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Theorem expnlbnd 12015
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 11018 . . . 4  |-  ( A  e.  RR+  ->  A  e.  RR )
2 rpne0 11027 . . . 4  |-  ( A  e.  RR+  ->  A  =/=  0 )
31, 2rereccld 10179 . . 3  |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR )
4 expnbnd 12014 . . 3  |-  ( ( ( 1  /  A
)  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  E. k  e.  NN  ( 1  /  A )  <  ( B ^ k ) )
53, 4syl3an1 1251 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. k  e.  NN  ( 1  /  A )  <  ( B ^ k ) )
6 rpregt0 11025 . . . . . 6  |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <  A ) )
763ad2ant1 1009 . . . . 5  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  ( A  e.  RR  /\  0  <  A ) )
87adantr 465 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( A  e.  RR  /\  0  <  A ) )
9 nnnn0 10607 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
10 reexpcl 11903 . . . . . . . 8  |-  ( ( B  e.  RR  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  RR )
119, 10sylan2 474 . . . . . . 7  |-  ( ( B  e.  RR  /\  k  e.  NN )  ->  ( B ^ k
)  e.  RR )
1211adantlr 714 . . . . . 6  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( B ^
k )  e.  RR )
13 simpll 753 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  B  e.  RR )
14 nnz 10689 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  ZZ )
1514adantl 466 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  k  e.  ZZ )
16 0lt1 9883 . . . . . . . . . 10  |-  0  <  1
17 0re 9407 . . . . . . . . . . 11  |-  0  e.  RR
18 1re 9406 . . . . . . . . . . 11  |-  1  e.  RR
19 lttr 9472 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
2017, 18, 19mp3an12 1304 . . . . . . . . . 10  |-  ( B  e.  RR  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
2116, 20mpani 676 . . . . . . . . 9  |-  ( B  e.  RR  ->  (
1  <  B  ->  0  <  B ) )
2221imp 429 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
0  <  B )
2322adantr 465 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  0  <  B
)
24 expgt0 11918 . . . . . . 7  |-  ( ( B  e.  RR  /\  k  e.  ZZ  /\  0  <  B )  ->  0  <  ( B ^ k
) )
2513, 15, 23, 24syl3anc 1218 . . . . . 6  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  0  <  ( B ^ k ) )
2612, 25jca 532 . . . . 5  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( ( B ^ k )  e.  RR  /\  0  < 
( B ^ k
) ) )
27263adantl1 1144 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( ( B ^
k )  e.  RR  /\  0  <  ( B ^ k ) ) )
28 ltrec1 10240 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( ( B ^ k )  e.  RR  /\  0  < 
( B ^ k
) ) )  -> 
( ( 1  /  A )  <  ( B ^ k )  <->  ( 1  /  ( B ^
k ) )  < 
A ) )
298, 27, 28syl2anc 661 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( ( 1  /  A )  <  ( B ^ k )  <->  ( 1  /  ( B ^
k ) )  < 
A ) )
3029rexbidva 2753 . 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 210 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 184    /\ wa 369    /\ w3a 965    e. wcel 1756   E.wrex 2737   class class class wbr 4313  (class class class)co 6112   RRcr 9302   0cc0 9303   1c1 9304    < clt 9439    / cdiv 10014   NNcn 10343   NN0cn0 10600   ZZcz 10667   RR+crp 11012   ^cexp 11886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-fl 11663  df-seq 11828  df-exp 11887
This theorem is referenced by:  expnlbnd2  12016  opnmbllem  21103  opnmbllem0  28453  heiborlem7  28742
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