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Theorem expnlbnd2 10757
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 10756 . 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 8025 . . . . . . . . . 10 |- 1 e. RR
5 ltle 8114 . . . . . . . . . 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 10684 . . . . . . . 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 8027 . . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 0 e. RR )
12 1red 8041 . . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 1 e. RR )
13 0lt1 8153 . . . . . . . . . . . 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 8152 . . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 0 < B )
162, 15elrpd 9768 . . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> B e. RR+ )
17 nnz 9345 . . . . . . . . . 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 10650 . . . . . . . . 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 9610 . . . . . . . . . 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 10650 . . . . . . . . 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 9791 . . . . . . 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 9783 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( 1 / ( B ^ k ) ) e. RR )
2820rprecred 9783 . . . . . . 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 9771 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> A e. RR )
31 lelttr 8115 . . . . . . 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 2577 . . 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 2599 . 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 2167   A.wral 2475   E.wrex 2476   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   RRcr 7878   0cc0 7879   1c1 7880   < clt 8061   <_ cle 8062   / cdiv 8699   NNcn 8990   ZZcz 9326   ZZ>=cuz 9601   RR+crp 9728   ^cexp 10630
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998
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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-rp 9729  df-seqfrec 10540  df-exp 10631
This theorem is referenced by: (None)
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