Theorem expnlbnd 10600

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 9617	. . . 4 |- ( A e. RR+ -> A e. RR )
2		rpap0 9627	. . . 4 |- ( A e. RR+ -> A # 0 )
3	1, 2	rerecclapd 8751	. . 3 |- ( A e. RR+ -> ( 1 / A ) e. RR )
4		expnbnd 10599	. . 3 |- ( ( ( 1 / A ) e. RR /\ B e. RR /\ 1 < B ) -> E. k e. NN ( 1 / A ) < ( B ^ k ) )
5	3, 4	syl3an1 1266	. 2 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> E. k e. NN ( 1 / A ) < ( B ^ k ) )
6		rpregt0 9624	. . . . . 6 |- ( A e. RR+ -> ( A e. RR /\ 0 < A ) )
7	6	3ad2ant1 1013	. . . . 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 9142	. . . . . . . 8 |- ( k e. NN -> k e. NN0 )
10		reexpcl 10493	. . . . . . . 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 474	. . . . . 6 |- ( ( ( B e. RR /\ 1 < B ) /\ k e. NN ) -> ( B ^ k ) e. RR )
13		simpll 524	. . . . . . 7 |- ( ( ( B e. RR /\ 1 < B ) /\ k e. NN ) -> B e. RR )
14		nnz 9231	. . . . . . . 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 8046	. . . . . . . . . 10 |- 0 < 1
17		0re 7920	. . . . . . . . . . 11 |- 0 e. RR
18		1re 7919	. . . . . . . . . . 11 |- 1 e. RR
19		lttr 7993	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
20	17, 18, 19	mp3an12 1322	. . . . . . . . . 10 |- ( B e. RR -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
21	16, 20	mpani 428	. . . . . . . . 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 10509	. . . . . . 7 |- ( ( B e. RR /\ k e. ZZ /\ 0 < B ) -> 0 < ( B ^ k ) )
25	13, 15, 23, 24	syl3anc 1233	. . . . . 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 1148	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ k e. NN ) -> ( ( B ^ k ) e. RR /\ 0 < ( B ^ k ) ) )
28		ltrec1 8804	. . . 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 2467	. 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 973   e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775   < clt 7954   / cdiv 8589   NNcn 8878   NN0cn0 9135   ZZcz 9212   RR+crp 9610   ^cexp 10475

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-seqfrec 10402  df-exp 10476

This theorem is referenced by:  expnlbnd2  10601