Theorem expnbnd 10192

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 946	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> A e. RR )
2	1	adantr 271	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> A e. RR )
3		simp2 947	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> B e. RR )
4	3	adantr 271	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> B e. RR )
5		simpr 109	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> 1 < A )
6		simp3 948	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> 1 < B )
7	6	adantr 271	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> 1 < B )
8		1red 7600	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> 1 e. RR )
9	1, 8	resubcld 7956	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( A - 1 ) e. RR )
10	3, 8	resubcld 7956	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( B - 1 ) e. RR )
11	8, 3	posdifd 8106	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( 1 < B <-> 0 < ( B - 1 ) ) )
12	6, 11	mpbid 146	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> 0 < ( B - 1 ) )
13	10, 12	gt0ap0d 8202	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( B - 1 ) # 0 )
14	9, 10, 13	redivclapd 8398	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( ( A - 1 ) / ( B - 1 ) ) e. RR )
15		arch 8768	. . . . . . 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 1146	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ 1 < B ) -> E. k e. NN ( ( A - 1 ) / ( B - 1 ) ) < k )
18	17	adantrl 463	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> E. k e. NN ( ( A - 1 ) / ( B - 1 ) ) < k )
19		simplll 501	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> A e. RR )
20	19	adantr 271	. . . . . . 7 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> A e. RR )
21		simpllr 502	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> B e. RR )
22		1red 7600	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 1 e. RR )
23	21, 22	resubcld 7956	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( B - 1 ) e. RR )
24		simpr 109	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. NN )
25	24	nnred 8533	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. RR )
26	23, 25	remulcld 7615	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( ( B - 1 ) x. k ) e. RR )
27	26, 22	readdcld 7614	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( ( ( B - 1 ) x. k ) + 1 ) e. RR )
28	27	adantr 271	. . . . . . 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 8824	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> k e. NN0 )
30		reexpcl 10087	. . . . . . . . 9 |- ( ( B e. RR /\ k e. NN0 ) -> ( B ^ k ) e. RR )
31	21, 29, 30	syl2anc 404	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> ( B ^ k ) e. RR )
32	31	adantr 271	. . . . . . 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 109	. . . . . . . . 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 7600	. . . . . . . . . . 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 7956	. . . . . . . . . 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 498	. . . . . . . . . . 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 8533	. . . . . . . . . 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 271	. . . . . . . . . . 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 7956	. . . . . . . . . 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 504	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 1 < B )
41	40	adantr 271	. . . . . . . . . . 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 8106	. . . . . . . . . . 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 146	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> 0 < ( B - 1 ) )
44		ltdivmul 8434	. . . . . . . . . 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 1185	. . . . . . . . 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 146	. . . . . . . 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 7615	. . . . . . . . 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 8115	. . . . . . . 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 146	. . . . . . 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 8824	. . . . . . . 8 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> k e. NN0 )
51		0red 7586	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 0 e. RR )
52		0lt1 7707	. . . . . . . . . . . 12 |- 0 < 1
53		0re 7585	. . . . . . . . . . . . 13 |- 0 e. RR
54		1re 7584	. . . . . . . . . . . . 13 |- 1 e. RR
55		lttr 7656	. . . . . . . . . . . . 13 |- ( ( 0 e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
56	53, 54, 55	mp3an12 1270	. . . . . . . . . . . 12 |- ( B e. RR -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
57	52, 56	mpani 422	. . . . . . . . . . 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 7699	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) -> 0 <_ B )
60	59	adantr 271	. . . . . . . 8 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) /\ k e. NN ) /\ ( ( A - 1 ) / ( B - 1 ) ) < k ) -> 0 <_ B )
61		bernneq2 10190	. . . . . . . 8 |- ( ( B e. RR /\ k e. NN0 /\ 0 <_ B ) -> ( ( ( B - 1 ) x. k ) + 1 ) <_ ( B ^ k ) )
62	38, 50, 60, 61	syl3anc 1181	. . . . . . 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 7998	. . . . . 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 114	. . . . 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 2487	. . . 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 1182	. 2 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 < A ) -> E. k e. NN A < ( B ^ k ) )
68		1nn 8531	. . 3 |- 1 e. NN
69		simpr 109	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> A < B )
70		simpl2 950	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> B e. RR )
71	70	recnd 7613	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> B e. CC )
72		exp1 10076	. . . . 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 3893	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> A < ( B ^ 1 ) )
75		oveq2 5698	. . . . 5 |- ( k = 1 -> ( B ^ k ) = ( B ^ 1 ) )
76	75	breq2d 3879	. . . 4 |- ( k = 1 -> ( A < ( B ^ k ) <-> A < ( B ^ 1 ) ) )
77	76	rspcev 2736	. . 3 |- ( ( 1 e. NN /\ A < ( B ^ 1 ) ) -> E. k e. NN A < ( B ^ k ) )
78	68, 74, 77	sylancr 406	. 2 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < B ) -> E. k e. NN A < ( B ^ k ) )
79		axltwlin 7651	. . . . 5 |- ( ( 1 e. RR /\ B e. RR /\ A e. RR ) -> ( 1 < B -> ( 1 < A \/ A < B ) ) )
80	54, 79	mp3an1 1267	. . . 4 |- ( ( B e. RR /\ A e. RR ) -> ( 1 < B -> ( 1 < A \/ A < B ) ) )
81	80	ancoms 265	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < B -> ( 1 < A \/ A < B ) ) )
82	81	3impia 1143	. 2 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( 1 < A \/ A < B ) )
83	67, 78, 82	mpjaodan 750	1 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> E. k e. NN A < ( B ^ k ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 667   /\ w3a 927   = wceq 1296   e. wcel 1445   E.wrex 2371   class class class wbr 3867  (class class class)co 5690   CCcc 7445   RRcr 7446   0cc0 7447   1c1 7448   + caddc 7450   x. cmul 7452   < clt 7619   <_ cle 7620   - cmin 7750   / cdiv 8236   NNcn 8520   NN0cn0 8771   ^cexp 10069

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560  ax-arch 7561

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-n0 8772  df-z 8849  df-uz 9119  df-seqfrec 10001  df-exp 10070

This theorem is referenced by:  expnlbnd  10193