MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  expnlbnd Structured version   Unicode version

Theorem expnlbnd 12399
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 11308 . . . 4  |-  ( A  e.  RR+  ->  A  e.  RR )
2 rpne0 11317 . . . 4  |-  ( A  e.  RR+  ->  A  =/=  0 )
31, 2rereccld 10433 . . 3  |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR )
4 expnbnd 12398 . . 3  |-  ( ( ( 1  /  A
)  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  E. k  e.  NN  ( 1  /  A )  <  ( B ^ k ) )
53, 4syl3an1 1297 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. k  e.  NN  ( 1  /  A )  <  ( B ^ k ) )
6 rpregt0 11315 . . . . . 6  |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <  A ) )
763ad2ant1 1026 . . . . 5  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  ( A  e.  RR  /\  0  <  A ) )
87adantr 466 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( A  e.  RR  /\  0  <  A ) )
9 nnnn0 10876 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
10 reexpcl 12286 . . . . . . . 8  |-  ( ( B  e.  RR  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  RR )
119, 10sylan2 476 . . . . . . 7  |-  ( ( B  e.  RR  /\  k  e.  NN )  ->  ( B ^ k
)  e.  RR )
1211adantlr 719 . . . . . 6  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( B ^
k )  e.  RR )
13 simpll 758 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  B  e.  RR )
14 nnz 10959 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  ZZ )
1514adantl 467 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  k  e.  ZZ )
16 0lt1 10135 . . . . . . . . . 10  |-  0  <  1
17 0re 9642 . . . . . . . . . . 11  |-  0  e.  RR
18 1re 9641 . . . . . . . . . . 11  |-  1  e.  RR
19 lttr 9709 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
2017, 18, 19mp3an12 1350 . . . . . . . . . 10  |-  ( B  e.  RR  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
2116, 20mpani 680 . . . . . . . . 9  |-  ( B  e.  RR  ->  (
1  <  B  ->  0  <  B ) )
2221imp 430 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
0  <  B )
2322adantr 466 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  0  <  B
)
24 expgt0 12302 . . . . . . 7  |-  ( ( B  e.  RR  /\  k  e.  ZZ  /\  0  <  B )  ->  0  <  ( B ^ k
) )
2513, 15, 23, 24syl3anc 1264 . . . . . 6  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  0  <  ( B ^ k ) )
2612, 25jca 534 . . . . 5  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( ( B ^ k )  e.  RR  /\  0  < 
( B ^ k
) ) )
27263adantl1 1161 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( ( B ^
k )  e.  RR  /\  0  <  ( B ^ k ) ) )
28 ltrec1 10493 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( ( B ^ k )  e.  RR  /\  0  < 
( B ^ k
) ) )  -> 
( ( 1  /  A )  <  ( B ^ k )  <->  ( 1  /  ( B ^
k ) )  < 
A ) )
298, 27, 28syl2anc 665 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( ( 1  /  A )  <  ( B ^ k )  <->  ( 1  /  ( B ^
k ) )  < 
A ) )
3029rexbidva 2943 . 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 213 1  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. k  e.  NN  ( 1  / 
( B ^ k
) )  <  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    e. wcel 1870   E.wrex 2783   class class class wbr 4426  (class class class)co 6305   RRcr 9537   0cc0 9538   1c1 9539    < clt 9674    / cdiv 10268   NNcn 10609   NN0cn0 10869   ZZcz 10937   RR+crp 11302   ^cexp 12269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fl 12025  df-seq 12211  df-exp 12270
This theorem is referenced by:  expnlbnd2  12400  opnmbllem  22436  opnmbllem0  31680  heiborlem7  31853
  Copyright terms: Public domain W3C validator