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Theorem expubnd 12183
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 991 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  A  e.  RR )
2 2re 10596 . . . . 5  |-  2  e.  RR
3 peano2rem 9877 . . . . 5  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
4 remulcl 9568 . . . . 5  |-  ( ( 2  e.  RR  /\  ( A  -  1
)  e.  RR )  ->  ( 2  x.  ( A  -  1 ) )  e.  RR )
52, 3, 4sylancr 663 . . . 4  |-  ( A  e.  RR  ->  (
2  x.  ( A  -  1 ) )  e.  RR )
653ad2ant1 1012 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  (
2  x.  ( A  -  1 ) )  e.  RR )
7 simp2 992 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  N  e.  NN0 )
8 0le2 10617 . . . . . . 7  |-  0  <_  2
9 0re 9587 . . . . . . . 8  |-  0  e.  RR
10 letr 9669 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  2  e.  RR  /\  A  e.  RR )  ->  (
( 0  <_  2  /\  2  <_  A )  ->  0  <_  A
) )
119, 2, 10mp3an12 1309 . . . . . . 7  |-  ( A  e.  RR  ->  (
( 0  <_  2  /\  2  <_  A )  ->  0  <_  A
) )
128, 11mpani 676 . . . . . 6  |-  ( A  e.  RR  ->  (
2  <_  A  ->  0  <_  A ) )
1312imp 429 . . . . 5  |-  ( ( A  e.  RR  /\  2  <_  A )  -> 
0  <_  A )
14 resubcl 9874 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  2  e.  RR )  ->  ( A  -  2 )  e.  RR )
152, 14mpan2 671 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  -  2 )  e.  RR )
16 leadd2 10012 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  A  e.  RR  /\  ( A  -  2 )  e.  RR )  -> 
( 2  <_  A  <->  ( ( A  -  2 )  +  2 )  <_  ( ( A  -  2 )  +  A ) ) )
172, 16mp3an1 1306 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( A  -  2
)  e.  RR )  ->  ( 2  <_  A 
<->  ( ( A  - 
2 )  +  2 )  <_  ( ( A  -  2 )  +  A ) ) )
1815, 17mpdan 668 . . . . . . 7  |-  ( A  e.  RR  ->  (
2  <_  A  <->  ( ( A  -  2 )  +  2 )  <_ 
( ( A  - 
2 )  +  A
) ) )
1918biimpa 484 . . . . . 6  |-  ( ( A  e.  RR  /\  2  <_  A )  -> 
( ( A  - 
2 )  +  2 )  <_  ( ( A  -  2 )  +  A ) )
20 recn 9573 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
21 2cn 10597 . . . . . . . 8  |-  2  e.  CC
22 npcan 9820 . . . . . . . 8  |-  ( ( A  e.  CC  /\  2  e.  CC )  ->  ( ( A  - 
2 )  +  2 )  =  A )
2320, 21, 22sylancl 662 . . . . . . 7  |-  ( A  e.  RR  ->  (
( A  -  2 )  +  2 )  =  A )
2423adantr 465 . . . . . 6  |-  ( ( A  e.  RR  /\  2  <_  A )  -> 
( ( A  - 
2 )  +  2 )  =  A )
25 ax-1cn 9541 . . . . . . . . . 10  |-  1  e.  CC
26 subdi 9981 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  A  e.  CC  /\  1  e.  CC )  ->  (
2  x.  ( A  -  1 ) )  =  ( ( 2  x.  A )  -  ( 2  x.  1 ) ) )
2721, 25, 26mp3an13 1310 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
2  x.  ( A  -  1 ) )  =  ( ( 2  x.  A )  -  ( 2  x.  1 ) ) )
28 2times 10645 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
29 2t1e2 10675 . . . . . . . . . . 11  |-  ( 2  x.  1 )  =  2
3029a1i 11 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
2  x.  1 )  =  2 )
3128, 30oveq12d 6295 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( 2  x.  A
)  -  ( 2  x.  1 ) )  =  ( ( A  +  A )  - 
2 ) )
32 addsub 9822 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  2  e.  CC )  ->  (
( A  +  A
)  -  2 )  =  ( ( A  -  2 )  +  A ) )
3321, 32mp3an3 1308 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  e.  CC )  ->  ( ( A  +  A )  -  2 )  =  ( ( A  -  2 )  +  A ) )
3433anidms 645 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A  +  A
)  -  2 )  =  ( ( A  -  2 )  +  A ) )
3527, 31, 343eqtrrd 2508 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A  -  2 )  +  A )  =  ( 2  x.  ( A  -  1 ) ) )
3620, 35syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  (
( A  -  2 )  +  A )  =  ( 2  x.  ( A  -  1 ) ) )
3736adantr 465 . . . . . 6  |-  ( ( A  e.  RR  /\  2  <_  A )  -> 
( ( A  - 
2 )  +  A
)  =  ( 2  x.  ( A  - 
1 ) ) )
3819, 24, 373brtr3d 4471 . . . . 5  |-  ( ( A  e.  RR  /\  2  <_  A )  ->  A  <_  ( 2  x.  ( A  -  1 ) ) )
3913, 38jca 532 . . . 4  |-  ( ( A  e.  RR  /\  2  <_  A )  -> 
( 0  <_  A  /\  A  <_  ( 2  x.  ( A  - 
1 ) ) ) )
40393adant2 1010 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  (
0  <_  A  /\  A  <_  ( 2  x.  ( A  -  1 ) ) ) )
41 leexp1a 12181 . . 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 1226 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  ( A ^ N )  <_ 
( ( 2  x.  ( A  -  1 ) ) ^ N
) )
433recnd 9613 . . . 4  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  CC )
44 mulexp 12162 . . . . 5  |-  ( ( 2  e.  CC  /\  ( A  -  1
)  e.  CC  /\  N  e.  NN0 )  -> 
( ( 2  x.  ( A  -  1 ) ) ^ N
)  =  ( ( 2 ^ N )  x.  ( ( A  -  1 ) ^ N ) ) )
4521, 44mp3an1 1306 . . . 4  |-  ( ( ( A  -  1 )  e.  CC  /\  N  e.  NN0 )  -> 
( ( 2  x.  ( A  -  1 ) ) ^ N
)  =  ( ( 2 ^ N )  x.  ( ( A  -  1 ) ^ N ) ) )
4643, 45sylan 471 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0 )  -> 
( ( 2  x.  ( A  -  1 ) ) ^ N
)  =  ( ( 2 ^ N )  x.  ( ( A  -  1 ) ^ N ) ) )
47463adant3 1011 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  (
( 2  x.  ( A  -  1 ) ) ^ N )  =  ( ( 2 ^ N )  x.  ( ( A  - 
1 ) ^ N
) ) )
4842, 47breqtrd 4466 1  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  ( A ^ N )  <_ 
( ( 2 ^ N )  x.  (
( A  -  1 ) ^ N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4442  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488    <_ cle 9620    - cmin 9796   2c2 10576   NN0cn0 10786   ^cexp 12124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-seq 12066  df-exp 12125
This theorem is referenced by:  faclbnd4lem1  12328
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