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Theorem iscmet3lem3 18820
Description: Lemma for iscmet3 18823. (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 10330 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  ZZ )
54, 1eleq2s 2450 . . . . 5  |-  ( k  e.  Z  ->  k  e.  ZZ )
65adantl 452 . . . 4  |-  ( ( ( M  e.  ZZ  /\  R  e.  RR+ )  /\  k  e.  Z
)  ->  k  e.  ZZ )
7 oveq2 5953 . . . . 5  |-  ( n  =  k  ->  (
( 1  /  2
) ^ n )  =  ( ( 1  /  2 ) ^
k ) )
8 eqid 2358 . . . . 5  |-  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^ n ) )  =  ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )
9 ovex 5970 . . . . 5  |-  ( ( 1  /  2 ) ^ k )  e. 
_V
107, 8, 9fvmpt 5685 . . . 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 10354 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
1312reseq2i 5034 . . . . . 6  |-  ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  NN0 )  =  ( ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )  |`  ( ZZ>= `  0 )
)
14 nn0ssz 10136 . . . . . . 7  |-  NN0  C_  ZZ
15 resmpt 5082 . . . . . . 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 2380 . . . . 5  |-  ( ( n  e.  ZZ  |->  ( ( 1  /  2
) ^ n ) )  |`  ( ZZ>= ` 
0 ) )  =  ( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) )
18 1rp 10450 . . . . . . . . . 10  |-  1  e.  RR+
19 rphalfcl 10470 . . . . . . . . . 10  |-  ( 1  e.  RR+  ->  ( 1  /  2 )  e.  RR+ )
2018, 19ax-mp 8 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR+
21 rpre 10452 . . . . . . . . 9  |-  ( ( 1  /  2 )  e.  RR+  ->  ( 1  /  2 )  e.  RR )
2220, 21ax-mp 8 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
2322recni 8939 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
2423a1i 10 . . . . . 6  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( 1  /  2
)  e.  CC )
25 rpge0 10458 . . . . . . . . . 10  |-  ( ( 1  /  2 )  e.  RR+  ->  0  <_ 
( 1  /  2
) )
2620, 25ax-mp 8 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
27 absid 11877 . . . . . . . . 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 10025 . . . . . . . 8  |-  ( 1  /  2 )  <  1
3028, 29eqbrtri 4123 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
3130a1i 10 . . . . . 6  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( abs `  (
1  /  2 ) )  <  1 )
3224, 31expcnv 12419 . . . . 5  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( n  e.  NN0  |->  ( ( 1  / 
2 ) ^ n
) )  ~~>  0 )
3317, 32syl5eqbr 4137 . . . 4  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  -> 
( ( n  e.  ZZ  |->  ( ( 1  /  2 ) ^
n ) )  |`  ( ZZ>= `  0 )
)  ~~>  0 )
34 0z 10127 . . . . 5  |-  0  e.  ZZ
35 zex 10125 . . . . . . 7  |-  ZZ  e.  _V
3635mptex 5832 . . . . . 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 12145 . . . . 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 12082 . 2  |-  ( ( M  e.  ZZ  /\  R  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( ( 1  /  2 ) ^ k ) )  <  R )
421uztrn2 10337 . . . . . 6  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
43 rpexpcl 11215 . . . . . . . . 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 10452 . . . . . . . . 9  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  ( ( 1  /  2 ) ^ k )  e.  RR )
46 rpge0 10458 . . . . . . . . 9  |-  ( ( ( 1  /  2
) ^ k )  e.  RR+  ->  0  <_ 
( ( 1  / 
2 ) ^ k
) )
4745, 46absidd 12001 . . . . . . . 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 4114 . . . . . 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 2635 . . 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 2636 . 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 1642    e. wcel 1710   A.wral 2619   E.wrex 2620   _Vcvv 2864    C_ wss 3228   class class class wbr 4104    e. cmpt 4158    |` cres 4773   ` cfv 5337  (class class class)co 5945   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828    < clt 8957    <_ cle 8958    / cdiv 9513   2c2 9885   NN0cn0 10057   ZZcz 10116   ZZ>=cuz 10322   RR+crp 10446   ^cexp 11197   abscabs 11815    ~~> cli 12054
This theorem is referenced by:  iscmet3lem1  18821  iscmet3lem2  18822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-pm 6863  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-fl 11017  df-seq 11139  df-exp 11198  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-rlim 12059
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