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Theorem expnlbnd2 11987
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 11986 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. j  e.  NN  ( 1  / 
( B ^ j
) )  <  A
)
2 simpl2 992 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  B  e.  RR )
3 simpl3 993 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  1  <  B
)
4 1re 9377 . . . . . . . . . 10  |-  1  e.  RR
5 ltle 9455 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  B  e.  RR )  ->  ( 1  <  B  ->  1  <_  B )
)
64, 2, 5sylancr 663 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  < 
B  ->  1  <_  B ) )
73, 6mpd 15 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  1  <_  B
)
8 simprr 756 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
9 leexp2a 11911 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  <_  B  /\  k  e.  ( ZZ>= `  j )
)  ->  ( B ^ j )  <_ 
( B ^ k
) )
102, 7, 8, 9syl3anc 1218 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
j )  <_  ( B ^ k ) )
11 0red 9379 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  e.  RR )
12 1red 9393 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  1  e.  RR )
13 0lt1 9854 . . . . . . . . . . . 12  |-  0  <  1
1413a1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  <  1
)
1511, 12, 2, 14, 3lttrd 9524 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  <  B
)
162, 15elrpd 11017 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  B  e.  RR+ )
17 nnz 10660 . . . . . . . . . 10  |-  ( j  e.  NN  ->  j  e.  ZZ )
1817ad2antrl 727 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  j  e.  ZZ )
19 rpexpcl 11876 . . . . . . . . 9  |-  ( ( B  e.  RR+  /\  j  e.  ZZ )  ->  ( B ^ j )  e.  RR+ )
2016, 18, 19syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
j )  e.  RR+ )
21 eluzelz 10862 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  j
)  ->  k  e.  ZZ )
2221ad2antll 728 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  ZZ )
23 rpexpcl 11876 . . . . . . . . 9  |-  ( ( B  e.  RR+  /\  k  e.  ZZ )  ->  ( B ^ k )  e.  RR+ )
2416, 22, 23syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
k )  e.  RR+ )
2520, 24lerecd 11038 . . . . . . 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 210 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ k
) )  <_  (
1  /  ( B ^ j ) ) )
2724rprecred 11030 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ k
) )  e.  RR )
2820rprecred 11030 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ j
) )  e.  RR )
29 simpl1 991 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A  e.  RR+ )
3029rpred 11019 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A  e.  RR )
31 lelttr 9457 . . . . . . 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 1218 . . . . . 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 675 . . . . 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 648 . . . 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 2801 . . 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 2823 . 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 15 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1756   A.wral 2710   E.wrex 2711   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   RRcr 9273   0cc0 9274   1c1 9275    < clt 9410    <_ cle 9411    / cdiv 9985   NNcn 10314   ZZcz 10638   ZZ>=cuz 10853   RR+crp 10983   ^cexp 11857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fl 11634  df-seq 11799  df-exp 11858
This theorem is referenced by: (None)
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