Theorem expnbnd 9372

Description: Exponentiation with a mantissa greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.)

Assertion

Ref	Expression
expnbnd	|- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> E. k e. NN A < ( B ^ k ) )

Distinct variable groups:   A, k   B, k

Proof of Theorem expnbnd

Step	Hyp	Ref	Expression
1		simp1 904	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> A e. RR )
2	1	adantr 261	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> A e. RR )
3		simp2 905	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> B e. RR )
4	3	adantr 261	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> B e. RR )
5		simpr 103	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> 1 < A )
6		simp3 906	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> 1 < B )
7	6	adantr 261	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> 1 < B )
8		1red 7042	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> 1 e. RR )
9	1, 8	resubcld 7379	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( A - 1 ) e. RR )
10	3, 8	resubcld 7379	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( B - 1 ) e. RR )
11	8, 3	posdifd 7523	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( 1 < B <-> 0 < ( B - 1 ) ) )
12	6, 11	mpbid 135	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> 0 < ( B - 1 ) )
13	10, 12	gt0ap0d 7619	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( B - 1 ) # 0 )
14	9, 10, 13	redivclapd 7808	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( ( A - 1 ) / ( B - 1 ) ) e. RR )
15		arch 8178	. . . . . . 7 |- ( ( ( A - 1 ) / ( B - 1 ) ) e. RR -> E. k e. NN ( ( A - 1 ) / ( B - 1 ) ) < k )
16	14, 15	syl 14	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> E. k e. NN ( ( A - 1 ) / ( B - 1 ) ) < k )
17	16	3expa 1104	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ 1 < B ) -> E. k e. NN ( ( A - 1 ) / ( B - 1 ) ) < k )
18	17	adantrl 447	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> E. k e. NN ( ( A - 1 ) / ( B - 1 ) ) < k )
19		simplll 485	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> A e. RR )
20	19	adantr 261	. . . . . . 7 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> A e. RR )
21		simpllr 486	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> B e. RR )
22		1red 7042	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 1 e. RR )
23	21, 22	resubcld 7379	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( B - 1 ) e. RR )
24		simpr 103	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. NN )
25	24	nnred 7927	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. RR )
26	23, 25	remulcld 7056	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( ( B - 1 ) x. k ) e. RR )
27	26, 22	readdcld 7055	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( ( ( B - 1 ) x. k ) + 1 ) e. RR )
28	27	adantr 261	. . . . . . 7 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> ( ( ( B - 1 ) x. k ) + 1 ) e. RR )
29	24	nnnn0d 8235	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. NN0 )
30		reexpcl 9272	. . . . . . . . 9 |- ( ( B e. RR /\ k e. NN0 ) -> ( B ^ k ) e. RR )
31	21, 29, 30	syl2anc 391	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( B ^ k ) e. RR )
32	31	adantr 261	. . . . . . 7 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> ( B ^ k ) e. RR )
33		simpr 103	. . . . . . . . 9 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> ( ( A - 1 ) / ( B - 1 ) ) < k )
34		1red 7042	. . . . . . . . . . 11 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> 1 e. RR )
35	20, 34	resubcld 7379	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> ( A - 1 ) e. RR )
36		simplr 482	. . . . . . . . . . 11 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> k e. NN )
37	36	nnred 7927	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> k e. RR )
38	21	adantr 261	. . . . . . . . . . 11 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> B e. RR )
39	38, 34	resubcld 7379	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> ( B - 1 ) e. RR )
40		simplrr 488	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 1 < B )
41	40	adantr 261	. . . . . . . . . . 11 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> 1 < B )
42	34, 38	posdifd 7523	. . . . . . . . . . 11 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> ( 1 < B <-> 0 < ( B - 1 ) ) )
43	41, 42	mpbid 135	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> 0 < ( B - 1 ) )
44		ltdivmul 7842	. . . . . . . . . 10 |- ( ( ( A - 1 ) e. RR /\ k e. RR /\ ( ( B - 1 ) e. RR /\ 0 < ( B - 1 ) ) ) -> ( ( ( A - 1 ) / ( B - 1 ) ) < k <-> ( A - 1 ) < ( ( B - 1 ) x. k ) ) )
45	35, 37, 39, 43, 44	syl112anc 1139	. . . . . . . . 9 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> ( ( ( A - 1 ) / ( B - 1 ) ) < k <-> ( A - 1 ) < ( ( B - 1 ) x. k ) ) )
46	33, 45	mpbid 135	. . . . . . . 8 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> ( A - 1 ) < ( ( B - 1 ) x. k ) )
47	39, 37	remulcld 7056	. . . . . . . . 9 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> ( ( B - 1 ) x. k ) e. RR )
48	20, 34, 47	ltsubaddd 7532	. . . . . . . 8 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> ( ( A - 1 ) < ( ( B - 1 ) x. k ) <-> A < ( ( ( B - 1 ) x. k ) + 1 ) ) )
49	46, 48	mpbid 135	. . . . . . 7 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> A < ( ( ( B - 1 ) x. k ) + 1 ) )
50	36	nnnn0d 8235	. . . . . . . 8 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> k e. NN0 )
51		0red 7028	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 0 e. RR )
52		0lt1 7141	. . . . . . . . . . . 12 |- 0 < 1
53		0re 7027	. . . . . . . . . . . . 13 |- 0 e. RR
54		1re 7026	. . . . . . . . . . . . 13 |- 1 e. RR
55		lttr 7092	. . . . . . . . . . . . 13 |- ( ( 0 e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
56	53, 54, 55	mp3an12 1222	. . . . . . . . . . . 12 |- ( B e. RR -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
57	52, 56	mpani 406	. . . . . . . . . . 11 |- ( B e. RR -> ( 1 < B -> 0 < B ) )
58	21, 40, 57	sylc 56	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 0 < B )
59	51, 21, 58	ltled 7135	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 0 <_ B )
60	59	adantr 261	. . . . . . . 8 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> 0 <_ B )
61		bernneq2 9370	. . . . . . . 8 |- ( ( B e. RR /\ k e. NN0 /\ 0 <_ B ) -> ( ( ( B - 1 ) x. k ) + 1 ) <_ ( B ^ k ) )
62	38, 50, 60, 61	syl3anc 1135	. . . . . . 7 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> ( ( ( B - 1 ) x. k ) + 1 ) <_ ( B ^ k ) )
63	20, 28, 32, 49, 62	ltletrd 7420	. . . . . 6 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> A < ( B ^ k ) )
64	63	ex 108	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( ( ( A - 1 ) / ( B - 1 ) ) < k -> A < ( B ^ k ) ) )
65	64	reximdva 2421	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> ( E. k e. NN ( ( A - 1 ) / ( B - 1 ) ) < k -> E. k e. NN A < ( B ^ k ) ) )
66	18, 65	mpd 13	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> E. k e. NN A < ( B ^ k ) )
67	2, 4, 5, 7, 66	syl22anc 1136	. 2 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> E. k e. NN A < ( B ^ k ) )
68		1nn 7925	. . 3 |- 1 e. NN
69		simpr 103	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> A < B )
70		simpl2 908	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> B e. RR )
71	70	recnd 7054	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> B e. CC )
72		exp1 9261	. . . . 5 |- ( B e. CC -> ( B ^ 1 ) = B )
73	71, 72	syl 14	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> ( B ^ 1 ) = B )
74	69, 73	breqtrrd 3790	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> A < ( B ^ 1 ) )
75		oveq2 5520	. . . . 5 |- ( k = 1 -> ( B ^ k ) = ( B ^ 1 ) )
76	75	breq2d 3776	. . . 4 |- ( k = 1 -> ( A < ( B ^ k ) <-> A < ( B ^ 1 ) ) )
77	76	rspcev 2656	. . 3 |- ( ( 1 e. NN /\ A < ( B ^ 1 ) ) -> E. k e. NN A < ( B ^ k ) )
78	68, 74, 77	sylancr 393	. 2 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> E. k e. NN A < ( B ^ k ) )
79		axltwlin 7087	. . . . 5 |- ( ( 1 e. RR /\ B e. RR /\ A e. RR ) -> ( 1 < B -> ( 1 < A \/ A < B ) ) )
80	54, 79	mp3an1 1219	. . . 4 |- ( ( B e. RR /\ A e. RR ) -> ( 1 < B -> ( 1 < A \/ A < B ) ) )
81	80	ancoms 255	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < B -> ( 1 < A \/ A < B ) ) )
82	81	3impia 1101	. 2 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( 1 < A \/ A < B ) )
83	67, 78, 82	mpjaodan 711	1 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> E. k e. NN A < ( B ^ k ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   /\ w3a 885   = wceq 1243   e. wcel 1393   E.wrex 2307   class class class wbr 3764  (class class class)co 5512   CCcc 6887   RRcr 6888   0cc0 6889   1c1 6890   + caddc 6892   x. cmul 6894   < clt 7060   <_ cle 7061   - cmin 7182   / cdiv 7651   NNcn 7914   NN0cn0 8181   ^cexp 9254

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 7002  ax-arch 7003

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-div 7652  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-iseq 9212  df-iexp 9255

This theorem is referenced by:  expnlbnd  9373