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Theorem expnlbnd2 11500
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 11499 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. j  e.  NN  ( 1  / 
( B ^ j
) )  <  A
)
2 simpl2 961 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  B  e.  RR )
3 simpl3 962 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  1  <  B
)
4 1re 9080 . . . . . . . . . 10  |-  1  e.  RR
5 ltle 9153 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  B  e.  RR )  ->  ( 1  <  B  ->  1  <_  B )
)
64, 2, 5sylancr 645 . . . . . . . . 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 734 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
9 leexp2a 11425 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  <_  B  /\  k  e.  ( ZZ>= `  j )
)  ->  ( B ^ j )  <_ 
( B ^ k
) )
102, 7, 8, 9syl3anc 1184 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
j )  <_  ( B ^ k ) )
11 0re 9081 . . . . . . . . . . . 12  |-  0  e.  RR
1211a1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  e.  RR )
134a1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  1  e.  RR )
14 0lt1 9540 . . . . . . . . . . . 12  |-  0  <  1
1514a1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  <  1
)
1612, 13, 2, 15, 3lttrd 9221 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  <  B
)
172, 16elrpd 10636 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  B  e.  RR+ )
18 nnz 10293 . . . . . . . . . 10  |-  ( j  e.  NN  ->  j  e.  ZZ )
1918ad2antrl 709 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  j  e.  ZZ )
20 rpexpcl 11390 . . . . . . . . 9  |-  ( ( B  e.  RR+  /\  j  e.  ZZ )  ->  ( B ^ j )  e.  RR+ )
2117, 19, 20syl2anc 643 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
j )  e.  RR+ )
22 eluzelz 10486 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  j
)  ->  k  e.  ZZ )
2322ad2antll 710 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  ZZ )
24 rpexpcl 11390 . . . . . . . . 9  |-  ( ( B  e.  RR+  /\  k  e.  ZZ )  ->  ( B ^ k )  e.  RR+ )
2517, 23, 24syl2anc 643 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
k )  e.  RR+ )
2621, 25lerecd 10657 . . . . . . 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 ) ) ) )
2710, 26mpbid 202 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ k
) )  <_  (
1  /  ( B ^ j ) ) )
2825rprecred 10649 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ k
) )  e.  RR )
2921rprecred 10649 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ j
) )  e.  RR )
30 simpl1 960 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A  e.  RR+ )
3130rpred 10638 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A  e.  RR )
32 lelttr 9155 . . . . . . 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 ) )
3328, 29, 31, 32syl3anc 1184 . . . . . 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 ) )
3427, 33mpand 657 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( 1  /  ( B ^
j ) )  < 
A  ->  ( 1  /  ( B ^
k ) )  < 
A ) )
3534anassrs 630 . . . 4  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  j  e.  NN )  /\  k  e.  (
ZZ>= `  j ) )  ->  ( ( 1  /  ( B ^
j ) )  < 
A  ->  ( 1  /  ( B ^
k ) )  < 
A ) )
3635ralrimdva 2788 . . 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 ) )
3736reximdva 2810 . 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 ) )
381, 37mpd 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725   A.wral 2697   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   RRcr 8979   0cc0 8980   1c1 8981    < clt 9110    <_ cle 9111    / cdiv 9667   NNcn 9990   ZZcz 10272   ZZ>=cuz 10478   RR+crp 10602   ^cexp 11372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-fl 11192  df-seq 11314  df-exp 11373
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