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Theorem expnlbnd2 9598
Description: The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
Assertion
Ref Expression
expnlbnd2 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> E. j e. NN A. k e. ( ZZ>= ` j ) ( 1 / ( B ^ k ) ) < A )
Distinct variable groups:   j, k, A   B, j, k
Proof of Theorem expnlbnd2
StepHypRef Expression
1 expnlbnd 9597 . 2 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> E. j e. NN ( 1 / ( B ^ j ) ) < A )
2 simpl2 942 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> B e. RR )
3 simpl3 943 . . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 1 < B )
4 1re 7118 . . . . . . . . . 10 |- 1 e. RR
5 ltle 7198 . . . . . . . . . 10 |- ( ( 1 e. RR /\ B e. RR ) -> ( 1 < B -> 1 <_ B ) )
64, 2, 5sylancr 405 . . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( 1 < B -> 1 <_ B ) )
73, 6mpd 13 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 1 <_ B )
8 simprr 498 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> k e. ( ZZ>= ` j ) )
9 leexp2a 9529 . . . . . . . 8 |- ( ( B e. RR /\ 1 <_ B /\ k e. ( ZZ>= ` j ) ) -> ( B ^ j ) <_ ( B ^ k ) )
102, 7, 8, 9syl3anc 1169 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( B ^ j ) <_ ( B ^ k ) )
11 0red 7120 . . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 0 e. RR )
12 1red 7134 . . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 1 e. RR )
13 0lt1 7236 . . . . . . . . . . . 12 |- 0 < 1
1413a1i 9 . . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 0 < 1 )
1511, 12, 2, 14, 3lttrd 7235 . . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 0 < B )
162, 15elrpd 8771 . . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> B e. RR+ )
17 nnz 8370 . . . . . . . . . 10 |- ( j e. NN -> j e. ZZ )
1817ad2antrl 473 . . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> j e. ZZ )
19 rpexpcl 9495 . . . . . . . . 9 |- ( ( B e. RR+ /\ j e. ZZ ) -> ( B ^ j ) e. RR+ )
2016, 18, 19syl2anc 403 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( B ^ j ) e. RR+ )
21 eluzelz 8628 . . . . . . . . . 10 |- ( k e. ( ZZ>= ` j ) -> k e. ZZ )
2221ad2antll 474 . . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> k e. ZZ )
23 rpexpcl 9495 . . . . . . . . 9 |- ( ( B e. RR+ /\ k e. ZZ ) -> ( B ^ k ) e. RR+ )
2416, 22, 23syl2anc 403 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( B ^ k ) e. RR+ )
2520, 24lerecd 8793 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( ( B ^ j ) <_ ( B ^ k ) <-> ( 1 / ( B ^ k ) ) <_ ( 1 / ( B ^ j ) ) ) )
2610, 25mpbid 145 . . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( 1 / ( B ^ k ) ) <_ ( 1 / ( B ^ j ) ) )
2724rprecred 8785 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( 1 / ( B ^ k ) ) e. RR )
2820rprecred 8785 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( 1 / ( B ^ j ) ) e. RR )
29 simpl1 941 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> A e. RR+ )
3029rpred 8773 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> A e. RR )
31 lelttr 7199 . . . . . . 7 |- ( ( ( 1 / ( B ^ k ) ) e. RR /\ ( 1 / ( B ^ j ) ) e. RR /\ A e. RR ) -> ( ( ( 1 / ( B ^ k ) ) <_ ( 1 / ( B ^ j ) ) /\ ( 1 / ( B ^ j ) ) < A ) -> ( 1 / ( B ^ k ) ) < A ) )
3227, 28, 30, 31syl3anc 1169 . . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( ( ( 1 / ( B ^ k ) ) <_ ( 1 / ( B ^ j ) ) /\ ( 1 / ( B ^ j ) ) < A ) -> ( 1 / ( B ^ k ) ) < A ) )
3326, 32mpand 419 . . . . 5 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( ( 1 / ( B ^ j ) ) < A -> ( 1 / ( B ^ k ) ) < A ) )
3433anassrs 392 . . . 4 |- ( ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ j e. NN ) /\ k e. ( ZZ>= ` j ) ) -> ( ( 1 / ( B ^ j ) ) < A -> ( 1 / ( B ^ k ) ) < A ) )
3534ralrimdva 2441 . . 3 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ j e. NN ) -> ( ( 1 / ( B ^ j ) ) < A -> A. k e. ( ZZ>= ` j ) ( 1 / ( B ^ k ) ) < A ) )
3635reximdva 2463 . 2 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> ( E. j e. NN ( 1 / ( B ^ j ) ) < A -> E. j e. NN A. k e. ( ZZ>= ` j ) ( 1 / ( B ^ k ) ) < A ) )
371, 36mpd 13 1 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> E. j e. NN A. k e. ( ZZ>= ` j ) ( 1 / ( B ^ k ) ) < A )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   e. wcel 1433   A.wral 2348   E.wrex 2349   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   RRcr 6980   0cc0 6981   1c1 6982   < clt 7153   <_ cle 7154   / cdiv 7760   NNcn 8039   ZZcz 8351   ZZ>=cuz 8619   RR+crp 8734   ^cexp 9475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095
This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-rp 8735  df-iseq 9432  df-iexp 9476
This theorem is referenced by: (None)
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