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Theorem expubnd 10958
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 1024 . . 3 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> A e. RR )
2 2re 9307 . . . . 5 |- 2 e. RR
3 peano2rem 8540 . . . . 5 |- ( A e. RR -> ( A - 1 ) e. RR )
4 remulcl 8255 . . . . 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 1045 . . 3 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> ( 2 x. ( A - 1 ) ) e. RR )
7 simp2 1025 . . 3 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> N e. NN0 )
8 0le2 9327 . . . . . . 7 |- 0 <_ 2
9 0re 8274 . . . . . . . 8 |- 0 e. RR
10 letr 8356 . . . . . . . 8 |- ( ( 0 e. RR /\ 2 e. RR /\ A e. RR ) -> ( ( 0 <_ 2 /\ 2 <_ A ) -> 0 <_ A ) )
119, 2, 10mp3an12 1364 . . . . . . 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 8537 . . . . . . . . 9 |- ( ( A e. RR /\ 2 e. RR ) -> ( A - 2 ) e. RR )
152, 14mpan2 425 . . . . . . . 8 |- ( A e. RR -> ( A - 2 ) e. RR )
16 leadd2 8705 . . . . . . . . 9 |- ( ( 2 e. RR /\ A e. RR /\ ( A - 2 ) e. RR ) -> ( 2 <_ A <-> ( ( A - 2 ) + 2 ) <_ ( ( A - 2 ) + A ) ) )
172, 16mp3an1 1361 . . . . . . . 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 8260 . . . . . . . 8 |- ( A e. RR -> A e. CC )
21 2cn 9308 . . . . . . . 8 |- 2 e. CC
22 npcan 8482 . . . . . . . 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 8220 . . . . . . . . . 10 |- 1 e. CC
26 subdi 8658 . . . . . . . . . 10 |- ( ( 2 e. CC /\ A e. CC /\ 1 e. CC ) -> ( 2 x. ( A - 1 ) ) = ( ( 2 x. A ) - ( 2 x. 1 ) ) )
2721, 25, 26mp3an13 1365 . . . . . . . . 9 |- ( A e. CC -> ( 2 x. ( A - 1 ) ) = ( ( 2 x. A ) - ( 2 x. 1 ) ) )
28 2times 9365 . . . . . . . . . 10 |- ( A e. CC -> ( 2 x. A ) = ( A + A ) )
29 2t1e2 9391 . . . . . . . . . . 11 |- ( 2 x. 1 ) = 2
3029a1i 9 . . . . . . . . . 10 |- ( A e. CC -> ( 2 x. 1 ) = 2 )
3128, 30oveq12d 6068 . . . . . . . . 9 |- ( A e. CC -> ( ( 2 x. A ) - ( 2 x. 1 ) ) = ( ( A + A ) - 2 ) )
32 addsub 8484 . . . . . . . . . . 11 |- ( ( A e. CC /\ A e. CC /\ 2 e. CC ) -> ( ( A + A ) - 2 ) = ( ( A - 2 ) + A ) )
3321, 32mp3an3 1363 . . . . . . . . . 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 2270 . . . . . . . 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 4140 . . . . 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 1043 . . 3 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> ( 0 <_ A /\ A <_ ( 2 x. ( A - 1 ) ) ) )
41 leexp1a 10956 . . 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 1277 . 2 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> ( A ^ N ) <_ ( ( 2 x. ( A - 1 ) ) ^ N ) )
433recnd 8302 . . . 4 |- ( A e. RR -> ( A - 1 ) e. CC )
44 mulexp 10940 . . . . 5 |- ( ( 2 e. CC /\ ( A - 1 ) e. CC /\ N e. NN0 ) -> ( ( 2 x. ( A - 1 ) ) ^ N ) = ( ( 2 ^ N ) x. ( ( A - 1 ) ^ N ) ) )
4521, 44mp3an1 1361 . . . 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 1044 . 2 |- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> ( ( 2 x. ( A - 1 ) ) ^ N ) = ( ( 2 ^ N ) x. ( ( A - 1 ) ^ N ) ) )
4842, 47breqtrd 4135 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 1005   = wceq 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   <_ cle 8309   - cmin 8444   2c2 9288   NN0cn0 9496   ^cexp 10900
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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245
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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854  df-seqfrec 10810  df-exp 10901
This theorem is referenced by: (None)
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