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Theorem expcan 10662
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
StepHypRef Expression
1 simpl1 1000 . . . . 5  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  A  e.  RR )
2 simpl2 1001 . . . . 5  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  M  e.  ZZ )
3 simpl3 1002 . . . . 5  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  N  e.  ZZ )
4 simpr 110 . . . . 5  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  1  <  A
)
51, 2, 3, 4expcanlem 10661 . . . 4  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( ( A ^ M )  <_ 
( A ^ N
)  ->  M  <_  N ) )
61, 3, 2, 4expcanlem 10661 . . . 4  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( ( A ^ N )  <_ 
( A ^ M
)  ->  N  <_  M ) )
75, 6anim12d 335 . . 3  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( ( ( A ^ M )  <_  ( A ^ N )  /\  ( A ^ N )  <_ 
( A ^ M
) )  ->  ( M  <_  N  /\  N  <_  M ) ) )
8 0red 7933 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  0  e.  RR )
9 1red 7947 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  1  e.  RR )
10 0lt1 8058 . . . . . . . 8  |-  0  <  1
1110a1i 9 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  0  <  1
)
128, 9, 1, 11, 4lttrd 8057 . . . . . 6  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  0  <  A
)
131, 12gt0ap0d 8560 . . . . 5  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  A #  0 )
141, 13, 2reexpclzapd 10646 . . . 4  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( A ^ M )  e.  RR )
151, 13, 3reexpclzapd 10646 . . . 4  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( A ^ N )  e.  RR )
1614, 15letri3d 8047 . . 3  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( ( A ^ M )  =  ( A ^ N
)  <->  ( ( A ^ M )  <_ 
( A ^ N
)  /\  ( A ^ N )  <_  ( A ^ M ) ) ) )
172zred 9346 . . . 4  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  M  e.  RR )
183zred 9346 . . . 4  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  N  e.  RR )
1917, 18letri3d 8047 . . 3  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( M  =  N  <->  ( M  <_  N  /\  N  <_  M
) ) )
207, 16, 193imtr4d 203 . 2  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( ( A ^ M )  =  ( A ^ N
)  ->  M  =  N ) )
21 oveq2 5873 . 2  |-  ( M  =  N  ->  ( A ^ M )  =  ( A ^ N
) )
2220, 21impbid1 142 1  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( ( A ^ M )  =  ( A ^ N
)  <->  M  =  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2146   class class class wbr 3998  (class class class)co 5865   RRcr 7785   0cc0 7786   1c1 7787    < clt 7966    <_ cle 7967   ZZcz 9224   ^cexp 10487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-n0 9148  df-z 9225  df-uz 9500  df-rp 9623  df-seqfrec 10414  df-exp 10488
This theorem is referenced by:  expcand  10663
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