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

Step	Hyp	Ref	Expression
1		rpre 9658	. . . 4 |- ( A e. RR+ -> A e. RR )
2		rpap0 9668	. . . 4 |- ( A e. RR+ -> A # 0 )
3	1, 2	rerecclapd 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 ) )
5	3, 4	syl3an1 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 ) )
7	6	3ad2ant1 1018	. . . . 5 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> ( A e. RR /\ 0 < A ) )
8	7	adantr 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 )
11	9, 10	sylan2 286	. . . . . . 7 |- ( ( B e. RR /\ k e. NN ) -> ( B ^ k ) e. RR )
12	11	adantlr 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 )
15	14	adantl 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 ) )
20	17, 18, 19	mp3an12 1327	. . . . . . . . . 10 |- ( B e. RR -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
21	16, 20	mpani 430	. . . . . . . . 9 |- ( B e. RR -> ( 1 < B -> 0 < B ) )
22	21	imp 124	. . . . . . . 8 |- ( ( B e. RR /\ 1 < B ) -> 0 < B )
23	22	adantr 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 ) )
25	13, 15, 23, 24	syl3anc 1238	. . . . . 6 |- ( ( ( B e. RR /\ 1 < B ) /\ k e. NN ) -> 0 < ( B ^ k ) )
26	12, 25	jca 306	. . . . 5 |- ( ( ( B e. RR /\ 1 < B ) /\ k e. NN ) -> ( ( B ^ k ) e. RR /\ 0 < ( B ^ k ) ) )
27	26	3adantl1 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 ) )
29	8, 27, 28	syl2anc 411	. . 3 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ k e. NN ) -> ( ( 1 / A ) < ( B ^ k ) <-> ( 1 / ( B ^ k ) ) < A ) )
30	29	rexbidva 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 ) )
31	5, 30	mpbid 147	1 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> E. k e. NN ( 1 / ( B ^ k ) ) < A )

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