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Theorem heiborlem7 28887
Description: Lemma for heibor 28891. 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 10510 . . . . . . 7  |-  3  e.  RR
2 3pos 10530 . . . . . . 7  |-  0  <  3
31, 2elrpii 11109 . . . . . 6  |-  3  e.  RR+
4 rpdivcl 11128 . . . . . 6  |-  ( ( r  e.  RR+  /\  3  e.  RR+ )  ->  (
r  /  3 )  e.  RR+ )
53, 4mpan2 671 . . . . 5  |-  ( r  e.  RR+  ->  ( r  /  3 )  e.  RR+ )
6 2re 10506 . . . . . 6  |-  2  e.  RR
7 1lt2 10603 . . . . . 6  |-  1  <  2
8 expnlbnd 12115 . . . . . 6  |-  ( ( ( r  /  3
)  e.  RR+  /\  2  e.  RR  /\  1  <  2 )  ->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) )
96, 7, 8mp3an23 1307 . . . . 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 10594 . . . . . . . . . . 11  |-  2  e.  NN
12 nnnn0 10701 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  NN0 )
13 nnexpcl 11999 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
1411, 12, 13sylancr 663 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  NN )
1514nnrpd 11141 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  RR+ )
16 rpcn 11114 . . . . . . . . . 10  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 2 ^ k )  e.  CC )
17 rpne0 11121 . . . . . . . . . 10  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 2 ^ k )  =/=  0 )
18 3cn 10511 . . . . . . . . . . 11  |-  3  e.  CC
19 divrec 10125 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  ( 2 ^ k
)  e.  CC  /\  ( 2 ^ k
)  =/=  0 )  ->  ( 3  / 
( 2 ^ k
) )  =  ( 3  x.  ( 1  /  ( 2 ^ k ) ) ) )
2018, 19mp3an1 1302 . . . . . . . . . 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 4413 . . . . . 6  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  /  (
2 ^ k ) )  <  r  <->  ( 3  x.  ( 1  / 
( 2 ^ k
) ) )  < 
r ) )
2514nnrecred 10482 . . . . . . 7  |-  ( k  e.  NN  ->  (
1  /  ( 2 ^ k ) )  e.  RR )
26 rpre 11112 . . . . . . 7  |-  ( r  e.  RR+  ->  r  e.  RR )
271, 2pm3.2i 455 . . . . . . . 8  |-  ( 3  e.  RR  /\  0  <  3 )
28 ltmuldiv2 10318 . . . . . . . 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 1304 . . . . . . 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 2865 . . . 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 5802 . . . . . . . . 9  |-  ( n  =  k  ->  ( S `  n )  =  ( S `  k ) )
35 oveq2 6211 . . . . . . . . . 10  |-  ( n  =  k  ->  (
2 ^ n )  =  ( 2 ^ k ) )
3635oveq2d 6219 . . . . . . . . 9  |-  ( n  =  k  ->  (
3  /  ( 2 ^ n ) )  =  ( 3  / 
( 2 ^ k
) ) )
3734, 36opeq12d 4178 . . . . . . . 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 4667 . . . . . . . 8  |-  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >.  e.  _V
4037, 38, 39fvmpt 5886 . . . . . . 7  |-  ( k  e.  NN  ->  ( M `  k )  =  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )
4140fveq2d 5806 . . . . . 6  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 2nd `  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
)
42 fvex 5812 . . . . . . 7  |-  ( S `
 k )  e. 
_V
43 ovex 6228 . . . . . . 7  |-  ( 3  /  ( 2 ^ k ) )  e. 
_V
4442, 43op2nd 6699 . . . . . 6  |-  ( 2nd `  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )  =  (
3  /  ( 2 ^ k ) )
4541, 44syl6eq 2511 . . . . 5  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 3  /  (
2 ^ k ) ) )
4645breq1d 4413 . . . 4  |-  ( k  e.  NN  ->  (
( 2nd `  ( M `  k )
)  <  r  <->  ( 3  /  ( 2 ^ k ) )  < 
r ) )
4746rexbiia 2861 . . 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 2899 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 965    = wceq 1370    e. wcel 1758   {cab 2439    =/= wne 2648   A.wral 2799   E.wrex 2800    i^i cin 3438    C_ wss 3439   ifcif 3902   ~Pcpw 3971   <.cop 3994   U.cuni 4202   U_ciun 4282   class class class wbr 4403   {copab 4460    |-> cmpt 4461   -->wf 5525   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   2ndc2nd 6689   Fincfn 7423   CCcc 9395   RRcr 9396   0cc0 9397   1c1 9398    + caddc 9400    x. cmul 9402    < clt 9533    - cmin 9710    / cdiv 10108   NNcn 10437   2c2 10486   3c3 10487   NN0cn0 10694   RR+crp 11106    seqcseq 11927   ^cexp 11986   ballcbl 17938   MetOpencmopn 17941   CMetcms 20907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-fl 11763  df-seq 11928  df-exp 11987
This theorem is referenced by:  heiborlem8  28888  heiborlem9  28889
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