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Theorem ltexp2a 12181
Description: Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
ltexp2a  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( A ^ M )  <  ( A ^ N ) )

Proof of Theorem ltexp2a
StepHypRef Expression
1 simpl1 999 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  A  e.  RR )
2 0red 9593 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  0  e.  RR )
3 1red 9607 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  1  e.  RR )
4 0lt1 10071 . . . . . . . . 9  |-  0  <  1
54a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  0  <  1 )
6 simprl 755 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  1  <  A )
72, 3, 1, 5, 6lttrd 9738 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  0  <  A )
81, 7elrpd 11250 . . . . . 6  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  A  e.  RR+ )
9 simpl2 1000 . . . . . 6  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  M  e.  ZZ )
10 rpexpcl 12149 . . . . . 6  |-  ( ( A  e.  RR+  /\  M  e.  ZZ )  ->  ( A ^ M )  e.  RR+ )
118, 9, 10syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( A ^ M )  e.  RR+ )
1211rpred 11252 . . . 4  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( A ^ M )  e.  RR )
1312recnd 9618 . . 3  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( A ^ M )  e.  CC )
1413mulid2d 9610 . 2  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( 1  x.  ( A ^ M ) )  =  ( A ^ M
) )
15 simprr 756 . . . . . 6  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  M  <  N )
16 simpl3 1001 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  N  e.  ZZ )
17 znnsub 10905 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  ( N  -  M )  e.  NN ) )
189, 16, 17syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( M  <  N  <->  ( N  -  M )  e.  NN ) )
1915, 18mpbid 210 . . . . 5  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( N  -  M )  e.  NN )
20 expgt1 12168 . . . . 5  |-  ( ( A  e.  RR  /\  ( N  -  M
)  e.  NN  /\  1  <  A )  -> 
1  <  ( A ^ ( N  -  M ) ) )
211, 19, 6, 20syl3anc 1228 . . . 4  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  1  <  ( A ^ ( N  -  M ) ) )
221recnd 9618 . . . . 5  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  A  e.  CC )
237gt0ne0d 10113 . . . . 5  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  A  =/=  0 )
24 expsub 12177 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( N  e.  ZZ  /\  M  e.  ZZ ) )  -> 
( A ^ ( N  -  M )
)  =  ( ( A ^ N )  /  ( A ^ M ) ) )
2522, 23, 16, 9, 24syl22anc 1229 . . . 4  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( A ^ ( N  -  M ) )  =  ( ( A ^ N )  /  ( A ^ M ) ) )
2621, 25breqtrd 4471 . . 3  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  1  <  ( ( A ^ N
)  /  ( A ^ M ) ) )
27 rpexpcl 12149 . . . . . 6  |-  ( ( A  e.  RR+  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  RR+ )
288, 16, 27syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( A ^ N )  e.  RR+ )
2928rpred 11252 . . . 4  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( A ^ N )  e.  RR )
303, 29, 11ltmuldivd 11295 . . 3  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( (
1  x.  ( A ^ M ) )  <  ( A ^ N )  <->  1  <  ( ( A ^ N
)  /  ( A ^ M ) ) ) )
3126, 30mpbird 232 . 2  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( 1  x.  ( A ^ M ) )  < 
( A ^ N
) )
3214, 31eqbrtrrd 4469 1  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( A ^ M )  <  ( A ^ N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    x. cmul 9493    < clt 9624    - cmin 9801    / cdiv 10202   NNcn 10532   ZZcz 10860   RR+crp 11216   ^cexp 12130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-seq 12072  df-exp 12131
This theorem is referenced by:  expcan  12182  ltexp2  12183  expnass  12237  perfectlem2  23233  2sqblem  23380
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