Theorem expnlbnd 10524

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

Step	Hyp	Ref	Expression
1		rpre 9549	. . . 4 |- ( A e. RR+ -> A e. RR )
2		rpap0 9559	. . . 4 |- ( A e. RR+ -> A # 0 )
3	1, 2	rerecclapd 8689	. . 3 |- ( A e. RR+ -> ( 1 / A ) e. RR )
4		expnbnd 10523	. . 3 |- ( ( ( 1 / A ) e. RR /\ B e. RR /\ 1 < B ) -> E. k e. NN ( 1 / A ) < ( B ^ k ) )
5	3, 4	syl3an1 1253	. 2 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> E. k e. NN ( 1 / A ) < ( B ^ k ) )
6		rpregt0 9556	. . . . . 6 |- ( A e. RR+ -> ( A e. RR /\ 0 < A ) )
7	6	3ad2ant1 1003	. . . . 5 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> ( A e. RR /\ 0 < A ) )
8	7	adantr 274	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ k e. NN ) -> ( A e. RR /\ 0 < A ) )
9		nnnn0 9080	. . . . . . . 8 |- ( k e. NN -> k e. NN0 )
10		reexpcl 10418	. . . . . . . 8 |- ( ( B e. RR /\ k e. NN0 ) -> ( B ^ k ) e. RR )
11	9, 10	sylan2 284	. . . . . . 7 |- ( ( B e. RR /\ k e. NN ) -> ( B ^ k ) e. RR )
12	11	adantlr 469	. . . . . 6 |- ( ( ( B e. RR /\ 1 < B ) /\ k e. NN ) -> ( B ^ k ) e. RR )
13		simpll 519	. . . . . . 7 |- ( ( ( B e. RR /\ 1 < B ) /\ k e. NN ) -> B e. RR )
14		nnz 9169	. . . . . . . 8 |- ( k e. NN -> k e. ZZ )
15	14	adantl 275	. . . . . . 7 |- ( ( ( B e. RR /\ 1 < B ) /\ k e. NN ) -> k e. ZZ )
16		0lt1 7985	. . . . . . . . . 10 |- 0 < 1
17		0re 7861	. . . . . . . . . . 11 |- 0 e. RR
18		1re 7860	. . . . . . . . . . 11 |- 1 e. RR
19		lttr 7934	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
20	17, 18, 19	mp3an12 1309	. . . . . . . . . 10 |- ( B e. RR -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
21	16, 20	mpani 427	. . . . . . . . 9 |- ( B e. RR -> ( 1 < B -> 0 < B ) )
22	21	imp 123	. . . . . . . 8 |- ( ( B e. RR /\ 1 < B ) -> 0 < B )
23	22	adantr 274	. . . . . . 7 |- ( ( ( B e. RR /\ 1 < B ) /\ k e. NN ) -> 0 < B )
24		expgt0 10434	. . . . . . 7 |- ( ( B e. RR /\ k e. ZZ /\ 0 < B ) -> 0 < ( B ^ k ) )
25	13, 15, 23, 24	syl3anc 1220	. . . . . 6 |- ( ( ( B e. RR /\ 1 < B ) /\ k e. NN ) -> 0 < ( B ^ k ) )
26	12, 25	jca 304	. . . . 5 |- ( ( ( B e. RR /\ 1 < B ) /\ k e. NN ) -> ( ( B ^ k ) e. RR /\ 0 < ( B ^ k ) ) )
27	26	3adantl1 1138	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ k e. NN ) -> ( ( B ^ k ) e. RR /\ 0 < ( B ^ k ) ) )
28		ltrec1 8742	. . . 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 409	. . 3 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ k e. NN ) -> ( ( 1 / A ) < ( B ^ k ) <-> ( 1 / ( B ^ k ) ) < A ) )
30	29	rexbidva 2454	. 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 146	1 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> E. k e. NN ( 1 / ( B ^ k ) ) < A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   e. wcel 2128   E.wrex 2436   class class class wbr 3965  (class class class)co 5818   RRcr 7714   0cc0 7715   1c1 7716   < clt 7895   / cdiv 8528   NNcn 8816   NN0cn0 9073   ZZcz 9150   RR+crp 9542   ^cexp 10400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-frec 6332  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-n0 9074  df-z 9151  df-uz 9423  df-rp 9543  df-seqfrec 10327  df-exp 10401

This theorem is referenced by:  expnlbnd2  10525