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Theorem ostth2lem1 24456
Description: Lemma for ostth2 24475, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 24475. 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 10679 . . . . . 6  |-  2  e.  RR
2 ostth2lem1.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
32adantr 467 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  B  e.  RR )
4 remulcl 9624 . . . . . 6  |-  ( ( 2  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  B
)  e.  RR )
51, 3, 4sylancr 669 . . . . 5  |-  ( (
ph  /\  1  <  A )  ->  ( 2  x.  B )  e.  RR )
6 simpr 463 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  1  <  A )
7 1re 9642 . . . . . . 7  |-  1  e.  RR
8 ostth2lem1.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
98adantr 467 . . . . . . 7  |-  ( (
ph  /\  1  <  A )  ->  A  e.  RR )
10 difrp 11337 . . . . . . 7  |-  ( ( 1  e.  RR  /\  A  e.  RR )  ->  ( 1  <  A  <->  ( A  -  1 )  e.  RR+ ) )
117, 9, 10sylancr 669 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  ( 1  <  A  <->  ( A  -  1 )  e.  RR+ ) )
126, 11mpbid 214 . . . . 5  |-  ( (
ph  /\  1  <  A )  ->  ( A  -  1 )  e.  RR+ )
135, 12rerpdivcld 11369 . . . 4  |-  ( (
ph  /\  1  <  A )  ->  ( (
2  x.  B )  /  ( A  - 
1 ) )  e.  RR )
14 expnbnd 12401 . . . 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 1268 . . 3  |-  ( (
ph  /\  1  <  A )  ->  E. k  e.  NN  ( ( 2  x.  B )  / 
( A  -  1 ) )  <  ( A ^ k ) )
16 nnnn0 10876 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
17 reexpcl 12289 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  RR )
189, 16, 17syl2an 480 . . . . 5  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  RR )
1913adantr 467 . . . . 5  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  B
)  /  ( A  -  1 ) )  e.  RR )
2012rpred 11341 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  A )  ->  ( A  -  1 )  e.  RR )
2120adantr 467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A  -  1 )  e.  RR )
22 nnre 10616 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  RR )
2322adantl 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  RR )
2421, 23remulcld 9671 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  e.  RR )
2524, 18remulcld 9671 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  e.  RR )
268ad2antrr 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  RR )
27 2nn 10767 . . . . . . . . . . . 12  |-  2  e.  NN
28 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  NN )
29 nnmulcl 10632 . . . . . . . . . . . 12  |-  ( ( 2  e.  NN  /\  k  e.  NN )  ->  ( 2  x.  k
)  e.  NN )
3027, 28, 29sylancr 669 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  e.  NN )
31 nnnn0 10876 . . . . . . . . . . 11  |-  ( ( 2  x.  k )  e.  NN  ->  (
2  x.  k )  e.  NN0 )
3230, 31syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  e.  NN0 )
3326, 32reexpcld 12433 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  e.  RR )
3430nnred 10624 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  e.  RR )
352ad2antrr 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  B  e.  RR )
3634, 35remulcld 9671 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  k
)  x.  B )  e.  RR )
37 0red 9644 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  0  e.  RR )
387a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  1  e.  RR )
39 0lt1 10136 . . . . . . . . . . . . . . 15  |-  0  <  1
4039a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  0  <  1 )
4137, 38, 9, 40, 6lttrd 9796 . . . . . . . . . . . . 13  |-  ( (
ph  /\  1  <  A )  ->  0  <  A )
429, 41elrpd 11338 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  A )  ->  A  e.  RR+ )
43 nnz 10959 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  ZZ )
44 rpexpcl 12291 . . . . . . . . . . . 12  |-  ( ( A  e.  RR+  /\  k  e.  ZZ )  ->  ( A ^ k )  e.  RR+ )
4542, 43, 44syl2an 480 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  RR+ )
46 peano2re 9806 . . . . . . . . . . . . 13  |-  ( ( ( A  -  1 )  x.  k )  e.  RR  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  e.  RR )
4724, 46syl 17 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  e.  RR )
4824ltp1d 10537 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  <  ( ( ( A  -  1 )  x.  k )  +  1 ) )
4916adantl 468 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  NN0 )
5042adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  RR+ )
5150rpge0d 11345 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <_  A )
52 bernneq2 12399 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  k  e.  NN0  /\  0  <_  A )  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  <_  ( A ^
k ) )
5326, 49, 51, 52syl3anc 1268 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  <_  ( A ^
k ) )
5424, 47, 18, 48, 53ltletrd 9795 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  <  ( A ^
k ) )
5524, 18, 45, 54ltmul1dd 11393 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  <  ( ( A ^ k )  x.  ( A ^ k
) ) )
5623recnd 9669 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  CC )
57562timesd 10855 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  =  ( k  +  k ) )
5857oveq2d 6306 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  =  ( A ^ (
k  +  k ) ) )
5926recnd 9669 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  CC )
6059, 49, 49expaddd 12418 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( k  +  k ) )  =  ( ( A ^
k )  x.  ( A ^ k ) ) )
6158, 60eqtrd 2485 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  =  ( ( A ^
k )  x.  ( A ^ k ) ) )
6255, 61breqtrrd 4429 . . . . . . . . 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 2802 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( A ^ n )  <_  ( n  x.  B ) )
6564ad2antrr 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A. n  e.  NN  ( A ^
n )  <_  (
n  x.  B ) )
66 oveq2 6298 . . . . . . . . . . . 12  |-  ( n  =  ( 2  x.  k )  ->  ( A ^ n )  =  ( A ^ (
2  x.  k ) ) )
67 oveq1 6297 . . . . . . . . . . . 12  |-  ( n  =  ( 2  x.  k )  ->  (
n  x.  B )  =  ( ( 2  x.  k )  x.  B ) )
6866, 67breq12d 4415 . . . . . . . . . . 11  |-  ( n  =  ( 2  x.  k )  ->  (
( A ^ n
)  <_  ( n  x.  B )  <->  ( A ^ ( 2  x.  k ) )  <_ 
( ( 2  x.  k )  x.  B
) ) )
6968rspcv 3146 . . . . . . . . . 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 62 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  <_ 
( ( 2  x.  k )  x.  B
) )
7125, 33, 36, 62, 70ltletrd 9795 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  <  ( ( 2  x.  k )  x.  B ) )
7221recnd 9669 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A  -  1 )  e.  CC )
7318recnd 9669 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  CC )
7472, 73, 56mul32d 9843 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  x.  k )  =  ( ( ( A  -  1 )  x.  k )  x.  ( A ^ k ) ) )
75 2cnd 10682 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  2  e.  CC )
7635recnd 9669 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  B  e.  CC )
7775, 76, 56mul32d 9843 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  B
)  x.  k )  =  ( ( 2  x.  k )  x.  B ) )
7871, 74, 773brtr4d 4433 . . . . . . 7  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  x.  k )  < 
( ( 2  x.  B )  x.  k
) )
7921, 18remulcld 9671 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  ( A ^ k ) )  e.  RR )
805adantr 467 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  B )  e.  RR )
81 nngt0 10638 . . . . . . . . 9  |-  ( k  e.  NN  ->  0  <  k )
8281adantl 468 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <  k )
83 ltmul1 10455 . . . . . . . 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 ) ) )
8479, 80, 23, 82, 83syl112anc 1272 . . . . . . 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
) ) )
8578, 84mpbird 236 . . . . . 6  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  ( A ^ k ) )  <  ( 2  x.  B ) )
8612rpgt0d 11344 . . . . . . . 8  |-  ( (
ph  /\  1  <  A )  ->  0  <  ( A  -  1 ) )
8786adantr 467 . . . . . . 7  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <  ( A  -  1 ) )
88 ltmuldiv2 10479 . . . . . . 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 ) ) ) )
8918, 80, 21, 87, 88syl112anc 1272 . . . . . 6  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  <  ( 2  x.  B )  <->  ( A ^ k )  < 
( ( 2  x.  B )  /  ( A  -  1 ) ) ) )
9085, 89mpbid 214 . . . . 5  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  < 
( ( 2  x.  B )  /  ( A  -  1 ) ) )
9118, 19, 90ltnsymd 9784 . . . 4  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  -.  ( ( 2  x.  B )  /  ( A  -  1 ) )  <  ( A ^ k ) )
9291nrexdv 2843 . . 3  |-  ( (
ph  /\  1  <  A )  ->  -.  E. k  e.  NN  ( ( 2  x.  B )  / 
( A  -  1 ) )  <  ( A ^ k ) )
9315, 92pm2.65da 580 . 2  |-  ( ph  ->  -.  1  <  A
)
94 lenlt 9712 . . 3  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <_  1  <->  -.  1  <  A ) )
958, 7, 94sylancl 668 . 2  |-  ( ph  ->  ( A  <_  1  <->  -.  1  <  A ) )
9693, 95mpbird 236 1  |-  ( ph  ->  A  <_  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738   class class class wbr 4402  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   RR+crp 11302   ^cexp 12272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fl 12028  df-seq 12214  df-exp 12273
This theorem is referenced by:  ostth2lem4  24474
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