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Theorem heiborlem8 32214
Description: Lemma for heibor 32217. 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 22334 . . . 4  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
3 metxmet 21427 . . . 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 3419 . . 3  |-  ( ph  ->  Z  e.  J )
8 heibor.15 . . 3  |-  ( ph  ->  Y  e.  Z )
9 heibor.1 . . . 4  |-  J  =  ( MetOpen `  D )
109mopni2 21586 . . 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 1292 . 2  |-  ( ph  ->  E. x  e.  RR+  ( Y ( ball `  D
) x )  C_  Z )
12 rphalfcl 11350 . . . . . 6  |-  ( x  e.  RR+  ->  ( x  /  2 )  e.  RR+ )
13 breq2 4399 . . . . . . . 8  |-  ( r  =  ( x  / 
2 )  ->  (
( 2nd `  ( M `  k )
)  <  r  <->  ( 2nd `  ( M `  k
) )  <  (
x  /  2 ) ) )
1413rexbidv 2892 . . . . . . 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 32213 . . . . . . 7  |-  A. r  e.  RR+  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  r
2514, 24vtoclri 3110 . . . . . 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 473 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  ( x  /  2 ) )
28 nnnn0 10900 . . . . . . 7  |-  ( k  e.  NN  ->  k  e.  NN0 )
299, 15, 16, 17, 1, 18, 19, 20, 21, 22heiborlem4 32210 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( S `  k ) G k )
30 fvex 5889 . . . . . . . . . 10  |-  ( S `
 k )  e. 
_V
31 vex 3034 . . . . . . . . . 10  |-  k  e. 
_V
329, 15, 16, 30, 31heiborlem2 32208 . . . . . . . . 9  |-  ( ( S `  k ) G k  <->  ( k  e.  NN0  /\  ( S `
 k )  e.  ( F `  k
)  /\  ( ( S `  k ) B k )  e.  K ) )
3332simp3bi 1047 . . . . . . . 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 482 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( S `  k ) B k )  e.  K )
3635ad2ant2r 761 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( S `
 k ) B k )  e.  K
)
374ad2antrr 740 . . . . . . . . . 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 32211 . . . . . . . . . . . . 13  |-  ( ph  ->  M : NN --> ( X  X.  RR+ ) )
3938ffvelrnda 6037 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  ( M `
 k )  e.  ( X  X.  RR+ ) )
4039ad2ant2r 761 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( M `  k )  e.  ( X  X.  RR+ )
)
41 xp1st 6842 . . . . . . . . . . 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 10790 . . . . . . . . . . . . . . 15  |-  2  e.  NN
44 nnexpcl 12323 . . . . . . . . . . . . . . 15  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
4543, 28, 44sylancr 676 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  NN )
4645nnrpd 11362 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  RR+ )
4746rpreccld 11374 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  (
1  /  ( 2 ^ k ) )  e.  RR+ )
4847ad2antrl 742 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1  / 
( 2 ^ k
) )  e.  RR+ )
4948rpxrd 11365 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1  / 
( 2 ^ k
) )  e.  RR* )
50 xp2nd 6843 . . . . . . . . . . . 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 11365 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 2nd `  ( M `  k )
)  e.  RR* )
53 1le3 10849 . . . . . . . . . . . . . 14  |-  1  <_  3
54 elrp 11327 . . . . . . . . . . . . . . 15  |-  ( ( 2 ^ k )  e.  RR+  <->  ( ( 2 ^ k )  e.  RR  /\  0  < 
( 2 ^ k
) ) )
55 1re 9660 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
56 3re 10705 . . . . . . . . . . . . . . . 16  |-  3  e.  RR
57 lediv1 10492 . . . . . . . . . . . . . . . 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 1380 . . . . . . . . . . . . . . 15  |-  ( ( ( 2 ^ k
)  e.  RR  /\  0  <  ( 2 ^ k ) )  -> 
( 1  <_  3  <->  ( 1  /  ( 2 ^ k ) )  <_  ( 3  / 
( 2 ^ k
) ) ) )
5954, 58sylbi 200 . . . . . . . . . . . . . 14  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 1  <_  3  <->  ( 1  /  ( 2 ^ k ) )  <_ 
( 3  /  (
2 ^ k ) ) ) )
6053, 59mpbii 216 . . . . . . . . . . . . 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 742 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1  / 
( 2 ^ k
) )  <_  (
3  /  ( 2 ^ k ) ) )
63 fveq2 5879 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  ( S `  n )  =  ( S `  k ) )
64 oveq2 6316 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  (
2 ^ n )  =  ( 2 ^ k ) )
6564oveq2d 6324 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  (
3  /  ( 2 ^ n ) )  =  ( 3  / 
( 2 ^ k
) ) )
6663, 65opeq12d 4166 . . . . . . . . . . . . . . 15  |-  ( n  =  k  ->  <. ( S `  n ) ,  ( 3  / 
( 2 ^ n
) ) >.  =  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
67 opex 4664 . . . . . . . . . . . . . . 15  |-  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >.  e.  _V
6866, 23, 67fvmpt 5963 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( M `  k )  =  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )
6968fveq2d 5883 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 2nd `  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
)
70 ovex 6336 . . . . . . . . . . . . . 14  |-  ( 3  /  ( 2 ^ k ) )  e. 
_V
7130, 70op2nd 6821 . . . . . . . . . . . . 13  |-  ( 2nd `  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )  =  (
3  /  ( 2 ^ k ) )
7269, 71syl6eq 2521 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 3  /  (
2 ^ k ) ) )
7372ad2antrl 742 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 2nd `  ( M `  k )
)  =  ( 3  /  ( 2 ^ k ) ) )
7462, 73breqtrrd 4422 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1  / 
( 2 ^ k
) )  <_  ( 2nd `  ( M `  k ) ) )
75 ssbl 21516 . . . . . . . . . 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 1303 . . . . . . . . 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 742 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  k  e.  NN0 )
78 oveq1 6315 . . . . . . . . . . . 12  |-  ( z  =  ( 1st `  ( M `  k )
)  ->  ( z
( ball `  D )
( 1  /  (
2 ^ m ) ) )  =  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( 1  / 
( 2 ^ m
) ) ) )
79 oveq2 6316 . . . . . . . . . . . . . 14  |-  ( m  =  k  ->  (
2 ^ m )  =  ( 2 ^ k ) )
8079oveq2d 6324 . . . . . . . . . . . . 13  |-  ( m  =  k  ->  (
1  /  ( 2 ^ m ) )  =  ( 1  / 
( 2 ^ k
) ) )
8180oveq2d 6324 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
( 1st `  ( M `  k )
) ( ball `  D
) ( 1  / 
( 2 ^ m
) ) )  =  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( 1  / 
( 2 ^ k
) ) ) )
82 ovex 6336 . . . . . . . . . . . 12  |-  ( ( 1st `  ( M `
 k ) ) ( ball `  D
) ( 1  / 
( 2 ^ k
) ) )  e. 
_V
8378, 81, 17, 82ovmpt2 6451 . . . . . . . . . . 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 673 . . . . . . . . . 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 5883 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  ( 1st `  ( M `  k ) )  =  ( 1st `  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
)
8630, 70op1st 6820 . . . . . . . . . . . . 13  |-  ( 1st `  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )  =  ( S `  k )
8785, 86syl6eq 2521 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  ( 1st `  ( M `  k ) )  =  ( S `  k
) )
8887ad2antrl 742 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1st `  ( M `  k )
)  =  ( S `
 k ) )
8988oveq1d 6323 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) B k )  =  ( ( S `  k ) B k ) )
9084, 89eqtr3d 2507 . . . . . . . . 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 6849 . . . . . . . . . . . 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 5883 . . . . . . . . . 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 6311 . . . . . . . . . 10  |-  ( ( 1st `  ( M `
 k ) ) ( ball `  D
) ( 2nd `  ( M `  k )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( M `  k
) ) ,  ( 2nd `  ( M `
 k ) )
>. )
9593, 94syl6reqr 2524 . . . . . . . . 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 3460 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( S `
 k ) B k )  C_  (
( ball `  D ) `  ( M `  k
) ) )
979mopntop 21533 . . . . . . . . . . 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 21511 . . . . . . . . . . . 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 1292 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 2nd `  ( M `  k
) ) )  C_  X )
1019mopnuni 21534 . . . . . . . . . . . 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 3460 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( ball `  D ) `  ( M `  k )
)  C_  U. J )
104 eqid 2471 . . . . . . . . . . 11  |-  U. J  =  U. J
105104sscls 20148 . . . . . . . . . 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 673 . . . . . . . . 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 5883 . . . . . . . . . . 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 741 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( x  / 
2 )  e.  RR+ )
109108rpxrd 11365 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( x  / 
2 )  e.  RR* )
110 simprr 774 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 2nd `  ( M `  k )
)  <  ( x  /  2 ) )
1119blsscls 21600 . . . . . . . . . . . 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 1299 . . . . . . . . . . 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 3453 . . . . . . . . . 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 11331 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  x  e.  RR )
115114ad2antlr 741 . . . . . . . . . . 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 32212 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A. t  e.  NN  ( ( ball `  D
) `  ( M `  ( t  +  1 ) ) )  C_  ( ( ball `  D
) `  ( M `  t ) ) )
1184, 38, 117, 9caublcls 22356 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( 1st  o.  M ) ( ~~> t `  J ) Y  /\  k  e.  NN )  ->  Y  e.  ( ( cls `  J ) `
 ( ( ball `  D ) `  ( M `  k )
) ) )
1191183expia 1233 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 1st  o.  M ) ( ~~> t `  J ) Y )  ->  ( k  e.  NN  ->  Y  e.  ( ( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) ) )
120116, 119mpdan 681 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( k  e.  NN  ->  Y  e.  ( ( cls `  J ) `
 ( ( ball `  D ) `  ( M `  k )
) ) ) )
121120imp 436 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  NN )  ->  Y  e.  ( ( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) )
122121ad2ant2r 761 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  Y  e.  ( ( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) )
123113, 122sseldd 3419 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  Y  e.  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( x  / 
2 ) ) )
124 blhalf 21498 . . . . . . . . . . 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 1293 . . . . . . . . . 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 3428 . . . . . . . . 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 3428 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( ball `  D ) `  ( M `  k )
)  C_  ( Y
( ball `  D )
x ) )
12896, 127sstrd 3428 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( S `
 k ) B k )  C_  ( Y ( ball `  D
) x ) )
129 sstr2 3425 . . . . . . 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 4206 . . . . . . . . . . . . 13  |-  ( Z  e.  U  ->  U. { Z }  =  Z
)
1326, 131syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  U. { Z }  =  Z )
133132sseq2d 3446 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( S `
 k ) B k )  C_  U. { Z }  <->  ( ( S `
 k ) B k )  C_  Z
) )
134133biimpar 493 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( S `  k ) B k )  C_  Z )  ->  (
( S `  k
) B k ) 
C_  U. { Z }
)
1356snssd 4108 . . . . . . . . . . . . 13  |-  ( ph  ->  { Z }  C_  U )
136 snex 4641 . . . . . . . . . . . . . 14  |-  { Z }  e.  _V
137136elpw 3948 . . . . . . . . . . . . 13  |-  ( { Z }  e.  ~P U 
<->  { Z }  C_  U )
138135, 137sylibr 217 . . . . . . . . . . . 12  |-  ( ph  ->  { Z }  e.  ~P U )
139 snfi 7668 . . . . . . . . . . . . 13  |-  { Z }  e.  Fin
140139a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  { Z }  e.  Fin )
141138, 140elind 3609 . . . . . . . . . . 11  |-  ( ph  ->  { Z }  e.  ( ~P U  i^i  Fin ) )
142 unieq 4198 . . . . . . . . . . . . 13  |-  ( v  =  { Z }  ->  U. v  =  U. { Z } )
143142sseq2d 3446 . . . . . . . . . . . 12  |-  ( v  =  { Z }  ->  ( ( ( S `
 k ) B k )  C_  U. v  <->  ( ( S `  k
) B k ) 
C_  U. { Z }
) )
144143rspcev 3136 . . . . . . . . . . 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 479 . . . . . . . . . 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 478 . . . . . . . . 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 6336 . . . . . . . . . . 11  |-  ( ( S `  k ) B k )  e. 
_V
148 sseq1 3439 . . . . . . . . . . . . 13  |-  ( u  =  ( ( S `
 k ) B k )  ->  (
u  C_  U. v  <->  ( ( S `  k
) B k ) 
C_  U. v ) )
149148rexbidv 2892 . . . . . . . . . . . 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 301 . . . . . . . . . . 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 3176 . . . . . . . . . 10  |-  ( ( ( S `  k
) B k )  e.  K  <->  -.  E. v  e.  ( ~P U  i^i  Fin ) ( ( S `
 k ) B k )  C_  U. v
)
152151con2bii 339 . . . . . . . . 9  |-  ( E. v  e.  ( ~P U  i^i  Fin )
( ( S `  k ) B k )  C_  U. v  <->  -.  ( ( S `  k ) B k )  e.  K )
153146, 152sylib 201 . . . . . . . 8  |-  ( (
ph  /\  ( ( S `  k ) B k )  C_  Z )  ->  -.  ( ( S `  k ) B k )  e.  K )
154153ex 441 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 k ) B k )  C_  Z  ->  -.  ( ( S `
 k ) B k )  e.  K
) )
155154ad2antrr 740 . . . . . 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 44 . . . . 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 121 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  -.  ( Y
( ball `  D )
x )  C_  Z
)
15827, 157rexlimddv 2875 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  -.  ( Y ( ball `  D
) x )  C_  Z )
159158nrexdv 2842 . 2  |-  ( ph  ->  -.  E. x  e.  RR+  ( Y ( ball `  D ) x ) 
C_  Z )
16011, 159pm2.21dd 179 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756   E.wrex 2757   _Vcvv 3031    i^i cin 3389    C_ wss 3390   ifcif 3872   ~Pcpw 3942   {csn 3959   <.cop 3965   U.cuni 4190   U_ciun 4269   class class class wbr 4395   {copab 4453    |-> cmpt 4454    X. cxp 4837    o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811   Fincfn 7587   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   2c2 10681   3c3 10682   NN0cn0 10893   RR+crp 11325    seqcseq 12251   ^cexp 12310   *Metcxmt 19032   Metcme 19033   ballcbl 19034   MetOpencmopn 19037   Topctop 19994   clsccl 20110   ~~> tclm 20319   CMetcms 22302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-fl 12061  df-seq 12252  df-exp 12311  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-top 19998  df-bases 19999  df-topon 20000  df-cld 20111  df-ntr 20112  df-cls 20113  df-lm 20322  df-cmet 22305
This theorem is referenced by:  heiborlem9  32215
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