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Theorem ltexp2r 12198
Description: The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
Assertion
Ref Expression
ltexp2r  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( M  <  N  <->  ( A ^ N )  <  ( A ^ M ) ) )

Proof of Theorem ltexp2r
StepHypRef Expression
1 simpl1 998 . . . . 5  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  A  e.  RR+ )
21rpcnd 11264 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  A  e.  CC )
31rpne0d 11267 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  A  =/=  0 )
4 simpl2 999 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  M  e.  ZZ )
5 exprec 12183 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  M  e.  ZZ )  ->  (
( 1  /  A
) ^ M )  =  ( 1  / 
( A ^ M
) ) )
62, 3, 4, 5syl3anc 1227 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( (
1  /  A ) ^ M )  =  ( 1  /  ( A ^ M ) ) )
7 simpl3 1000 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  N  e.  ZZ )
8 exprec 12183 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
( 1  /  A
) ^ N )  =  ( 1  / 
( A ^ N
) ) )
92, 3, 7, 8syl3anc 1227 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( (
1  /  A ) ^ N )  =  ( 1  /  ( A ^ N ) ) )
106, 9breq12d 4447 . 2  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( (
( 1  /  A
) ^ M )  <  ( ( 1  /  A ) ^ N )  <->  ( 1  /  ( A ^ M ) )  < 
( 1  /  ( A ^ N ) ) ) )
111rprecred 11273 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( 1  /  A )  e.  RR )
12 simpr 461 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  A  <  1 )
131reclt1d 11275 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( A  <  1  <->  1  <  (
1  /  A ) ) )
1412, 13mpbid 210 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  1  <  ( 1  /  A ) )
15 ltexp2 12195 . . 3  |-  ( ( ( ( 1  /  A )  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  ( 1  /  A ) )  ->  ( M  < 
N  <->  ( ( 1  /  A ) ^ M )  <  (
( 1  /  A
) ^ N ) ) )
1611, 4, 7, 14, 15syl31anc 1230 . 2  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( M  <  N  <->  ( ( 1  /  A ) ^ M )  <  (
( 1  /  A
) ^ N ) ) )
17 rpexpcl 12161 . . . 4  |-  ( ( A  e.  RR+  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  RR+ )
181, 7, 17syl2anc 661 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( A ^ N )  e.  RR+ )
19 rpexpcl 12161 . . . 4  |-  ( ( A  e.  RR+  /\  M  e.  ZZ )  ->  ( A ^ M )  e.  RR+ )
201, 4, 19syl2anc 661 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( A ^ M )  e.  RR+ )
2118, 20ltrecd 11280 . 2  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( ( A ^ N )  < 
( A ^ M
)  <->  ( 1  / 
( A ^ M
) )  <  (
1  /  ( A ^ N ) ) ) )
2210, 16, 213bitr4d 285 1  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( M  <  N  <->  ( A ^ N )  <  ( A ^ M ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4434  (class class class)co 6278   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    < clt 9628    / cdiv 10209   ZZcz 10867   RR+crp 11226   ^cexp 12142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-2nd 6783  df-recs 7041  df-rdg 7075  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-div 10210  df-nn 10540  df-n0 10799  df-z 10868  df-uz 11088  df-rp 11227  df-seq 12084  df-exp 12143
This theorem is referenced by:  ltexp2rd  12310
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