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Theorem iscmet3lem3 21457
Description: Lemma for iscmet3 21460. (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 457 . . 3  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  M  e.  ZZ )
3 simpr 461 . . 3  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  R  e.  RR+ )
4 eluzelz 11080 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
54, 1eleq2s 2568 . . . . 5  |-  ( k  e.  Z  ->  k  e.  ZZ )
65adantl 466 . . . 4  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  k  e.  ZZ )
7 oveq2 6283 . . . . 5  |-  ( n  =  k  ->  (
( 1  /  2
) ^ n )  =  ( ( 1  /  2 ) ^
k ) )
8 eqid 2460 . . . . 5  |-  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^ n ) )  =  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )
9 ovex 6300 . . . . 5  |-  ( ( 1  /  2 ) ^ k )  e. 
_V
107, 8, 9fvmpt 5941 . . . 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 11105 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
1312reseq2i 5261 . . . . . 6  |-  ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  NN0 )  =  ( ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )  |`  ( ZZ>= `  0 )
)
14 nn0ssz 10874 . . . . . . 7  |-  NN0  C_  ZZ
15 resmpt 5314 . . . . . . 7  |-  ( NN0  C_  ZZ  ->  ( (
n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  NN0 )  =  ( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) ) )
1614, 15ax-mp 5 . . . . . 6  |-  ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  NN0 )  =  ( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) )
1713, 16eqtr3i 2491 . . . . 5  |-  ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  ( ZZ>= ` 
0 ) )  =  ( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) )
18 halfcn 10744 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
1918a1i 11 . . . . . 6  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( 1  /  2
)  e.  CC )
20 halfre 10743 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
21 1rp 11213 . . . . . . . . . . 11  |-  1  e.  RR+
22 rphalfcl 11233 . . . . . . . . . . 11  |-  ( 1  e.  RR+  ->  ( 1  /  2 )  e.  RR+ )
2321, 22ax-mp 5 . . . . . . . . . 10  |-  ( 1  /  2 )  e.  RR+
24 rpge0 11221 . . . . . . . . . 10  |-  ( ( 1  /  2 )  e.  RR+  ->  0  <_ 
( 1  /  2
) )
2523, 24ax-mp 5 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
26 absid 13079 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  0  <_  ( 1  / 
2 ) )  -> 
( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
2720, 25, 26mp2an 672 . . . . . . . 8  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
28 halflt1 10746 . . . . . . . 8  |-  ( 1  /  2 )  <  1
2927, 28eqbrtri 4459 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
3029a1i 11 . . . . . 6  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( abs `  (
1  /  2 ) )  <  1 )
3119, 30expcnv 13627 . . . . 5  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) )  ~~>  0 )
3217, 31syl5eqbr 4473 . . . 4  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )  |`  ( ZZ>= `  0 )
)  ~~>  0 )
33 0z 10864 . . . . 5  |-  0  e.  ZZ
34 zex 10862 . . . . . . 7  |-  ZZ  e.  _V
3534mptex 6122 . . . . . 6  |-  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^ n ) )  e.  _V
3635a1i 11 . . . . 5  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( n  e.  ZZ  |->  ( ( 1  / 
2 ) ^ n
) )  e.  _V )
37 climres 13347 . . . . 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 ) )
3833, 36, 37sylancr 663 . . . 4  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( ( ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^ n ) )  |`  ( ZZ>= `  0 )
)  ~~>  0  <->  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )  ~~>  0 ) )
3932, 38mpbid 210 . . 3  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( n  e.  ZZ  |->  ( ( 1  / 
2 ) ^ n
) )  ~~>  0 )
401, 2, 3, 11, 39climi0 13284 . 2  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( ( 1  /  2 ) ^ k ) )  <  R )
411uztrn2 11088 . . . . . 6  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
42 rpexpcl 12141 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR+  /\  k  e.  ZZ )  ->  (
( 1  /  2
) ^ k )  e.  RR+ )
4323, 6, 42sylancr 663 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  ( (
1  /  2 ) ^ k )  e.  RR+ )
44 rpre 11215 . . . . . . . . 9  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  ( ( 1  /  2 ) ^ k )  e.  RR )
45 rpge0 11221 . . . . . . . . 9  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  0  <_ 
( ( 1  / 
2 ) ^ k
) )
4644, 45absidd 13203 . . . . . . . 8  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  ( abs `  ( ( 1  / 
2 ) ^ k
) )  =  ( ( 1  /  2
) ^ k ) )
4743, 46syl 16 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  ( abs `  ( ( 1  / 
2 ) ^ k
) )  =  ( ( 1  /  2
) ^ k ) )
4847breq1d 4450 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  ( ( abs `  ( ( 1  /  2 ) ^
k ) )  < 
R  <->  ( ( 1  /  2 ) ^
k )  <  R
) )
4941, 48sylan2 474 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  ( j  e.  Z  /\  k  e.  ( ZZ>=
`  j ) ) )  ->  ( ( abs `  ( ( 1  /  2 ) ^
k ) )  < 
R  <->  ( ( 1  /  2 ) ^
k )  <  R
) )
5049anassrs 648 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( ( 1  /  2 ) ^
k ) )  < 
R  <->  ( ( 1  /  2 ) ^
k )  <  R
) )
5150ralbidva 2893 . . 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 ) )
5251rexbidva 2963 . 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 )
)
5340, 52mpbid 210 1  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( ( 1  /  2
) ^ k )  <  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   _Vcvv 3106    C_ wss 3469   class class class wbr 4440    |-> cmpt 4498    |` cres 4994   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    < clt 9617    <_ cle 9618    / cdiv 10195   2c2 10574   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   RR+crp 11209   ^cexp 12122   abscabs 13017    ~~> cli 13256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fl 11886  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-rlim 13261
This theorem is referenced by:  iscmet3lem1  21458  iscmet3lem2  21459
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