Theorem expnbnd 10971

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 1024	. . . 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 1025	. . . 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 1026	. . . 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 8237	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> 1 e. RR )
9	1, 8	resubcld 8602	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( A - 1 ) e. RR )
10	3, 8	resubcld 8602	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( B - 1 ) e. RR )
11	8, 3	posdifd 8754	. . . . . . . . . 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 8851	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( B - 1 ) # 0 )
14	9, 10, 13	redivclapd 9057	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( ( A - 1 ) / ( B - 1 ) ) e. RR )
15		arch 9441	. . . . . . 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 1230	. . . . 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 535	. . . . . . . 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 536	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> B e. RR )
22		1red 8237	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 1 e. RR )
23	21, 22	resubcld 8602	. . . . . . . . . 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 9198	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. RR )
26	23, 25	remulcld 8252	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( ( B - 1 ) x. k ) e. RR )
27	26, 22	readdcld 8251	. . . . . . . 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 9499	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. NN0 )
30		reexpcl 10864	. . . . . . . . 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 8237	. . . . . . . . . . 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 8602	. . . . . . . . . 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 529	. . . . . . . . . . 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 9198	. . . . . . . . . 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 8602	. . . . . . . . . 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 538	. . . . . . . . . . . 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 8754	. . . . . . . . . . 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 9098	. . . . . . . . . 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 1278	. . . . . . . . 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 8252	. . . . . . . . 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 8763	. . . . . . . 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 9499	. . . . . . . 8 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> k e. NN0 )
51		0red 8223	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 0 e. RR )
52		0lt1 8348	. . . . . . . . . . . 12 |- 0 < 1
53		0re 8222	. . . . . . . . . . . . 13 |- 0 e. RR
54		1re 8221	. . . . . . . . . . . . 13 |- 1 e. RR
55		lttr 8295	. . . . . . . . . . . . 13 |- ( ( 0 e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
56	53, 54, 55	mp3an12 1364	. . . . . . . . . . . 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 8340	. . . . . . . . 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 10969	. . . . . . . 8 |- ( ( B e. RR /\ k e. NN0 /\ 0 <_ B ) -> ( ( ( B - 1 ) x. k ) + 1 ) <_ ( B ^ k ) )
62	38, 50, 60, 61	syl3anc 1274	. . . . . . 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 8645	. . . . . 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 2635	. . . 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 1275	. 2 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> E. k e. NN A < ( B ^ k ) )
68		1nn 9196	. . 3 |- 1 e. NN
69		simpr 110	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> A < B )
70		simpl2 1028	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> B e. RR )
71	70	recnd 8250	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> B e. CC )
72		exp1 10853	. . . . 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 4121	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> A < ( B ^ 1 ) )
75		oveq2 6036	. . . . 5 |- ( k = 1 -> ( B ^ k ) = ( B ^ 1 ) )
76	75	breq2d 4105	. . . 4 |- ( k = 1 -> ( A < ( B ^ k ) <-> A < ( B ^ 1 ) ) )
77	76	rspcev 2911	. . 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 8289	. . . . 5 |- ( ( 1 e. RR /\ B e. RR /\ A e. RR ) -> ( 1 < B -> ( 1 < A \/ A < B ) ) )
80	54, 79	mp3an1 1361	. . . 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 1227	. 2 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( 1 < A \/ A < B ) )
83	67, 78, 82	mpjaodan 806	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 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   E.wrex 2512   class class class wbr 4093  (class class class)co 6028   CCcc 8073   RRcr 8074   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   < clt 8256   <_ cle 8257   - cmin 8392   / cdiv 8894   NNcn 9185   NN0cn0 9444   ^cexp 10846

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-seqfrec 10756  df-exp 10847

This theorem is referenced by:  expnlbnd  10972  bitsfzolem  12578  bitsfi  12581  pclemub  12923