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Theorem ltexp2 12197
Description: Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
Assertion
Ref Expression
ltexp2  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( M  < 
N  <->  ( A ^ M )  <  ( A ^ N ) ) )

Proof of Theorem ltexp2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6302 . . . . . 6  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
2 oveq2 6302 . . . . . 6  |-  ( x  =  M  ->  ( A ^ x )  =  ( A ^ M
) )
3 oveq2 6302 . . . . . 6  |-  ( x  =  N  ->  ( A ^ x )  =  ( A ^ N
) )
4 zssre 10881 . . . . . 6  |-  ZZ  C_  RR
5 simpl 457 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR )
6 0red 9607 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  e.  RR )
7 1red 9621 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  e.  RR )
8 0lt1 10085 . . . . . . . . . . 11  |-  0  <  1
98a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  1 )
10 simpr 461 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  <  A )
116, 7, 5, 9, 10lttrd 9752 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  A )
125, 11elrpd 11264 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR+ )
13 rpexpcl 12163 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  x  e.  ZZ )  ->  ( A ^ x )  e.  RR+ )
1412, 13sylan 471 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  RR+ )
1514rpred 11266 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  RR )
16 simpll 753 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  A  e.  RR )
17 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  x  e.  ZZ )
18 simprr 756 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
y  e.  ZZ )
19 simplr 754 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
1  <  A )
20 ltexp2a 12195 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  x  e.  ZZ  /\  y  e.  ZZ )  /\  ( 1  <  A  /\  x  <  y ) )  ->  ( A ^ x )  < 
( A ^ y
) )
2120expr 615 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  x  e.  ZZ  /\  y  e.  ZZ )  /\  1  <  A )  ->  ( x  < 
y  ->  ( A ^ x )  < 
( A ^ y
) ) )
2216, 17, 18, 19, 21syl31anc 1231 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( x  <  y  ->  ( A ^ x
)  <  ( A ^ y ) ) )
231, 2, 3, 4, 15, 22ltord1 10089 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  <  N  <->  ( A ^ M )  <  ( A ^ N ) ) )
2423ancom2s 800 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( N  e.  ZZ  /\  M  e.  ZZ ) )  -> 
( M  <  N  <->  ( A ^ M )  <  ( A ^ N ) ) )
2524exp43 612 . . 3  |-  ( A  e.  RR  ->  (
1  <  A  ->  ( N  e.  ZZ  ->  ( M  e.  ZZ  ->  ( M  <  N  <->  ( A ^ M )  <  ( A ^ N ) ) ) ) ) )
2625com24 87 . 2  |-  ( A  e.  RR  ->  ( M  e.  ZZ  ->  ( N  e.  ZZ  ->  ( 1  <  A  -> 
( M  <  N  <->  ( A ^ M )  <  ( A ^ N ) ) ) ) ) )
27263imp1 1209 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    e. wcel 1767   class class class wbr 4452  (class class class)co 6294   RRcr 9501   0cc0 9502   1c1 9503    < clt 9638   ZZcz 10874   RR+crp 11230   ^cexp 12144
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 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-seq 12086  df-exp 12145
This theorem is referenced by:  leexp2  12198  ltexp2r  12200  ltexp2d  12317
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