Theorem expnlbnd 10846

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 9817	. . . 4 |- ( A e. RR+ -> A e. RR )
2		rpap0 9827	. . . 4 |- ( A e. RR+ -> A # 0 )
3	1, 2	rerecclapd 8942	. . 3 |- ( A e. RR+ -> ( 1 / A ) e. RR )
4		expnbnd 10845	. . 3 |- ( ( ( 1 / A ) e. RR /\ B e. RR /\ 1 < B ) -> E. k e. NN ( 1 / A ) < ( B ^ k ) )
5	3, 4	syl3an1 1283	. 2 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> E. k e. NN ( 1 / A ) < ( B ^ k ) )
6		rpregt0 9824	. . . . . 6 |- ( A e. RR+ -> ( A e. RR /\ 0 < A ) )
7	6	3ad2ant1 1021	. . . . 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 9337	. . . . . . . 8 |- ( k e. NN -> k e. NN0 )
10		reexpcl 10738	. . . . . . . 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 9426	. . . . . . . 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 8234	. . . . . . . . . 10 |- 0 < 1
17		0re 8107	. . . . . . . . . . 11 |- 0 e. RR
18		1re 8106	. . . . . . . . . . 11 |- 1 e. RR
19		lttr 8181	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
20	17, 18, 19	mp3an12 1340	. . . . . . . . . 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 10754	. . . . . . 7 |- ( ( B e. RR /\ k e. ZZ /\ 0 < B ) -> 0 < ( B ^ k ) )
25	13, 15, 23, 24	syl3anc 1250	. . . . . 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 1156	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ k e. NN ) -> ( ( B ^ k ) e. RR /\ 0 < ( B ^ k ) ) )
28		ltrec1 8996	. . . 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 2505	. 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 981   e. wcel 2178   E.wrex 2487   class class class wbr 4059  (class class class)co 5967   RRcr 7959   0cc0 7960   1c1 7961   < clt 8142   / cdiv 8780   NNcn 9071   NN0cn0 9330   ZZcz 9407   RR+crp 9810   ^cexp 10720

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-seqfrec 10630  df-exp 10721

This theorem is referenced by:  expnlbnd2  10847