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Theorem heiborlem7 26526
Description: Lemma for heibor 26530. 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 10071 . . . . . . 7  |-  3  e.  RR
2 3pos 10084 . . . . . . 7  |-  0  <  3
31, 2elrpii 10615 . . . . . 6  |-  3  e.  RR+
4 rpdivcl 10634 . . . . . 6  |-  ( ( r  e.  RR+  /\  3  e.  RR+ )  ->  (
r  /  3 )  e.  RR+ )
53, 4mpan2 653 . . . . 5  |-  ( r  e.  RR+  ->  ( r  /  3 )  e.  RR+ )
6 2re 10069 . . . . . 6  |-  2  e.  RR
7 1lt2 10142 . . . . . 6  |-  1  <  2
8 expnlbnd 11509 . . . . . 6  |-  ( ( ( r  /  3
)  e.  RR+  /\  2  e.  RR  /\  1  <  2 )  ->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) )
96, 7, 8mp3an23 1271 . . . . 5  |-  ( ( r  /  3 )  e.  RR+  ->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) )
105, 9syl 16 . . . 4  |-  ( r  e.  RR+  ->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) )
11 2nn 10133 . . . . . . . . . . 11  |-  2  e.  NN
12 nnnn0 10228 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  NN0 )
13 nnexpcl 11394 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
1411, 12, 13sylancr 645 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  NN )
1514nnrpd 10647 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  RR+ )
16 rpcn 10620 . . . . . . . . . 10  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 2 ^ k )  e.  CC )
17 rpne0 10627 . . . . . . . . . 10  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 2 ^ k )  =/=  0 )
18 3cn 10072 . . . . . . . . . . 11  |-  3  e.  CC
19 divrec 9694 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  ( 2 ^ k
)  e.  CC  /\  ( 2 ^ k
)  =/=  0 )  ->  ( 3  / 
( 2 ^ k
) )  =  ( 3  x.  ( 1  /  ( 2 ^ k ) ) ) )
2018, 19mp3an1 1266 . . . . . . . . . 10  |-  ( ( ( 2 ^ k
)  e.  CC  /\  ( 2 ^ k
)  =/=  0 )  ->  ( 3  / 
( 2 ^ k
) )  =  ( 3  x.  ( 1  /  ( 2 ^ k ) ) ) )
2116, 17, 20syl2anc 643 . . . . . . . . 9  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 3  /  ( 2 ^ k ) )  =  ( 3  x.  (
1  /  ( 2 ^ k ) ) ) )
2215, 21syl 16 . . . . . . . 8  |-  ( k  e.  NN  ->  (
3  /  ( 2 ^ k ) )  =  ( 3  x.  ( 1  /  (
2 ^ k ) ) ) )
2322adantl 453 . . . . . . 7  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
3  /  ( 2 ^ k ) )  =  ( 3  x.  ( 1  /  (
2 ^ k ) ) ) )
2423breq1d 4222 . . . . . 6  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  /  (
2 ^ k ) )  <  r  <->  ( 3  x.  ( 1  / 
( 2 ^ k
) ) )  < 
r ) )
2514nnrecred 10045 . . . . . . 7  |-  ( k  e.  NN  ->  (
1  /  ( 2 ^ k ) )  e.  RR )
26 rpre 10618 . . . . . . 7  |-  ( r  e.  RR+  ->  r  e.  RR )
271, 2pm3.2i 442 . . . . . . . 8  |-  ( 3  e.  RR  /\  0  <  3 )
28 ltmuldiv2 9881 . . . . . . . 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 1268 . . . . . . 7  |-  ( ( ( 1  /  (
2 ^ k ) )  e.  RR  /\  r  e.  RR )  ->  ( ( 3  x.  ( 1  /  (
2 ^ k ) ) )  <  r  <->  ( 1  /  ( 2 ^ k ) )  <  ( r  / 
3 ) ) )
3025, 26, 29syl2anr 465 . . . . . 6  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  x.  (
1  /  ( 2 ^ k ) ) )  <  r  <->  ( 1  /  ( 2 ^ k ) )  < 
( r  /  3
) ) )
3124, 30bitrd 245 . . . . 5  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  /  (
2 ^ k ) )  <  r  <->  ( 1  /  ( 2 ^ k ) )  < 
( r  /  3
) ) )
3231rexbidva 2722 . . . 4  |-  ( r  e.  RR+  ->  ( E. k  e.  NN  (
3  /  ( 2 ^ k ) )  <  r  <->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) ) )
3310, 32mpbird 224 . . 3  |-  ( r  e.  RR+  ->  E. k  e.  NN  ( 3  / 
( 2 ^ k
) )  <  r
)
34 fveq2 5728 . . . . . . . . 9  |-  ( n  =  k  ->  ( S `  n )  =  ( S `  k ) )
35 oveq2 6089 . . . . . . . . . 10  |-  ( n  =  k  ->  (
2 ^ n )  =  ( 2 ^ k ) )
3635oveq2d 6097 . . . . . . . . 9  |-  ( n  =  k  ->  (
3  /  ( 2 ^ n ) )  =  ( 3  / 
( 2 ^ k
) ) )
3734, 36opeq12d 3992 . . . . . . . 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 4427 . . . . . . . 8  |-  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >.  e.  _V
4037, 38, 39fvmpt 5806 . . . . . . 7  |-  ( k  e.  NN  ->  ( M `  k )  =  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )
4140fveq2d 5732 . . . . . 6  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 2nd `  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
)
42 fvex 5742 . . . . . . 7  |-  ( S `
 k )  e. 
_V
43 ovex 6106 . . . . . . 7  |-  ( 3  /  ( 2 ^ k ) )  e. 
_V
4442, 43op2nd 6356 . . . . . 6  |-  ( 2nd `  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )  =  (
3  /  ( 2 ^ k ) )
4541, 44syl6eq 2484 . . . . 5  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 3  /  (
2 ^ k ) ) )
4645breq1d 4222 . . . 4  |-  ( k  e.  NN  ->  (
( 2nd `  ( M `  k )
)  <  r  <->  ( 3  /  ( 2 ^ k ) )  < 
r ) )
4746rexbiia 2738 . . 3  |-  ( E. k  e.  NN  ( 2nd `  ( M `  k ) )  < 
r  <->  E. k  e.  NN  ( 3  /  (
2 ^ k ) )  <  r )
4833, 47sylibr 204 . 2  |-  ( r  e.  RR+  ->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  r )
4948rgen 2771 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   A.wral 2705   E.wrex 2706    i^i cin 3319    C_ wss 3320   ifcif 3739   ~Pcpw 3799   <.cop 3817   U.cuni 4015   U_ciun 4093   class class class wbr 4212   {copab 4265    e. cmpt 4266   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   2ndc2nd 6348   Fincfn 7109   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    - cmin 9291    / cdiv 9677   NNcn 10000   2c2 10049   3c3 10050   NN0cn0 10221   RR+crp 10612    seq cseq 11323   ^cexp 11382   ballcbl 16688   MetOpencmopn 16691   CMetcms 19207
This theorem is referenced by:  heiborlem8  26527  heiborlem9  26528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fl 11202  df-seq 11324  df-exp 11383
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