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Theorem expnlbnd 10641
Description: The reciprocal of exponentiation with a base greater than 1 has no positive 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 9658 . . . 4  |-  ( A  e.  RR+  ->  A  e.  RR )
2 rpap0 9668 . . . 4  |-  ( A  e.  RR+  ->  A #  0 )
31, 2rerecclapd 8789 . . 3  |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR )
4 expnbnd 10640 . . 3  |-  ( ( ( 1  /  A
)  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  E. k  e.  NN  ( 1  /  A )  <  ( B ^ k ) )
53, 4syl3an1 1271 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. k  e.  NN  ( 1  /  A )  <  ( B ^ k ) )
6 rpregt0 9665 . . . . . 6  |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <  A ) )
763ad2ant1 1018 . . . . 5  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  ( A  e.  RR  /\  0  <  A ) )
87adantr 276 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( A  e.  RR  /\  0  <  A ) )
9 nnnn0 9181 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
10 reexpcl 10534 . . . . . . . 8  |-  ( ( B  e.  RR  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  RR )
119, 10sylan2 286 . . . . . . 7  |-  ( ( B  e.  RR  /\  k  e.  NN )  ->  ( B ^ k
)  e.  RR )
1211adantlr 477 . . . . . 6  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( B ^
k )  e.  RR )
13 simpll 527 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  B  e.  RR )
14 nnz 9270 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  ZZ )
1514adantl 277 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  k  e.  ZZ )
16 0lt1 8082 . . . . . . . . . 10  |-  0  <  1
17 0re 7956 . . . . . . . . . . 11  |-  0  e.  RR
18 1re 7955 . . . . . . . . . . 11  |-  1  e.  RR
19 lttr 8029 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
2017, 18, 19mp3an12 1327 . . . . . . . . . 10  |-  ( B  e.  RR  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
2116, 20mpani 430 . . . . . . . . 9  |-  ( B  e.  RR  ->  (
1  <  B  ->  0  <  B ) )
2221imp 124 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
0  <  B )
2322adantr 276 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  0  <  B
)
24 expgt0 10550 . . . . . . 7  |-  ( ( B  e.  RR  /\  k  e.  ZZ  /\  0  <  B )  ->  0  <  ( B ^ k
) )
2513, 15, 23, 24syl3anc 1238 . . . . . 6  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  0  <  ( B ^ k ) )
2612, 25jca 306 . . . . 5  |-  ( ( ( B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( ( B ^ k )  e.  RR  /\  0  < 
( B ^ k
) ) )
27263adantl1 1153 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( ( B ^
k )  e.  RR  /\  0  <  ( B ^ k ) ) )
28 ltrec1 8843 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( ( B ^ k )  e.  RR  /\  0  < 
( B ^ k
) ) )  -> 
( ( 1  /  A )  <  ( B ^ k )  <->  ( 1  /  ( B ^
k ) )  < 
A ) )
298, 27, 28syl2anc 411 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  k  e.  NN )  ->  ( ( 1  /  A )  <  ( B ^ k )  <->  ( 1  /  ( B ^
k ) )  < 
A ) )
3029rexbidva 2474 . 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 147 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    /\ wa 104    <-> wb 105    /\ w3a 978    e. wcel 2148   E.wrex 2456   class class class wbr 4003  (class class class)co 5874   RRcr 7809   0cc0 7810   1c1 7811    < clt 7990    / cdiv 8627   NNcn 8917   NN0cn0 9174   ZZcz 9251   RR+crp 9651   ^cexp 10516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-n0 9175  df-z 9252  df-uz 9527  df-rp 9652  df-seqfrec 10443  df-exp 10517
This theorem is referenced by:  expnlbnd2  10642
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