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Theorem heiborlem8 31854
Description: Lemma for heibor 31857. The previous lemmas establish that the sequence  M is Cauchy, so using completeness we now consider the convergent point 
Y. By assumption,  U is an open cover, so  Y is an element of some  Z  e.  U, and some ball centered at  Y is contained in  Z. But the sequence contains arbitrarily small balls close to  Y, so some element  ball ( M `  n ) of the sequence is contained in  Z. And finally we arrive at a contradiction, because  { Z } is a finite subcover of  U that covers  ball ( M `  n ), yet  ball ( M `  n )  e.  K. For convenience, we write this contradiction as 
ph  ->  ps where  ph is all the accumulated hypotheses and  ps is anything at all. (Contributed by Jeff Madsen, 22-Jan-2014.)
Hypotheses
Ref Expression
heibor.1  |-  J  =  ( MetOpen `  D )
heibor.3  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
heibor.4  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
heibor.5  |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D ) ( 1  /  ( 2 ^ m ) ) ) )
heibor.6  |-  ( ph  ->  D  e.  ( CMet `  X ) )
heibor.7  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
heibor.8  |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
heibor.9  |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
heibor.10  |-  ( ph  ->  C G 0 )
heibor.11  |-  S  =  seq 0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) )
heibor.12  |-  M  =  ( n  e.  NN  |->  <. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.
)
heibor.13  |-  ( ph  ->  U  C_  J )
heibor.14  |-  Y  e. 
_V
heibor.15  |-  ( ph  ->  Y  e.  Z )
heibor.16  |-  ( ph  ->  Z  e.  U )
heibor.17  |-  ( ph  ->  ( 1st  o.  M
) ( ~~> t `  J ) Y )
Assertion
Ref Expression
heiborlem8  |-  ( ph  ->  ps )
Distinct variable groups:    x, n, y, u, F    x, G    ph, x    m, n, u, v, x, y, z, D    m, M, u, x, y, z    T, m, n, x, y, z    B, n, u, v, y   
m, J, n, u, v, x, y, z    U, n, u, v, x, y, z    ps, y,
z    S, m, n, u, v, x, y, z   
m, X, n, u, v, x, y, z    C, m, n, u, v, y    n, K, x, y, z    x, Y   
v, Z, x    x, B
Allowed substitution hints:    ph( y, z, v, u, m, n)    ps( x, v, u, m, n)    B( z, m)    C( x, z)    T( v, u)    U( m)    F( z, v, m)    G( y, z, v, u, m, n)    K( v, u, m)    M( v, n)    Y( y, z, v, u, m, n)    Z( y, z, u, m, n)

Proof of Theorem heiborlem8
Dummy variables  t 
k  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 heibor.6 . . . 4  |-  ( ph  ->  D  e.  ( CMet `  X ) )
2 cmetmet 22142 . . . 4  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
3 metxmet 21273 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
41, 2, 33syl 18 . . 3  |-  ( ph  ->  D  e.  ( *Met `  X ) )
5 heibor.13 . . . 4  |-  ( ph  ->  U  C_  J )
6 heibor.16 . . . 4  |-  ( ph  ->  Z  e.  U )
75, 6sseldd 3462 . . 3  |-  ( ph  ->  Z  e.  J )
8 heibor.15 . . 3  |-  ( ph  ->  Y  e.  Z )
9 heibor.1 . . . 4  |-  J  =  ( MetOpen `  D )
109mopni2 21432 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  Z  e.  J  /\  Y  e.  Z
)  ->  E. x  e.  RR+  ( Y (
ball `  D )
x )  C_  Z
)
114, 7, 8, 10syl3anc 1264 . 2  |-  ( ph  ->  E. x  e.  RR+  ( Y ( ball `  D
) x )  C_  Z )
12 rphalfcl 11316 . . . . . 6  |-  ( x  e.  RR+  ->  ( x  /  2 )  e.  RR+ )
13 breq2 4421 . . . . . . . 8  |-  ( r  =  ( x  / 
2 )  ->  (
( 2nd `  ( M `  k )
)  <  r  <->  ( 2nd `  ( M `  k
) )  <  (
x  /  2 ) ) )
1413rexbidv 2937 . . . . . . 7  |-  ( r  =  ( x  / 
2 )  ->  ( E. k  e.  NN  ( 2nd `  ( M `
 k ) )  <  r  <->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  ( x  /  2 ) ) )
15 heibor.3 . . . . . . . 8  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
16 heibor.4 . . . . . . . 8  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
17 heibor.5 . . . . . . . 8  |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D ) ( 1  /  ( 2 ^ m ) ) ) )
18 heibor.7 . . . . . . . 8  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
19 heibor.8 . . . . . . . 8  |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
20 heibor.9 . . . . . . . 8  |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
21 heibor.10 . . . . . . . 8  |-  ( ph  ->  C G 0 )
22 heibor.11 . . . . . . . 8  |-  S  =  seq 0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) )
23 heibor.12 . . . . . . . 8  |-  M  =  ( n  e.  NN  |->  <. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.
)
249, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23heiborlem7 31853 . . . . . . 7  |-  A. r  e.  RR+  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  r
2514, 24vtoclri 3153 . . . . . 6  |-  ( ( x  /  2 )  e.  RR+  ->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  ( x  /  2 ) )
2612, 25syl 17 . . . . 5  |-  ( x  e.  RR+  ->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  ( x  /  2 ) )
2726adantl 467 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  ( x  /  2 ) )
28 nnnn0 10865 . . . . . . 7  |-  ( k  e.  NN  ->  k  e.  NN0 )
299, 15, 16, 17, 1, 18, 19, 20, 21, 22heiborlem4 31850 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( S `  k ) G k )
30 fvex 5882 . . . . . . . . . 10  |-  ( S `
 k )  e. 
_V
31 vex 3081 . . . . . . . . . 10  |-  k  e. 
_V
329, 15, 16, 30, 31heiborlem2 31848 . . . . . . . . 9  |-  ( ( S `  k ) G k  <->  ( k  e.  NN0  /\  ( S `
 k )  e.  ( F `  k
)  /\  ( ( S `  k ) B k )  e.  K ) )
3332simp3bi 1022 . . . . . . . 8  |-  ( ( S `  k ) G k  ->  (
( S `  k
) B k )  e.  K )
3429, 33syl 17 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( S `  k ) B k )  e.  K )
3528, 34sylan2 476 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( S `  k ) B k )  e.  K )
3635ad2ant2r 751 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( S `
 k ) B k )  e.  K
)
374ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  D  e.  ( *Met `  X
) )
389, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23heiborlem5 31851 . . . . . . . . . . . . 13  |-  ( ph  ->  M : NN --> ( X  X.  RR+ ) )
3938ffvelrnda 6028 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  ( M `
 k )  e.  ( X  X.  RR+ ) )
4039ad2ant2r 751 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( M `  k )  e.  ( X  X.  RR+ )
)
41 xp1st 6828 . . . . . . . . . . 11  |-  ( ( M `  k )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( M `  k
) )  e.  X
)
4240, 41syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1st `  ( M `  k )
)  e.  X )
43 2nn 10756 . . . . . . . . . . . . . . 15  |-  2  e.  NN
44 nnexpcl 12271 . . . . . . . . . . . . . . 15  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
4543, 28, 44sylancr 667 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  NN )
4645nnrpd 11328 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  RR+ )
4746rpreccld 11340 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  (
1  /  ( 2 ^ k ) )  e.  RR+ )
4847ad2antrl 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1  / 
( 2 ^ k
) )  e.  RR+ )
4948rpxrd 11331 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1  / 
( 2 ^ k
) )  e.  RR* )
50 xp2nd 6829 . . . . . . . . . . . 12  |-  ( ( M `  k )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( M `  k
) )  e.  RR+ )
5140, 50syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 2nd `  ( M `  k )
)  e.  RR+ )
5251rpxrd 11331 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 2nd `  ( M `  k )
)  e.  RR* )
53 1le3 10815 . . . . . . . . . . . . . 14  |-  1  <_  3
54 elrp 11293 . . . . . . . . . . . . . . 15  |-  ( ( 2 ^ k )  e.  RR+  <->  ( ( 2 ^ k )  e.  RR  /\  0  < 
( 2 ^ k
) ) )
55 1re 9631 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
56 3re 10672 . . . . . . . . . . . . . . . 16  |-  3  e.  RR
57 lediv1 10459 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  RR  /\  3  e.  RR  /\  (
( 2 ^ k
)  e.  RR  /\  0  <  ( 2 ^ k ) ) )  ->  ( 1  <_ 
3  <->  ( 1  / 
( 2 ^ k
) )  <_  (
3  /  ( 2 ^ k ) ) ) )
5855, 56, 57mp3an12 1350 . . . . . . . . . . . . . . 15  |-  ( ( ( 2 ^ k
)  e.  RR  /\  0  <  ( 2 ^ k ) )  -> 
( 1  <_  3  <->  ( 1  /  ( 2 ^ k ) )  <_  ( 3  / 
( 2 ^ k
) ) ) )
5954, 58sylbi 198 . . . . . . . . . . . . . 14  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 1  <_  3  <->  ( 1  /  ( 2 ^ k ) )  <_ 
( 3  /  (
2 ^ k ) ) ) )
6053, 59mpbii 214 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 1  /  ( 2 ^ k ) )  <_ 
( 3  /  (
2 ^ k ) ) )
6146, 60syl 17 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  (
1  /  ( 2 ^ k ) )  <_  ( 3  / 
( 2 ^ k
) ) )
6261ad2antrl 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1  / 
( 2 ^ k
) )  <_  (
3  /  ( 2 ^ k ) ) )
63 fveq2 5872 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  ( S `  n )  =  ( S `  k ) )
64 oveq2 6304 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  (
2 ^ n )  =  ( 2 ^ k ) )
6564oveq2d 6312 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  (
3  /  ( 2 ^ n ) )  =  ( 3  / 
( 2 ^ k
) ) )
6663, 65opeq12d 4189 . . . . . . . . . . . . . . 15  |-  ( n  =  k  ->  <. ( S `  n ) ,  ( 3  / 
( 2 ^ n
) ) >.  =  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
67 opex 4677 . . . . . . . . . . . . . . 15  |-  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >.  e.  _V
6866, 23, 67fvmpt 5955 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( M `  k )  =  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )
6968fveq2d 5876 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 2nd `  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
)
70 ovex 6324 . . . . . . . . . . . . . 14  |-  ( 3  /  ( 2 ^ k ) )  e. 
_V
7130, 70op2nd 6807 . . . . . . . . . . . . 13  |-  ( 2nd `  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )  =  (
3  /  ( 2 ^ k ) )
7269, 71syl6eq 2477 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 3  /  (
2 ^ k ) ) )
7372ad2antrl 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 2nd `  ( M `  k )
)  =  ( 3  /  ( 2 ^ k ) ) )
7462, 73breqtrrd 4443 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1  / 
( 2 ^ k
) )  <_  ( 2nd `  ( M `  k ) ) )
75 ssbl 21362 . . . . . . . . . 10  |-  ( ( ( D  e.  ( *Met `  X
)  /\  ( 1st `  ( M `  k
) )  e.  X
)  /\  ( (
1  /  ( 2 ^ k ) )  e.  RR*  /\  ( 2nd `  ( M `  k ) )  e. 
RR* )  /\  (
1  /  ( 2 ^ k ) )  <_  ( 2nd `  ( M `  k )
) )  ->  (
( 1st `  ( M `  k )
) ( ball `  D
) ( 1  / 
( 2 ^ k
) ) )  C_  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( 2nd `  ( M `  k )
) ) )
7637, 42, 49, 52, 74, 75syl221anc 1275 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 1  /  ( 2 ^ k ) ) ) 
C_  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 2nd `  ( M `  k
) ) ) )
7728ad2antrl 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  k  e.  NN0 )
78 oveq1 6303 . . . . . . . . . . . 12  |-  ( z  =  ( 1st `  ( M `  k )
)  ->  ( z
( ball `  D )
( 1  /  (
2 ^ m ) ) )  =  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( 1  / 
( 2 ^ m
) ) ) )
79 oveq2 6304 . . . . . . . . . . . . . 14  |-  ( m  =  k  ->  (
2 ^ m )  =  ( 2 ^ k ) )
8079oveq2d 6312 . . . . . . . . . . . . 13  |-  ( m  =  k  ->  (
1  /  ( 2 ^ m ) )  =  ( 1  / 
( 2 ^ k
) ) )
8180oveq2d 6312 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
( 1st `  ( M `  k )
) ( ball `  D
) ( 1  / 
( 2 ^ m
) ) )  =  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( 1  / 
( 2 ^ k
) ) ) )
82 ovex 6324 . . . . . . . . . . . 12  |-  ( ( 1st `  ( M `
 k ) ) ( ball `  D
) ( 1  / 
( 2 ^ k
) ) )  e. 
_V
8378, 81, 17, 82ovmpt2 6437 . . . . . . . . . . 11  |-  ( ( ( 1st `  ( M `  k )
)  e.  X  /\  k  e.  NN0 )  -> 
( ( 1st `  ( M `  k )
) B k )  =  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 1  /  ( 2 ^ k ) ) ) )
8442, 77, 83syl2anc 665 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) B k )  =  ( ( 1st `  ( M `
 k ) ) ( ball `  D
) ( 1  / 
( 2 ^ k
) ) ) )
8568fveq2d 5876 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  ( 1st `  ( M `  k ) )  =  ( 1st `  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
)
8630, 70op1st 6806 . . . . . . . . . . . . 13  |-  ( 1st `  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )  =  ( S `  k )
8785, 86syl6eq 2477 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  ( 1st `  ( M `  k ) )  =  ( S `  k
) )
8887ad2antrl 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1st `  ( M `  k )
)  =  ( S `
 k ) )
8988oveq1d 6311 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) B k )  =  ( ( S `  k ) B k ) )
9084, 89eqtr3d 2463 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 1  /  ( 2 ^ k ) ) )  =  ( ( S `
 k ) B k ) )
91 1st2nd2 6835 . . . . . . . . . . . 12  |-  ( ( M `  k )  e.  ( X  X.  RR+ )  ->  ( M `  k )  =  <. ( 1st `  ( M `
 k ) ) ,  ( 2nd `  ( M `  k )
) >. )
9240, 91syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( M `  k )  =  <. ( 1st `  ( M `
 k ) ) ,  ( 2nd `  ( M `  k )
) >. )
9392fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( ball `  D ) `  ( M `  k )
)  =  ( (
ball `  D ) `  <. ( 1st `  ( M `  k )
) ,  ( 2nd `  ( M `  k
) ) >. )
)
94 df-ov 6299 . . . . . . . . . 10  |-  ( ( 1st `  ( M `
 k ) ) ( ball `  D
) ( 2nd `  ( M `  k )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( M `  k
) ) ,  ( 2nd `  ( M `
 k ) )
>. )
9593, 94syl6reqr 2480 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 2nd `  ( M `  k
) ) )  =  ( ( ball `  D
) `  ( M `  k ) ) )
9676, 90, 953sstr3d 3503 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( S `
 k ) B k )  C_  (
( ball `  D ) `  ( M `  k
) ) )
979mopntop 21379 . . . . . . . . . . 11  |-  ( D  e.  ( *Met `  X )  ->  J  e.  Top )
9837, 97syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  J  e.  Top )
99 blssm 21357 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  X )  /\  ( 1st `  ( M `  k )
)  e.  X  /\  ( 2nd `  ( M `
 k ) )  e.  RR* )  ->  (
( 1st `  ( M `  k )
) ( ball `  D
) ( 2nd `  ( M `  k )
) )  C_  X
)
10037, 42, 52, 99syl3anc 1264 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 2nd `  ( M `  k
) ) )  C_  X )
1019mopnuni 21380 . . . . . . . . . . . 12  |-  ( D  e.  ( *Met `  X )  ->  X  =  U. J )
10237, 101syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  X  =  U. J )
103100, 95, 1023sstr3d 3503 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( ball `  D ) `  ( M `  k )
)  C_  U. J )
104 eqid 2420 . . . . . . . . . . 11  |-  U. J  =  U. J
105104sscls 19995 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( ( ball `  D
) `  ( M `  k ) )  C_  U. J )  ->  (
( ball `  D ) `  ( M `  k
) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) )
10698, 103, 105syl2anc 665 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( ball `  D ) `  ( M `  k )
)  C_  ( ( cls `  J ) `  ( ( ball `  D
) `  ( M `  k ) ) ) )
10795fveq2d 5876 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( cls `  J ) `  (
( 1st `  ( M `  k )
) ( ball `  D
) ( 2nd `  ( M `  k )
) ) )  =  ( ( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) )
10812ad2antlr 731 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( x  / 
2 )  e.  RR+ )
109108rpxrd 11331 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( x  / 
2 )  e.  RR* )
110 simprr 764 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 2nd `  ( M `  k )
)  <  ( x  /  2 ) )
1119blsscls 21446 . . . . . . . . . . . 12  |-  ( ( ( D  e.  ( *Met `  X
)  /\  ( 1st `  ( M `  k
) )  e.  X
)  /\  ( ( 2nd `  ( M `  k ) )  e. 
RR*  /\  ( x  /  2 )  e. 
RR*  /\  ( 2nd `  ( M `  k
) )  <  (
x  /  2 ) ) )  ->  (
( cls `  J
) `  ( ( 1st `  ( M `  k ) ) (
ball `  D )
( 2nd `  ( M `  k )
) ) )  C_  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( x  / 
2 ) ) )
11237, 42, 52, 109, 110, 111syl23anc 1271 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( cls `  J ) `  (
( 1st `  ( M `  k )
) ( ball `  D
) ( 2nd `  ( M `  k )
) ) )  C_  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( x  / 
2 ) ) )
113107, 112eqsstr3d 3496 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( cls `  J ) `  (
( ball `  D ) `  ( M `  k
) ) )  C_  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( x  / 
2 ) ) )
114 rpre 11297 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  x  e.  RR )
115114ad2antlr 731 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  x  e.  RR )
116 heibor.17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st  o.  M
) ( ~~> t `  J ) Y )
1179, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23heiborlem6 31852 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A. t  e.  NN  ( ( ball `  D
) `  ( M `  ( t  +  1 ) ) )  C_  ( ( ball `  D
) `  ( M `  t ) ) )
1184, 38, 117, 9caublcls 22164 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( 1st  o.  M ) ( ~~> t `  J ) Y  /\  k  e.  NN )  ->  Y  e.  ( ( cls `  J ) `
 ( ( ball `  D ) `  ( M `  k )
) ) )
1191183expia 1207 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 1st  o.  M ) ( ~~> t `  J ) Y )  ->  ( k  e.  NN  ->  Y  e.  ( ( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) ) )
120116, 119mpdan 672 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( k  e.  NN  ->  Y  e.  ( ( cls `  J ) `
 ( ( ball `  D ) `  ( M `  k )
) ) ) )
121120imp 430 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  NN )  ->  Y  e.  ( ( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) )
122121ad2ant2r 751 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  Y  e.  ( ( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) )
123113, 122sseldd 3462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  Y  e.  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( x  / 
2 ) ) )
124 blhalf 21344 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( *Met `  X
)  /\  ( 1st `  ( M `  k
) )  e.  X
)  /\  ( x  e.  RR  /\  Y  e.  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( x  / 
2 ) ) ) )  ->  ( ( 1st `  ( M `  k ) ) (
ball `  D )
( x  /  2
) )  C_  ( Y ( ball `  D
) x ) )
12537, 42, 115, 123, 124syl22anc 1265 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( x  /  2 ) ) 
C_  ( Y (
ball `  D )
x ) )
126113, 125sstrd 3471 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( cls `  J ) `  (
( ball `  D ) `  ( M `  k
) ) )  C_  ( Y ( ball `  D
) x ) )
127106, 126sstrd 3471 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( ball `  D ) `  ( M `  k )
)  C_  ( Y
( ball `  D )
x ) )
12896, 127sstrd 3471 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( S `
 k ) B k )  C_  ( Y ( ball `  D
) x ) )
129 sstr2 3468 . . . . . . 7  |-  ( ( ( S `  k
) B k ) 
C_  ( Y (
ball `  D )
x )  ->  (
( Y ( ball `  D ) x ) 
C_  Z  ->  (
( S `  k
) B k ) 
C_  Z ) )
130128, 129syl 17 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( Y ( ball `  D
) x )  C_  Z  ->  ( ( S `
 k ) B k )  C_  Z
) )
131 unisng 4229 . . . . . . . . . . . . 13  |-  ( Z  e.  U  ->  U. { Z }  =  Z
)
1326, 131syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  U. { Z }  =  Z )
133132sseq2d 3489 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( S `
 k ) B k )  C_  U. { Z }  <->  ( ( S `
 k ) B k )  C_  Z
) )
134133biimpar 487 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( S `  k ) B k )  C_  Z )  ->  (
( S `  k
) B k ) 
C_  U. { Z }
)
1356snssd 4139 . . . . . . . . . . . . 13  |-  ( ph  ->  { Z }  C_  U )
136 snex 4654 . . . . . . . . . . . . . 14  |-  { Z }  e.  _V
137136elpw 3982 . . . . . . . . . . . . 13  |-  ( { Z }  e.  ~P U 
<->  { Z }  C_  U )
138135, 137sylibr 215 . . . . . . . . . . . 12  |-  ( ph  ->  { Z }  e.  ~P U )
139 snfi 7648 . . . . . . . . . . . . 13  |-  { Z }  e.  Fin
140139a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  { Z }  e.  Fin )
141138, 140elind 3647 . . . . . . . . . . 11  |-  ( ph  ->  { Z }  e.  ( ~P U  i^i  Fin ) )
142 unieq 4221 . . . . . . . . . . . . 13  |-  ( v  =  { Z }  ->  U. v  =  U. { Z } )
143142sseq2d 3489 . . . . . . . . . . . 12  |-  ( v  =  { Z }  ->  ( ( ( S `
 k ) B k )  C_  U. v  <->  ( ( S `  k
) B k ) 
C_  U. { Z }
) )
144143rspcev 3179 . . . . . . . . . . 11  |-  ( ( { Z }  e.  ( ~P U  i^i  Fin )  /\  ( ( S `
 k ) B k )  C_  U. { Z } )  ->  E. v  e.  ( ~P U  i^i  Fin ) ( ( S `
 k ) B k )  C_  U. v
)
145141, 144sylan 473 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( S `  k ) B k )  C_  U. { Z } )  ->  E. v  e.  ( ~P U  i^i  Fin ) ( ( S `
 k ) B k )  C_  U. v
)
146134, 145syldan 472 . . . . . . . . 9  |-  ( (
ph  /\  ( ( S `  k ) B k )  C_  Z )  ->  E. v  e.  ( ~P U  i^i  Fin ) ( ( S `
 k ) B k )  C_  U. v
)
147 ovex 6324 . . . . . . . . . . 11  |-  ( ( S `  k ) B k )  e. 
_V
148 sseq1 3482 . . . . . . . . . . . . 13  |-  ( u  =  ( ( S `
 k ) B k )  ->  (
u  C_  U. v  <->  ( ( S `  k
) B k ) 
C_  U. v ) )
149148rexbidv 2937 . . . . . . . . . . . 12  |-  ( u  =  ( ( S `
 k ) B k )  ->  ( E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v  <->  E. v  e.  ( ~P U  i^i  Fin )
( ( S `  k ) B k )  C_  U. v
) )
150149notbid 295 . . . . . . . . . . 11  |-  ( u  =  ( ( S `
 k ) B k )  ->  ( -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v  <->  -. 
E. v  e.  ( ~P U  i^i  Fin ) ( ( S `
 k ) B k )  C_  U. v
) )
151147, 150, 15elab2 3218 . . . . . . . . . 10  |-  ( ( ( S `  k
) B k )  e.  K  <->  -.  E. v  e.  ( ~P U  i^i  Fin ) ( ( S `
 k ) B k )  C_  U. v
)
152151con2bii 333 . . . . . . . . 9  |-  ( E. v  e.  ( ~P U  i^i  Fin )
( ( S `  k ) B k )  C_  U. v  <->  -.  ( ( S `  k ) B k )  e.  K )
153146, 152sylib 199 . . . . . . . 8  |-  ( (
ph  /\  ( ( S `  k ) B k )  C_  Z )  ->  -.  ( ( S `  k ) B k )  e.  K )
154153ex 435 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 k ) B k )  C_  Z  ->  -.  ( ( S `
 k ) B k )  e.  K
) )
155154ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( ( S `  k ) B k )  C_  Z  ->  -.  ( ( S `  k ) B k )  e.  K ) )
156130, 155syld 45 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( Y ( ball `  D
) x )  C_  Z  ->  -.  ( ( S `  k ) B k )  e.  K ) )
15736, 156mt2d 120 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  -.  ( Y
( ball `  D )
x )  C_  Z
)
15827, 157rexlimddv 2919 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  -.  ( Y ( ball `  D
) x )  C_  Z )
159158nrexdv 2879 . 2  |-  ( ph  ->  -.  E. x  e.  RR+  ( Y ( ball `  D ) x ) 
C_  Z )
16011, 159pm2.21dd 177 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   {cab 2405   A.wral 2773   E.wrex 2774   _Vcvv 3078    i^i cin 3432    C_ wss 3433   ifcif 3906   ~Pcpw 3976   {csn 3993   <.cop 3999   U.cuni 4213   U_ciun 4293   class class class wbr 4417   {copab 4474    |-> cmpt 4475    X. cxp 4843    o. ccom 4849   -->wf 5588   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6796   2ndc2nd 6797   Fincfn 7568   RRcr 9527   0cc0 9528   1c1 9529    + caddc 9531   RR*cxr 9663    < clt 9664    <_ cle 9665    - cmin 9849    / cdiv 10258   NNcn 10598   2c2 10648   3c3 10649   NN0cn0 10858   RR+crp 11291    seqcseq 12199   ^cexp 12258   *Metcxmt 18883   Metcme 18884   ballcbl 18885   MetOpencmopn 18888   Topctop 19841   clsccl 19957   ~~> tclm 20166   CMetcms 22110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-er 7362  df-map 7473  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-fl 12014  df-seq 12200  df-exp 12259  df-topgen 15294  df-psmet 18890  df-xmet 18891  df-met 18892  df-bl 18893  df-mopn 18894  df-top 19845  df-bases 19846  df-topon 19847  df-cld 19958  df-ntr 19959  df-cls 19960  df-lm 20169  df-cmet 22113
This theorem is referenced by:  heiborlem9  31855
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