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Theorem heiborlem7 32213
Description: Lemma for heibor 32217. 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 10705 . . . . . . 7  |-  3  e.  RR
2 3pos 10725 . . . . . . 7  |-  0  <  3
31, 2elrpii 11328 . . . . . 6  |-  3  e.  RR+
4 rpdivcl 11348 . . . . . 6  |-  ( ( r  e.  RR+  /\  3  e.  RR+ )  ->  (
r  /  3 )  e.  RR+ )
53, 4mpan2 685 . . . . 5  |-  ( r  e.  RR+  ->  ( r  /  3 )  e.  RR+ )
6 2re 10701 . . . . . 6  |-  2  e.  RR
7 1lt2 10799 . . . . . 6  |-  1  <  2
8 expnlbnd 12440 . . . . . 6  |-  ( ( ( r  /  3
)  e.  RR+  /\  2  e.  RR  /\  1  <  2 )  ->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) )
96, 7, 8mp3an23 1382 . . . . 5  |-  ( ( r  /  3 )  e.  RR+  ->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) )
105, 9syl 17 . . . 4  |-  ( r  e.  RR+  ->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) )
11 2nn 10790 . . . . . . . . . . 11  |-  2  e.  NN
12 nnnn0 10900 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  NN0 )
13 nnexpcl 12323 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
1411, 12, 13sylancr 676 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  NN )
1514nnrpd 11362 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  RR+ )
16 rpcn 11333 . . . . . . . . . 10  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 2 ^ k )  e.  CC )
17 rpne0 11340 . . . . . . . . . 10  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 2 ^ k )  =/=  0 )
18 3cn 10706 . . . . . . . . . . 11  |-  3  e.  CC
19 divrec 10308 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  ( 2 ^ k
)  e.  CC  /\  ( 2 ^ k
)  =/=  0 )  ->  ( 3  / 
( 2 ^ k
) )  =  ( 3  x.  ( 1  /  ( 2 ^ k ) ) ) )
2018, 19mp3an1 1377 . . . . . . . . . 10  |-  ( ( ( 2 ^ k
)  e.  CC  /\  ( 2 ^ k
)  =/=  0 )  ->  ( 3  / 
( 2 ^ k
) )  =  ( 3  x.  ( 1  /  ( 2 ^ k ) ) ) )
2116, 17, 20syl2anc 673 . . . . . . . . 9  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 3  /  ( 2 ^ k ) )  =  ( 3  x.  (
1  /  ( 2 ^ k ) ) ) )
2215, 21syl 17 . . . . . . . 8  |-  ( k  e.  NN  ->  (
3  /  ( 2 ^ k ) )  =  ( 3  x.  ( 1  /  (
2 ^ k ) ) ) )
2322adantl 473 . . . . . . 7  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
3  /  ( 2 ^ k ) )  =  ( 3  x.  ( 1  /  (
2 ^ k ) ) ) )
2423breq1d 4405 . . . . . 6  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  /  (
2 ^ k ) )  <  r  <->  ( 3  x.  ( 1  / 
( 2 ^ k
) ) )  < 
r ) )
2514nnrecred 10677 . . . . . . 7  |-  ( k  e.  NN  ->  (
1  /  ( 2 ^ k ) )  e.  RR )
26 rpre 11331 . . . . . . 7  |-  ( r  e.  RR+  ->  r  e.  RR )
271, 2pm3.2i 462 . . . . . . . 8  |-  ( 3  e.  RR  /\  0  <  3 )
28 ltmuldiv2 10501 . . . . . . . 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 1379 . . . . . . 7  |-  ( ( ( 1  /  (
2 ^ k ) )  e.  RR  /\  r  e.  RR )  ->  ( ( 3  x.  ( 1  /  (
2 ^ k ) ) )  <  r  <->  ( 1  /  ( 2 ^ k ) )  <  ( r  / 
3 ) ) )
3025, 26, 29syl2anr 486 . . . . . 6  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  x.  (
1  /  ( 2 ^ k ) ) )  <  r  <->  ( 1  /  ( 2 ^ k ) )  < 
( r  /  3
) ) )
3124, 30bitrd 261 . . . . 5  |-  ( ( r  e.  RR+  /\  k  e.  NN )  ->  (
( 3  /  (
2 ^ k ) )  <  r  <->  ( 1  /  ( 2 ^ k ) )  < 
( r  /  3
) ) )
3231rexbidva 2889 . . . 4  |-  ( r  e.  RR+  ->  ( E. k  e.  NN  (
3  /  ( 2 ^ k ) )  <  r  <->  E. k  e.  NN  ( 1  / 
( 2 ^ k
) )  <  (
r  /  3 ) ) )
3310, 32mpbird 240 . . 3  |-  ( r  e.  RR+  ->  E. k  e.  NN  ( 3  / 
( 2 ^ k
) )  <  r
)
34 fveq2 5879 . . . . . . . . 9  |-  ( n  =  k  ->  ( S `  n )  =  ( S `  k ) )
35 oveq2 6316 . . . . . . . . . 10  |-  ( n  =  k  ->  (
2 ^ n )  =  ( 2 ^ k ) )
3635oveq2d 6324 . . . . . . . . 9  |-  ( n  =  k  ->  (
3  /  ( 2 ^ n ) )  =  ( 3  / 
( 2 ^ k
) ) )
3734, 36opeq12d 4166 . . . . . . . 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 4664 . . . . . . . 8  |-  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >.  e.  _V
4037, 38, 39fvmpt 5963 . . . . . . 7  |-  ( k  e.  NN  ->  ( M `  k )  =  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )
4140fveq2d 5883 . . . . . 6  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 2nd `  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
)
42 fvex 5889 . . . . . . 7  |-  ( S `
 k )  e. 
_V
43 ovex 6336 . . . . . . 7  |-  ( 3  /  ( 2 ^ k ) )  e. 
_V
4442, 43op2nd 6821 . . . . . 6  |-  ( 2nd `  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )  =  (
3  /  ( 2 ^ k ) )
4541, 44syl6eq 2521 . . . . 5  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 3  /  (
2 ^ k ) ) )
4645breq1d 4405 . . . 4  |-  ( k  e.  NN  ->  (
( 2nd `  ( M `  k )
)  <  r  <->  ( 3  /  ( 2 ^ k ) )  < 
r ) )
4746rexbiia 2880 . . 3  |-  ( E. k  e.  NN  ( 2nd `  ( M `  k ) )  < 
r  <->  E. k  e.  NN  ( 3  /  (
2 ^ k ) )  <  r )
4833, 47sylibr 217 . 2  |-  ( r  e.  RR+  ->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  r )
4948rgen 2766 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {cab 2457    =/= wne 2641   A.wral 2756   E.wrex 2757    i^i cin 3389    C_ wss 3390   ifcif 3872   ~Pcpw 3942   <.cop 3965   U.cuni 4190   U_ciun 4269   class class class wbr 4395   {copab 4453    |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   2ndc2nd 6811   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    - cmin 9880    / cdiv 10291   NNcn 10631   2c2 10681   3c3 10682   NN0cn0 10893   RR+crp 11325    seqcseq 12251   ^cexp 12310   ballcbl 19034   MetOpencmopn 19037   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-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-iun 4271  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-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  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-rp 11326  df-fl 12061  df-seq 12252  df-exp 12311
This theorem is referenced by:  heiborlem8  32214  heiborlem9  32215
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