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Theorem ostth2lem1 20767
Description: Lemma for ostth2 20786, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 20786. 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 9815 . . . . . 6  |-  2  e.  RR
2 ostth2lem1.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
32adantr 451 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  B  e.  RR )
4 remulcl 8822 . . . . . 6  |-  ( ( 2  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  B
)  e.  RR )
51, 3, 4sylancr 644 . . . . 5  |-  ( (
ph  /\  1  <  A )  ->  ( 2  x.  B )  e.  RR )
6 simpr 447 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  1  <  A )
7 1re 8837 . . . . . . 7  |-  1  e.  RR
8 ostth2lem1.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
98adantr 451 . . . . . . 7  |-  ( (
ph  /\  1  <  A )  ->  A  e.  RR )
10 difrp 10387 . . . . . . 7  |-  ( ( 1  e.  RR  /\  A  e.  RR )  ->  ( 1  <  A  <->  ( A  -  1 )  e.  RR+ ) )
117, 9, 10sylancr 644 . . . . . 6  |-  ( (
ph  /\  1  <  A )  ->  ( 1  <  A  <->  ( A  -  1 )  e.  RR+ ) )
126, 11mpbid 201 . . . . 5  |-  ( (
ph  /\  1  <  A )  ->  ( A  -  1 )  e.  RR+ )
135, 12rerpdivcld 10417 . . . 4  |-  ( (
ph  /\  1  <  A )  ->  ( (
2  x.  B )  /  ( A  - 
1 ) )  e.  RR )
14 expnbnd 11230 . . . 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 1182 . . 3  |-  ( (
ph  /\  1  <  A )  ->  E. k  e.  NN  ( ( 2  x.  B )  / 
( A  -  1 ) )  <  ( A ^ k ) )
16 nnnn0 9972 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
17 reexpcl 11120 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  RR )
189, 16, 17syl2an 463 . . . . 5  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  RR )
1913adantr 451 . . . . 5  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  B
)  /  ( A  -  1 ) )  e.  RR )
2012rpred 10390 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  A )  ->  ( A  -  1 )  e.  RR )
2120adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A  -  1 )  e.  RR )
22 nnre 9753 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  RR )
2322adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  RR )
2421, 23remulcld 8863 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  e.  RR )
2524, 18remulcld 8863 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  e.  RR )
268ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  RR )
27 2nn 9877 . . . . . . . . . . . 12  |-  2  e.  NN
28 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  NN )
29 nnmulcl 9769 . . . . . . . . . . . 12  |-  ( ( 2  e.  NN  /\  k  e.  NN )  ->  ( 2  x.  k
)  e.  NN )
3027, 28, 29sylancr 644 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  e.  NN )
31 nnnn0 9972 . . . . . . . . . . 11  |-  ( ( 2  x.  k )  e.  NN  ->  (
2  x.  k )  e.  NN0 )
3230, 31syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  e.  NN0 )
3326, 32reexpcld 11262 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  e.  RR )
3430nnred 9761 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  e.  RR )
352ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  B  e.  RR )
3634, 35remulcld 8863 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  k
)  x.  B )  e.  RR )
37 0re 8838 . . . . . . . . . . . . . . 15  |-  0  e.  RR
3837a1i 10 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  0  e.  RR )
397a1i 10 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  1  e.  RR )
40 0lt1 9296 . . . . . . . . . . . . . . 15  |-  0  <  1
4140a1i 10 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  1  <  A )  ->  0  <  1 )
4238, 39, 9, 41, 6lttrd 8977 . . . . . . . . . . . . 13  |-  ( (
ph  /\  1  <  A )  ->  0  <  A )
439, 42elrpd 10388 . . . . . . . . . . . 12  |-  ( (
ph  /\  1  <  A )  ->  A  e.  RR+ )
44 nnz 10045 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  ZZ )
45 rpexpcl 11122 . . . . . . . . . . . 12  |-  ( ( A  e.  RR+  /\  k  e.  ZZ )  ->  ( A ^ k )  e.  RR+ )
4643, 44, 45syl2an 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  RR+ )
47 peano2re 8985 . . . . . . . . . . . . 13  |-  ( ( ( A  -  1 )  x.  k )  e.  RR  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  e.  RR )
4824, 47syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  e.  RR )
4924ltp1d 9687 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  <  ( ( ( A  -  1 )  x.  k )  +  1 ) )
5016adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  NN0 )
5143adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  RR+ )
5251rpge0d 10394 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <_  A )
53 bernneq2 11228 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  k  e.  NN0  /\  0  <_  A )  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  <_  ( A ^
k ) )
5426, 50, 52, 53syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  +  1 )  <_  ( A ^
k ) )
5524, 48, 18, 49, 54ltletrd 8976 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  k )  <  ( A ^
k ) )
5624, 18, 46, 55ltmul1dd 10441 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  <  ( ( A ^ k )  x.  ( A ^ k
) ) )
5723recnd 8861 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  k  e.  CC )
58572timesd 9954 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  k )  =  ( k  +  k ) )
5958oveq2d 5874 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  =  ( A ^ (
k  +  k ) ) )
6026recnd 8861 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A  e.  CC )
6160, 50, 50expaddd 11247 . . . . . . . . . . 11  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( k  +  k ) )  =  ( ( A ^
k )  x.  ( A ^ k ) ) )
6259, 61eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  =  ( ( A ^
k )  x.  ( A ^ k ) ) )
6356, 62breqtrrd 4049 . . . . . . . . 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 2626 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( A ^ n )  <_  ( n  x.  B ) )
6665ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  A. n  e.  NN  ( A ^
n )  <_  (
n  x.  B ) )
67 oveq2 5866 . . . . . . . . . . . 12  |-  ( n  =  ( 2  x.  k )  ->  ( A ^ n )  =  ( A ^ (
2  x.  k ) ) )
68 oveq1 5865 . . . . . . . . . . . 12  |-  ( n  =  ( 2  x.  k )  ->  (
n  x.  B )  =  ( ( 2  x.  k )  x.  B ) )
6967, 68breq12d 4036 . . . . . . . . . . 11  |-  ( n  =  ( 2  x.  k )  ->  (
( A ^ n
)  <_  ( n  x.  B )  <->  ( A ^ ( 2  x.  k ) )  <_ 
( ( 2  x.  k )  x.  B
) ) )
7069rspcv 2880 . . . . . . . . . 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 56 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ ( 2  x.  k ) )  <_ 
( ( 2  x.  k )  x.  B
) )
7225, 33, 36, 63, 71ltletrd 8976 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  k
)  x.  ( A ^ k ) )  <  ( ( 2  x.  k )  x.  B ) )
7321recnd 8861 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A  -  1 )  e.  CC )
7418recnd 8861 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  e.  CC )
7573, 74, 57mul32d 9022 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  x.  k )  =  ( ( ( A  -  1 )  x.  k )  x.  ( A ^ k ) ) )
76 2cn 9816 . . . . . . . . . . 11  |-  2  e.  CC
7776a1i 10 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  2  e.  CC )
7835recnd 8861 . . . . . . . . . 10  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  B  e.  CC )
7977, 78, 57mul32d 9022 . . . . . . . . 9  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( 2  x.  B
)  x.  k )  =  ( ( 2  x.  k )  x.  B ) )
8075, 79breq12d 4036 . . . . . . . 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 223 . . . . . . 7  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  x.  k )  < 
( ( 2  x.  B )  x.  k
) )
8221, 18remulcld 8863 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  ( A ^ k ) )  e.  RR )
835adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
2  x.  B )  e.  RR )
84 nngt0 9775 . . . . . . . . 9  |-  ( k  e.  NN  ->  0  <  k )
8584adantl 452 . . . . . . . 8  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <  k )
86 ltmul1 9606 . . . . . . . 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 1186 . . . . . . 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 223 . . . . . 6  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( A  -  1 )  x.  ( A ^ k ) )  <  ( 2  x.  B ) )
8912rpgt0d 10393 . . . . . . . 8  |-  ( (
ph  /\  1  <  A )  ->  0  <  ( A  -  1 ) )
9089adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  0  <  ( A  -  1 ) )
91 ltmuldiv2 9627 . . . . . . 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 1186 . . . . . 6  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  (
( ( A  - 
1 )  x.  ( A ^ k ) )  <  ( 2  x.  B )  <->  ( A ^ k )  < 
( ( 2  x.  B )  /  ( A  -  1 ) ) ) )
9388, 92mpbid 201 . . . . 5  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  ( A ^ k )  < 
( ( 2  x.  B )  /  ( A  -  1 ) ) )
9418, 19, 93ltnsymd 8968 . . . 4  |-  ( ( ( ph  /\  1  <  A )  /\  k  e.  NN )  ->  -.  ( ( 2  x.  B )  /  ( A  -  1 ) )  <  ( A ^ k ) )
9594nrexdv 2646 . . 3  |-  ( (
ph  /\  1  <  A )  ->  -.  E. k  e.  NN  ( ( 2  x.  B )  / 
( A  -  1 ) )  <  ( A ^ k ) )
9615, 95pm2.65da 559 . 2  |-  ( ph  ->  -.  1  <  A
)
97 lenlt 8901 . . 3  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <_  1  <->  -.  1  <  A ) )
988, 7, 97sylancl 643 . 2  |-  ( ph  ->  ( A  <_  1  <->  -.  1  <  A ) )
9996, 98mpbird 223 1  |-  ( ph  ->  A  <_  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   RR+crp 10354   ^cexp 11104
This theorem is referenced by:  ostth2lem4  20785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-seq 11047  df-exp 11105
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