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Theorem iscmet3lem3 20917
Description: Lemma for iscmet3 20920. (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 10971 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
54, 1eleq2s 2559 . . . . 5  |-  ( k  e.  Z  ->  k  e.  ZZ )
65adantl 466 . . . 4  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  k  e.  ZZ )
7 oveq2 6198 . . . . 5  |-  ( n  =  k  ->  (
( 1  /  2
) ^ n )  =  ( ( 1  /  2 ) ^
k ) )
8 eqid 2451 . . . . 5  |-  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^ n ) )  =  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )
9 ovex 6215 . . . . 5  |-  ( ( 1  /  2 ) ^ k )  e. 
_V
107, 8, 9fvmpt 5873 . . . 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 10996 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
1312reseq2i 5205 . . . . . 6  |-  ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  NN0 )  =  ( ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )  |`  ( ZZ>= `  0 )
)
14 nn0ssz 10768 . . . . . . 7  |-  NN0  C_  ZZ
15 resmpt 5254 . . . . . . 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 2482 . . . . 5  |-  ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  ( ZZ>= ` 
0 ) )  =  ( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) )
18 halfcn 10642 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
1918a1i 11 . . . . . 6  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( 1  /  2
)  e.  CC )
20 halfre 10641 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
21 1rp 11096 . . . . . . . . . . 11  |-  1  e.  RR+
22 rphalfcl 11116 . . . . . . . . . . 11  |-  ( 1  e.  RR+  ->  ( 1  /  2 )  e.  RR+ )
2321, 22ax-mp 5 . . . . . . . . . 10  |-  ( 1  /  2 )  e.  RR+
24 rpge0 11104 . . . . . . . . . 10  |-  ( ( 1  /  2 )  e.  RR+  ->  0  <_ 
( 1  /  2
) )
2523, 24ax-mp 5 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
26 absid 12887 . . . . . . . . 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 10644 . . . . . . . 8  |-  ( 1  /  2 )  <  1
2927, 28eqbrtri 4409 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
3029a1i 11 . . . . . 6  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( abs `  (
1  /  2 ) )  <  1 )
3119, 30expcnv 13428 . . . . 5  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) )  ~~>  0 )
3217, 31syl5eqbr 4423 . . . 4  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )  |`  ( ZZ>= `  0 )
)  ~~>  0 )
33 0z 10758 . . . . 5  |-  0  e.  ZZ
34 zex 10756 . . . . . . 7  |-  ZZ  e.  _V
3534mptex 6047 . . . . . 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 13155 . . . . 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 13092 . 2  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( ( 1  /  2 ) ^ k ) )  <  R )
411uztrn2 10979 . . . . . 6  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
42 rpexpcl 11985 . . . . . . . . 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 11098 . . . . . . . . 9  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  ( ( 1  /  2 ) ^ k )  e.  RR )
45 rpge0 11104 . . . . . . . . 9  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  0  <_ 
( ( 1  / 
2 ) ^ k
) )
4644, 45absidd 13011 . . . . . . . 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 4400 . . . . . 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 2837 . . 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 2842 . 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 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   _Vcvv 3068    C_ wss 3426   class class class wbr 4390    |-> cmpt 4448    |` cres 4940   ` cfv 5516  (class class class)co 6190   CCcc 9381   RRcr 9382   0cc0 9383   1c1 9384    < clt 9519    <_ cle 9520    / cdiv 10094   2c2 10472   NN0cn0 10680   ZZcz 10747   ZZ>=cuz 10962   RR+crp 11092   ^cexp 11966   abscabs 12825    ~~> cli 13064
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 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-fl 11743  df-seq 11908  df-exp 11967  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-rlim 13069
This theorem is referenced by:  iscmet3lem1  20918  iscmet3lem2  20919
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