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Theorem expcan 11170
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 5882 . . . . . . 7  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
2 oveq2 5882 . . . . . . 7  |-  ( x  =  M  ->  ( A ^ x )  =  ( A ^ M
) )
3 oveq2 5882 . . . . . . 7  |-  ( x  =  N  ->  ( A ^ x )  =  ( A ^ N
) )
4 zssre 10047 . . . . . . 7  |-  ZZ  C_  RR
5 simpl 443 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR )
6 0re 8854 . . . . . . . . . . . 12  |-  0  e.  RR
76a1i 10 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  e.  RR )
8 1re 8853 . . . . . . . . . . . 12  |-  1  e.  RR
98a1i 10 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  e.  RR )
10 0lt1 9312 . . . . . . . . . . . 12  |-  0  <  1
1110a1i 10 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  1 )
12 simpr 447 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  <  A )
137, 9, 5, 11, 12lttrd 8993 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  A )
145, 13elrpd 10404 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR+ )
15 rpexpcl 11138 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  x  e.  ZZ )  ->  ( A ^ x )  e.  RR+ )
1614, 15sylan 457 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  RR+ )
1716rpred 10406 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  RR )
18 simpll 730 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  A  e.  RR )
19 simprl 732 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  x  e.  ZZ )
20 simprr 733 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
y  e.  ZZ )
21 simplr 731 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
1  <  A )
22 ltexp2a 11169 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  x  e.  ZZ  /\  y  e.  ZZ )  /\  ( 1  <  A  /\  x  <  y ) )  ->  ( A ^ x )  < 
( A ^ y
) )
2322expr 598 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  x  e.  ZZ  /\  y  e.  ZZ )  /\  1  <  A )  ->  ( x  < 
y  ->  ( A ^ x )  < 
( A ^ y
) ) )
2418, 19, 20, 21, 23syl31anc 1185 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( x  <  y  ->  ( A ^ x
)  <  ( A ^ y ) ) )
251, 2, 3, 4, 17, 24eqord1 9317 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  =  N  <-> 
( A ^ M
)  =  ( A ^ N ) ) )
2625ancom2s 777 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( N  e.  ZZ  /\  M  e.  ZZ ) )  -> 
( M  =  N  <-> 
( A ^ M
)  =  ( A ^ N ) ) )
2726exp43 595 . . . 4  |-  ( A  e.  RR  ->  (
1  <  A  ->  ( N  e.  ZZ  ->  ( M  e.  ZZ  ->  ( M  =  N  <->  ( A ^ M )  =  ( A ^ N ) ) ) ) ) )
2827com24 81 . . 3  |-  ( A  e.  RR  ->  ( M  e.  ZZ  ->  ( N  e.  ZZ  ->  ( 1  <  A  -> 
( M  =  N  <-> 
( A ^ M
)  =  ( A ^ N ) ) ) ) ) )
29283imp1 1164 . 2  |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( M  =  N  <->  ( A ^ M )  =  ( A ^ N ) ) )
3029bicomd 192 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    < clt 8883   ZZcz 10040   RR+crp 10370   ^cexp 11120
This theorem is referenced by:  expcand  11292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121
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