Theorem expnlbnd 10647

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 9662	. . . 4 |- ( A e. RR+ -> A e. RR )
2		rpap0 9672	. . . 4 |- ( A e. RR+ -> A # 0 )
3	1, 2	rerecclapd 8793	. . 3 |- ( A e. RR+ -> ( 1 / A ) e. RR )
4		expnbnd 10646	. . 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 9669	. . . . . 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 9185	. . . . . . . 8 |- ( k e. NN -> k e. NN0 )
10		reexpcl 10539	. . . . . . . 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 9274	. . . . . . . 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 8086	. . . . . . . . . 10 |- 0 < 1
17		0re 7959	. . . . . . . . . . 11 |- 0 e. RR
18		1re 7958	. . . . . . . . . . 11 |- 1 e. RR
19		lttr 8033	. . . . . . . . . . 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 10555	. . . . . . 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 8847	. . . 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 4005  (class class class)co 5877   RRcr 7812   0cc0 7813   1c1 7814   < clt 7994   / cdiv 8631   NNcn 8921   NN0cn0 9178   ZZcz 9255   RR+crp 9655   ^cexp 10521

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-rp 9656  df-seqfrec 10448  df-exp 10522

This theorem is referenced by:  expnlbnd2  10648