Theorem expnbnd 10806

Description: Exponentiation with a base 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 999	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> A e. RR )
2	1	adantr 276	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> A e. RR )
3		simp2 1000	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> B e. RR )
4	3	adantr 276	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> B e. RR )
5		simpr 110	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> 1 < A )
6		simp3 1001	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> 1 < B )
7	6	adantr 276	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> 1 < B )
8		1red 8086	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> 1 e. RR )
9	1, 8	resubcld 8452	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( A - 1 ) e. RR )
10	3, 8	resubcld 8452	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( B - 1 ) e. RR )
11	8, 3	posdifd 8604	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( 1 < B <-> 0 < ( B - 1 ) ) )
12	6, 11	mpbid 147	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> 0 < ( B - 1 ) )
13	10, 12	gt0ap0d 8701	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( B - 1 ) # 0 )
14	9, 10, 13	redivclapd 8907	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( ( A - 1 ) / ( B - 1 ) ) e. RR )
15		arch 9291	. . . . . . 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 1205	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ 1 < B ) -> E. k e. NN ( ( A - 1 ) / ( B - 1 ) ) < k )
18	17	adantrl 478	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> E. k e. NN ( ( A - 1 ) / ( B - 1 ) ) < k )
19		simplll 533	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> A e. RR )
20	19	adantr 276	. . . . . . 7 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> A e. RR )
21		simpllr 534	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> B e. RR )
22		1red 8086	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 1 e. RR )
23	21, 22	resubcld 8452	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( B - 1 ) e. RR )
24		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. NN )
25	24	nnred 9048	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. RR )
26	23, 25	remulcld 8102	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( ( B - 1 ) x. k ) e. RR )
27	26, 22	readdcld 8101	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( ( ( B - 1 ) x. k ) + 1 ) e. RR )
28	27	adantr 276	. . . . . . 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 9347	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. NN0 )
30		reexpcl 10699	. . . . . . . . 9 |- ( ( B e. RR /\ k e. NN0 ) -> ( B ^ k ) e. RR )
31	21, 29, 30	syl2anc 411	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( B ^ k ) e. RR )
32	31	adantr 276	. . . . . . 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 110	. . . . . . . . 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 8086	. . . . . . . . . . 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 8452	. . . . . . . . . 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 528	. . . . . . . . . . 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 9048	. . . . . . . . . 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 276	. . . . . . . . . . 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 8452	. . . . . . . . . 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 536	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 1 < B )
41	40	adantr 276	. . . . . . . . . . 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 8604	. . . . . . . . . . 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 147	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> 0 < ( B - 1 ) )
44		ltdivmul 8948	. . . . . . . . . 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 1253	. . . . . . . . 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 147	. . . . . . . 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 8102	. . . . . . . . 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 8613	. . . . . . . 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 147	. . . . . . 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 9347	. . . . . . . 8 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> k e. NN0 )
51		0red 8072	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 0 e. RR )
52		0lt1 8198	. . . . . . . . . . . 12 |- 0 < 1
53		0re 8071	. . . . . . . . . . . . 13 |- 0 e. RR
54		1re 8070	. . . . . . . . . . . . 13 |- 1 e. RR
55		lttr 8145	. . . . . . . . . . . . 13 |- ( ( 0 e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
56	53, 54, 55	mp3an12 1339	. . . . . . . . . . . 12 |- ( B e. RR -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
57	52, 56	mpani 430	. . . . . . . . . . 11 |- ( B e. RR -> ( 1 < B -> 0 < B ) )
58	21, 40, 57	sylc 62	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 0 < B )
59	51, 21, 58	ltled 8190	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 0 <_ B )
60	59	adantr 276	. . . . . . . 8 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> 0 <_ B )
61		bernneq2 10804	. . . . . . . 8 |- ( ( B e. RR /\ k e. NN0 /\ 0 <_ B ) -> ( ( ( B - 1 ) x. k ) + 1 ) <_ ( B ^ k ) )
62	38, 50, 60, 61	syl3anc 1249	. . . . . . 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 8495	. . . . . 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 115	. . . . 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 2607	. . . 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 1250	. 2 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> E. k e. NN A < ( B ^ k ) )
68		1nn 9046	. . 3 |- 1 e. NN
69		simpr 110	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> A < B )
70		simpl2 1003	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> B e. RR )
71	70	recnd 8100	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> B e. CC )
72		exp1 10688	. . . . 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 4071	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> A < ( B ^ 1 ) )
75		oveq2 5951	. . . . 5 |- ( k = 1 -> ( B ^ k ) = ( B ^ 1 ) )
76	75	breq2d 4055	. . . 4 |- ( k = 1 -> ( A < ( B ^ k ) <-> A < ( B ^ 1 ) ) )
77	76	rspcev 2876	. . 3 |- ( ( 1 e. NN /\ A < ( B ^ 1 ) ) -> E. k e. NN A < ( B ^ k ) )
78	68, 74, 77	sylancr 414	. 2 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> E. k e. NN A < ( B ^ k ) )
79		axltwlin 8139	. . . . 5 |- ( ( 1 e. RR /\ B e. RR /\ A e. RR ) -> ( 1 < B -> ( 1 < A \/ A < B ) ) )
80	54, 79	mp3an1 1336	. . . 4 |- ( ( B e. RR /\ A e. RR ) -> ( 1 < B -> ( 1 < A \/ A < B ) ) )
81	80	ancoms 268	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < B -> ( 1 < A \/ A < B ) ) )
82	81	3impia 1202	. 2 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( 1 < A \/ A < B ) )
83	67, 78, 82	mpjaodan 799	1 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> E. k e. NN A < ( B ^ k ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1372   e. wcel 2175   E.wrex 2484   class class class wbr 4043  (class class class)co 5943   CCcc 7922   RRcr 7923   0cc0 7924   1c1 7925   + caddc 7927   x. cmul 7929   < clt 8106   <_ cle 8107   - cmin 8242   / cdiv 8744   NNcn 9035   NN0cn0 9294   ^cexp 10681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-n0 9295  df-z 9372  df-uz 9648  df-seqfrec 10591  df-exp 10682

This theorem is referenced by:  expnlbnd  10807  bitsfzolem  12236  bitsfi  12239  pclemub  12581