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Theorem expnlbnd2 12279
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 12278 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  1  < 
B )  ->  E. j  e.  NN  ( 1  / 
( B ^ j
) )  <  A
)
2 simpl2 998 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  B  e.  RR )
3 simpl3 999 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  1  <  B
)
4 1re 9584 . . . . . . . . . 10  |-  1  e.  RR
5 ltle 9662 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  B  e.  RR )  ->  ( 1  <  B  ->  1  <_  B )
)
64, 2, 5sylancr 661 . . . . . . . . 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 755 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  (
ZZ>= `  j ) )
9 leexp2a 12203 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  <_  B  /\  k  e.  ( ZZ>= `  j )
)  ->  ( B ^ j )  <_ 
( B ^ k
) )
102, 7, 8, 9syl3anc 1226 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
j )  <_  ( B ^ k ) )
11 0red 9586 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  e.  RR )
12 1red 9600 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  1  e.  RR )
13 0lt1 10071 . . . . . . . . . . . 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 9732 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  0  <  B
)
162, 15elrpd 11256 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  B  e.  RR+ )
17 nnz 10882 . . . . . . . . . 10  |-  ( j  e.  NN  ->  j  e.  ZZ )
1817ad2antrl 725 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  j  e.  ZZ )
19 rpexpcl 12167 . . . . . . . . 9  |-  ( ( B  e.  RR+  /\  j  e.  ZZ )  ->  ( B ^ j )  e.  RR+ )
2016, 18, 19syl2anc 659 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
j )  e.  RR+ )
21 eluzelz 11091 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  j
)  ->  k  e.  ZZ )
2221ad2antll 726 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  k  e.  ZZ )
23 rpexpcl 12167 . . . . . . . . 9  |-  ( ( B  e.  RR+  /\  k  e.  ZZ )  ->  ( B ^ k )  e.  RR+ )
2416, 22, 23syl2anc 659 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( B ^
k )  e.  RR+ )
2520, 24lerecd 11278 . . . . . . 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 11270 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ k
) )  e.  RR )
2820rprecred 11270 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( 1  / 
( B ^ j
) )  e.  RR )
29 simpl1 997 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A  e.  RR+ )
3029rpred 11259 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR  /\  1  <  B )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) ) )  ->  A  e.  RR )
31 lelttr 9664 . . . . . . 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 1226 . . . . . 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 673 . . . . 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 646 . . . 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 2872 . . 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 2929 . 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 367    /\ w3a 971    e. wcel 1823   A.wral 2804   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482    < clt 9617    <_ cle 9618    / cdiv 10202   NNcn 10531   ZZcz 10860   ZZ>=cuz 11082   RR+crp 11221   ^cexp 12148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fl 11910  df-seq 12090  df-exp 12149
This theorem is referenced by: (None)
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