
1 )  ->  (
g `  n )  =  ( g `  ( m  +  1
) ) )
4948fveq2d 5735 . . . . . . . . . . 11  |-  ( n  =  ( m  + 
1 )  ->  ( 2nd `  ( g `  n ) )  =  ( 2nd `  (
g `  ( m  +  1 ) ) ) )
5049breq1d 4225 . . . . . . . . . 10  |-  ( n  =  ( m  + 
1 )  ->  (
( 2nd `  (
g `  n )
)  <  r  <->  ( 2nd `  ( g `  (
m  +  1 ) ) )  <  r
) )
5150rspcev 3054 . . . . . . . . 9  |-  ( ( ( m  +  1 )  e.  NN  /\  ( 2nd `  ( g `
 ( m  + 
1 ) ) )  <  r )  ->  E. n  e.  NN  ( 2nd `  ( g `
 n ) )  <  r )
5220, 47, 51ee12an 1373 . . . . . . . 8  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
( 1  /  m
)  <  r  ->  E. n  e.  NN  ( 2nd `  ( g `  n ) )  < 
r ) )
5352rexlimdva 2832 . . . . . . 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 2791 . . . . 5  |-  ( ph  ->  A. r  e.  RR+  E. n  e.  NN  ( 2nd `  ( g `  n ) )  < 
r )
565, 6, 14, 55caubl 19265 . . . 4  |-  ( ph  ->  ( 1st  o.  g
)  e.  ( Cau `  D ) )
577cmetcau 19247 . . . 4  |-  ( ( D  e.  ( CMet `  X )  /\  ( 1st  o.  g )  e.  ( Cau `  D
) )  ->  ( 1st  o.  g )  e. 
dom  ( ~~> t `  J ) )
581, 56, 57syl2anc 644 . . 3  |-  ( ph  ->  ( 1st  o.  g
)  e.  dom  ( ~~> t `  J )
)
59 fo1st 6369 . . . . . 6  |-  1st : _V -onto-> _V
60 fofun 5657 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
6159, 60ax-mp 5 . . . . 5  |-  Fun  1st
62 vex 2961 . . . . 5  |-  g  e. 
_V
63 cofunexg 5962 . . . . 5  |-  ( ( Fun  1st  /\  g  e.  _V )  ->  ( 1st  o.  g )  e. 
_V )
6461, 62, 63mp2an 655 . . . 4  |-  ( 1st 
o.  g )  e. 
_V
6564eldm 5070 . . 3  |-  ( ( 1st  o.  g )  e.  dom  ( ~~> t `  J )  <->  E. x
( 1st  o.  g
) ( ~~> t `  J ) x )
6658, 65sylib 190 . 2  |-  ( ph  ->  E. x ( 1st 
o.  g ) ( ~~> t `  J ) x )
67 1nn 10016 . . . . . 6  |-  1  e.  NN
687, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 19284 . . . . . 6  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  1  e.  NN )  ->  x  e.  ( (
ball `  D ) `  ( g `  1
) ) )
6967, 68mp3an3 1269 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  x  e.  ( ( ball `  D
) `  ( g `  1 ) ) )
7012fveq2d 5735 . . . . . . 7  |-  ( ph  ->  ( ( ball `  D
) `  ( g `  1 ) )  =  ( ( ball `  D ) `  <. C ,  R >. )
)
71 df-ov 6087 . . . . . . 7  |-  ( C ( ball `  D
) R )  =  ( ( ball `  D
) `  <. C ,  R >. )
7270, 71syl6eqr 2488 . . . . . 6  |-  ( ph  ->  ( ( ball `  D
) `  ( g `  1 ) )  =  ( C (
ball `  D ) R ) )
7372adantr 453 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  ( ( ball `  D ) `  (
g `  1 )
)  =  ( C ( ball `  D
) R ) )
7469, 73eleqtrd 2514 . . . 4  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  x  e.  ( C ( ball `  D
) R ) )
757mopntop 18475 . . . . . . . . . . . . . 14  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Top )
765, 75syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  Top )
7776adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  J  e. 
Top )
785adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  D  e.  ( * Met `  X
) )
79 xp1st 6379 . . . . . . . . . . . . . . 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 10654 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  e.  RR* )
82 blssm 18453 . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( 1st `  ( g `
 ( m  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  C_  X
)
84 1st2nd2 6389 . . . . . . . . . . . . . . . 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 5735 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
m  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( m  + 
1 ) ) )
>. ) )
87 df-ov 6087 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( g `
 ( m  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
m  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( m  + 
1 ) ) )
>. )
8886, 87syl6reqr 2489 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( 1st `  ( g `
 ( m  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  =  ( ( ball `  D
) `  ( g `  ( m  +  1 ) ) ) )
897mopnuni 18476 . . . . . . . . . . . . . . 15  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
905, 89syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  X  =  U. J
)
9190adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  X  = 
U. J )
9283, 88, 913sstr3d 3392 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  U. J
)
93 eqid 2438 . . . . . . . . . . . . 13  |-  U. J  =  U. J
9493sscls 17125 . . . . . . . . . . . 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 644 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
m  +  1 ) ) ) ) )
9633simp3d 972 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( cls `  J ) `
 ( ( ball `  D ) `  (
g `  ( m  +  1 ) ) ) )  C_  (
( ( ball `  D
) `  ( g `  m ) )  \ 
( M `  m
) ) )
9795, 96sstrd 3360 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  (
( ( ball `  D
) `  ( g `  m ) )  \ 
( M `  m
) ) )
98973adant2 977 . . . . . . . . 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 19284 . . . . . . . . . 10  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  ( m  +  1
)  e.  NN )  ->  x  e.  ( ( ball `  D
) `  ( g `  ( m  +  1 ) ) ) )
10019, 99syl3an3 1220 . . . . . . . . 9  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  x  e.  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) ) )
10198, 100sseldd 3351 . . . . . . . 8  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  x  e.  ( ( ( ball `  D
) `  ( g `  m ) )  \ 
( M `  m
) ) )
102101eldifbd 3335 . . . . . . 7  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  -.  x  e.  ( M `  m ) )
1031023expa 1154 . . . . . 6  |-  ( ( ( ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  /\  m  e.  NN )  ->  -.  x  e.  ( M `  m ) )
104103ralrimiva 2791 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  A. m  e.  NN  -.  x  e.  ( M `  m )
)
105 eluni2 4021 . . . . . . . . 9  |-  ( x  e.  U. ran  M  <->  E. y  e.  ran  M  x  e.  y )
106 ffn 5594 . . . . . . . . . . 11  |-  ( M : NN --> ( Clsd `  J )  ->  M  Fn  NN )
1079, 106syl 16 . . . . . . . . . 10  |-  ( ph  ->  M  Fn  NN )
108 eleq2 2499 . . . . . . . . . . 11  |-  ( y  =  ( M `  m )  ->  (
x  e.  y  <->  x  e.  ( M `  m ) ) )
109108rexrn 5875 . . . . . . . . . 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 250 . . . . . . . 8  |-  ( ph  ->  ( x  e.  U. ran  M  <->  E. m  e.  NN  x  e.  ( M `  m ) ) )
112111notbid 287 . . . . . . 7  |-  ( ph  ->  ( -.  x  e. 
U. ran  M  <->  -.  E. m  e.  NN  x  e.  ( M `  m ) ) )
113 ralnex 2717 . . . . . . 7  |-  ( A. m  e.  NN  -.  x  e.  ( M `  m )  <->  -.  E. m  e.  NN  x  e.  ( M `  m ) )
114112, 113syl6bbr 256 . . . . . 6  |-  ( ph  ->  ( -.  x  e. 
U. ran  M  <->  A. m  e.  NN  -.  x  e.  ( M `  m
) ) )
115114biimpar 473 . . . . 5  |-  ( (
ph  /\  A. m  e.  NN  -.  x  e.  ( M `  m
) )  ->  -.  x  e.  U. ran  M
)
116104, 115syldan 458 . . . 4  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  -.  x  e.  U.
ran  M )
11774, 116eldifd 3333 . . 3  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  x  e.  ( ( C ( ball `  D ) R ) 
\  U. ran  M ) )
118 ne0i 3636 . . 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 1649 1  |-  ( ph  ->  ( ( C (
ball `  D ) R )  \  U. ran  M )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958    \ cdif 3319    C_ wss 3322   (/)c0 3630   <.cop 3819   U.cuni 4017   class class class wbr 4215   {copab 4268    X. cxp 4879   dom cdm 4881   ran crn 4882    o. ccom 4885   Fun wfun 5451    Fn wfn 5452   -->wf 5453   -onto->wfo 5455   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998   RR*cxr 9124    < clt 9125    / cdiv 9682   NNcn 10005   RR+crp 10617   * Metcxmt 16691   Metcme 16692   ballcbl 16693   MetOpencmopn 16696   Topctop 16963   Clsdccld 17085   clsccl 17087   ~~> tclm 17295   Caucca 19211   CMetcms 19212
This theorem is referenced by:  bcthlem5  19286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ico 10927  df-rest 13655  df-topgen 13672  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-top 16968  df-bases 16970  df-topon 16971  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lm 17298  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-cfil 19213  df-cau 19214  df-cmet 19215
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