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Theorem bcthlem4 18847
Description: Lemma for bcth 18849. 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 18810 . . . . . . 7  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
31, 2syl 15 . . . . . 6  |-  ( ph  ->  D  e.  ( Met `  X ) )
4 metxmet 17995 . . . . . 6  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
53, 4syl 15 . . . . 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 18845 . . . . 5  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) )
15 elrp 10445 . . . . . . . . 9  |-  ( r  e.  RR+  <->  ( r  e.  RR  /\  0  < 
r ) )
16 nnrecl 10052 . . . . . . . . 9  |-  ( ( r  e.  RR  /\  0  <  r )  ->  E. m  e.  NN  ( 1  /  m
)  <  r )
1715, 16sylbi 187 . . . . . . . 8  |-  ( r  e.  RR+  ->  E. m  e.  NN  ( 1  /  m )  <  r
)
1817adantl 452 . . . . . . 7  |-  ( (
ph  /\  r  e.  RR+ )  ->  E. m  e.  NN  ( 1  /  m )  <  r
)
19 peano2nn 9845 . . . . . . . . . 10  |-  ( m  e.  NN  ->  (
m  +  1 )  e.  NN )
2019adantl 452 . . . . . . . . 9  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
m  +  1 )  e.  NN )
21 oveq1 5949 . . . . . . . . . . . . . . . . 17  |-  ( k  =  m  ->  (
k  +  1 )  =  ( m  + 
1 ) )
2221fveq2d 5609 . . . . . . . . . . . . . . . 16  |-  ( k  =  m  ->  (
g `  ( k  +  1 ) )  =  ( g `  ( m  +  1
) ) )
23 id 19 . . . . . . . . . . . . . . . . 17  |-  ( k  =  m  ->  k  =  m )
24 fveq2 5605 . . . . . . . . . . . . . . . . 17  |-  ( k  =  m  ->  (
g `  k )  =  ( g `  m ) )
2523, 24oveq12d 5960 . . . . . . . . . . . . . . . 16  |-  ( k  =  m  ->  (
k F ( g `
 k ) )  =  ( m F ( g `  m
) ) )
2622, 25eleq12d 2426 . . . . . . . . . . . . . . 15  |-  ( k  =  m  ->  (
( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  <->  ( g `  ( m  +  1 ) )  e.  ( m F ( g `
 m ) ) ) )
2726rspccva 2959 . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 ( m  + 
1 ) )  e.  ( m F ( g `  m ) ) )
29 ffvelrn 5743 . . . . . . . . . . . . . . 15  |-  ( ( g : NN --> ( X  X.  RR+ )  /\  m  e.  NN )  ->  (
g `  m )  e.  ( X  X.  RR+ ) )
306, 29sylan 457 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 m )  e.  ( X  X.  RR+ ) )
317, 1, 8bcthlem1 18844 . . . . . . . . . . . . . . 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 ) ) ) ) )
3231expr 598 . . . . . . . . . . . . . 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 ) ) ) ) ) )
3330, 32mpd 14 . . . . . . . . . . . . 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 ) ) ) ) )
3428, 33mpbid 201 . . . . . . . . . . . 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 ) ) ) )
3534simp2d 968 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  <  (
1  /  m ) )
3635adantlr 695 . . . . . . . . . 10  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  ( 2nd `  ( g `  ( m  +  1
) ) )  < 
( 1  /  m
) )
3734simp1d 967 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 ( m  + 
1 ) )  e.  ( X  X.  RR+ ) )
38 xp2nd 6234 . . . . . . . . . . . . . 14  |-  ( ( g `  ( m  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  e.  RR+ )
3937, 38syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  e.  RR+ )
4039rpred 10479 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  e.  RR )
4140adantlr 695 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  ( 2nd `  ( g `  ( m  +  1
) ) )  e.  RR )
42 nnrecre 9869 . . . . . . . . . . . 12  |-  ( m  e.  NN  ->  (
1  /  m )  e.  RR )
4342adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
1  /  m )  e.  RR )
44 rpre 10449 . . . . . . . . . . . 12  |-  ( r  e.  RR+  ->  r  e.  RR )
4544ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  r  e.  RR )
46 lttr 8986 . . . . . . . . . . 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
) )
4741, 43, 45, 46syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
( ( 2nd `  (
g `  ( m  +  1 ) ) )  <  ( 1  /  m )  /\  ( 1  /  m
)  <  r )  ->  ( 2nd `  (
g `  ( m  +  1 ) ) )  <  r ) )
4836, 47mpand 656 . . . . . . . . 9  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
( 1  /  m
)  <  r  ->  ( 2nd `  ( g `
 ( m  + 
1 ) ) )  <  r ) )
49 fveq2 5605 . . . . . . . . . . . 12  |-  ( n  =  ( m  + 
1 )  ->  (
g `  n )  =  ( g `  ( m  +  1
) ) )
5049fveq2d 5609 . . . . . . . . . . 11  |-  ( n  =  ( m  + 
1 )  ->  ( 2nd `  ( g `  n ) )  =  ( 2nd `  (
g `  ( m  +  1 ) ) ) )
5150breq1d 4112 . . . . . . . . . 10  |-  ( n  =  ( m  + 
1 )  ->  (
( 2nd `  (
g `  n )
)  <  r  <->  ( 2nd `  ( g `  (
m  +  1 ) ) )  <  r
) )
5251rspcev 2960 . . . . . . . . 9  |-  ( ( ( m  +  1 )  e.  NN  /\  ( 2nd `  ( g `
 ( m  + 
1 ) ) )  <  r )  ->  E. n  e.  NN  ( 2nd `  ( g `
 n ) )  <  r )
5320, 48, 52ee12an 1363 . . . . . . . 8  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  m  e.  NN )  ->  (
( 1  /  m
)  <  r  ->  E. n  e.  NN  ( 2nd `  ( g `  n ) )  < 
r ) )
5453rexlimdva 2743 . . . . . . 7  |-  ( (
ph  /\  r  e.  RR+ )  ->  ( E. m  e.  NN  (
1  /  m )  <  r  ->  E. n  e.  NN  ( 2nd `  (
g `  n )
)  <  r )
)
5518, 54mpd 14 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  E. n  e.  NN  ( 2nd `  (
g `  n )
)  <  r )
5655ralrimiva 2702 . . . . 5  |-  ( ph  ->  A. r  e.  RR+  E. n  e.  NN  ( 2nd `  ( g `  n ) )  < 
r )
575, 6, 14, 56caubl 18831 . . . 4  |-  ( ph  ->  ( 1st  o.  g
)  e.  ( Cau `  D ) )
587cmetcau 18813 . . . 4  |-  ( ( D  e.  ( CMet `  X )  /\  ( 1st  o.  g )  e.  ( Cau `  D
) )  ->  ( 1st  o.  g )  e. 
dom  ( ~~> t `  J ) )
591, 57, 58syl2anc 642 . . 3  |-  ( ph  ->  ( 1st  o.  g
)  e.  dom  ( ~~> t `  J )
)
60 fo1st 6223 . . . . . 6  |-  1st : _V -onto-> _V
61 fofun 5532 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
6260, 61ax-mp 8 . . . . 5  |-  Fun  1st
63 vex 2867 . . . . 5  |-  g  e. 
_V
64 cofunexg 5822 . . . . 5  |-  ( ( Fun  1st  /\  g  e.  _V )  ->  ( 1st  o.  g )  e. 
_V )
6562, 63, 64mp2an 653 . . . 4  |-  ( 1st 
o.  g )  e. 
_V
6665eldm 4955 . . 3  |-  ( ( 1st  o.  g )  e.  dom  ( ~~> t `  J )  <->  E. x
( 1st  o.  g
) ( ~~> t `  J ) x )
6759, 66sylib 188 . 2  |-  ( ph  ->  E. x ( 1st 
o.  g ) ( ~~> t `  J ) x )
68 1nn 9844 . . . . . . . 8  |-  1  e.  NN
697, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 18846 . . . . . . . 8  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  1  e.  NN )  ->  x  e.  ( (
ball `  D ) `  ( g `  1
) ) )
7068, 69mp3an3 1266 . . . . . . 7  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  x  e.  ( ( ball `  D
) `  ( g `  1 ) ) )
7112fveq2d 5609 . . . . . . . . 9  |-  ( ph  ->  ( ( ball `  D
) `  ( g `  1 ) )  =  ( ( ball `  D ) `  <. C ,  R >. )
)
72 df-ov 5945 . . . . . . . . 9  |-  ( C ( ball `  D
) R )  =  ( ( ball `  D
) `  <. C ,  R >. )
7371, 72syl6eqr 2408 . . . . . . . 8  |-  ( ph  ->  ( ( ball `  D
) `  ( g `  1 ) )  =  ( C (
ball `  D ) R ) )
7473adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  ( ( ball `  D ) `  (
g `  1 )
)  =  ( C ( ball `  D
) R ) )
7570, 74eleqtrd 2434 . . . . . 6  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  x  e.  ( C ( ball `  D
) R ) )
767mopntop 18082 . . . . . . . . . . . . . . . 16  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Top )
775, 76syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  J  e.  Top )
7877adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  J  e. 
Top )
795adantr 451 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  NN )  ->  D  e.  ( * Met `  X
) )
80 xp1st 6233 . . . . . . . . . . . . . . . . 17  |-  ( ( g `  ( m  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( g `  (
m  +  1 ) ) )  e.  X
)
8137, 80syl 15 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  NN )  ->  ( 1st `  ( g `  (
m  +  1 ) ) )  e.  X
)
8239rpxrd 10480 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  NN )  ->  ( 2nd `  ( g `  (
m  +  1 ) ) )  e.  RR* )
83 blssm 18064 . . . . . . . . . . . . . . . 16  |-  ( ( 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
)
8479, 81, 82, 83syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( 1st `  ( g `
 ( m  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  C_  X
)
85 1st2nd2 6243 . . . . . . . . . . . . . . . . . 18  |-  ( ( g `  ( m  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( g `  ( m  +  1 ) )  =  <. ( 1st `  ( g `
 ( m  + 
1 ) ) ) ,  ( 2nd `  (
g `  ( m  +  1 ) ) ) >. )
8637, 85syl 15 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  m  e.  NN )  ->  ( g `
 ( m  + 
1 ) )  = 
<. ( 1st `  (
g `  ( m  +  1 ) ) ) ,  ( 2nd `  ( g `  (
m  +  1 ) ) ) >. )
8786fveq2d 5609 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
m  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( m  + 
1 ) ) )
>. ) )
88 df-ov 5945 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  ( g `
 ( m  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
m  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( m  + 
1 ) ) )
>. )
8987, 88syl6reqr 2409 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( 1st `  ( g `
 ( m  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( m  +  1 ) ) ) )  =  ( ( ball `  D
) `  ( g `  ( m  +  1 ) ) ) )
907mopnuni 18083 . . . . . . . . . . . . . . . . 17  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
915, 90syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X  =  U. J
)
9291adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN )  ->  X  = 
U. J )
9384, 89, 923sstr3d 3296 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  U. J
)
94 eqid 2358 . . . . . . . . . . . . . . 15  |-  U. J  =  U. J
9594sscls 16893 . . . . . . . . . . . . . 14  |-  ( ( J  e.  Top  /\  ( ( ball `  D
) `  ( g `  ( m  +  1 ) ) )  C_  U. J )  ->  (
( ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
m  +  1 ) ) ) ) )
9678, 93, 95syl2anc 642 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
m  +  1 ) ) ) ) )
9734simp3d 969 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( cls `  J ) `
 ( ( ball `  D ) `  (
g `  ( m  +  1 ) ) ) )  C_  (
( ( ball `  D
) `  ( g `  m ) )  \ 
( M `  m
) ) )
9896, 97sstrd 3265 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) )  C_  (
( ( ball `  D
) `  ( g `  m ) )  \ 
( M `  m
) ) )
99983adant2 974 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  ( ( ball `  D
) `  ( g `  ( m  +  1 ) ) )  C_  ( ( ( ball `  D ) `  (
g `  m )
)  \  ( M `  m ) ) )
1007, 1, 8, 9, 10, 11, 6, 12, 13bcthlem3 18846 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  ( m  +  1
)  e.  NN )  ->  x  e.  ( ( ball `  D
) `  ( g `  ( m  +  1 ) ) ) )
10119, 100syl3an3 1217 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  x  e.  ( (
ball `  D ) `  ( g `  (
m  +  1 ) ) ) )
10299, 101sseldd 3257 . . . . . . . . . 10  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  x  e.  ( ( ( ball `  D
) `  ( g `  m ) )  \ 
( M `  m
) ) )
103 eldifn 3375 . . . . . . . . . 10  |-  ( x  e.  ( ( (
ball `  D ) `  ( g `  m
) )  \  ( M `  m )
)  ->  -.  x  e.  ( M `  m
) )
104102, 103syl 15 . . . . . . . . 9  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  m  e.  NN )  ->  -.  x  e.  ( M `  m ) )
1051043expa 1151 . . . . . . . 8  |-  ( ( ( ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  /\  m  e.  NN )  ->  -.  x  e.  ( M `  m ) )
106105ralrimiva 2702 . . . . . . 7  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  A. m  e.  NN  -.  x  e.  ( M `  m )
)
107 eluni2 3910 . . . . . . . . . . 11  |-  ( x  e.  U. ran  M  <->  E. y  e.  ran  M  x  e.  y )
108 ffn 5469 . . . . . . . . . . . . 13  |-  ( M : NN --> ( Clsd `  J )  ->  M  Fn  NN )
1099, 108syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  M  Fn  NN )
110 eleq2 2419 . . . . . . . . . . . . 13  |-  ( y  =  ( M `  m )  ->  (
x  e.  y  <->  x  e.  ( M `  m ) ) )
111110rexrn 5747 . . . . . . . . . . . 12  |-  ( M  Fn  NN  ->  ( E. y  e.  ran  M  x  e.  y  <->  E. m  e.  NN  x  e.  ( M `  m ) ) )
112109, 111syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( E. y  e. 
ran  M  x  e.  y 
<->  E. m  e.  NN  x  e.  ( M `  m ) ) )
113107, 112syl5bb 248 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  U. ran  M  <->  E. m  e.  NN  x  e.  ( M `  m ) ) )
114113notbid 285 . . . . . . . . 9  |-  ( ph  ->  ( -.  x  e. 
U. ran  M  <->  -.  E. m  e.  NN  x  e.  ( M `  m ) ) )
115 ralnex 2629 . . . . . . . . 9  |-  ( A. m  e.  NN  -.  x  e.  ( M `  m )  <->  -.  E. m  e.  NN  x  e.  ( M `  m ) )
116114, 115syl6bbr 254 . . . . . . . 8  |-  ( ph  ->  ( -.  x  e. 
U. ran  M  <->  A. m  e.  NN  -.  x  e.  ( M `  m
) ) )
117116biimpar 471 . . . . . . 7  |-  ( (
ph  /\  A. m  e.  NN  -.  x  e.  ( M `  m
) )  ->  -.  x  e.  U. ran  M
)
118106, 117syldan 456 . . . . . 6  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  -.  x  e.  U.
ran  M )
119 eldif 3238 . . . . . 6  |-  ( x  e.  ( ( C ( ball `  D
) R )  \  U. ran  M )  <->  ( x  e.  ( C ( ball `  D ) R )  /\  -.  x  e. 
U. ran  M )
)
12075, 118, 119sylanbrc 645 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  x  e.  ( ( C ( ball `  D ) R ) 
\  U. ran  M ) )
121 ne0i 3537 . . . . 5  |-  ( x  e.  ( ( C ( ball `  D
) R )  \  U. ran  M )  -> 
( ( C (
ball `  D ) R )  \  U. ran  M )  =/=  (/) )
122120, 121syl 15 . . . 4  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x )  ->  ( ( C ( ball `  D
) R )  \  U. ran  M )  =/=  (/) )
123122ex 423 . . 3  |-  ( ph  ->  ( ( 1st  o.  g ) ( ~~> t `  J ) x  -> 
( ( C (
ball `  D ) R )  \  U. ran  M )  =/=  (/) ) )
124123exlimdv 1636 . 2  |-  ( ph  ->  ( E. x ( 1st  o.  g ) ( ~~> t `  J
) x  ->  (
( C ( ball `  D ) R ) 
\  U. ran  M )  =/=  (/) ) )
12567, 124mpd 14 1  |-  ( ph  ->  ( ( C (
ball `  D ) R )  \  U. ran  M )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1541    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620   _Vcvv 2864    \ cdif 3225    C_ wss 3228   (/)c0 3531   <.cop 3719   U.cuni 3906   class class class wbr 4102   {copab 4155    X. cxp 4766   dom cdm 4768   ran crn 4769    o. ccom 4772   Fun wfun 5328    Fn wfn 5329   -->wf 5330   -onto->wfo 5332   ` cfv 5334  (class class class)co 5942    e. cmpt2 5944   1stc1st 6204   2ndc2nd 6205   RRcr 8823   0cc0 8824   1c1 8825    + caddc 8827   RR*cxr 8953    < clt 8954    / cdiv 9510   NNcn 9833   RR+crp 10443   * Metcxmt 16462   Metcme 16463   ballcbl 16464   MetOpencmopn 16467   Topctop 16731   Clsdccld 16853   clsccl 16855   ~~> tclm 17056   Caucca 18777   CMetcms 18778
This theorem is referenced by:  bcthlem5  18848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-map 6859  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-n0 10055  df-z 10114  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-ico 10751  df-rest 13420  df-topgen 13437  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-fbas 16473  df-fg 16474  df-top 16736  df-bases 16738  df-topon 16739  df-cld 16856  df-ntr 16857  df-cls 16858  df-nei 16935  df-lm 17059  df-fil 17637  df-fm 17729  df-flim 17730  df-flf 17731  df-cfil 18779  df-cau 18780  df-cmet 18781
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