Theorem ostth2lem1 22826

Description: Lemma for ostth2 22845, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 22845. 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.

Step	Hyp	Ref	Expression
1		2re 10387	. . . . . 6 |- 2 e. RR
2		ostth2lem1.2	. . . . . . 7 |- ( ph -> B e. RR )
3	2	adantr 462	. . . . . 6 |- ( ( ph /\ 1 < A ) -> B e. RR )
4		remulcl 9363	. . . . . 6 |- ( ( 2 e. RR /\ B e. RR ) -> ( 2 x. B ) e. RR )
5	1, 3, 4	sylancr 658	. . . . 5 |- ( ( ph /\ 1 < A ) -> ( 2 x. B ) e. RR )
6		simpr 458	. . . . . 6 |- ( ( ph /\ 1 < A ) -> 1 < A )
7		1re 9381	. . . . . . 7 |- 1 e. RR
8		ostth2lem1.1	. . . . . . . 8 |- ( ph -> A e. RR )
9	8	adantr 462	. . . . . . 7 |- ( ( ph /\ 1 < A ) -> A e. RR )
10		difrp 11020	. . . . . . 7 |- ( ( 1 e. RR /\ A e. RR ) -> ( 1 < A <-> ( A - 1 ) e. RR+ ) )
11	7, 9, 10	sylancr 658	. . . . . 6 |- ( ( ph /\ 1 < A ) -> ( 1 < A <-> ( A - 1 ) e. RR+ ) )
12	6, 11	mpbid 210	. . . . 5 |- ( ( ph /\ 1 < A ) -> ( A - 1 ) e. RR+ )
13	5, 12	rerpdivcld 11050	. . . 4 |- ( ( ph /\ 1 < A ) -> ( ( 2 x. B ) / ( A - 1 ) ) e. RR )
14		expnbnd 11989	. . . 4 |- ( ( ( ( 2 x. B ) / ( A - 1 ) ) e. RR /\ A e. RR /\ 1 < A ) -> E. k e. NN ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
15	13, 9, 6, 14	syl3anc 1213	. . 3 |- ( ( ph /\ 1 < A ) -> E. k e. NN ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
16		nnnn0 10582	. . . . . 6 |- ( k e. NN -> k e. NN0 )
17		reexpcl 11878	. . . . . 6 |- ( ( A e. RR /\ k e. NN0 ) -> ( A ^ k ) e. RR )
18	9, 16, 17	syl2an 474	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. RR )
19	13	adantr 462	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. B ) / ( A - 1 ) ) e. RR )
20	12	rpred 11023	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> ( A - 1 ) e. RR )
21	20	adantr 462	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A - 1 ) e. RR )
22		nnre 10325	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
23	22	adantl 463	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. RR )
24	21, 23	remulcld 9410	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) e. RR )
25	24, 18	remulcld 9410	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) e. RR )
26	8	ad2antrr 720	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. RR )
27		2nn 10475	. . . . . . . . . . . 12 |- 2 e. NN
28		simpr 458	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. NN )
29		nnmulcl 10341	. . . . . . . . . . . 12 |- ( ( 2 e. NN /\ k e. NN ) -> ( 2 x. k ) e. NN )
30	27, 28, 29	sylancr 658	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) e. NN )
31		nnnn0 10582	. . . . . . . . . . 11 |- ( ( 2 x. k ) e. NN -> ( 2 x. k ) e. NN0 )
32	30, 31	syl 16	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) e. NN0 )
33	26, 32	reexpcld 12021	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) e. RR )
34	30	nnred 10333	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) e. RR )
35	2	ad2antrr 720	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> B e. RR )
36	34, 35	remulcld 9410	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. k ) x. B ) e. RR )
37		0red 9383	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 0 e. RR )
38	7	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 1 e. RR )
39		0lt1 9858	. . . . . . . . . . . . . . 15 |- 0 < 1
40	39	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 0 < 1 )
41	37, 38, 9, 40, 6	lttrd 9528	. . . . . . . . . . . . 13 |- ( ( ph /\ 1 < A ) -> 0 < A )
42	9, 41	elrpd 11021	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> A e. RR+ )
43		nnz 10664	. . . . . . . . . . . 12 |- ( k e. NN -> k e. ZZ )
44		rpexpcl 11880	. . . . . . . . . . . 12 |- ( ( A e. RR+ /\ k e. ZZ ) -> ( A ^ k ) e. RR+ )
45	42, 43, 44	syl2an 474	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. RR+ )
46		peano2re 9538	. . . . . . . . . . . . 13 |- ( ( ( A - 1 ) x. k ) e. RR -> ( ( ( A - 1 ) x. k ) + 1 ) e. RR )
47	24, 46	syl 16	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) + 1 ) e. RR )
48	24	ltp1d 10259	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) < ( ( ( A - 1 ) x. k ) + 1 ) )
49	16	adantl 463	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. NN0 )
50	42	adantr 462	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. RR+ )
51	50	rpge0d 11027	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 <_ A )
52		bernneq2 11987	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ k e. NN0 /\ 0 <_ A ) -> ( ( ( A - 1 ) x. k ) + 1 ) <_ ( A ^ k ) )
53	26, 49, 51, 52	syl3anc 1213	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) + 1 ) <_ ( A ^ k ) )
54	24, 47, 18, 48, 53	ltletrd 9527	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) < ( A ^ k ) )
55	24, 18, 45, 54	ltmul1dd 11074	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) < ( ( A ^ k ) x. ( A ^ k ) ) )
56	23	recnd 9408	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. CC )
57	56	2timesd 10563	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) = ( k + k ) )
58	57	oveq2d 6106	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) = ( A ^ ( k + k ) ) )
59	26	recnd 9408	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. CC )
60	59, 49, 49	expaddd 12006	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( k + k ) ) = ( ( A ^ k ) x. ( A ^ k ) ) )
61	58, 60	eqtrd 2473	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) = ( ( A ^ k ) x. ( A ^ k ) ) )
62	55, 61	breqtrrd 4315	. . . . . . . . 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 ) )
64	63	ralrimiva 2797	. . . . . . . . . . 11 |- ( ph -> A. n e. NN ( A ^ n ) <_ ( n x. B ) )
65	64	ad2antrr 720	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A. n e. NN ( A ^ n ) <_ ( n x. B ) )
66		oveq2 6098	. . . . . . . . . . . 12 |- ( n = ( 2 x. k ) -> ( A ^ n ) = ( A ^ ( 2 x. k ) ) )
67		oveq1 6097	. . . . . . . . . . . 12 |- ( n = ( 2 x. k ) -> ( n x. B ) = ( ( 2 x. k ) x. B ) )
68	66, 67	breq12d 4302	. . . . . . . . . . 11 |- ( n = ( 2 x. k ) -> ( ( A ^ n ) <_ ( n x. B ) <-> ( A ^ ( 2 x. k ) ) <_ ( ( 2 x. k ) x. B ) ) )
69	68	rspcv 3066	. . . . . . . . . 10 |- ( ( 2 x. k ) e. NN -> ( A. n e. NN ( A ^ n ) <_ ( n x. B ) -> ( A ^ ( 2 x. k ) ) <_ ( ( 2 x. k ) x. B ) ) )
70	30, 65, 69	sylc 60	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) <_ ( ( 2 x. k ) x. B ) )
71	25, 33, 36, 62, 70	ltletrd 9527	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) < ( ( 2 x. k ) x. B ) )
72	21	recnd 9408	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A - 1 ) e. CC )
73	18	recnd 9408	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. CC )
74	72, 73, 56	mul32d 9575	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. ( A ^ k ) ) x. k ) = ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) )
75		2cnd 10390	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 2 e. CC )
76	35	recnd 9408	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> B e. CC )
77	75, 76, 56	mul32d 9575	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. B ) x. k ) = ( ( 2 x. k ) x. B ) )
78	74, 77	breq12d 4302	. . . . . . . 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 ) ) )
79	71, 78	mpbird 232	. . . . . . 7 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. ( A ^ k ) ) x. k ) < ( ( 2 x. B ) x. k ) )
80	21, 18	remulcld 9410	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. ( A ^ k ) ) e. RR )
81	5	adantr 462	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. B ) e. RR )
82		nngt0 10347	. . . . . . . . 9 |- ( k e. NN -> 0 < k )
83	82	adantl 463	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 < k )
84		ltmul1 10175	. . . . . . . 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 ) ) )
85	80, 81, 23, 83, 84	syl112anc 1217	. . . . . . 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 ) ) )
86	79, 85	mpbird 232	. . . . . 6 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. ( A ^ k ) ) < ( 2 x. B ) )
87	12	rpgt0d 11026	. . . . . . . 8 |- ( ( ph /\ 1 < A ) -> 0 < ( A - 1 ) )
88	87	adantr 462	. . . . . . 7 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 < ( A - 1 ) )
89		ltmuldiv2 10199	. . . . . . 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 ) ) ) )
90	18, 81, 21, 88, 89	syl112anc 1217	. . . . . 6 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. ( A ^ k ) ) < ( 2 x. B ) <-> ( A ^ k ) < ( ( 2 x. B ) / ( A - 1 ) ) ) )
91	86, 90	mpbid 210	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) < ( ( 2 x. B ) / ( A - 1 ) ) )
92	18, 19, 91	ltnsymd 9519	. . . 4 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> -. ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
93	92	nrexdv 2817	. . 3 |- ( ( ph /\ 1 < A ) -> -. E. k e. NN ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
94	15, 93	pm2.65da 573	. 2 |- ( ph -> -. 1 < A )
95		lenlt 9449	. . 3 |- ( ( A e. RR /\ 1 e. RR ) -> ( A <_ 1 <-> -. 1 < A ) )
96	8, 7, 95	sylancl 657	. 2 |- ( ph -> ( A <_ 1 <-> -. 1 < A ) )
97	94, 96	mpbird 232	1 |- ( ph -> A <_ 1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1364   e. wcel 1761   A.wral 2713   E.wrex 2714   class class class wbr 4289  (class class class)co 6090   RRcr 9277   0cc0 9278   1c1 9279   + caddc 9281   x. cmul 9283   < clt 9414   <_ cle 9415   - cmin 9591   / cdiv 9989   NNcn 10318   2c2 10367   NN0cn0 10575   ZZcz 10642   RR+crp 10987   ^cexp 11861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fl 11638  df-seq 11803  df-exp 11862

This theorem is referenced by:  ostth2lem4  22844