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Theorem ostth2lem1 21179
Description: Lemma for ostth2 21198, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 21198. 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 10001 . . . . . 6  |-  2  e.  RR
2 ostth2lem1.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
32adantr 452 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  B  e.  RR )
4 remulcl 9008 . . . . . 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 9023 . . . . . . 7  |-  1  e.  RR
8 ostth2lem1.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
98adantr 452 . . . . . . 7  |-  ( (
ph  /\  1  <  A )  ->  A  e.  RR )
10 difrp 10577 . . . . . . 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 10607 . . . 4  |-  ( (
ph  /\  1  <  A )  ->  ( (
2  x.  B )  /  ( A  - 
1 ) )  e.  RR )
14 expnbnd 11435 . . . 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 10160 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
17 reexpcl 11325 . . . . . 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 10580 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  A )  ->  ( A  -  1 )  e.  RR )
2120adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A  -  1 )  e.  RR )
22 nnre 9939 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  RR )
2322adantl 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  RR )
2421, 23remulcld 9049 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  e.  RR )
2524, 18remulcld 9049 . . . . . . . . 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 10065 . . . . . . . . . . . 12  |-  2  e.  NN
28 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  NN )
29 nnmulcl 9955 . . . . . . . . . . . 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 10160 . . . . . . . . . . 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 11467 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  e.  RR )
3430nnred 9947 . . . . . . . . . 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 9049 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  k
)  x.  B )  e.  RR )
37 0re 9024 . . . . . . . . . . . . . . 15  |-  0  e.  RR
3837a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  0  e.  RR )
397a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  1  e.  RR )
40 0lt1 9482 . . . . . . . . . . . . . . 15  |-  0  <  1
4140a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  0  <  1 )
4238, 39, 9, 41, 6lttrd 9163 . . . . . . . . . . . . 13  |-  ( (
ph  /\  1  <  A )  ->  0  <  A )
439, 42elrpd 10578 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  A )  ->  A  e.  RR+ )
44 nnz 10235 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  ZZ )
45 rpexpcl 11327 . . . . . . . . . . . 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 9171 . . . . . . . . . . . . 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 9873 . . . . . . . . . . . 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 10584 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <_  A )
53 bernneq2 11433 . . . . . . . . . . . . 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 9162 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  <  ( A ^
k ) )
5624, 18, 46, 55ltmul1dd 10631 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  <  ( ( A ^ k )  x.  ( A ^ k
) ) )
5723recnd 9047 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  CC )
58572timesd 10142 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  =  ( k  +  k ) )
5958oveq2d 6036 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  =  ( A ^ (
k  +  k ) ) )
6026recnd 9047 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  CC )
6160, 50, 50expaddd 11452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( k  +  k ) )  =  ( ( A ^
k )  x.  ( A ^ k ) ) )
6259, 61eqtrd 2419 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  =  ( ( A ^
k )  x.  ( A ^ k ) ) )
6356, 62breqtrrd 4179 . . . . . . . . 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 2732 . . . . . . . . . . 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 6028 . . . . . . . . . . . 12  |-  ( n  =  ( 2  x.  k )  ->  ( A ^ n )  =  ( A ^ (
2  x.  k ) ) )
68 oveq1 6027 . . . . . . . . . . . 12  |-  ( n  =  ( 2  x.  k )  ->  (
n  x.  B )  =  ( ( 2  x.  k )  x.  B ) )
6967, 68breq12d 4166 . . . . . . . . . . 11  |-  ( n  =  ( 2  x.  k )  ->  (
( A ^ n
)  <_  ( n  x.  B )  <->  ( A ^ ( 2  x.  k ) )  <_ 
( ( 2  x.  k )  x.  B
) ) )
7069rspcv 2991 . . . . . . . . . 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 9162 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  <  ( ( 2  x.  k )  x.  B ) )
7321recnd 9047 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A  -  1 )  e.  CC )
7418recnd 9047 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  CC )
7573, 74, 57mul32d 9208 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  x.  k )  =  ( ( ( A  -  1 )  x.  k )  x.  ( A ^ k ) ) )
76 2cn 10002 . . . . . . . . . . 11  |-  2  e.  CC
7776a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  2  e.  CC )
7835recnd 9047 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  B  e.  CC )
7977, 78, 57mul32d 9208 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  B
)  x.  k )  =  ( ( 2  x.  k )  x.  B ) )
8075, 79breq12d 4166 . . . . . . . 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 9049 . . . . . . . 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 9961 . . . . . . . . 9  |-  ( k  e.  NN  ->  0  <  k )
8584adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <  k )
86 ltmul1 9792 . . . . . . . 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 10583 . . . . . . . 8  |-  ( (
ph  /\  1  <  A )  ->  0  <  ( A  -  1 ) )
9089adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <  ( A  -  1 ) )
91 ltmuldiv2 9813 . . . . . . 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 9154 . . . 4  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  -.  ( ( 2  x.  B )  /  ( A  -  1 ) )  <  ( A ^ k ) )
9594nrexdv 2752 . . 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 9087 . . 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 1649    e. wcel 1717   A.wral 2649   E.wrex 2650   class class class wbr 4153  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    < clt 9053    <_ cle 9054    - cmin 9223    / cdiv 9609   NNcn 9932   2c2 9981   NN0cn0 10153   ZZcz 10214   RR+crp 10544   ^cexp 11309
This theorem is referenced by:  ostth2lem4  21197
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fl 11129  df-seq 11251  df-exp 11310
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