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Theorem expnlbnd2 10736
Description: The reciprocal of exponentiation with a base greater than 1 has no positive 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 10735 . 2 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> E. j e. NN ( 1 / ( B ^ j ) ) < A )
2 simpl2 1003 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> B e. RR )
3 simpl3 1004 . . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 1 < B )
4 1re 8018 . . . . . . . . . 10 |- 1 e. RR
5 ltle 8107 . . . . . . . . . 10 |- ( ( 1 e. RR /\ B e. RR ) -> ( 1 < B -> 1 <_ B ) )
64, 2, 5sylancr 414 . . . . . . . . 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 531 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> k e. ( ZZ>= ` j ) )
9 leexp2a 10663 . . . . . . . 8 |- ( ( B e. RR /\ 1 <_ B /\ k e. ( ZZ>= ` j ) ) -> ( B ^ j ) <_ ( B ^ k ) )
102, 7, 8, 9syl3anc 1249 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( B ^ j ) <_ ( B ^ k ) )
11 0red 8020 . . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 0 e. RR )
12 1red 8034 . . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 1 e. RR )
13 0lt1 8146 . . . . . . . . . . . 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 8145 . . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 0 < B )
162, 15elrpd 9759 . . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> B e. RR+ )
17 nnz 9336 . . . . . . . . . 10 |- ( j e. NN -> j e. ZZ )
1817ad2antrl 490 . . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> j e. ZZ )
19 rpexpcl 10629 . . . . . . . . 9 |- ( ( B e. RR+ /\ j e. ZZ ) -> ( B ^ j ) e. RR+ )
2016, 18, 19syl2anc 411 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( B ^ j ) e. RR+ )
21 eluzelz 9601 . . . . . . . . . 10 |- ( k e. ( ZZ>= ` j ) -> k e. ZZ )
2221ad2antll 491 . . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> k e. ZZ )
23 rpexpcl 10629 . . . . . . . . 9 |- ( ( B e. RR+ /\ k e. ZZ ) -> ( B ^ k ) e. RR+ )
2416, 22, 23syl2anc 411 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( B ^ k ) e. RR+ )
2520, 24lerecd 9782 . . . . . . 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 147 . . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( 1 / ( B ^ k ) ) <_ ( 1 / ( B ^ j ) ) )
2724rprecred 9774 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( 1 / ( B ^ k ) ) e. RR )
2820rprecred 9774 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( 1 / ( B ^ j ) ) e. RR )
29 simpl1 1002 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> A e. RR+ )
3029rpred 9762 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> A e. RR )
31 lelttr 8108 . . . . . . 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 1249 . . . . . 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 429 . . . . 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 400 . . . 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 2574 . . 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 2596 . 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 104   /\ w3a 980   e. wcel 2164   A.wral 2472   E.wrex 2473   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   RRcr 7871   0cc0 7872   1c1 7873   < clt 8054   <_ cle 8055   / cdiv 8691   NNcn 8982   ZZcz 9317   ZZ>=cuz 9592   RR+crp 9719   ^cexp 10609
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-seqfrec 10519  df-exp 10610
This theorem is referenced by: (None)
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