MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ostth2lem1 Structured version   Unicode version

Theorem ostth2lem1 22983
Description: Lemma for ostth2 23002, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 23002. If a power is upper bounded by a linear term, the exponent must be less than one. Or in big-O notation, 
n  e.  o ( A ^ n ) for any 
1  <  A. (Contributed by Mario Carneiro, 10-Sep-2014.)
Hypotheses
Ref Expression
ostth2lem1.1  |-  ( ph  ->  A  e.  RR )
ostth2lem1.2  |-  ( ph  ->  B  e.  RR )
ostth2lem1.3  |-  ( (
ph  /\  n  e.  NN )  ->  ( A ^ n )  <_ 
( n  x.  B
) )
Assertion
Ref Expression
ostth2lem1  |-  ( ph  ->  A  <_  1 )
Distinct variable groups:    A, n    B, n    ph, n

Proof of Theorem ostth2lem1
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 2re 10492 . . . . . 6  |-  2  e.  RR
2 ostth2lem1.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
32adantr 465 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  B  e.  RR )
4 remulcl 9468 . . . . . 6  |-  ( ( 2  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  B
)  e.  RR )
51, 3, 4sylancr 663 . . . . 5  |-  ( (
ph  /\  1  <  A )  ->  ( 2  x.  B )  e.  RR )
6 simpr 461 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  1  <  A )
7 1re 9486 . . . . . . 7  |-  1  e.  RR
8 ostth2lem1.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
98adantr 465 . . . . . . 7  |-  ( (
ph  /\  1  <  A )  ->  A  e.  RR )
10 difrp 11125 . . . . . . 7  |-  ( ( 1  e.  RR  /\  A  e.  RR )  ->  ( 1  <  A  <->  ( A  -  1 )  e.  RR+ ) )
117, 9, 10sylancr 663 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  ( 1  <  A  <->  ( A  -  1 )  e.  RR+ ) )
126, 11mpbid 210 . . . . 5  |-  ( (
ph  /\  1  <  A )  ->  ( A  -  1 )  e.  RR+ )
135, 12rerpdivcld 11155 . . . 4  |-  ( (
ph  /\  1  <  A )  ->  ( (
2  x.  B )  /  ( A  - 
1 ) )  e.  RR )
14 expnbnd 12094 . . . 4  |-  ( ( ( ( 2  x.  B )  /  ( A  -  1 ) )  e.  RR  /\  A  e.  RR  /\  1  <  A )  ->  E. k  e.  NN  ( ( 2  x.  B )  / 
( A  -  1 ) )  <  ( A ^ k ) )
1513, 9, 6, 14syl3anc 1219 . . 3  |-  ( (
ph  /\  1  <  A )  ->  E. k  e.  NN  ( ( 2  x.  B )  / 
( A  -  1 ) )  <  ( A ^ k ) )
16 nnnn0 10687 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
17 reexpcl 11983 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  RR )
189, 16, 17syl2an 477 . . . . 5  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  RR )
1913adantr 465 . . . . 5  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  B
)  /  ( A  -  1 ) )  e.  RR )
2012rpred 11128 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  A )  ->  ( A  -  1 )  e.  RR )
2120adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A  -  1 )  e.  RR )
22 nnre 10430 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  RR )
2322adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  RR )
2421, 23remulcld 9515 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  e.  RR )
2524, 18remulcld 9515 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  e.  RR )
268ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  RR )
27 2nn 10580 . . . . . . . . . . . 12  |-  2  e.  NN
28 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  NN )
29 nnmulcl 10446 . . . . . . . . . . . 12  |-  ( ( 2  e.  NN  /\  k  e.  NN )  ->  ( 2  x.  k
)  e.  NN )
3027, 28, 29sylancr 663 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  e.  NN )
31 nnnn0 10687 . . . . . . . . . . 11  |-  ( ( 2  x.  k )  e.  NN  ->  (
2  x.  k )  e.  NN0 )
3230, 31syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  e.  NN0 )
3326, 32reexpcld 12126 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  e.  RR )
3430nnred 10438 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  e.  RR )
352ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  B  e.  RR )
3634, 35remulcld 9515 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  k
)  x.  B )  e.  RR )
37 0red 9488 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  0  e.  RR )
387a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  1  e.  RR )
39 0lt1 9963 . . . . . . . . . . . . . . 15  |-  0  <  1
4039a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  0  <  1 )
4137, 38, 9, 40, 6lttrd 9633 . . . . . . . . . . . . 13  |-  ( (
ph  /\  1  <  A )  ->  0  <  A )
429, 41elrpd 11126 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  A )  ->  A  e.  RR+ )
43 nnz 10769 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  ZZ )
44 rpexpcl 11985 . . . . . . . . . . . 12  |-  ( ( A  e.  RR+  /\  k  e.  ZZ )  ->  ( A ^ k )  e.  RR+ )
4542, 43, 44syl2an 477 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  RR+ )
46 peano2re 9643 . . . . . . . . . . . . 13  |-  ( ( ( A  -  1 )  x.  k )  e.  RR  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  e.  RR )
4724, 46syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  e.  RR )
4824ltp1d 10364 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  <  ( ( ( A  -  1 )  x.  k )  +  1 ) )
4916adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  NN0 )
5042adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  RR+ )
5150rpge0d 11132 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <_  A )
52 bernneq2 12092 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  k  e.  NN0  /\  0  <_  A )  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  <_  ( A ^
k ) )
5326, 49, 51, 52syl3anc 1219 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  <_  ( A ^
k ) )
5424, 47, 18, 48, 53ltletrd 9632 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  <  ( A ^
k ) )
5524, 18, 45, 54ltmul1dd 11179 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  <  ( ( A ^ k )  x.  ( A ^ k
) ) )
5623recnd 9513 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  CC )
57562timesd 10668 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  =  ( k  +  k ) )
5857oveq2d 6206 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  =  ( A ^ (
k  +  k ) ) )
5926recnd 9513 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  CC )
6059, 49, 49expaddd 12111 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( k  +  k ) )  =  ( ( A ^
k )  x.  ( A ^ k ) ) )
6158, 60eqtrd 2492 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  =  ( ( A ^
k )  x.  ( A ^ k ) ) )
6255, 61breqtrrd 4416 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  <  ( A ^
( 2  x.  k
) ) )
63 ostth2lem1.3 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( A ^ n )  <_ 
( n  x.  B
) )
6463ralrimiva 2822 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( A ^ n )  <_  ( n  x.  B ) )
6564ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A. n  e.  NN  ( A ^
n )  <_  (
n  x.  B ) )
66 oveq2 6198 . . . . . . . . . . . 12  |-  ( n  =  ( 2  x.  k )  ->  ( A ^ n )  =  ( A ^ (
2  x.  k ) ) )
67 oveq1 6197 . . . . . . . . . . . 12  |-  ( n  =  ( 2  x.  k )  ->  (
n  x.  B )  =  ( ( 2  x.  k )  x.  B ) )
6866, 67breq12d 4403 . . . . . . . . . . 11  |-  ( n  =  ( 2  x.  k )  ->  (
( A ^ n
)  <_  ( n  x.  B )  <->  ( A ^ ( 2  x.  k ) )  <_ 
( ( 2  x.  k )  x.  B
) ) )
6968rspcv 3165 . . . . . . . . . 10  |-  ( ( 2  x.  k )  e.  NN  ->  ( A. n  e.  NN  ( A ^ n )  <_  ( n  x.  B )  ->  ( A ^ ( 2  x.  k ) )  <_ 
( ( 2  x.  k )  x.  B
) ) )
7030, 65, 69sylc 60 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  <_ 
( ( 2  x.  k )  x.  B
) )
7125, 33, 36, 62, 70ltletrd 9632 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  <  ( ( 2  x.  k )  x.  B ) )
7221recnd 9513 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A  -  1 )  e.  CC )
7318recnd 9513 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  CC )
7472, 73, 56mul32d 9680 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  x.  k )  =  ( ( ( A  -  1 )  x.  k )  x.  ( A ^ k ) ) )
75 2cnd 10495 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  2  e.  CC )
7635recnd 9513 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  B  e.  CC )
7775, 76, 56mul32d 9680 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  B
)  x.  k )  =  ( ( 2  x.  k )  x.  B ) )
7874, 77breq12d 4403 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( ( A  -  1 )  x.  ( A ^ k
) )  x.  k
)  <  ( (
2  x.  B )  x.  k )  <->  ( (
( A  -  1 )  x.  k )  x.  ( A ^
k ) )  < 
( ( 2  x.  k )  x.  B
) ) )
7971, 78mpbird 232 . . . . . . 7  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  x.  k )  < 
( ( 2  x.  B )  x.  k
) )
8021, 18remulcld 9515 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  ( A ^ k ) )  e.  RR )
815adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  B )  e.  RR )
82 nngt0 10452 . . . . . . . . 9  |-  ( k  e.  NN  ->  0  <  k )
8382adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <  k )
84 ltmul1 10280 . . . . . . . 8  |-  ( ( ( ( A  - 
1 )  x.  ( A ^ k ) )  e.  RR  /\  (
2  x.  B )  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( ( A  -  1 )  x.  ( A ^
k ) )  < 
( 2  x.  B
)  <->  ( ( ( A  -  1 )  x.  ( A ^
k ) )  x.  k )  <  (
( 2  x.  B
)  x.  k ) ) )
8580, 81, 23, 83, 84syl112anc 1223 . . . . . . 7  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  <  ( 2  x.  B )  <->  ( (
( A  -  1 )  x.  ( A ^ k ) )  x.  k )  < 
( ( 2  x.  B )  x.  k
) ) )
8679, 85mpbird 232 . . . . . 6  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  ( A ^ k ) )  <  ( 2  x.  B ) )
8712rpgt0d 11131 . . . . . . . 8  |-  ( (
ph  /\  1  <  A )  ->  0  <  ( A  -  1 ) )
8887adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <  ( A  -  1 ) )
89 ltmuldiv2 10304 . . . . . . 7  |-  ( ( ( A ^ k
)  e.  RR  /\  ( 2  x.  B
)  e.  RR  /\  ( ( A  - 
1 )  e.  RR  /\  0  <  ( A  -  1 ) ) )  ->  ( (
( A  -  1 )  x.  ( A ^ k ) )  <  ( 2  x.  B )  <->  ( A ^ k )  < 
( ( 2  x.  B )  /  ( A  -  1 ) ) ) )
9018, 81, 21, 88, 89syl112anc 1223 . . . . . 6  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  <  ( 2  x.  B )  <->  ( A ^ k )  < 
( ( 2  x.  B )  /  ( A  -  1 ) ) ) )
9186, 90mpbid 210 . . . . 5  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  < 
( ( 2  x.  B )  /  ( A  -  1 ) ) )
9218, 19, 91ltnsymd 9624 . . . 4  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  -.  ( ( 2  x.  B )  /  ( A  -  1 ) )  <  ( A ^ k ) )
9392nrexdv 2915 . . 3  |-  ( (
ph  /\  1  <  A )  ->  -.  E. k  e.  NN  ( ( 2  x.  B )  / 
( A  -  1 ) )  <  ( A ^ k ) )
9415, 93pm2.65da 576 . 2  |-  ( ph  ->  -.  1  <  A
)
95 lenlt 9554 . . 3  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <_  1  <->  -.  1  <  A ) )
968, 7, 95sylancl 662 . 2  |-  ( ph  ->  ( A  <_  1  <->  -.  1  <  A ) )
9794, 96mpbird 232 1  |-  ( ph  ->  A  <_  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   class class class wbr 4390  (class class class)co 6190   RRcr 9382   0cc0 9383   1c1 9384    + caddc 9386    x. cmul 9388    < clt 9519    <_ cle 9520    - cmin 9696    / cdiv 10094   NNcn 10423   2c2 10472   NN0cn0 10680   ZZcz 10747   RR+crp 11092   ^cexp 11966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-fl 11743  df-seq 11908  df-exp 11967
This theorem is referenced by:  ostth2lem4  23001
  Copyright terms: Public domain W3C validator