Theorem ostth2lem1 24001

Description: Lemma for ostth2 24020, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 24020. 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 10601	. . . . . 6 |- 2 e. RR
2		ostth2lem1.2	. . . . . . 7 |- ( ph -> B e. RR )
3	2	adantr 463	. . . . . 6 |- ( ( ph /\ 1 < A ) -> B e. RR )
4		remulcl 9566	. . . . . 6 |- ( ( 2 e. RR /\ B e. RR ) -> ( 2 x. B ) e. RR )
5	1, 3, 4	sylancr 661	. . . . 5 |- ( ( ph /\ 1 < A ) -> ( 2 x. B ) e. RR )
6		simpr 459	. . . . . 6 |- ( ( ph /\ 1 < A ) -> 1 < A )
7		1re 9584	. . . . . . 7 |- 1 e. RR
8		ostth2lem1.1	. . . . . . . 8 |- ( ph -> A e. RR )
9	8	adantr 463	. . . . . . 7 |- ( ( ph /\ 1 < A ) -> A e. RR )
10		difrp 11255	. . . . . . 7 |- ( ( 1 e. RR /\ A e. RR ) -> ( 1 < A <-> ( A - 1 ) e. RR+ ) )
11	7, 9, 10	sylancr 661	. . . . . 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 11286	. . . 4 |- ( ( ph /\ 1 < A ) -> ( ( 2 x. B ) / ( A - 1 ) ) e. RR )
14		expnbnd 12277	. . . 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 1226	. . 3 |- ( ( ph /\ 1 < A ) -> E. k e. NN ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
16		nnnn0 10798	. . . . . 6 |- ( k e. NN -> k e. NN0 )
17		reexpcl 12165	. . . . . 6 |- ( ( A e. RR /\ k e. NN0 ) -> ( A ^ k ) e. RR )
18	9, 16, 17	syl2an 475	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. RR )
19	13	adantr 463	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. B ) / ( A - 1 ) ) e. RR )
20	12	rpred 11259	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> ( A - 1 ) e. RR )
21	20	adantr 463	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A - 1 ) e. RR )
22		nnre 10538	. . . . . . . . . . . 12 |- ( k e. NN -> k e. RR )
23	22	adantl 464	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. RR )
24	21, 23	remulcld 9613	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) e. RR )
25	24, 18	remulcld 9613	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) e. RR )
26	8	ad2antrr 723	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. RR )
27		2nn 10689	. . . . . . . . . . . 12 |- 2 e. NN
28		simpr 459	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. NN )
29		nnmulcl 10554	. . . . . . . . . . . 12 |- ( ( 2 e. NN /\ k e. NN ) -> ( 2 x. k ) e. NN )
30	27, 28, 29	sylancr 661	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) e. NN )
31		nnnn0 10798	. . . . . . . . . . 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 12309	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) e. RR )
34	30	nnred 10546	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) e. RR )
35	2	ad2antrr 723	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> B e. RR )
36	34, 35	remulcld 9613	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. k ) x. B ) e. RR )
37		0red 9586	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 0 e. RR )
38	7	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 1 e. RR )
39		0lt1 10071	. . . . . . . . . . . . . . 15 |- 0 < 1
40	39	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < A ) -> 0 < 1 )
41	37, 38, 9, 40, 6	lttrd 9732	. . . . . . . . . . . . 13 |- ( ( ph /\ 1 < A ) -> 0 < A )
42	9, 41	elrpd 11256	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> A e. RR+ )
43		nnz 10882	. . . . . . . . . . . 12 |- ( k e. NN -> k e. ZZ )
44		rpexpcl 12167	. . . . . . . . . . . 12 |- ( ( A e. RR+ /\ k e. ZZ ) -> ( A ^ k ) e. RR+ )
45	42, 43, 44	syl2an 475	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. RR+ )
46		peano2re 9742	. . . . . . . . . . . . 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 10471	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) < ( ( ( A - 1 ) x. k ) + 1 ) )
49	16	adantl 464	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. NN0 )
50	42	adantr 463	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. RR+ )
51	50	rpge0d 11263	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 <_ A )
52		bernneq2 12275	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ k e. NN0 /\ 0 <_ A ) -> ( ( ( A - 1 ) x. k ) + 1 ) <_ ( A ^ k ) )
53	26, 49, 51, 52	syl3anc 1226	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) + 1 ) <_ ( A ^ k ) )
54	24, 47, 18, 48, 53	ltletrd 9731	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. k ) < ( A ^ k ) )
55	24, 18, 45, 54	ltmul1dd 11310	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) < ( ( A ^ k ) x. ( A ^ k ) ) )
56	23	recnd 9611	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> k e. CC )
57	56	2timesd 10777	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. k ) = ( k + k ) )
58	57	oveq2d 6286	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) = ( A ^ ( k + k ) ) )
59	26	recnd 9611	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A e. CC )
60	59, 49, 49	expaddd 12294	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( k + k ) ) = ( ( A ^ k ) x. ( A ^ k ) ) )
61	58, 60	eqtrd 2495	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ ( 2 x. k ) ) = ( ( A ^ k ) x. ( A ^ k ) ) )
62	55, 61	breqtrrd 4465	. . . . . . . . 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 2868	. . . . . . . . . . 11 |- ( ph -> A. n e. NN ( A ^ n ) <_ ( n x. B ) )
65	64	ad2antrr 723	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> A. n e. NN ( A ^ n ) <_ ( n x. B ) )
66		oveq2 6278	. . . . . . . . . . . 12 |- ( n = ( 2 x. k ) -> ( A ^ n ) = ( A ^ ( 2 x. k ) ) )
67		oveq1 6277	. . . . . . . . . . . 12 |- ( n = ( 2 x. k ) -> ( n x. B ) = ( ( 2 x. k ) x. B ) )
68	66, 67	breq12d 4452	. . . . . . . . . . 11 |- ( n = ( 2 x. k ) -> ( ( A ^ n ) <_ ( n x. B ) <-> ( A ^ ( 2 x. k ) ) <_ ( ( 2 x. k ) x. B ) ) )
69	68	rspcv 3203	. . . . . . . . . 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 9731	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) < ( ( 2 x. k ) x. B ) )
72	21	recnd 9611	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A - 1 ) e. CC )
73	18	recnd 9611	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) e. CC )
74	72, 73, 56	mul32d 9779	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. ( A ^ k ) ) x. k ) = ( ( ( A - 1 ) x. k ) x. ( A ^ k ) ) )
75		2cnd 10604	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 2 e. CC )
76	35	recnd 9611	. . . . . . . . 9 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> B e. CC )
77	75, 76, 56	mul32d 9779	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( 2 x. B ) x. k ) = ( ( 2 x. k ) x. B ) )
78	71, 74, 77	3brtr4d 4469	. . . . . . 7 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( ( A - 1 ) x. ( A ^ k ) ) x. k ) < ( ( 2 x. B ) x. k ) )
79	21, 18	remulcld 9613	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. ( A ^ k ) ) e. RR )
80	5	adantr 463	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( 2 x. B ) e. RR )
81		nngt0 10560	. . . . . . . . 9 |- ( k e. NN -> 0 < k )
82	81	adantl 464	. . . . . . . 8 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 < k )
83		ltmul1 10388	. . . . . . . 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 1230	. . . . . . 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 232	. . . . . 6 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( ( A - 1 ) x. ( A ^ k ) ) < ( 2 x. B ) )
86	12	rpgt0d 11262	. . . . . . . 8 |- ( ( ph /\ 1 < A ) -> 0 < ( A - 1 ) )
87	86	adantr 463	. . . . . . 7 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> 0 < ( A - 1 ) )
88		ltmuldiv2 10412	. . . . . . 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 1230	. . . . . 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 210	. . . . 5 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> ( A ^ k ) < ( ( 2 x. B ) / ( A - 1 ) ) )
91	18, 19, 90	ltnsymd 9723	. . . 4 |- ( ( ( ph /\ 1 < A ) /\ k e. NN ) -> -. ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
92	91	nrexdv 2910	. . 3 |- ( ( ph /\ 1 < A ) -> -. E. k e. NN ( ( 2 x. B ) / ( A - 1 ) ) < ( A ^ k ) )
93	15, 92	pm2.65da 574	. 2 |- ( ph -> -. 1 < A )
94		lenlt 9652	. . 3 |- ( ( A e. RR /\ 1 e. RR ) -> ( A <_ 1 <-> -. 1 < A ) )
95	8, 7, 94	sylancl 660	. 2 |- ( ph -> ( A <_ 1 <-> -. 1 < A ) )
96	93, 95	mpbird 232	1 |- ( ph -> A <_ 1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   A.wral 2804   E.wrex 2805   class class class wbr 4439  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482   + caddc 9484   x. cmul 9486   < clt 9617   <_ cle 9618   - cmin 9796   / cdiv 10202   NNcn 10531   2c2 10581   NN0cn0 10791   ZZcz 10860   RR+crp 11221   ^cexp 12148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fl 11910  df-seq 12090  df-exp 12149

This theorem is referenced by:  ostth2lem4  24019