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Theorem heiborlem7 26644
Description: Lemma for heibor 26648. Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff Madsen, 23-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 ) ) >.
)
Assertion
Ref Expression
heiborlem7  |-  A. r  e.  RR+  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  r
Distinct variable groups:    x, n, y, k, r, u, F   
k, G, x    ph, k,
r, x    k, m, v, z, D, n, r, u, x, y    k, M, m, r, u, x, y, z    T, m, n, x, y, z    B, n, u, v, y   
k, J, m, n, r, u, v, x, y, z    U, n, u, v, x, y, z    S, k, m, n, u, v, x, y, z    k, X, m, n, r, u, v, x, y, z    C, m, n, u, v, y   
n, K, x, y, z    x, B
Allowed substitution hints:    ph( y, z, v, u, m, n)    B( z, k, m, r)    C( x, z, k, r)    S( r)    T( v, u, k, r)    U( k, m, r)    F( z, v, m)    G( y,
z, v, u, m, n, r)    K( v, u, k, m, r)    M( v, n)

Proof of Theorem heiborlem7
StepHypRef Expression
1 3re 9833 . . . . . . 7  |-  3  e.  RR
2 3pos 9846 . . . . . . 7  |-  0  <  3
31, 2elrpii 10373 . . . . . 6  |-  3  e.  RR+
4 rpdivcl 10392 . . . . . 6  |-  ( ( r  e.  RR+  /\  3  e.  RR+ )  ->  (
r  /  3 )  e.  RR+ )
53, 4mpan2 652 . . . . 5  |-  ( r  e.  RR+  ->  ( r  /  3 )  e.  RR+ )
6 2re 9831 . . . . . 6  |-  2  e.  RR
7 1lt2 9902 . . . . . 6  |-  1  <  2
8 expnlbnd 11247 . . . . . 6  |-  ( ( ( r  /  3
)  e.  RR+  /\  2  e.  RR  /\  1  <  2 )  ->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) )
96, 7, 8mp3an23 1269 . . . . 5  |-  ( ( r  /  3 )  e.  RR+  ->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) )
105, 9syl 15 . . . 4  |-  ( r  e.  RR+  ->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) )
11 2nn 9893 . . . . . . . . . . 11  |-  2  e.  NN
12 nnnn0 9988 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  NN0 )
13 nnexpcl 11132 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
1411, 12, 13sylancr 644 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  NN )
1514nnrpd 10405 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  RR+ )
16 rpcn 10378 . . . . . . . . . 10  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 2 ^ k )  e.  CC )
17 rpne0 10385 . . . . . . . . . 10  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 2 ^ k )  =/=  0 )
18 3cn 9834 . . . . . . . . . . 11  |-  3  e.  CC
19 divrec 9456 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  ( 2 ^ k
)  e.  CC  /\  ( 2 ^ k
)  =/=  0 )  ->  ( 3  / 
( 2 ^ k
) )  =  ( 3  x.  ( 1  /  ( 2 ^ k ) ) ) )
2018, 19mp3an1 1264 . . . . . . . . . 10  |-  ( ( ( 2 ^ k
)  e.  CC  /\  ( 2 ^ k
)  =/=  0 )  ->  ( 3  / 
( 2 ^ k
) )  =  ( 3  x.  ( 1  /  ( 2 ^ k ) ) ) )
2116, 17, 20syl2anc 642 . . . . . . . . 9  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 3  /  ( 2 ^ k ) )  =  ( 3  x.  (
1  /  ( 2 ^ k ) ) ) )
2215, 21syl 15 . . . . . . . 8  |-  ( k  e.  NN  ->  (
3  /  ( 2 ^ k ) )  =  ( 3  x.  ( 1  /  (
2 ^ k ) ) ) )
2322adantl 452 . . . . . . 7  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
3  /  ( 2 ^ k ) )  =  ( 3  x.  ( 1  /  (
2 ^ k ) ) ) )
2423breq1d 4049 . . . . . 6  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  /  (
2 ^ k ) )  <  r  <->  ( 3  x.  ( 1  / 
( 2 ^ k
) ) )  < 
r ) )
2514nnrecred 9807 . . . . . . 7  |-  ( k  e.  NN  ->  (
1  /  ( 2 ^ k ) )  e.  RR )
26 rpre 10376 . . . . . . 7  |-  ( r  e.  RR+  ->  r  e.  RR )
271, 2pm3.2i 441 . . . . . . . 8  |-  ( 3  e.  RR  /\  0  <  3 )
28 ltmuldiv2 9643 . . . . . . . 8  |-  ( ( ( 1  /  (
2 ^ k ) )  e.  RR  /\  r  e.  RR  /\  (
3  e.  RR  /\  0  <  3 ) )  ->  ( ( 3  x.  ( 1  / 
( 2 ^ k
) ) )  < 
r  <->  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) ) )
2927, 28mp3an3 1266 . . . . . . 7  |-  ( ( ( 1  /  (
2 ^ k ) )  e.  RR  /\  r  e.  RR )  ->  ( ( 3  x.  ( 1  /  (
2 ^ k ) ) )  <  r  <->  ( 1  /  ( 2 ^ k ) )  <  ( r  / 
3 ) ) )
3025, 26, 29syl2anr 464 . . . . . 6  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  x.  (
1  /  ( 2 ^ k ) ) )  <  r  <->  ( 1  /  ( 2 ^ k ) )  < 
( r  /  3
) ) )
3124, 30bitrd 244 . . . . 5  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  /  (
2 ^ k ) )  <  r  <->  ( 1  /  ( 2 ^ k ) )  < 
( r  /  3
) ) )
3231rexbidva 2573 . . . 4  |-  ( r  e.  RR+  ->  ( E. k  e.  NN  (
3  /  ( 2 ^ k ) )  <  r  <->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) ) )
3310, 32mpbird 223 . . 3  |-  ( r  e.  RR+  ->  E. k  e.  NN  ( 3  / 
( 2 ^ k
) )  <  r
)
34 fveq2 5541 . . . . . . . . 9  |-  ( n  =  k  ->  ( S `  n )  =  ( S `  k ) )
35 oveq2 5882 . . . . . . . . . 10  |-  ( n  =  k  ->  (
2 ^ n )  =  ( 2 ^ k ) )
3635oveq2d 5890 . . . . . . . . 9  |-  ( n  =  k  ->  (
3  /  ( 2 ^ n ) )  =  ( 3  / 
( 2 ^ k
) ) )
3734, 36opeq12d 3820 . . . . . . . 8  |-  ( n  =  k  ->  <. ( S `  n ) ,  ( 3  / 
( 2 ^ n
) ) >.  =  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
38 heibor.12 . . . . . . . 8  |-  M  =  ( n  e.  NN  |->  <. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.
)
39 opex 4253 . . . . . . . 8  |-  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >.  e.  _V
4037, 38, 39fvmpt 5618 . . . . . . 7  |-  ( k  e.  NN  ->  ( M `  k )  =  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )
4140fveq2d 5545 . . . . . 6  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 2nd `  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
)
42 fvex 5555 . . . . . . 7  |-  ( S `
 k )  e. 
_V
43 ovex 5899 . . . . . . 7  |-  ( 3  /  ( 2 ^ k ) )  e. 
_V
4442, 43op2nd 6145 . . . . . 6  |-  ( 2nd `  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )  =  (
3  /  ( 2 ^ k ) )
4541, 44syl6eq 2344 . . . . 5  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 3  /  (
2 ^ k ) ) )
4645breq1d 4049 . . . 4  |-  ( k  e.  NN  ->  (
( 2nd `  ( M `  k )
)  <  r  <->  ( 3  /  ( 2 ^ k ) )  < 
r ) )
4746rexbiia 2589 . . 3  |-  ( E. k  e.  NN  ( 2nd `  ( M `  k ) )  < 
r  <->  E. k  e.  NN  ( 3  /  (
2 ^ k ) )  <  r )
4833, 47sylibr 203 . 2  |-  ( r  e.  RR+  ->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  r )
4948rgen 2621 1  |-  A. r  e.  RR+  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  r
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   ifcif 3578   ~Pcpw 3638   <.cop 3656   U.cuni 3843   U_ciun 3921   class class class wbr 4039   {copab 4092    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2ndc2nd 6137   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   3c3 9812   NN0cn0 9981   RR+crp 10370    seq cseq 11062   ^cexp 11120   ballcbl 16387   MetOpencmopn 16388   CMetcms 18696
This theorem is referenced by:  heiborlem8  26645  heiborlem9  26646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-seq 11063  df-exp 11121
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