Theorem expnbnd 10658

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 998	. . . 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 999	. . . 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 1000	. . . 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 7986	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> 1 e. RR )
9	1, 8	resubcld 8352	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( A - 1 ) e. RR )
10	3, 8	resubcld 8352	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( B - 1 ) e. RR )
11	8, 3	posdifd 8503	. . . . . . . . . 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 8600	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( B - 1 ) # 0 )
14	9, 10, 13	redivclapd 8806	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( ( A - 1 ) / ( B - 1 ) ) e. RR )
15		arch 9187	. . . . . . 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 1204	. . . . 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 7986	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 1 e. RR )
23	21, 22	resubcld 8352	. . . . . . . . . 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 8946	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. RR )
26	23, 25	remulcld 8002	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( ( B - 1 ) x. k ) e. RR )
27	26, 22	readdcld 8001	. . . . . . . 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 9243	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. NN0 )
30		reexpcl 10551	. . . . . . . . 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 7986	. . . . . . . . . . 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 8352	. . . . . . . . . 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 8946	. . . . . . . . . 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 8352	. . . . . . . . . 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 8503	. . . . . . . . . . 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 8847	. . . . . . . . . 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 1252	. . . . . . . . 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 8002	. . . . . . . . 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 8512	. . . . . . . 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 9243	. . . . . . . 8 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> k e. NN0 )
51		0red 7972	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 0 e. RR )
52		0lt1 8098	. . . . . . . . . . . 12 |- 0 < 1
53		0re 7971	. . . . . . . . . . . . 13 |- 0 e. RR
54		1re 7970	. . . . . . . . . . . . 13 |- 1 e. RR
55		lttr 8045	. . . . . . . . . . . . 13 |- ( ( 0 e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
56	53, 54, 55	mp3an12 1337	. . . . . . . . . . . 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 8090	. . . . . . . . 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 10656	. . . . . . . 8 |- ( ( B e. RR /\ k e. NN0 /\ 0 <_ B ) -> ( ( ( B - 1 ) x. k ) + 1 ) <_ ( B ^ k ) )
62	38, 50, 60, 61	syl3anc 1248	. . . . . . 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 8394	. . . . . 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 2589	. . . 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 1249	. 2 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> E. k e. NN A < ( B ^ k ) )
68		1nn 8944	. . 3 |- 1 e. NN
69		simpr 110	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> A < B )
70		simpl2 1002	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> B e. RR )
71	70	recnd 8000	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> B e. CC )
72		exp1 10540	. . . . 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 4043	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> A < ( B ^ 1 ) )
75		oveq2 5896	. . . . 5 |- ( k = 1 -> ( B ^ k ) = ( B ^ 1 ) )
76	75	breq2d 4027	. . . 4 |- ( k = 1 -> ( A < ( B ^ k ) <-> A < ( B ^ 1 ) ) )
77	76	rspcev 2853	. . 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 8039	. . . . 5 |- ( ( 1 e. RR /\ B e. RR /\ A e. RR ) -> ( 1 < B -> ( 1 < A \/ A < B ) ) )
80	54, 79	mp3an1 1334	. . . 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 1201	. 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 979   = wceq 1363   e. wcel 2158   E.wrex 2466   class class class wbr 4015  (class class class)co 5888   CCcc 7823   RRcr 7824   0cc0 7825   1c1 7826   + caddc 7828   x. cmul 7830   < clt 8006   <_ cle 8007   - cmin 8142   / cdiv 8643   NNcn 8933   NN0cn0 9190   ^cexp 10533

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-n0 9191  df-z 9268  df-uz 9543  df-seqfrec 10460  df-exp 10534

This theorem is referenced by:  expnlbnd  10659  pclemub  12301