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Theorem iscmet3lem3 19204
Description: Lemma for iscmet3 19207. (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 444 . . 3  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  M  e.  ZZ )
3 simpr 448 . . 3  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  R  e.  RR+ )
4 eluzelz 10460 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
54, 1eleq2s 2504 . . . . 5  |-  ( k  e.  Z  ->  k  e.  ZZ )
65adantl 453 . . . 4  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  k  e.  ZZ )
7 oveq2 6056 . . . . 5  |-  ( n  =  k  ->  (
( 1  /  2
) ^ n )  =  ( ( 1  /  2 ) ^
k ) )
8 eqid 2412 . . . . 5  |-  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^ n ) )  =  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )
9 ovex 6073 . . . . 5  |-  ( ( 1  /  2 ) ^ k )  e. 
_V
107, 8, 9fvmpt 5773 . . . 4  |-  ( k  e.  ZZ  ->  (
( n  e.  ZZ  |->  ( ( 1  / 
2 ) ^ n
) ) `  k
)  =  ( ( 1  /  2 ) ^ k ) )
116, 10syl 16 . . 3  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  ( (
n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) ) `  k )  =  ( ( 1  /  2 ) ^
k ) )
12 nn0uz 10484 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
1312reseq2i 5110 . . . . . 6  |-  ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  NN0 )  =  ( ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )  |`  ( ZZ>= `  0 )
)
14 nn0ssz 10266 . . . . . . 7  |-  NN0  C_  ZZ
15 resmpt 5158 . . . . . . 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 2434 . . . . 5  |-  ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  ( ZZ>= ` 
0 ) )  =  ( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) )
18 1rp 10580 . . . . . . . . . 10  |-  1  e.  RR+
19 rphalfcl 10600 . . . . . . . . . 10  |-  ( 1  e.  RR+  ->  ( 1  /  2 )  e.  RR+ )
2018, 19ax-mp 8 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR+
21 rpre 10582 . . . . . . . . 9  |-  ( ( 1  /  2 )  e.  RR+  ->  ( 1  /  2 )  e.  RR )
2220, 21ax-mp 8 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
2322recni 9066 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
2423a1i 11 . . . . . 6  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( 1  /  2
)  e.  CC )
25 rpge0 10588 . . . . . . . . . 10  |-  ( ( 1  /  2 )  e.  RR+  ->  0  <_ 
( 1  /  2
) )
2620, 25ax-mp 8 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
27 absid 12064 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  0  <_  ( 1  / 
2 ) )  -> 
( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
2822, 26, 27mp2an 654 . . . . . . . 8  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
29 halflt1 10153 . . . . . . . 8  |-  ( 1  /  2 )  <  1
3028, 29eqbrtri 4199 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
3130a1i 11 . . . . . 6  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( abs `  (
1  /  2 ) )  <  1 )
3224, 31expcnv 12606 . . . . 5  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) )  ~~>  0 )
3317, 32syl5eqbr 4213 . . . 4  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )  |`  ( ZZ>= `  0 )
)  ~~>  0 )
34 0z 10257 . . . . 5  |-  0  e.  ZZ
35 zex 10255 . . . . . . 7  |-  ZZ  e.  _V
3635mptex 5933 . . . . . 6  |-  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^ n ) )  e.  _V
3736a1i 11 . . . . 5  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( n  e.  ZZ  |->  ( ( 1  / 
2 ) ^ n
) )  e.  _V )
38 climres 12332 . . . . 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 645 . . . 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 202 . . 3  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( n  e.  ZZ  |->  ( ( 1  / 
2 ) ^ n
) )  ~~>  0 )
411, 2, 3, 11, 40climi0 12269 . 2  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( ( 1  /  2 ) ^ k ) )  <  R )
421uztrn2 10467 . . . . . 6  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
43 rpexpcl 11363 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR+  /\  k  e.  ZZ )  ->  (
( 1  /  2
) ^ k )  e.  RR+ )
4420, 6, 43sylancr 645 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  ( (
1  /  2 ) ^ k )  e.  RR+ )
45 rpre 10582 . . . . . . . . 9  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  ( ( 1  /  2 ) ^ k )  e.  RR )
46 rpge0 10588 . . . . . . . . 9  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  0  <_ 
( ( 1  / 
2 ) ^ k
) )
4745, 46absidd 12188 . . . . . . . 8  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  ( abs `  ( ( 1  / 
2 ) ^ k
) )  =  ( ( 1  /  2
) ^ k ) )
4844, 47syl 16 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  ( abs `  ( ( 1  / 
2 ) ^ k
) )  =  ( ( 1  /  2
) ^ k ) )
4948breq1d 4190 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  ( ( abs `  ( ( 1  /  2 ) ^
k ) )  < 
R  <->  ( ( 1  /  2 ) ^
k )  <  R
) )
5042, 49sylan2 461 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  ->  ( ( abs `  ( ( 1  /  2 ) ^
k ) )  < 
R  <->  ( ( 1  /  2 ) ^
k )  <  R
) )
5150anassrs 630 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( ( 1  /  2 ) ^
k ) )  < 
R  <->  ( ( 1  /  2 ) ^
k )  <  R
) )
5251ralbidva 2690 . . 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 2691 . 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 202 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   E.wrex 2675   _Vcvv 2924    C_ wss 3288   class class class wbr 4180    e. cmpt 4234    |` cres 4847   ` cfv 5421  (class class class)co 6048   CCcc 8952   RRcr 8953   0cc0 8954   1c1 8955    < clt 9084    <_ cle 9085    / cdiv 9641   2c2 10013   NN0cn0 10185   ZZcz 10246   ZZ>=cuz 10452   RR+crp 10576   ^cexp 11345   abscabs 12002    ~~> cli 12241
This theorem is referenced by:  iscmet3lem1  19205  iscmet3lem2  19206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fl 11165  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246
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