Theorem ostth2lem1 25307

Description: Lemma for ostth2 25326, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 25326. 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 11090	. . . . . 6 |- 2 e. RR
2		ostth2lem1.2	. . . . . . 7 |- ( ph -> B e. RR )
3	2	adantr 481	. . . . . 6 |- ( ( ph /\ 1 < A ) -> B e. RR )
4		remulcl 10021	. . . . . 6 |- ( ( 2 e. RR /\ B e. RR ) -> ( 2 x. B ) e. RR )
5	1, 3, 4	sylancr 695	. . . . 5 |- ( ( ph /\ 1 < A ) -> ( 2 x. B ) e. RR )
6		simpr 477	. . . . . 6 |- ( ( ph /\ 1 < A ) -> 1 < A )
7		1re 10039	. . . . . . 7 |- 1 e. RR
8		ostth2lem1.1	. . . . . . . 8 |- ( ph -> A e. RR )
9	8	adantr 481	. . . . . . 7 |- ( ( ph /\ 1 < A ) -> A e. RR )
10		difrp 11868	. . . . . . 7 |- ( ( 1 e. RR /\ A e. RR ) -> ( 1 < A <-> ( A - 1 ) e. RR+ ) )
11	7, 9, 10	sylancr 695	. . . . . 6 |- ( ( ph /\ 1 < A ) -> ( 1 < A <-> ( A - 1 ) e. RR+ ) )
12	6, 11	mpbid 222	. . . . 5 |- ( ( ph /\ 1 < A ) -> ( A - 1 ) e. RR+ )
13	5, 12	rerpdivcld 11903	. . . 4 |- ( ( ph /\ 1 < A ) -> ( ( 2 x. B ) / ( A - 1 ) ) e. RR )
14		expnbnd 12993	. . . 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 1326	. . 3 |- ( ( ph /\ 1 < A ) -> E. k e. NN ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
16		nnnn0 11299	. . . . . 6 |- ( k e. NN -> k e. NN0 )
17		reexpcl 12877	. . . . . 6 |- ( ( A e. RR /\ k e. NN0 ) -> ( A ^ k ) e. RR )
18	9, 16, 17	syl2an 494	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. RR )
19	13	adantr 481	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. B ) / ( A - 1 ) ) e. RR )
20	12	rpred 11872	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> ( A - 1 ) e. RR )
21	20	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A - 1 ) e. RR )
22		nnre 11027	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
23	22	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. RR )
24	21, 23	remulcld 10070	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) e. RR )
25	24, 18	remulcld 10070	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) e. RR )
26	8	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. RR )
27		2nn 11185	. . . . . . . . . . . 12 |- 2 e. NN
28		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. NN )
29		nnmulcl 11043	. . . . . . . . . . . 12 |- ( ( 2 e. NN /\ k e. NN ) -> ( 2 x. k ) e. NN )
30	27, 28, 29	sylancr 695	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) e. NN )
31		nnnn0 11299	. . . . . . . . . . 11 |- ( ( 2 x. k ) e. NN -> ( 2 x. k ) e. NN0 )
32	30, 31	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) e. NN0 )
33	26, 32	reexpcld 13025	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) e. RR )
34	30	nnred 11035	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) e. RR )
35	2	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> B e. RR )
36	34, 35	remulcld 10070	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. k ) x. B ) e. RR )
37		0red 10041	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 0 e. RR )
38	7	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 1 e. RR )
39		0lt1 10550	. . . . . . . . . . . . . . 15 |- 0 < 1
40	39	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 0 < 1 )
41	37, 38, 9, 40, 6	lttrd 10198	. . . . . . . . . . . . 13 |- ( ( ph /\ 1 < A ) -> 0 < A )
42	9, 41	elrpd 11869	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> A e. RR+ )
43		nnz 11399	. . . . . . . . . . . 12 |- ( k e. NN -> k e. ZZ )
44		rpexpcl 12879	. . . . . . . . . . . 12 |- ( ( A e. RR+ /\ k e. ZZ ) -> ( A ^ k ) e. RR+ )
45	42, 43, 44	syl2an 494	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. RR+ )
46		peano2re 10209	. . . . . . . . . . . . 13 |- ( ( ( A - 1 ) x. k ) e. RR -> ( ( ( A - 1 ) x. k ) + 1 ) e. RR )
47	24, 46	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) + 1 ) e. RR )
48	24	ltp1d 10954	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) < ( ( ( A - 1 ) x. k ) + 1 ) )
49	16	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. NN0 )
50	42	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. RR+ )
51	50	rpge0d 11876	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 <_ A )
52		bernneq2 12991	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ k e. NN0 /\ 0 <_ A ) -> ( ( ( A - 1 ) x. k ) + 1 ) <_ ( A ^ k ) )
53	26, 49, 51, 52	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) + 1 ) <_ ( A ^ k ) )
54	24, 47, 18, 48, 53	ltletrd 10197	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) < ( A ^ k ) )
55	24, 18, 45, 54	ltmul1dd 11927	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) < ( ( A ^ k ) x. ( A ^ k ) ) )
56	23	recnd 10068	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. CC )
57	56	2timesd 11275	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) = ( k + k ) )
58	57	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) = ( A ^ ( k + k ) ) )
59	26	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. CC )
60	59, 49, 49	expaddd 13010	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( k + k ) ) = ( ( A ^ k ) x. ( A ^ k ) ) )
61	58, 60	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) = ( ( A ^ k ) x. ( A ^ k ) ) )
62	55, 61	breqtrrd 4681	. . . . . . . . 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 2966	. . . . . . . . . . 11 |- ( ph -> A. n e. NN ( A ^ n ) <_ ( n x. B ) )
65	64	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A. n e. NN ( A ^ n ) <_ ( n x. B ) )
66		oveq2 6658	. . . . . . . . . . . 12 |- ( n = ( 2 x. k ) -> ( A ^ n ) = ( A ^ ( 2 x. k ) ) )
67		oveq1 6657	. . . . . . . . . . . 12 |- ( n = ( 2 x. k ) -> ( n x. B ) = ( ( 2 x. k ) x. B ) )
68	66, 67	breq12d 4666	. . . . . . . . . . 11 |- ( n = ( 2 x. k ) -> ( ( A ^ n ) <_ ( n x. B ) <-> ( A ^ ( 2 x. k ) ) <_ ( ( 2 x. k ) x. B ) ) )
69	68	rspcv 3305	. . . . . . . . . 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 65	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) <_ ( ( 2 x. k ) x. B ) )
71	25, 33, 36, 62, 70	ltletrd 10197	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) < ( ( 2 x. k ) x. B ) )
72	21	recnd 10068	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A - 1 ) e. CC )
73	18	recnd 10068	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. CC )
74	72, 73, 56	mul32d 10246	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. ( A ^ k ) ) x. k ) = ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) )
75		2cnd 11093	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 2 e. CC )
76	35	recnd 10068	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> B e. CC )
77	75, 76, 56	mul32d 10246	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. B ) x. k ) = ( ( 2 x. k ) x. B ) )
78	71, 74, 77	3brtr4d 4685	. . . . . . 7 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. ( A ^ k ) ) x. k ) < ( ( 2 x. B ) x. k ) )
79	21, 18	remulcld 10070	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. ( A ^ k ) ) e. RR )
80	5	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. B ) e. RR )
81		nngt0 11049	. . . . . . . . 9 |- ( k e. NN -> 0 < k )
82	81	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 < k )
83		ltmul1 10873	. . . . . . . 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 ) ) )
84	79, 80, 23, 82, 83	syl112anc 1330	. . . . . . 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 ) ) )
85	78, 84	mpbird 247	. . . . . 6 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. ( A ^ k ) ) < ( 2 x. B ) )
86	12	rpgt0d 11875	. . . . . . . 8 |- ( ( ph /\ 1 < A ) -> 0 < ( A - 1 ) )
87	86	adantr 481	. . . . . . 7 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 < ( A - 1 ) )
88		ltmuldiv2 10897	. . . . . . 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 ) ) ) )
89	18, 80, 21, 87, 88	syl112anc 1330	. . . . . 6 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. ( A ^ k ) ) < ( 2 x. B ) <-> ( A ^ k ) < ( ( 2 x. B ) / ( A - 1 ) ) ) )
90	85, 89	mpbid 222	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) < ( ( 2 x. B ) / ( A - 1 ) ) )
91	18, 19, 90	ltnsymd 10186	. . . 4 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> -. ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
92	91	nrexdv 3001	. . 3 |- ( ( ph /\ 1 < A ) -> -. E. k e. NN ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
93	15, 92	pm2.65da 600	. 2 |- ( ph -> -. 1 < A )
94		lenlt 10116	. . 3 |- ( ( A e. RR /\ 1 e. RR ) -> ( A <_ 1 <-> -. 1 < A ) )
95	8, 7, 94	sylancl 694	. 2 |- ( ph -> ( A <_ 1 <-> -. 1 < A ) )
96	93, 95	mpbird 247	1 |- ( ph -> A <_ 1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   RR+crp 11832   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-seq 12802  df-exp 12861

This theorem is referenced by:  ostth2lem4  25325