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Theorem expnlbnd 11472
Description: The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.)
Assertion
Ref Expression
expnlbnd  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. k  e.  NN  ( 1  / 
( B ^ k
) )  <  A
)
Distinct variable groups:    A, k    B, k

Proof of Theorem expnlbnd
StepHypRef Expression
1 rpre 10582 . . . 4  |-  ( A  e.  RR+  ->  A  e.  RR )
2 rpne0 10591 . . . 4  |-  ( A  e.  RR+  ->  A  =/=  0 )
31, 2rereccld 9805 . . 3  |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR )
4 expnbnd 11471 . . 3  |-  ( ( ( 1  /  A
)  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  E. k  e.  NN  ( 1  /  A )  <  ( B ^ k ) )
53, 4syl3an1 1217 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. k  e.  NN  ( 1  /  A )  <  ( B ^ k ) )
6 rpregt0 10589 . . . . . 6  |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <  A ) )
763ad2ant1 978 . . . . 5  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  ( A  e.  RR  /\  0  <  A ) )
87adantr 452 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( A  e.  RR  /\  0  <  A ) )
9 nnnn0 10192 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
10 reexpcl 11361 . . . . . . . 8  |-  ( ( B  e.  RR  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  RR )
119, 10sylan2 461 . . . . . . 7  |-  ( ( B  e.  RR  /\  k  e.  NN )  ->  ( B ^ k
)  e.  RR )
1211adantlr 696 . . . . . 6  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( B ^
k )  e.  RR )
13 simpll 731 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  B  e.  RR )
14 nnz 10267 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  ZZ )
1514adantl 453 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  k  e.  ZZ )
16 0lt1 9514 . . . . . . . . . 10  |-  0  <  1
17 0re 9055 . . . . . . . . . . 11  |-  0  e.  RR
18 1re 9054 . . . . . . . . . . 11  |-  1  e.  RR
19 lttr 9116 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
2017, 18, 19mp3an12 1269 . . . . . . . . . 10  |-  ( B  e.  RR  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
2116, 20mpani 658 . . . . . . . . 9  |-  ( B  e.  RR  ->  (
1  <  B  ->  0  <  B ) )
2221imp 419 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
0  <  B )
2322adantr 452 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  0  <  B
)
24 expgt0 11376 . . . . . . 7  |-  ( ( B  e.  RR  /\  k  e.  ZZ  /\  0  <  B )  ->  0  <  ( B ^ k
) )
2513, 15, 23, 24syl3anc 1184 . . . . . 6  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  0  <  ( B ^ k ) )
2612, 25jca 519 . . . . 5  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( ( B ^ k )  e.  RR  /\  0  < 
( B ^ k
) ) )
27263adantl1 1113 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( ( B ^
k )  e.  RR  /\  0  <  ( B ^ k ) ) )
28 ltrec1 9861 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( ( B ^ k )  e.  RR  /\  0  < 
( B ^ k
) ) )  -> 
( ( 1  /  A )  <  ( B ^ k )  <->  ( 1  /  ( B ^
k ) )  < 
A ) )
298, 27, 28syl2anc 643 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( ( 1  /  A )  <  ( B ^ k )  <->  ( 1  /  ( B ^
k ) )  < 
A ) )
3029rexbidva 2691 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  ( E. k  e.  NN  ( 1  /  A
)  <  ( B ^ k )  <->  E. k  e.  NN  ( 1  / 
( B ^ k
) )  <  A
) )
315, 30mpbid 202 1  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. k  e.  NN  ( 1  / 
( B ^ k
) )  <  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1721   E.wrex 2675   class class class wbr 4180  (class class class)co 6048   RRcr 8953   0cc0 8954   1c1 8955    < clt 9084    / cdiv 9641   NNcn 9964   NN0cn0 10185   ZZcz 10246   RR+crp 10576   ^cexp 11345
This theorem is referenced by:  expnlbnd2  11473  opnmbllem  19454  mblfinlem  26151  heiborlem7  26424
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-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-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-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fl 11165  df-seq 11287  df-exp 11346
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