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Theorem expcan 11900
Description: Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expcan  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( ( A ^ M )  =  ( A ^ N
)  <->  M  =  N
) )

Proof of Theorem expcan
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6088 . . . . . . 7  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
2 oveq2 6088 . . . . . . 7  |-  ( x  =  M  ->  ( A ^ x )  =  ( A ^ M
) )
3 oveq2 6088 . . . . . . 7  |-  ( x  =  N  ->  ( A ^ x )  =  ( A ^ N
) )
4 zssre 10641 . . . . . . 7  |-  ZZ  C_  RR
5 simpl 454 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR )
6 0red 9375 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  e.  RR )
7 1red 9389 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  e.  RR )
8 0lt1 9850 . . . . . . . . . . . 12  |-  0  <  1
98a1i 11 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  1 )
10 simpr 458 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  <  A )
116, 7, 5, 9, 10lttrd 9520 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  A )
125, 11elrpd 11013 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR+ )
13 rpexpcl 11868 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  x  e.  ZZ )  ->  ( A ^ x )  e.  RR+ )
1412, 13sylan 468 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  RR+ )
1514rpred 11015 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  RR )
16 simpll 746 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  A  e.  RR )
17 simprl 748 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  x  e.  ZZ )
18 simprr 749 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
y  e.  ZZ )
19 simplr 747 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
1  <  A )
20 ltexp2a 11899 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  x  e.  ZZ  /\  y  e.  ZZ )  /\  ( 1  <  A  /\  x  <  y ) )  ->  ( A ^ x )  < 
( A ^ y
) )
2120expr 610 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  x  e.  ZZ  /\  y  e.  ZZ )  /\  1  <  A )  ->  ( x  < 
y  ->  ( A ^ x )  < 
( A ^ y
) ) )
2216, 17, 18, 19, 21syl31anc 1214 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( x  <  y  ->  ( A ^ x
)  <  ( A ^ y ) ) )
231, 2, 3, 4, 15, 22eqord1 9856 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  =  N  <-> 
( A ^ M
)  =  ( A ^ N ) ) )
2423ancom2s 793 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( N  e.  ZZ  /\  M  e.  ZZ ) )  -> 
( M  =  N  <-> 
( A ^ M
)  =  ( A ^ N ) ) )
2524exp43 607 . . . 4  |-  ( A  e.  RR  ->  (
1  <  A  ->  ( N  e.  ZZ  ->  ( M  e.  ZZ  ->  ( M  =  N  <->  ( A ^ M )  =  ( A ^ N ) ) ) ) ) )
2625com24 87 . . 3  |-  ( A  e.  RR  ->  ( M  e.  ZZ  ->  ( N  e.  ZZ  ->  ( 1  <  A  -> 
( M  =  N  <-> 
( A ^ M
)  =  ( A ^ N ) ) ) ) ) )
27263imp1 1193 . 2  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( M  =  N  <->  ( A ^ M )  =  ( A ^ N ) ) )
2827bicomd 201 1  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( ( A ^ M )  =  ( A ^ N
)  <->  M  =  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   class class class wbr 4280  (class class class)co 6080   RRcr 9269   0cc0 9270   1c1 9271    < clt 9406   ZZcz 10634   RR+crp 10979   ^cexp 11849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-seq 11791  df-exp 11850
This theorem is referenced by:  expcand  12023
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