Theorem ostth2lem1 22983

Description: Lemma for ostth2 23002, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 23002. 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 10492	. . . . . 6 |- 2 e. RR
2		ostth2lem1.2	. . . . . . 7 |- ( ph -> B e. RR )
3	2	adantr 465	. . . . . 6 |- ( ( ph /\ 1 < A ) -> B e. RR )
4		remulcl 9468	. . . . . 6 |- ( ( 2 e. RR /\ B e. RR ) -> ( 2 x. B ) e. RR )
5	1, 3, 4	sylancr 663	. . . . 5 |- ( ( ph /\ 1 < A ) -> ( 2 x. B ) e. RR )
6		simpr 461	. . . . . 6 |- ( ( ph /\ 1 < A ) -> 1 < A )
7		1re 9486	. . . . . . 7 |- 1 e. RR
8		ostth2lem1.1	. . . . . . . 8 |- ( ph -> A e. RR )
9	8	adantr 465	. . . . . . 7 |- ( ( ph /\ 1 < A ) -> A e. RR )
10		difrp 11125	. . . . . . 7 |- ( ( 1 e. RR /\ A e. RR ) -> ( 1 < A <-> ( A - 1 ) e. RR+ ) )
11	7, 9, 10	sylancr 663	. . . . . 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 11155	. . . 4 |- ( ( ph /\ 1 < A ) -> ( ( 2 x. B ) / ( A - 1 ) ) e. RR )
14		expnbnd 12094	. . . 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 1219	. . 3 |- ( ( ph /\ 1 < A ) -> E. k e. NN ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
16		nnnn0 10687	. . . . . 6 |- ( k e. NN -> k e. NN0 )
17		reexpcl 11983	. . . . . 6 |- ( ( A e. RR /\ k e. NN0 ) -> ( A ^ k ) e. RR )
18	9, 16, 17	syl2an 477	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. RR )
19	13	adantr 465	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. B ) / ( A - 1 ) ) e. RR )
20	12	rpred 11128	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> ( A - 1 ) e. RR )
21	20	adantr 465	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A - 1 ) e. RR )
22		nnre 10430	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
23	22	adantl 466	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. RR )
24	21, 23	remulcld 9515	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) e. RR )
25	24, 18	remulcld 9515	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) e. RR )
26	8	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. RR )
27		2nn 10580	. . . . . . . . . . . 12 |- 2 e. NN
28		simpr 461	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. NN )
29		nnmulcl 10446	. . . . . . . . . . . 12 |- ( ( 2 e. NN /\ k e. NN ) -> ( 2 x. k ) e. NN )
30	27, 28, 29	sylancr 663	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) e. NN )
31		nnnn0 10687	. . . . . . . . . . 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 12126	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) e. RR )
34	30	nnred 10438	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) e. RR )
35	2	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> B e. RR )
36	34, 35	remulcld 9515	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. k ) x. B ) e. RR )
37		0red 9488	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 0 e. RR )
38	7	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 1 e. RR )
39		0lt1 9963	. . . . . . . . . . . . . . 15 |- 0 < 1
40	39	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 0 < 1 )
41	37, 38, 9, 40, 6	lttrd 9633	. . . . . . . . . . . . 13 |- ( ( ph /\ 1 < A ) -> 0 < A )
42	9, 41	elrpd 11126	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> A e. RR+ )
43		nnz 10769	. . . . . . . . . . . 12 |- ( k e. NN -> k e. ZZ )
44		rpexpcl 11985	. . . . . . . . . . . 12 |- ( ( A e. RR+ /\ k e. ZZ ) -> ( A ^ k ) e. RR+ )
45	42, 43, 44	syl2an 477	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. RR+ )
46		peano2re 9643	. . . . . . . . . . . . 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 10364	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) < ( ( ( A - 1 ) x. k ) + 1 ) )
49	16	adantl 466	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. NN0 )
50	42	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. RR+ )
51	50	rpge0d 11132	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 <_ A )
52		bernneq2 12092	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ k e. NN0 /\ 0 <_ A ) -> ( ( ( A - 1 ) x. k ) + 1 ) <_ ( A ^ k ) )
53	26, 49, 51, 52	syl3anc 1219	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) + 1 ) <_ ( A ^ k ) )
54	24, 47, 18, 48, 53	ltletrd 9632	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) < ( A ^ k ) )
55	24, 18, 45, 54	ltmul1dd 11179	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) < ( ( A ^ k ) x. ( A ^ k ) ) )
56	23	recnd 9513	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. CC )
57	56	2timesd 10668	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) = ( k + k ) )
58	57	oveq2d 6206	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) = ( A ^ ( k + k ) ) )
59	26	recnd 9513	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. CC )
60	59, 49, 49	expaddd 12111	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( k + k ) ) = ( ( A ^ k ) x. ( A ^ k ) ) )
61	58, 60	eqtrd 2492	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) = ( ( A ^ k ) x. ( A ^ k ) ) )
62	55, 61	breqtrrd 4416	. . . . . . . . 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 2822	. . . . . . . . . . 11 |- ( ph -> A. n e. NN ( A ^ n ) <_ ( n x. B ) )
65	64	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A. n e. NN ( A ^ n ) <_ ( n x. B ) )
66		oveq2 6198	. . . . . . . . . . . 12 |- ( n = ( 2 x. k ) -> ( A ^ n ) = ( A ^ ( 2 x. k ) ) )
67		oveq1 6197	. . . . . . . . . . . 12 |- ( n = ( 2 x. k ) -> ( n x. B ) = ( ( 2 x. k ) x. B ) )
68	66, 67	breq12d 4403	. . . . . . . . . . 11 |- ( n = ( 2 x. k ) -> ( ( A ^ n ) <_ ( n x. B ) <-> ( A ^ ( 2 x. k ) ) <_ ( ( 2 x. k ) x. B ) ) )
69	68	rspcv 3165	. . . . . . . . . 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 9632	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) < ( ( 2 x. k ) x. B ) )
72	21	recnd 9513	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A - 1 ) e. CC )
73	18	recnd 9513	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. CC )
74	72, 73, 56	mul32d 9680	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. ( A ^ k ) ) x. k ) = ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) )
75		2cnd 10495	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 2 e. CC )
76	35	recnd 9513	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> B e. CC )
77	75, 76, 56	mul32d 9680	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. B ) x. k ) = ( ( 2 x. k ) x. B ) )
78	74, 77	breq12d 4403	. . . . . . . 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 9515	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. ( A ^ k ) ) e. RR )
81	5	adantr 465	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. B ) e. RR )
82		nngt0 10452	. . . . . . . . 9 |- ( k e. NN -> 0 < k )
83	82	adantl 466	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 < k )
84		ltmul1 10280	. . . . . . . 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 1223	. . . . . . 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 11131	. . . . . . . 8 |- ( ( ph /\ 1 < A ) -> 0 < ( A - 1 ) )
88	87	adantr 465	. . . . . . 7 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 < ( A - 1 ) )
89		ltmuldiv2 10304	. . . . . . 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 1223	. . . . . 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 9624	. . . 4 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> -. ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
93	92	nrexdv 2915	. . 3 |- ( ( ph /\ 1 < A ) -> -. E. k e. NN ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
94	15, 93	pm2.65da 576	. 2 |- ( ph -> -. 1 < A )
95		lenlt 9554	. . 3 |- ( ( A e. RR /\ 1 e. RR ) -> ( A <_ 1 <-> -. 1 < A ) )
96	8, 7, 95	sylancl 662	. 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 1370   e. wcel 1758   A.wral 2795   E.wrex 2796   class class class wbr 4390  (class class class)co 6190   RRcr 9382   0cc0 9383   1c1 9384   + caddc 9386   x. cmul 9388   < clt 9519   <_ cle 9520   - cmin 9696   / cdiv 10094   NNcn 10423   2c2 10472   NN0cn0 10680   ZZcz 10747   RR+crp 11092   ^cexp 11966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-fl 11743  df-seq 11908  df-exp 11967

This theorem is referenced by:  ostth2lem4  23001