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Theorem expcan 12173
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 6283 . . . . . . 7  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
2 oveq2 6283 . . . . . . 7  |-  ( x  =  M  ->  ( A ^ x )  =  ( A ^ M
) )
3 oveq2 6283 . . . . . . 7  |-  ( x  =  N  ->  ( A ^ x )  =  ( A ^ N
) )
4 zssre 10860 . . . . . . 7  |-  ZZ  C_  RR
5 simpl 457 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR )
6 0red 9586 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  e.  RR )
7 1red 9600 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  e.  RR )
8 0lt1 10064 . . . . . . . . . . . 12  |-  0  <  1
98a1i 11 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  1 )
10 simpr 461 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
1  <  A )
116, 7, 5, 9, 10lttrd 9731 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <  A )  -> 
0  <  A )
125, 11elrpd 11243 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <  A )  ->  A  e.  RR+ )
13 rpexpcl 12141 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  x  e.  ZZ )  ->  ( A ^ x )  e.  RR+ )
1412, 13sylan 471 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  RR+ )
1514rpred 11245 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  RR )
16 simpll 753 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  A  e.  RR )
17 simprl 755 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  x  e.  ZZ )
18 simprr 756 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
y  e.  ZZ )
19 simplr 754 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
1  <  A )
20 ltexp2a 12172 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  x  e.  ZZ  /\  y  e.  ZZ )  /\  ( 1  <  A  /\  x  <  y ) )  ->  ( A ^ x )  < 
( A ^ y
) )
2120expr 615 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  x  e.  ZZ  /\  y  e.  ZZ )  /\  1  <  A )  ->  ( x  < 
y  ->  ( A ^ x )  < 
( A ^ y
) ) )
2216, 17, 18, 19, 21syl31anc 1226 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( x  <  y  ->  ( A ^ x
)  <  ( A ^ y ) ) )
231, 2, 3, 4, 15, 22eqord1 10070 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  =  N  <-> 
( A ^ M
)  =  ( A ^ N ) ) )
2423ancom2s 800 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <  A )  /\  ( N  e.  ZZ  /\  M  e.  ZZ ) )  -> 
( M  =  N  <-> 
( A ^ M
)  =  ( A ^ N ) ) )
2524exp43 612 . . . 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 1204 . 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 968    = wceq 1374    e. wcel 1762   class class class wbr 4440  (class class class)co 6275   RRcr 9480   0cc0 9481   1c1 9482    < clt 9617   ZZcz 10853   RR+crp 11209   ^cexp 12122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-seq 12064  df-exp 12123
This theorem is referenced by:  expcand  12296
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