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Theorem expnlbnd2 9374
Description: The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
Assertion
Ref Expression
expnlbnd2  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( 1  /  ( B ^ k ) )  <  A )
Distinct variable groups:    j, k, A    B, j, k

Proof of Theorem expnlbnd2
StepHypRef Expression
1 expnlbnd 9373 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. j  e.  NN  ( 1  / 
( B ^ j
) )  <  A
)
2 simpl2 908 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  B  e.  RR )
3 simpl3 909 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  1  <  B
)
4 1re 7026 . . . . . . . . . 10  |-  1  e.  RR
5 ltle 7105 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  B  e.  RR )  ->  ( 1  <  B  ->  1  <_  B )
)
64, 2, 5sylancr 393 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  < 
B  ->  1  <_  B ) )
73, 6mpd 13 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  1  <_  B
)
8 simprr 484 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
9 leexp2a 9307 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  <_  B  /\  k  e.  ( ZZ>= `  j )
)  ->  ( B ^ j )  <_ 
( B ^ k
) )
102, 7, 8, 9syl3anc 1135 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
j )  <_  ( B ^ k ) )
11 0red 7028 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  e.  RR )
12 1red 7042 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  1  e.  RR )
13 0lt1 7141 . . . . . . . . . . . 12  |-  0  <  1
1413a1i 9 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  <  1
)
1511, 12, 2, 14, 3lttrd 7140 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  <  B
)
162, 15elrpd 8620 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  B  e.  RR+ )
17 nnz 8264 . . . . . . . . . 10  |-  ( j  e.  NN  ->  j  e.  ZZ )
1817ad2antrl 459 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  j  e.  ZZ )
19 rpexpcl 9274 . . . . . . . . 9  |-  ( ( B  e.  RR+  /\  j  e.  ZZ )  ->  ( B ^ j )  e.  RR+ )
2016, 18, 19syl2anc 391 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
j )  e.  RR+ )
21 eluzelz 8482 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  j
)  ->  k  e.  ZZ )
2221ad2antll 460 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  ZZ )
23 rpexpcl 9274 . . . . . . . . 9  |-  ( ( B  e.  RR+  /\  k  e.  ZZ )  ->  ( B ^ k )  e.  RR+ )
2416, 22, 23syl2anc 391 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
k )  e.  RR+ )
2520, 24lerecd 8642 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( B ^ j )  <_ 
( B ^ k
)  <->  ( 1  / 
( B ^ k
) )  <_  (
1  /  ( B ^ j ) ) ) )
2610, 25mpbid 135 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ k
) )  <_  (
1  /  ( B ^ j ) ) )
2724rprecred 8634 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ k
) )  e.  RR )
2820rprecred 8634 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ j
) )  e.  RR )
29 simpl1 907 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A  e.  RR+ )
3029rpred 8622 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A  e.  RR )
31 lelttr 7106 . . . . . . 7  |-  ( ( ( 1  /  ( B ^ k ) )  e.  RR  /\  (
1  /  ( B ^ j ) )  e.  RR  /\  A  e.  RR )  ->  (
( ( 1  / 
( B ^ k
) )  <_  (
1  /  ( B ^ j ) )  /\  ( 1  / 
( B ^ j
) )  <  A
)  ->  ( 1  /  ( B ^
k ) )  < 
A ) )
3227, 28, 30, 31syl3anc 1135 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( ( 1  /  ( B ^ k ) )  <_  ( 1  / 
( B ^ j
) )  /\  (
1  /  ( B ^ j ) )  <  A )  -> 
( 1  /  ( B ^ k ) )  <  A ) )
3326, 32mpand 405 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( 1  /  ( B ^
j ) )  < 
A  ->  ( 1  /  ( B ^
k ) )  < 
A ) )
3433anassrs 380 . . . 4  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  j  e.  NN )  /\  k  e.  (
ZZ>= `  j ) )  ->  ( ( 1  /  ( B ^
j ) )  < 
A  ->  ( 1  /  ( B ^
k ) )  < 
A ) )
3534ralrimdva 2399 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  j  e.  NN )  ->  ( ( 1  / 
( B ^ j
) )  <  A  ->  A. k  e.  (
ZZ>= `  j ) ( 1  /  ( B ^ k ) )  <  A ) )
3635reximdva 2421 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  ( E. j  e.  NN  ( 1  /  ( B ^ j ) )  <  A  ->  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( 1  /  ( B ^ k ) )  <  A ) )
371, 36mpd 13 1  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( 1  /  ( B ^ k ) )  <  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    /\ w3a 885    e. wcel 1393   A.wral 2306   E.wrex 2307   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   RRcr 6888   0cc0 6889   1c1 6890    < clt 7060    <_ cle 7061    / cdiv 7651   NNcn 7914   ZZcz 8245   ZZ>=cuz 8473   RR+crp 8583   ^cexp 9254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 7002  ax-arch 7003
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-div 7652  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-rp 8584  df-iseq 9212  df-iexp 9255
This theorem is referenced by: (None)
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