MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscmet3lem3 Unicode version

Theorem iscmet3lem3 18716
Description: Lemma for iscmet3 18719. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypothesis
Ref Expression
iscmet3.1  |-  Z  =  ( ZZ>= `  M )
Assertion
Ref Expression
iscmet3lem3  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( ( 1  /  2
) ^ k )  <  R )
Distinct variable groups:    j, k, R    j, Z, k    j, M, k

Proof of Theorem iscmet3lem3
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 iscmet3.1 . . 3  |-  Z  =  ( ZZ>= `  M )
2 simpl 443 . . 3  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  M  e.  ZZ )
3 simpr 447 . . 3  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  R  e.  RR+ )
4 eluzelz 10238 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
54, 1eleq2s 2375 . . . . 5  |-  ( k  e.  Z  ->  k  e.  ZZ )
65adantl 452 . . . 4  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  k  e.  ZZ )
7 oveq2 5866 . . . . 5  |-  ( n  =  k  ->  (
( 1  /  2
) ^ n )  =  ( ( 1  /  2 ) ^
k ) )
8 eqid 2283 . . . . 5  |-  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^ n ) )  =  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )
9 ovex 5883 . . . . 5  |-  ( ( 1  /  2 ) ^ k )  e. 
_V
107, 8, 9fvmpt 5602 . . . 4  |-  ( k  e.  ZZ  ->  (
( n  e.  ZZ  |->  ( ( 1  / 
2 ) ^ n
) ) `  k
)  =  ( ( 1  /  2 ) ^ k ) )
116, 10syl 15 . . 3  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  ( (
n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) ) `  k )  =  ( ( 1  /  2 ) ^
k ) )
12 nn0uz 10262 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
1312reseq2i 4952 . . . . . 6  |-  ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  NN0 )  =  ( ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )  |`  ( ZZ>= `  0 )
)
14 nn0ssz 10044 . . . . . . 7  |-  NN0  C_  ZZ
15 resmpt 5000 . . . . . . 7  |-  ( NN0  C_  ZZ  ->  ( (
n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  NN0 )  =  ( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) ) )
1614, 15ax-mp 8 . . . . . 6  |-  ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  NN0 )  =  ( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) )
1713, 16eqtr3i 2305 . . . . 5  |-  ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  ( ZZ>= ` 
0 ) )  =  ( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) )
18 1rp 10358 . . . . . . . . . 10  |-  1  e.  RR+
19 rphalfcl 10378 . . . . . . . . . 10  |-  ( 1  e.  RR+  ->  ( 1  /  2 )  e.  RR+ )
2018, 19ax-mp 8 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR+
21 rpre 10360 . . . . . . . . 9  |-  ( ( 1  /  2 )  e.  RR+  ->  ( 1  /  2 )  e.  RR )
2220, 21ax-mp 8 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
2322recni 8849 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
2423a1i 10 . . . . . 6  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( 1  /  2
)  e.  CC )
25 rpge0 10366 . . . . . . . . . 10  |-  ( ( 1  /  2 )  e.  RR+  ->  0  <_ 
( 1  /  2
) )
2620, 25ax-mp 8 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
27 absid 11781 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  0  <_  ( 1  / 
2 ) )  -> 
( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
2822, 26, 27mp2an 653 . . . . . . . 8  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
29 halflt1 9933 . . . . . . . 8  |-  ( 1  /  2 )  <  1
3028, 29eqbrtri 4042 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
3130a1i 10 . . . . . 6  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( abs `  (
1  /  2 ) )  <  1 )
3224, 31expcnv 12322 . . . . 5  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) )  ~~>  0 )
3317, 32syl5eqbr 4056 . . . 4  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )  |`  ( ZZ>= `  0 )
)  ~~>  0 )
34 0z 10035 . . . . 5  |-  0  e.  ZZ
35 zex 10033 . . . . . . 7  |-  ZZ  e.  _V
3635mptex 5746 . . . . . 6  |-  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^ n ) )  e.  _V
3736a1i 10 . . . . 5  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( n  e.  ZZ  |->  ( ( 1  / 
2 ) ^ n
) )  e.  _V )
38 climres 12049 . . . . 5  |-  ( ( 0  e.  ZZ  /\  ( n  e.  ZZ  |->  ( ( 1  / 
2 ) ^ n
) )  e.  _V )  ->  ( ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  ( ZZ>= ` 
0 ) )  ~~>  0  <->  (
n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  ~~>  0 ) )
3934, 37, 38sylancr 644 . . . 4  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( ( ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^ n ) )  |`  ( ZZ>= `  0 )
)  ~~>  0  <->  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )  ~~>  0 ) )
4033, 39mpbid 201 . . 3  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( n  e.  ZZ  |->  ( ( 1  / 
2 ) ^ n
) )  ~~>  0 )
411, 2, 3, 11, 40climi0 11986 . 2  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( ( 1  /  2 ) ^ k ) )  <  R )
421uztrn2 10245 . . . . . 6  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
43 rpexpcl 11122 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR+  /\  k  e.  ZZ )  ->  (
( 1  /  2
) ^ k )  e.  RR+ )
4420, 6, 43sylancr 644 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  ( (
1  /  2 ) ^ k )  e.  RR+ )
45 rpre 10360 . . . . . . . . 9  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  ( ( 1  /  2 ) ^ k )  e.  RR )
46 rpge0 10366 . . . . . . . . 9  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  0  <_ 
( ( 1  / 
2 ) ^ k
) )
4745, 46absidd 11905 . . . . . . . 8  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  ( abs `  ( ( 1  / 
2 ) ^ k
) )  =  ( ( 1  /  2
) ^ k ) )
4844, 47syl 15 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  ( abs `  ( ( 1  / 
2 ) ^ k
) )  =  ( ( 1  /  2
) ^ k ) )
4948breq1d 4033 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  ( ( abs `  ( ( 1  /  2 ) ^
k ) )  < 
R  <->  ( ( 1  /  2 ) ^
k )  <  R
) )
5042, 49sylan2 460 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  ->  ( ( abs `  ( ( 1  /  2 ) ^
k ) )  < 
R  <->  ( ( 1  /  2 ) ^
k )  <  R
) )
5150anassrs 629 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( ( 1  /  2 ) ^
k ) )  < 
R  <->  ( ( 1  /  2 ) ^
k )  <  R
) )
5251ralbidva 2559 . . 3  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  j  e.  Z
)  ->  ( A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( 1  / 
2 ) ^ k
) )  <  R  <->  A. k  e.  ( ZZ>= `  j ) ( ( 1  /  2 ) ^ k )  < 
R ) )
5352rexbidva 2560 . 2  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( 1  /  2
) ^ k ) )  <  R  <->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( ( 1  / 
2 ) ^ k
)  <  R )
)
5441, 53mpbid 201 1  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( ( 1  /  2
) ^ k )  <  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    e. cmpt 4077    |` cres 4691   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    < clt 8867    <_ cle 8868    / cdiv 9423   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   ^cexp 11104   abscabs 11719    ~~> cli 11958
This theorem is referenced by:  iscmet3lem1  18717  iscmet3lem2  18718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963
  Copyright terms: Public domain W3C validator