Theorem ostth2lem1 24456

Description: Lemma for ostth2 24475, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 24475. 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 10679	. . . . . 6 |- 2 e. RR
2		ostth2lem1.2	. . . . . . 7 |- ( ph -> B e. RR )
3	2	adantr 467	. . . . . 6 |- ( ( ph /\ 1 < A ) -> B e. RR )
4		remulcl 9624	. . . . . 6 |- ( ( 2 e. RR /\ B e. RR ) -> ( 2 x. B ) e. RR )
5	1, 3, 4	sylancr 669	. . . . 5 |- ( ( ph /\ 1 < A ) -> ( 2 x. B ) e. RR )
6		simpr 463	. . . . . 6 |- ( ( ph /\ 1 < A ) -> 1 < A )
7		1re 9642	. . . . . . 7 |- 1 e. RR
8		ostth2lem1.1	. . . . . . . 8 |- ( ph -> A e. RR )
9	8	adantr 467	. . . . . . 7 |- ( ( ph /\ 1 < A ) -> A e. RR )
10		difrp 11337	. . . . . . 7 |- ( ( 1 e. RR /\ A e. RR ) -> ( 1 < A <-> ( A - 1 ) e. RR+ ) )
11	7, 9, 10	sylancr 669	. . . . . 6 |- ( ( ph /\ 1 < A ) -> ( 1 < A <-> ( A - 1 ) e. RR+ ) )
12	6, 11	mpbid 214	. . . . 5 |- ( ( ph /\ 1 < A ) -> ( A - 1 ) e. RR+ )
13	5, 12	rerpdivcld 11369	. . . 4 |- ( ( ph /\ 1 < A ) -> ( ( 2 x. B ) / ( A - 1 ) ) e. RR )
14		expnbnd 12401	. . . 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 1268	. . 3 |- ( ( ph /\ 1 < A ) -> E. k e. NN ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
16		nnnn0 10876	. . . . . 6 |- ( k e. NN -> k e. NN0 )
17		reexpcl 12289	. . . . . 6 |- ( ( A e. RR /\ k e. NN0 ) -> ( A ^ k ) e. RR )
18	9, 16, 17	syl2an 480	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. RR )
19	13	adantr 467	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. B ) / ( A - 1 ) ) e. RR )
20	12	rpred 11341	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> ( A - 1 ) e. RR )
21	20	adantr 467	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A - 1 ) e. RR )
22		nnre 10616	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
23	22	adantl 468	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. RR )
24	21, 23	remulcld 9671	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) e. RR )
25	24, 18	remulcld 9671	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) e. RR )
26	8	ad2antrr 732	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. RR )
27		2nn 10767	. . . . . . . . . . . 12 |- 2 e. NN
28		simpr 463	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. NN )
29		nnmulcl 10632	. . . . . . . . . . . 12 |- ( ( 2 e. NN /\ k e. NN ) -> ( 2 x. k ) e. NN )
30	27, 28, 29	sylancr 669	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) e. NN )
31		nnnn0 10876	. . . . . . . . . . 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 12433	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) e. RR )
34	30	nnred 10624	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) e. RR )
35	2	ad2antrr 732	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> B e. RR )
36	34, 35	remulcld 9671	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. k ) x. B ) e. RR )
37		0red 9644	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 0 e. RR )
38	7	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 1 e. RR )
39		0lt1 10136	. . . . . . . . . . . . . . 15 |- 0 < 1
40	39	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 0 < 1 )
41	37, 38, 9, 40, 6	lttrd 9796	. . . . . . . . . . . . 13 |- ( ( ph /\ 1 < A ) -> 0 < A )
42	9, 41	elrpd 11338	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> A e. RR+ )
43		nnz 10959	. . . . . . . . . . . 12 |- ( k e. NN -> k e. ZZ )
44		rpexpcl 12291	. . . . . . . . . . . 12 |- ( ( A e. RR+ /\ k e. ZZ ) -> ( A ^ k ) e. RR+ )
45	42, 43, 44	syl2an 480	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. RR+ )
46		peano2re 9806	. . . . . . . . . . . . 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 10537	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) < ( ( ( A - 1 ) x. k ) + 1 ) )
49	16	adantl 468	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. NN0 )
50	42	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. RR+ )
51	50	rpge0d 11345	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 <_ A )
52		bernneq2 12399	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ k e. NN0 /\ 0 <_ A ) -> ( ( ( A - 1 ) x. k ) + 1 ) <_ ( A ^ k ) )
53	26, 49, 51, 52	syl3anc 1268	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) + 1 ) <_ ( A ^ k ) )
54	24, 47, 18, 48, 53	ltletrd 9795	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) < ( A ^ k ) )
55	24, 18, 45, 54	ltmul1dd 11393	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) < ( ( A ^ k ) x. ( A ^ k ) ) )
56	23	recnd 9669	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. CC )
57	56	2timesd 10855	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) = ( k + k ) )
58	57	oveq2d 6306	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) = ( A ^ ( k + k ) ) )
59	26	recnd 9669	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. CC )
60	59, 49, 49	expaddd 12418	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( k + k ) ) = ( ( A ^ k ) x. ( A ^ k ) ) )
61	58, 60	eqtrd 2485	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) = ( ( A ^ k ) x. ( A ^ k ) ) )
62	55, 61	breqtrrd 4429	. . . . . . . . 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 2802	. . . . . . . . . . 11 |- ( ph -> A. n e. NN ( A ^ n ) <_ ( n x. B ) )
65	64	ad2antrr 732	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A. n e. NN ( A ^ n ) <_ ( n x. B ) )
66		oveq2 6298	. . . . . . . . . . . 12 |- ( n = ( 2 x. k ) -> ( A ^ n ) = ( A ^ ( 2 x. k ) ) )
67		oveq1 6297	. . . . . . . . . . . 12 |- ( n = ( 2 x. k ) -> ( n x. B ) = ( ( 2 x. k ) x. B ) )
68	66, 67	breq12d 4415	. . . . . . . . . . 11 |- ( n = ( 2 x. k ) -> ( ( A ^ n ) <_ ( n x. B ) <-> ( A ^ ( 2 x. k ) ) <_ ( ( 2 x. k ) x. B ) ) )
69	68	rspcv 3146	. . . . . . . . . 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 62	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) <_ ( ( 2 x. k ) x. B ) )
71	25, 33, 36, 62, 70	ltletrd 9795	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) < ( ( 2 x. k ) x. B ) )
72	21	recnd 9669	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A - 1 ) e. CC )
73	18	recnd 9669	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. CC )
74	72, 73, 56	mul32d 9843	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. ( A ^ k ) ) x. k ) = ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) )
75		2cnd 10682	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 2 e. CC )
76	35	recnd 9669	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> B e. CC )
77	75, 76, 56	mul32d 9843	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. B ) x. k ) = ( ( 2 x. k ) x. B ) )
78	71, 74, 77	3brtr4d 4433	. . . . . . 7 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. ( A ^ k ) ) x. k ) < ( ( 2 x. B ) x. k ) )
79	21, 18	remulcld 9671	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. ( A ^ k ) ) e. RR )
80	5	adantr 467	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. B ) e. RR )
81		nngt0 10638	. . . . . . . . 9 |- ( k e. NN -> 0 < k )
82	81	adantl 468	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 < k )
83		ltmul1 10455	. . . . . . . 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 1272	. . . . . . 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 236	. . . . . 6 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. ( A ^ k ) ) < ( 2 x. B ) )
86	12	rpgt0d 11344	. . . . . . . 8 |- ( ( ph /\ 1 < A ) -> 0 < ( A - 1 ) )
87	86	adantr 467	. . . . . . 7 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 < ( A - 1 ) )
88		ltmuldiv2 10479	. . . . . . 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 1272	. . . . . 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 214	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) < ( ( 2 x. B ) / ( A - 1 ) ) )
91	18, 19, 90	ltnsymd 9784	. . . 4 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> -. ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
92	91	nrexdv 2843	. . 3 |- ( ( ph /\ 1 < A ) -> -. E. k e. NN ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
93	15, 92	pm2.65da 580	. 2 |- ( ph -> -. 1 < A )
94		lenlt 9712	. . 3 |- ( ( A e. RR /\ 1 e. RR ) -> ( A <_ 1 <-> -. 1 < A ) )
95	8, 7, 94	sylancl 668	. 2 |- ( ph -> ( A <_ 1 <-> -. 1 < A ) )
96	93, 95	mpbird 236	1 |- ( ph -> A <_ 1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   A.wral 2737   E.wrex 2738   class class class wbr 4402  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   x. cmul 9544   < clt 9675   <_ cle 9676   - cmin 9860   / cdiv 10269   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   RR+crp 11302   ^cexp 12272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fl 12028  df-seq 12214  df-exp 12273

This theorem is referenced by:  ostth2lem4  24474