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Theorem expubnd 10848
Description: An upper bound on A ^ N when 2 <_ A. (Contributed by NM, 19-Dec-2005.)
Assertion
Ref Expression
expubnd |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> ( A ^ N ) <_ ( ( 2 ^ N ) x. ( ( A - 1 ) ^ N ) ) )
Proof of Theorem expubnd
StepHypRef Expression
1 simp1 1021 . . 3 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> A e. RR )
2 2re 9203 . . . . 5 |- 2 e. RR
3 peano2rem 8436 . . . . 5 |- ( A e. RR -> ( A - 1 ) e. RR )
4 remulcl 8150 . . . . 5 |- ( ( 2 e. RR /\ ( A - 1 ) e. RR ) -> ( 2 x. ( A - 1 ) ) e. RR )
52, 3, 4sylancr 414 . . . 4 |- ( A e. RR -> ( 2 x. ( A - 1 ) ) e. RR )
653ad2ant1 1042 . . 3 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> ( 2 x. ( A - 1 ) ) e. RR )
7 simp2 1022 . . 3 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> N e. NN0 )
8 0le2 9223 . . . . . . 7 |- 0 <_ 2
9 0re 8169 . . . . . . . 8 |- 0 e. RR
10 letr 8252 . . . . . . . 8 |- ( ( 0 e. RR /\ 2 e. RR /\ A e. RR ) -> ( ( 0 <_ 2 /\ 2 <_ A ) -> 0 <_ A ) )
119, 2, 10mp3an12 1361 . . . . . . 7 |- ( A e. RR -> ( ( 0 <_ 2 /\ 2 <_ A ) -> 0 <_ A ) )
128, 11mpani 430 . . . . . 6 |- ( A e. RR -> ( 2 <_ A -> 0 <_ A ) )
1312imp 124 . . . . 5 |- ( ( A e. RR /\ 2 <_ A ) -> 0 <_ A )
14 resubcl 8433 . . . . . . . . 9 |- ( ( A e. RR /\ 2 e. RR ) -> ( A - 2 ) e. RR )
152, 14mpan2 425 . . . . . . . 8 |- ( A e. RR -> ( A - 2 ) e. RR )
16 leadd2 8601 . . . . . . . . 9 |- ( ( 2 e. RR /\ A e. RR /\ ( A - 2 ) e. RR ) -> ( 2 <_ A <-> ( ( A - 2 ) + 2 ) <_ ( ( A - 2 ) + A ) ) )
172, 16mp3an1 1358 . . . . . . . 8 |- ( ( A e. RR /\ ( A - 2 ) e. RR ) -> ( 2 <_ A <-> ( ( A - 2 ) + 2 ) <_ ( ( A - 2 ) + A ) ) )
1815, 17mpdan 421 . . . . . . 7 |- ( A e. RR -> ( 2 <_ A <-> ( ( A - 2 ) + 2 ) <_ ( ( A - 2 ) + A ) ) )
1918biimpa 296 . . . . . 6 |- ( ( A e. RR /\ 2 <_ A ) -> ( ( A - 2 ) + 2 ) <_ ( ( A - 2 ) + A ) )
20 recn 8155 . . . . . . . 8 |- ( A e. RR -> A e. CC )
21 2cn 9204 . . . . . . . 8 |- 2 e. CC
22 npcan 8378 . . . . . . . 8 |- ( ( A e. CC /\ 2 e. CC ) -> ( ( A - 2 ) + 2 ) = A )
2320, 21, 22sylancl 413 . . . . . . 7 |- ( A e. RR -> ( ( A - 2 ) + 2 ) = A )
2423adantr 276 . . . . . 6 |- ( ( A e. RR /\ 2 <_ A ) -> ( ( A - 2 ) + 2 ) = A )
25 ax-1cn 8115 . . . . . . . . . 10 |- 1 e. CC
26 subdi 8554 . . . . . . . . . 10 |- ( ( 2 e. CC /\ A e. CC /\ 1 e. CC ) -> ( 2 x. ( A - 1 ) ) = ( ( 2 x. A ) - ( 2 x. 1 ) ) )
2721, 25, 26mp3an13 1362 . . . . . . . . 9 |- ( A e. CC -> ( 2 x. ( A - 1 ) ) = ( ( 2 x. A ) - ( 2 x. 1 ) ) )
28 2times 9261 . . . . . . . . . 10 |- ( A e. CC -> ( 2 x. A ) = ( A + A ) )
29 2t1e2 9287 . . . . . . . . . . 11 |- ( 2 x. 1 ) = 2
3029a1i 9 . . . . . . . . . 10 |- ( A e. CC -> ( 2 x. 1 ) = 2 )
3128, 30oveq12d 6031 . . . . . . . . 9 |- ( A e. CC -> ( ( 2 x. A ) - ( 2 x. 1 ) ) = ( ( A + A ) - 2 ) )
32 addsub 8380 . . . . . . . . . . 11 |- ( ( A e. CC /\ A e. CC /\ 2 e. CC ) -> ( ( A + A ) - 2 ) = ( ( A - 2 ) + A ) )
3321, 32mp3an3 1360 . . . . . . . . . 10 |- ( ( A e. CC /\ A e. CC ) -> ( ( A + A ) - 2 ) = ( ( A - 2 ) + A ) )
3433anidms 397 . . . . . . . . 9 |- ( A e. CC -> ( ( A + A ) - 2 ) = ( ( A - 2 ) + A ) )
3527, 31, 343eqtrrd 2267 . . . . . . . 8 |- ( A e. CC -> ( ( A - 2 ) + A ) = ( 2 x. ( A - 1 ) ) )
3620, 35syl 14 . . . . . . 7 |- ( A e. RR -> ( ( A - 2 ) + A ) = ( 2 x. ( A - 1 ) ) )
3736adantr 276 . . . . . 6 |- ( ( A e. RR /\ 2 <_ A ) -> ( ( A - 2 ) + A ) = ( 2 x. ( A - 1 ) ) )
3819, 24, 373brtr3d 4117 . . . . 5 |- ( ( A e. RR /\ 2 <_ A ) -> A <_ ( 2 x. ( A - 1 ) ) )
3913, 38jca 306 . . . 4 |- ( ( A e. RR /\ 2 <_ A ) -> ( 0 <_ A /\ A <_ ( 2 x. ( A - 1 ) ) ) )
40393adant2 1040 . . 3 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> ( 0 <_ A /\ A <_ ( 2 x. ( A - 1 ) ) ) )
41 leexp1a 10846 . . 3 |- ( ( ( A e. RR /\ ( 2 x. ( A - 1 ) ) e. RR /\ N e. NN0 ) /\ ( 0 <_ A /\ A <_ ( 2 x. ( A - 1 ) ) ) ) -> ( A ^ N ) <_ ( ( 2 x. ( A - 1 ) ) ^ N ) )
421, 6, 7, 40, 41syl31anc 1274 . 2 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> ( A ^ N ) <_ ( ( 2 x. ( A - 1 ) ) ^ N ) )
433recnd 8198 . . . 4 |- ( A e. RR -> ( A - 1 ) e. CC )
44 mulexp 10830 . . . . 5 |- ( ( 2 e. CC /\ ( A - 1 ) e. CC /\ N e. NN0 ) -> ( ( 2 x. ( A - 1 ) ) ^ N ) = ( ( 2 ^ N ) x. ( ( A - 1 ) ^ N ) ) )
4521, 44mp3an1 1358 . . . 4 |- ( ( ( A - 1 ) e. CC /\ N e. NN0 ) -> ( ( 2 x. ( A - 1 ) ) ^ N ) = ( ( 2 ^ N ) x. ( ( A - 1 ) ^ N ) ) )
4643, 45sylan 283 . . 3 |- ( ( A e. RR /\ N e. NN0 ) -> ( ( 2 x. ( A - 1 ) ) ^ N ) = ( ( 2 ^ N ) x. ( ( A - 1 ) ^ N ) ) )
47463adant3 1041 . 2 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> ( ( 2 x. ( A - 1 ) ) ^ N ) = ( ( 2 ^ N ) x. ( ( A - 1 ) ^ N ) ) )
4842, 47breqtrd 4112 1 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> ( A ^ N ) <_ ( ( 2 ^ N ) x. ( ( A - 1 ) ^ N ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   1c1 8023   + caddc 8025   x. cmul 8027   <_ cle 8205   - cmin 8340   2c2 9184   NN0cn0 9392   ^cexp 10790
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746  df-seqfrec 10700  df-exp 10791
This theorem is referenced by: (None)
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