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Theorem ostth2lem1 21305
Description: Lemma for ostth2 21324, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 21324. 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 10062 . . . . . 6  |-  2  e.  RR
2 ostth2lem1.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
32adantr 452 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  B  e.  RR )
4 remulcl 9068 . . . . . 6  |-  ( ( 2  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  B
)  e.  RR )
51, 3, 4sylancr 645 . . . . 5  |-  ( (
ph  /\  1  <  A )  ->  ( 2  x.  B )  e.  RR )
6 simpr 448 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  1  <  A )
7 1re 9083 . . . . . . 7  |-  1  e.  RR
8 ostth2lem1.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
98adantr 452 . . . . . . 7  |-  ( (
ph  /\  1  <  A )  ->  A  e.  RR )
10 difrp 10638 . . . . . . 7  |-  ( ( 1  e.  RR  /\  A  e.  RR )  ->  ( 1  <  A  <->  ( A  -  1 )  e.  RR+ ) )
117, 9, 10sylancr 645 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  ( 1  <  A  <->  ( A  -  1 )  e.  RR+ ) )
126, 11mpbid 202 . . . . 5  |-  ( (
ph  /\  1  <  A )  ->  ( A  -  1 )  e.  RR+ )
135, 12rerpdivcld 10668 . . . 4  |-  ( (
ph  /\  1  <  A )  ->  ( (
2  x.  B )  /  ( A  - 
1 ) )  e.  RR )
14 expnbnd 11501 . . . 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 1184 . . 3  |-  ( (
ph  /\  1  <  A )  ->  E. k  e.  NN  ( ( 2  x.  B )  / 
( A  -  1 ) )  <  ( A ^ k ) )
16 nnnn0 10221 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
17 reexpcl 11391 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  RR )
189, 16, 17syl2an 464 . . . . 5  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  RR )
1913adantr 452 . . . . 5  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  B
)  /  ( A  -  1 ) )  e.  RR )
2012rpred 10641 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  A )  ->  ( A  -  1 )  e.  RR )
2120adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A  -  1 )  e.  RR )
22 nnre 10000 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  RR )
2322adantl 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  RR )
2421, 23remulcld 9109 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  e.  RR )
2524, 18remulcld 9109 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  e.  RR )
268ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  RR )
27 2nn 10126 . . . . . . . . . . . 12  |-  2  e.  NN
28 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  NN )
29 nnmulcl 10016 . . . . . . . . . . . 12  |-  ( ( 2  e.  NN  /\  k  e.  NN )  ->  ( 2  x.  k
)  e.  NN )
3027, 28, 29sylancr 645 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  e.  NN )
31 nnnn0 10221 . . . . . . . . . . 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 11533 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  e.  RR )
3430nnred 10008 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  e.  RR )
352ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  B  e.  RR )
3634, 35remulcld 9109 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  k
)  x.  B )  e.  RR )
37 0re 9084 . . . . . . . . . . . . . . 15  |-  0  e.  RR
3837a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  0  e.  RR )
397a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  1  e.  RR )
40 0lt1 9543 . . . . . . . . . . . . . . 15  |-  0  <  1
4140a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  0  <  1 )
4238, 39, 9, 41, 6lttrd 9224 . . . . . . . . . . . . 13  |-  ( (
ph  /\  1  <  A )  ->  0  <  A )
439, 42elrpd 10639 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  A )  ->  A  e.  RR+ )
44 nnz 10296 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  ZZ )
45 rpexpcl 11393 . . . . . . . . . . . 12  |-  ( ( A  e.  RR+  /\  k  e.  ZZ )  ->  ( A ^ k )  e.  RR+ )
4643, 44, 45syl2an 464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  RR+ )
47 peano2re 9232 . . . . . . . . . . . . 13  |-  ( ( ( A  -  1 )  x.  k )  e.  RR  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  e.  RR )
4824, 47syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  e.  RR )
4924ltp1d 9934 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  <  ( ( ( A  -  1 )  x.  k )  +  1 ) )
5016adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  NN0 )
5143adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  RR+ )
5251rpge0d 10645 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <_  A )
53 bernneq2 11499 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  k  e.  NN0  /\  0  <_  A )  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  <_  ( A ^
k ) )
5426, 50, 52, 53syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  <_  ( A ^
k ) )
5524, 48, 18, 49, 54ltletrd 9223 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  <  ( A ^
k ) )
5624, 18, 46, 55ltmul1dd 10692 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  <  ( ( A ^ k )  x.  ( A ^ k
) ) )
5723recnd 9107 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  CC )
58572timesd 10203 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  =  ( k  +  k ) )
5958oveq2d 6090 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  =  ( A ^ (
k  +  k ) ) )
6026recnd 9107 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  CC )
6160, 50, 50expaddd 11518 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( k  +  k ) )  =  ( ( A ^
k )  x.  ( A ^ k ) ) )
6259, 61eqtrd 2468 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  =  ( ( A ^
k )  x.  ( A ^ k ) ) )
6356, 62breqtrrd 4231 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  <  ( A ^
( 2  x.  k
) ) )
64 ostth2lem1.3 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( A ^ n )  <_ 
( n  x.  B
) )
6564ralrimiva 2782 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( A ^ n )  <_  ( n  x.  B ) )
6665ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A. n  e.  NN  ( A ^
n )  <_  (
n  x.  B ) )
67 oveq2 6082 . . . . . . . . . . . 12  |-  ( n  =  ( 2  x.  k )  ->  ( A ^ n )  =  ( A ^ (
2  x.  k ) ) )
68 oveq1 6081 . . . . . . . . . . . 12  |-  ( n  =  ( 2  x.  k )  ->  (
n  x.  B )  =  ( ( 2  x.  k )  x.  B ) )
6967, 68breq12d 4218 . . . . . . . . . . 11  |-  ( n  =  ( 2  x.  k )  ->  (
( A ^ n
)  <_  ( n  x.  B )  <->  ( A ^ ( 2  x.  k ) )  <_ 
( ( 2  x.  k )  x.  B
) ) )
7069rspcv 3041 . . . . . . . . . 10  |-  ( ( 2  x.  k )  e.  NN  ->  ( A. n  e.  NN  ( A ^ n )  <_  ( n  x.  B )  ->  ( A ^ ( 2  x.  k ) )  <_ 
( ( 2  x.  k )  x.  B
) ) )
7130, 66, 70sylc 58 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  <_ 
( ( 2  x.  k )  x.  B
) )
7225, 33, 36, 63, 71ltletrd 9223 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  <  ( ( 2  x.  k )  x.  B ) )
7321recnd 9107 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A  -  1 )  e.  CC )
7418recnd 9107 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  CC )
7573, 74, 57mul32d 9269 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  x.  k )  =  ( ( ( A  -  1 )  x.  k )  x.  ( A ^ k ) ) )
76 2cn 10063 . . . . . . . . . . 11  |-  2  e.  CC
7776a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  2  e.  CC )
7835recnd 9107 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  B  e.  CC )
7977, 78, 57mul32d 9269 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  B
)  x.  k )  =  ( ( 2  x.  k )  x.  B ) )
8075, 79breq12d 4218 . . . . . . . 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
) ) )
8172, 80mpbird 224 . . . . . . 7  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  x.  k )  < 
( ( 2  x.  B )  x.  k
) )
8221, 18remulcld 9109 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  ( A ^ k ) )  e.  RR )
835adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  B )  e.  RR )
84 nngt0 10022 . . . . . . . . 9  |-  ( k  e.  NN  ->  0  <  k )
8584adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <  k )
86 ltmul1 9853 . . . . . . . 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 ) ) )
8782, 83, 23, 85, 86syl112anc 1188 . . . . . . 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
) ) )
8881, 87mpbird 224 . . . . . 6  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  ( A ^ k ) )  <  ( 2  x.  B ) )
8912rpgt0d 10644 . . . . . . . 8  |-  ( (
ph  /\  1  <  A )  ->  0  <  ( A  -  1 ) )
9089adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <  ( A  -  1 ) )
91 ltmuldiv2 9874 . . . . . . 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 ) ) ) )
9218, 83, 21, 90, 91syl112anc 1188 . . . . . 6  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  <  ( 2  x.  B )  <->  ( A ^ k )  < 
( ( 2  x.  B )  /  ( A  -  1 ) ) ) )
9388, 92mpbid 202 . . . . 5  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  < 
( ( 2  x.  B )  /  ( A  -  1 ) ) )
9418, 19, 93ltnsymd 9215 . . . 4  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  -.  ( ( 2  x.  B )  /  ( A  -  1 ) )  <  ( A ^ k ) )
9594nrexdv 2802 . . 3  |-  ( (
ph  /\  1  <  A )  ->  -.  E. k  e.  NN  ( ( 2  x.  B )  / 
( A  -  1 ) )  <  ( A ^ k ) )
9615, 95pm2.65da 560 . 2  |-  ( ph  ->  -.  1  <  A
)
97 lenlt 9147 . . 3  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <_  1  <->  -.  1  <  A ) )
988, 7, 97sylancl 644 . 2  |-  ( ph  ->  ( A  <_  1  <->  -.  1  <  A ) )
9996, 98mpbird 224 1  |-  ( ph  ->  A  <_  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2698   E.wrex 2699   class class class wbr 4205  (class class class)co 6074   CCcc 8981   RRcr 8982   0cc0 8983   1c1 8984    + caddc 8986    x. cmul 8988    < clt 9113    <_ cle 9114    - cmin 9284    / cdiv 9670   NNcn 9993   2c2 10042   NN0cn0 10214   ZZcz 10275   RR+crp 10605   ^cexp 11375
This theorem is referenced by:  ostth2lem4  21323
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 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-sup 7439  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-n0 10215  df-z 10276  df-uz 10482  df-rp 10606  df-fl 11195  df-seq 11317  df-exp 11376
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