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Theorem ltexp2r 11935
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 991 . . . . 5  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  A  e.  RR+ )
21rpcnd 11044 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  A  e.  CC )
31rpne0d 11047 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  A  =/=  0 )
4 simpl2 992 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  M  e.  ZZ )
5 exprec 11920 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  M  e.  ZZ )  ->  (
( 1  /  A
) ^ M )  =  ( 1  / 
( A ^ M
) ) )
62, 3, 4, 5syl3anc 1218 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( (
1  /  A ) ^ M )  =  ( 1  /  ( A ^ M ) ) )
7 simpl3 993 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  N  e.  ZZ )
8 exprec 11920 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
( 1  /  A
) ^ N )  =  ( 1  / 
( A ^ N
) ) )
92, 3, 7, 8syl3anc 1218 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( (
1  /  A ) ^ N )  =  ( 1  /  ( A ^ N ) ) )
106, 9breq12d 4320 . 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 11053 . . 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 11055 . . . 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 11932 . . 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 1221 . 2  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( M  <  N  <->  ( ( 1  /  A ) ^ M )  <  (
( 1  /  A
) ^ N ) ) )
17 rpexpcl 11899 . . . 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 11899 . . . 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 11060 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   class class class wbr 4307  (class class class)co 6106   CCcc 9295   RRcr 9296   0cc0 9297   1c1 9298    < clt 9433    / cdiv 10008   ZZcz 10661   RR+crp 11006   ^cexp 11880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-2nd 6593  df-recs 6847  df-rdg 6881  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-n0 10595  df-z 10662  df-uz 10877  df-rp 11007  df-seq 11822  df-exp 11881
This theorem is referenced by:  ltexp2rd  12047
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