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Theorem expnlbnd2 11978
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 11977 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. j  e.  NN  ( 1  / 
( B ^ j
) )  <  A
)
2 simpl2 985 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  B  e.  RR )
3 simpl3 986 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  1  <  B
)
4 1re 9372 . . . . . . . . . 10  |-  1  e.  RR
5 ltle 9450 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  B  e.  RR )  ->  ( 1  <  B  ->  1  <_  B )
)
64, 2, 5sylancr 656 . . . . . . . . 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 749 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
9 leexp2a 11902 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  <_  B  /\  k  e.  ( ZZ>= `  j )
)  ->  ( B ^ j )  <_ 
( B ^ k
) )
102, 7, 8, 9syl3anc 1211 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
j )  <_  ( B ^ k ) )
11 0red 9374 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  e.  RR )
12 1red 9388 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  1  e.  RR )
13 0lt1 9849 . . . . . . . . . . . 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 9519 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  <  B
)
162, 15elrpd 11012 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  B  e.  RR+ )
17 nnz 10655 . . . . . . . . . 10  |-  ( j  e.  NN  ->  j  e.  ZZ )
1817ad2antrl 720 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  j  e.  ZZ )
19 rpexpcl 11867 . . . . . . . . 9  |-  ( ( B  e.  RR+  /\  j  e.  ZZ )  ->  ( B ^ j )  e.  RR+ )
2016, 18, 19syl2anc 654 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
j )  e.  RR+ )
21 eluzelz 10857 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  j
)  ->  k  e.  ZZ )
2221ad2antll 721 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  ZZ )
23 rpexpcl 11867 . . . . . . . . 9  |-  ( ( B  e.  RR+  /\  k  e.  ZZ )  ->  ( B ^ k )  e.  RR+ )
2416, 22, 23syl2anc 654 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
k )  e.  RR+ )
2520, 24lerecd 11033 . . . . . . 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 11025 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ k
) )  e.  RR )
2820rprecred 11025 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ j
) )  e.  RR )
29 simpl1 984 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A  e.  RR+ )
3029rpred 11014 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A  e.  RR )
31 lelttr 9452 . . . . . . 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 1211 . . . . . 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 668 . . . . 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 641 . . . 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 2796 . . 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 2818 . 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 958    e. wcel 1755   A.wral 2705   E.wrex 2706   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   RRcr 9268   0cc0 9269   1c1 9270    < clt 9405    <_ cle 9406    / cdiv 9980   NNcn 10309   ZZcz 10633   ZZ>=cuz 10848   RR+crp 10978   ^cexp 11848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-n0 10567  df-z 10634  df-uz 10849  df-rp 10979  df-fl 11625  df-seq 11790  df-exp 11849
This theorem is referenced by: (None)
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