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Theorem expubnd 11471
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 958 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  A  e.  RR )
2 2re 10100 . . . . 5  |-  2  e.  RR
3 peano2rem 9398 . . . . 5  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
4 remulcl 9106 . . . . 5  |-  ( ( 2  e.  RR  /\  ( A  -  1
)  e.  RR )  ->  ( 2  x.  ( A  -  1 ) )  e.  RR )
52, 3, 4sylancr 646 . . . 4  |-  ( A  e.  RR  ->  (
2  x.  ( A  -  1 ) )  e.  RR )
653ad2ant1 979 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  (
2  x.  ( A  -  1 ) )  e.  RR )
7 simp2 959 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  N  e.  NN0 )
8 0re 9122 . . . . . . . 8  |-  0  e.  RR
9 2pos 10113 . . . . . . . 8  |-  0  <  2
108, 2, 9ltleii 9227 . . . . . . 7  |-  0  <_  2
11 letr 9198 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  2  e.  RR  /\  A  e.  RR )  ->  (
( 0  <_  2  /\  2  <_  A )  ->  0  <_  A
) )
128, 2, 11mp3an12 1270 . . . . . . 7  |-  ( A  e.  RR  ->  (
( 0  <_  2  /\  2  <_  A )  ->  0  <_  A
) )
1310, 12mpani 659 . . . . . 6  |-  ( A  e.  RR  ->  (
2  <_  A  ->  0  <_  A ) )
1413imp 420 . . . . 5  |-  ( ( A  e.  RR  /\  2  <_  A )  -> 
0  <_  A )
15 resubcl 9396 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  2  e.  RR )  ->  ( A  -  2 )  e.  RR )
162, 15mpan2 654 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  -  2 )  e.  RR )
17 leadd2 9528 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  A  e.  RR  /\  ( A  -  2 )  e.  RR )  -> 
( 2  <_  A  <->  ( ( A  -  2 )  +  2 )  <_  ( ( A  -  2 )  +  A ) ) )
182, 17mp3an1 1267 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( A  -  2
)  e.  RR )  ->  ( 2  <_  A 
<->  ( ( A  - 
2 )  +  2 )  <_  ( ( A  -  2 )  +  A ) ) )
1916, 18mpdan 651 . . . . . . 7  |-  ( A  e.  RR  ->  (
2  <_  A  <->  ( ( A  -  2 )  +  2 )  <_ 
( ( A  - 
2 )  +  A
) ) )
2019biimpa 472 . . . . . 6  |-  ( ( A  e.  RR  /\  2  <_  A )  -> 
( ( A  - 
2 )  +  2 )  <_  ( ( A  -  2 )  +  A ) )
21 recn 9111 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
22 2cn 10101 . . . . . . . 8  |-  2  e.  CC
23 npcan 9345 . . . . . . . 8  |-  ( ( A  e.  CC  /\  2  e.  CC )  ->  ( ( A  - 
2 )  +  2 )  =  A )
2421, 22, 23sylancl 645 . . . . . . 7  |-  ( A  e.  RR  ->  (
( A  -  2 )  +  2 )  =  A )
2524adantr 453 . . . . . 6  |-  ( ( A  e.  RR  /\  2  <_  A )  -> 
( ( A  - 
2 )  +  2 )  =  A )
26 ax-1cn 9079 . . . . . . . . . 10  |-  1  e.  CC
27 subdi 9498 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  A  e.  CC  /\  1  e.  CC )  ->  (
2  x.  ( A  -  1 ) )  =  ( ( 2  x.  A )  -  ( 2  x.  1 ) ) )
2822, 26, 27mp3an13 1271 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
2  x.  ( A  -  1 ) )  =  ( ( 2  x.  A )  -  ( 2  x.  1 ) ) )
29 2times 10130 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
3022mulid1i 9123 . . . . . . . . . . 11  |-  ( 2  x.  1 )  =  2
3130a1i 11 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
2  x.  1 )  =  2 )
3229, 31oveq12d 6128 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( 2  x.  A
)  -  ( 2  x.  1 ) )  =  ( ( A  +  A )  - 
2 ) )
33 addsub 9347 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  2  e.  CC )  ->  (
( A  +  A
)  -  2 )  =  ( ( A  -  2 )  +  A ) )
3422, 33mp3an3 1269 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  e.  CC )  ->  ( ( A  +  A )  -  2 )  =  ( ( A  -  2 )  +  A ) )
3534anidms 628 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A  +  A
)  -  2 )  =  ( ( A  -  2 )  +  A ) )
3628, 32, 353eqtrrd 2479 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A  -  2 )  +  A )  =  ( 2  x.  ( A  -  1 ) ) )
3721, 36syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  (
( A  -  2 )  +  A )  =  ( 2  x.  ( A  -  1 ) ) )
3837adantr 453 . . . . . 6  |-  ( ( A  e.  RR  /\  2  <_  A )  -> 
( ( A  - 
2 )  +  A
)  =  ( 2  x.  ( A  - 
1 ) ) )
3920, 25, 383brtr3d 4266 . . . . 5  |-  ( ( A  e.  RR  /\  2  <_  A )  ->  A  <_  ( 2  x.  ( A  -  1 ) ) )
4014, 39jca 520 . . . 4  |-  ( ( A  e.  RR  /\  2  <_  A )  -> 
( 0  <_  A  /\  A  <_  ( 2  x.  ( A  - 
1 ) ) ) )
41403adant2 977 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  (
0  <_  A  /\  A  <_  ( 2  x.  ( A  -  1 ) ) ) )
42 leexp1a 11469 . . 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 ) )
431, 6, 7, 41, 42syl31anc 1188 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  ( A ^ N )  <_ 
( ( 2  x.  ( A  -  1 ) ) ^ N
) )
443recnd 9145 . . . 4  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  CC )
45 mulexp 11450 . . . . 5  |-  ( ( 2  e.  CC  /\  ( A  -  1
)  e.  CC  /\  N  e.  NN0 )  -> 
( ( 2  x.  ( A  -  1 ) ) ^ N
)  =  ( ( 2 ^ N )  x.  ( ( A  -  1 ) ^ N ) ) )
4622, 45mp3an1 1267 . . . 4  |-  ( ( ( A  -  1 )  e.  CC  /\  N  e.  NN0 )  -> 
( ( 2  x.  ( A  -  1 ) ) ^ N
)  =  ( ( 2 ^ N )  x.  ( ( A  -  1 ) ^ N ) ) )
4744, 46sylan 459 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0 )  -> 
( ( 2  x.  ( A  -  1 ) ) ^ N
)  =  ( ( 2 ^ N )  x.  ( ( A  -  1 ) ^ N ) ) )
48473adant3 978 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  (
( 2  x.  ( A  -  1 ) ) ^ N )  =  ( ( 2 ^ N )  x.  ( ( A  - 
1 ) ^ N
) ) )
4943, 48breqtrd 4261 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    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   class class class wbr 4237  (class class class)co 6110   CCcc 9019   RRcr 9020   0cc0 9021   1c1 9022    + caddc 9024    x. cmul 9026    <_ cle 9152    - cmin 9322   2c2 10080   NN0cn0 10252   ^cexp 11413
This theorem is referenced by:  faclbnd4lem1  11615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-n0 10253  df-z 10314  df-uz 10520  df-seq 11355  df-exp 11414
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