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Theorem ltexp2r 11158
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 958 . . . . 5  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  A  e.  RR+ )
21rpcnd 10392 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  A  e.  CC )
31rpne0d 10395 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  A  =/=  0 )
4 simpl2 959 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  M  e.  ZZ )
5 exprec 11143 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  M  e.  ZZ )  ->  (
( 1  /  A
) ^ M )  =  ( 1  / 
( A ^ M
) ) )
62, 3, 4, 5syl3anc 1182 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( (
1  /  A ) ^ M )  =  ( 1  /  ( A ^ M ) ) )
7 simpl3 960 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  N  e.  ZZ )
8 exprec 11143 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  (
( 1  /  A
) ^ N )  =  ( 1  / 
( A ^ N
) ) )
92, 3, 7, 8syl3anc 1182 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( (
1  /  A ) ^ N )  =  ( 1  /  ( A ^ N ) ) )
106, 9breq12d 4036 . 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 10401 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( 1  /  A )  e.  RR )
12 simpr 447 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  A  <  1 )
131reclt1d 10403 . . . 4  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( A  <  1  <->  1  <  (
1  /  A ) ) )
1412, 13mpbid 201 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  1  <  ( 1  /  A ) )
15 ltexp2 11155 . . 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 1185 . 2  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( M  <  N  <->  ( ( 1  /  A ) ^ M )  <  (
( 1  /  A
) ^ N ) ) )
17 rpexpcl 11122 . . . 4  |-  ( ( A  e.  RR+  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  RR+ )
181, 7, 17syl2anc 642 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( A ^ N )  e.  RR+ )
19 rpexpcl 11122 . . . 4  |-  ( ( A  e.  RR+  /\  M  e.  ZZ )  ->  ( A ^ M )  e.  RR+ )
201, 4, 19syl2anc 642 . . 3  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( A ^ M )  e.  RR+ )
2118, 20ltrecd 10408 . 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 276 1  |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
)  ->  ( M  <  N  <->  ( A ^ N )  <  ( A ^ M ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    < clt 8867    / cdiv 9423   ZZcz 10024   RR+crp 10354   ^cexp 11104
This theorem is referenced by:  ltexp2rd  11269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105
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