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Theorem expnbnd 12993
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
StepHypRef Expression
1 1nn 11031 . . 3 |- 1 e. NN
2 1re 10039 . . . . . . . 8 |- 1 e. RR
3 lttr 10114 . . . . . . . 8 |- ( ( A e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( A < 1 /\ 1 < B ) -> A < B ) )
42, 3mp3an2 1412 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( A < 1 /\ 1 < B ) -> A < B ) )
54exp4b 632 . . . . . 6 |- ( A e. RR -> ( B e. RR -> ( A < 1 -> ( 1 < B -> A < B ) ) ) )
65com34 91 . . . . 5 |- ( A e. RR -> ( B e. RR -> ( 1 < B -> ( A < 1 -> A < B ) ) ) )
763imp1 1280 . . . 4 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < 1 ) -> A < B )
8 recn 10026 . . . . . . 7 |- ( B e. RR -> B e. CC )
9 exp1 12866 . . . . . . 7 |- ( B e. CC -> ( B ^ 1 ) = B )
108, 9syl 17 . . . . . 6 |- ( B e. RR -> ( B ^ 1 ) = B )
11103ad2ant2 1083 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> ( B ^ 1 ) = B )
1211adantr 481 . . . 4 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < 1 ) -> ( B ^ 1 ) = B )
137, 12breqtrrd 4681 . . 3 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < 1 ) -> A < ( B ^ 1 ) )
14 oveq2 6658 . . . . 5 |- ( k = 1 -> ( B ^ k ) = ( B ^ 1 ) )
1514breq2d 4665 . . . 4 |- ( k = 1 -> ( A < ( B ^ k ) <-> A < ( B ^ 1 ) ) )
1615rspcev 3309 . . 3 |- ( ( 1 e. NN /\ A < ( B ^ 1 ) ) -> E. k e. NN A < ( B ^ k ) )
171, 13, 16sylancr 695 . 2 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ A < 1 ) -> E. k e. NN A < ( B ^ k ) )
18 peano2rem 10348 . . . . . . . . . . 11 |- ( A e. RR -> ( A - 1 ) e. RR )
1918adantr 481 . . . . . . . . . 10 |- ( ( A e. RR /\ ( B e. RR /\ 1 < B ) ) -> ( A - 1 ) e. RR )
20 peano2rem 10348 . . . . . . . . . . . 12 |- ( B e. RR -> ( B - 1 ) e. RR )
2120adantr 481 . . . . . . . . . . 11 |- ( ( B e. RR /\ 1 < B ) -> ( B - 1 ) e. RR )
2221adantl 482 . . . . . . . . . 10 |- ( ( A e. RR /\ ( B e. RR /\ 1 < B ) ) -> ( B - 1 ) e. RR )
23 posdif 10521 . . . . . . . . . . . . . 14 |- ( ( 1 e. RR /\ B e. RR ) -> ( 1 < B <-> 0 < ( B - 1 ) ) )
242, 23mpan 706 . . . . . . . . . . . . 13 |- ( B e. RR -> ( 1 < B <-> 0 < ( B - 1 ) ) )
2524biimpa 501 . . . . . . . . . . . 12 |- ( ( B e. RR /\ 1 < B ) -> 0 < ( B - 1 ) )
2625gt0ne0d 10592 . . . . . . . . . . 11 |- ( ( B e. RR /\ 1 < B ) -> ( B - 1 ) =/= 0 )
2726adantl 482 . . . . . . . . . 10 |- ( ( A e. RR /\ ( B e. RR /\ 1 < B ) ) -> ( B - 1 ) =/= 0 )
2819, 22, 27redivcld 10853 . . . . . . . . 9 |- ( ( A e. RR /\ ( B e. RR /\ 1 < B ) ) -> ( ( A - 1 ) / ( B - 1 ) ) e. RR )
2928adantll 750 . . . . . . . 8 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( ( A - 1 ) / ( B - 1 ) ) e. RR )
3018adantl 482 . . . . . . . . . 10 |- ( ( 1 <_ A /\ A e. RR ) -> ( A - 1 ) e. RR )
31 subge0 10541 . . . . . . . . . . . 12 |- ( ( A e. RR /\ 1 e. RR ) -> ( 0 <_ ( A - 1 ) <-> 1 <_ A ) )
322, 31mpan2 707 . . . . . . . . . . 11 |- ( A e. RR -> ( 0 <_ ( A - 1 ) <-> 1 <_ A ) )
3332biimparc 504 . . . . . . . . . 10 |- ( ( 1 <_ A /\ A e. RR ) -> 0 <_ ( A - 1 ) )
3430, 33jca 554 . . . . . . . . 9 |- ( ( 1 <_ A /\ A e. RR ) -> ( ( A - 1 ) e. RR /\ 0 <_ ( A - 1 ) ) )
3521, 25jca 554 . . . . . . . . 9 |- ( ( B e. RR /\ 1 < B ) -> ( ( B - 1 ) e. RR /\ 0 < ( B - 1 ) ) )
36 divge0 10892 . . . . . . . . 9 |- ( ( ( ( A - 1 ) e. RR /\ 0 <_ ( A - 1 ) ) /\ ( ( B - 1 ) e. RR /\ 0 < ( B - 1 ) ) ) -> 0 <_ ( ( A - 1 ) / ( B - 1 ) ) )
3734, 35, 36syl2an 494 . . . . . . . 8 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> 0 <_ ( ( A - 1 ) / ( B - 1 ) ) )
38 flge0nn0 12621 . . . . . . . 8 |- ( ( ( ( A - 1 ) / ( B - 1 ) ) e. RR /\ 0 <_ ( ( A - 1 ) / ( B - 1 ) ) ) -> ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) e. NN0 )
3929, 37, 38syl2anc 693 . . . . . . 7 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) e. NN0 )
40 nn0p1nn 11332 . . . . . . 7 |- ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) e. NN0 -> ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) e. NN )
4139, 40syl 17 . . . . . 6 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) e. NN )
42 simplr 792 . . . . . . 7 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> A e. RR )
4321adantl 482 . . . . . . . . 9 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( B - 1 ) e. RR )
44 peano2nn0 11333 . . . . . . . . . . 11 |- ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) e. NN0 -> ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) e. NN0 )
4539, 44syl 17 . . . . . . . . . 10 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) e. NN0 )
4645nn0red 11352 . . . . . . . . 9 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) e. RR )
4743, 46remulcld 10070 . . . . . . . 8 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) e. RR )
48 peano2re 10209 . . . . . . . 8 |- ( ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) e. RR -> ( ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) + 1 ) e. RR )
4947, 48syl 17 . . . . . . 7 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) + 1 ) e. RR )
50 simprl 794 . . . . . . . 8 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> B e. RR )
51 reexpcl 12877 . . . . . . . 8 |- ( ( B e. RR /\ ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) e. NN0 ) -> ( B ^ ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) e. RR )
5250, 45, 51syl2anc 693 . . . . . . 7 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( B ^ ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) e. RR )
53 flltp1 12601 . . . . . . . . . 10 |- ( ( ( A - 1 ) / ( B - 1 ) ) e. RR -> ( ( A - 1 ) / ( B - 1 ) ) < ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) )
5429, 53syl 17 . . . . . . . . 9 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( ( A - 1 ) / ( B - 1 ) ) < ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) )
5530adantr 481 . . . . . . . . . 10 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( A - 1 ) e. RR )
5625adantl 482 . . . . . . . . . 10 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> 0 < ( B - 1 ) )
57 ltdivmul 10898 . . . . . . . . . 10 |- ( ( ( A - 1 ) e. RR /\ ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) e. RR /\ ( ( B - 1 ) e. RR /\ 0 < ( B - 1 ) ) ) -> ( ( ( A - 1 ) / ( B - 1 ) ) < ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) <-> ( A - 1 ) < ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) ) )
5855, 46, 43, 56, 57syl112anc 1330 . . . . . . . . 9 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( ( ( A - 1 ) / ( B - 1 ) ) < ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) <-> ( A - 1 ) < ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) ) )
5954, 58mpbid 222 . . . . . . . 8 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( A - 1 ) < ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) )
60 ltsubadd 10498 . . . . . . . . . 10 |- ( ( A e. RR /\ 1 e. RR /\ ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) e. RR ) -> ( ( A - 1 ) < ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) <-> A < ( ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) + 1 ) ) )
612, 60mp3an2 1412 . . . . . . . . 9 |- ( ( A e. RR /\ ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) e. RR ) -> ( ( A - 1 ) < ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) <-> A < ( ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) + 1 ) ) )
6242, 47, 61syl2anc 693 . . . . . . . 8 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( ( A - 1 ) < ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) <-> A < ( ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) + 1 ) ) )
6359, 62mpbid 222 . . . . . . 7 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> A < ( ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) + 1 ) )
64 0lt1 10550 . . . . . . . . . . . 12 |- 0 < 1
65 0re 10040 . . . . . . . . . . . . 13 |- 0 e. RR
66 lttr 10114 . . . . . . . . . . . . 13 |- ( ( 0 e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
6765, 2, 66mp3an12 1414 . . . . . . . . . . . 12 |- ( B e. RR -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
6864, 67mpani 712 . . . . . . . . . . 11 |- ( B e. RR -> ( 1 < B -> 0 < B ) )
69 ltle 10126 . . . . . . . . . . . 12 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 < B -> 0 <_ B ) )
7065, 69mpan 706 . . . . . . . . . . 11 |- ( B e. RR -> ( 0 < B -> 0 <_ B ) )
7168, 70syld 47 . . . . . . . . . 10 |- ( B e. RR -> ( 1 < B -> 0 <_ B ) )
7271imp 445 . . . . . . . . 9 |- ( ( B e. RR /\ 1 < B ) -> 0 <_ B )
7372adantl 482 . . . . . . . 8 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> 0 <_ B )
74 bernneq2 12991 . . . . . . . 8 |- ( ( B e. RR /\ ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) e. NN0 /\ 0 <_ B ) -> ( ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) + 1 ) <_ ( B ^ ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) )
7550, 45, 73, 74syl3anc 1326 . . . . . . 7 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> ( ( ( B - 1 ) x. ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) + 1 ) <_ ( B ^ ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) )
7642, 49, 52, 63, 75ltletrd 10197 . . . . . 6 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> A < ( B ^ ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) )
77 oveq2 6658 . . . . . . . 8 |- ( k = ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) -> ( B ^ k ) = ( B ^ ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) )
7877breq2d 4665 . . . . . . 7 |- ( k = ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) -> ( A < ( B ^ k ) <-> A < ( B ^ ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) ) )
7978rspcev 3309 . . . . . 6 |- ( ( ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) e. NN /\ A < ( B ^ ( ( |_ ` ( ( A - 1 ) / ( B - 1 ) ) ) + 1 ) ) ) -> E. k e. NN A < ( B ^ k ) )
8041, 76, 79syl2anc 693 . . . . 5 |- ( ( ( 1 <_ A /\ A e. RR ) /\ ( B e. RR /\ 1 < B ) ) -> E. k e. NN A < ( B ^ k ) )
8180exp43 640 . . . 4 |- ( 1 <_ A -> ( A e. RR -> ( B e. RR -> ( 1 < B -> E. k e. NN A < ( B ^ k ) ) ) ) )
8281com4l 92 . . 3 |- ( A e. RR -> ( B e. RR -> ( 1 < B -> ( 1 <_ A -> E. k e. NN A < ( B ^ k ) ) ) ) )
83823imp1 1280 . 2 |- ( ( ( A e. RR /\ B e. RR /\ 1 < B ) /\ 1 <_ A ) -> E. k e. NN A < ( B ^ k ) )
84 simp1 1061 . 2 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> A e. RR )
85 1red 10055 . 2 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> 1 e. RR )
8617, 83, 84, 85ltlecasei 10145 1 |- ( ( A e. RR /\ B e. RR /\ 1 < B ) -> E. k e. NN A < ( B ^ k ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   |_cfl 12591   ^cexp 12860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fl 12593  df-seq 12802  df-exp 12861
This theorem is referenced by:  expnlbnd  12994  expmulnbnd  12996  bitsfzolem  15156  bitsfi  15159  pclem  15543  aaliou3lem8  24100  ostth2lem1  25307  ostth3  25327  knoppndvlem18  32520
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