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Theorem heiborlem7 30240
Description: Lemma for heibor 30244. 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 10621 . . . . . . 7  |-  3  e.  RR
2 3pos 10641 . . . . . . 7  |-  0  <  3
31, 2elrpii 11235 . . . . . 6  |-  3  e.  RR+
4 rpdivcl 11254 . . . . . 6  |-  ( ( r  e.  RR+  /\  3  e.  RR+ )  ->  (
r  /  3 )  e.  RR+ )
53, 4mpan2 671 . . . . 5  |-  ( r  e.  RR+  ->  ( r  /  3 )  e.  RR+ )
6 2re 10617 . . . . . 6  |-  2  e.  RR
7 1lt2 10714 . . . . . 6  |-  1  <  2
8 expnlbnd 12276 . . . . . 6  |-  ( ( ( r  /  3
)  e.  RR+  /\  2  e.  RR  /\  1  <  2 )  ->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) )
96, 7, 8mp3an23 1316 . . . . 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 10705 . . . . . . . . . . 11  |-  2  e.  NN
12 nnnn0 10814 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  NN0 )
13 nnexpcl 12159 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
1411, 12, 13sylancr 663 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  NN )
1514nnrpd 11267 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  RR+ )
16 rpcn 11240 . . . . . . . . . 10  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 2 ^ k )  e.  CC )
17 rpne0 11247 . . . . . . . . . 10  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 2 ^ k )  =/=  0 )
18 3cn 10622 . . . . . . . . . . 11  |-  3  e.  CC
19 divrec 10235 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  ( 2 ^ k
)  e.  CC  /\  ( 2 ^ k
)  =/=  0 )  ->  ( 3  / 
( 2 ^ k
) )  =  ( 3  x.  ( 1  /  ( 2 ^ k ) ) ) )
2018, 19mp3an1 1311 . . . . . . . . . 10  |-  ( ( ( 2 ^ k
)  e.  CC  /\  ( 2 ^ k
)  =/=  0 )  ->  ( 3  / 
( 2 ^ k
) )  =  ( 3  x.  ( 1  /  ( 2 ^ k ) ) ) )
2116, 17, 20syl2anc 661 . . . . . . . . 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 466 . . . . . . 7  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
3  /  ( 2 ^ k ) )  =  ( 3  x.  ( 1  /  (
2 ^ k ) ) ) )
2423breq1d 4463 . . . . . 6  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  /  (
2 ^ k ) )  <  r  <->  ( 3  x.  ( 1  / 
( 2 ^ k
) ) )  < 
r ) )
2514nnrecred 10593 . . . . . . 7  |-  ( k  e.  NN  ->  (
1  /  ( 2 ^ k ) )  e.  RR )
26 rpre 11238 . . . . . . 7  |-  ( r  e.  RR+  ->  r  e.  RR )
271, 2pm3.2i 455 . . . . . . . 8  |-  ( 3  e.  RR  /\  0  <  3 )
28 ltmuldiv2 10428 . . . . . . . 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 1313 . . . . . . 7  |-  ( ( ( 1  /  (
2 ^ k ) )  e.  RR  /\  r  e.  RR )  ->  ( ( 3  x.  ( 1  /  (
2 ^ k ) ) )  <  r  <->  ( 1  /  ( 2 ^ k ) )  <  ( r  / 
3 ) ) )
3025, 26, 29syl2anr 478 . . . . . 6  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  x.  (
1  /  ( 2 ^ k ) ) )  <  r  <->  ( 1  /  ( 2 ^ k ) )  < 
( r  /  3
) ) )
3124, 30bitrd 253 . . . . 5  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  /  (
2 ^ k ) )  <  r  <->  ( 1  /  ( 2 ^ k ) )  < 
( r  /  3
) ) )
3231rexbidva 2975 . . . 4  |-  ( r  e.  RR+  ->  ( E. k  e.  NN  (
3  /  ( 2 ^ k ) )  <  r  <->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) ) )
3310, 32mpbird 232 . . 3  |-  ( r  e.  RR+  ->  E. k  e.  NN  ( 3  / 
( 2 ^ k
) )  <  r
)
34 fveq2 5872 . . . . . . . . 9  |-  ( n  =  k  ->  ( S `  n )  =  ( S `  k ) )
35 oveq2 6303 . . . . . . . . . 10  |-  ( n  =  k  ->  (
2 ^ n )  =  ( 2 ^ k ) )
3635oveq2d 6311 . . . . . . . . 9  |-  ( n  =  k  ->  (
3  /  ( 2 ^ n ) )  =  ( 3  / 
( 2 ^ k
) ) )
3734, 36opeq12d 4227 . . . . . . . 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 4717 . . . . . . . 8  |-  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >.  e.  _V
4037, 38, 39fvmpt 5957 . . . . . . 7  |-  ( k  e.  NN  ->  ( M `  k )  =  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )
4140fveq2d 5876 . . . . . 6  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 2nd `  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
)
42 fvex 5882 . . . . . . 7  |-  ( S `
 k )  e. 
_V
43 ovex 6320 . . . . . . 7  |-  ( 3  /  ( 2 ^ k ) )  e. 
_V
4442, 43op2nd 6804 . . . . . 6  |-  ( 2nd `  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )  =  (
3  /  ( 2 ^ k ) )
4541, 44syl6eq 2524 . . . . 5  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 3  /  (
2 ^ k ) ) )
4645breq1d 4463 . . . 4  |-  ( k  e.  NN  ->  (
( 2nd `  ( M `  k )
)  <  r  <->  ( 3  /  ( 2 ^ k ) )  < 
r ) )
4746rexbiia 2968 . . 3  |-  ( E. k  e.  NN  ( 2nd `  ( M `  k ) )  < 
r  <->  E. k  e.  NN  ( 3  /  (
2 ^ k ) )  <  r )
4833, 47sylibr 212 . 2  |-  ( r  e.  RR+  ->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  r )
4948rgen 2827 1  |-  A. r  e.  RR+  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  r
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2817   E.wrex 2818    i^i cin 3480    C_ wss 3481   ifcif 3945   ~Pcpw 4016   <.cop 4039   U.cuni 4251   U_ciun 4331   class class class wbr 4453   {copab 4510    |-> cmpt 4511   -->wf 5590   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   2ndc2nd 6794   Fincfn 7528   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    < clt 9640    - cmin 9817    / cdiv 10218   NNcn 10548   2c2 10597   3c3 10598   NN0cn0 10807   RR+crp 11232    seqcseq 12087   ^cexp 12146   ballcbl 18275   MetOpencmopn 18278   CMetcms 21561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fl 11909  df-seq 12088  df-exp 12147
This theorem is referenced by:  heiborlem8  30241  heiborlem9  30242
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