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Theorem expnlbnd2 10899
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 10898 . 2 |- ( ( A e. RR+ /\ B e. RR /\ 1 < B ) -> E. j e. NN ( 1 / ( B ^ j ) ) < A )
2 simpl2 1025 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> B e. RR )
3 simpl3 1026 . . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 1 < B )
4 1re 8156 . . . . . . . . . 10 |- 1 e. RR
5 ltle 8245 . . . . . . . . . 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 10826 . . . . . . . 8 |- ( ( B e. RR /\ 1 <_ B /\ k e. ( ZZ>= ` j ) ) -> ( B ^ j ) <_ ( B ^ k ) )
102, 7, 8, 9syl3anc 1271 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( B ^ j ) <_ ( B ^ k ) )
11 0red 8158 . . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 0 e. RR )
12 1red 8172 . . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 1 e. RR )
13 0lt1 8284 . . . . . . . . . . . 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 8283 . . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> 0 < B )
162, 15elrpd 9901 . . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> B e. RR+ )
17 nnz 9476 . . . . . . . . . 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 10792 . . . . . . . . 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 9743 . . . . . . . . . 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 10792 . . . . . . . . 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 9924 . . . . . . 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 9916 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( 1 / ( B ^ k ) ) e. RR )
2820rprecred 9916 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> ( 1 / ( B ^ j ) ) e. RR )
29 simpl1 1024 . . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> A e. RR+ )
3029rpred 9904 . . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR /\ 1 < B ) /\ ( j e. NN /\ k e. ( ZZ>= ` j ) ) ) -> A e. RR )
31 lelttr 8246 . . . . . . 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 1271 . . . . . 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 2610 . . 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 2632 . 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 1002   e. wcel 2200   A.wral 2508   E.wrex 2509   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   RRcr 8009   0cc0 8010   1c1 8011   < clt 8192   <_ cle 8193   / cdiv 8830   NNcn 9121   ZZcz 9457   ZZ>=cuz 9733   RR+crp 9861   ^cexp 10772
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-rp 9862  df-seqfrec 10682  df-exp 10773
This theorem is referenced by: (None)
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