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Theorem expsub 11242
Description: Exponent subtraction law for nonnegative integer exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expsub  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ ( M  -  N )
)  =  ( ( A ^ M )  /  ( A ^ N ) ) )

Proof of Theorem expsub
StepHypRef Expression
1 znegcl 10147 . . 3  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
2 expaddz 11239 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  -u N  e.  ZZ ) )  -> 
( A ^ ( M  +  -u N ) )  =  ( ( A ^ M )  x.  ( A ^ -u N ) ) )
31, 2sylanr2 634 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ ( M  +  -u N ) )  =  ( ( A ^ M )  x.  ( A ^ -u N ) ) )
4 zcn 10121 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  CC )
5 zcn 10121 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  CC )
6 negsub 9185 . . . . 5  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  +  -u N )  =  ( M  -  N ) )
74, 5, 6syl2an 463 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  -u N )  =  ( M  -  N ) )
87adantl 452 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  +  -u N )  =  ( M  -  N ) )
98oveq2d 5961 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ ( M  +  -u N ) )  =  ( A ^ ( M  -  N ) ) )
10 expnegz 11229 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
11103expa 1151 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  N  e.  ZZ )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
1211adantrl 696 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
1312oveq2d 5961 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( A ^ M )  x.  ( A ^ -u N ) )  =  ( ( A ^ M )  x.  ( 1  / 
( A ^ N
) ) ) )
14 expclz 11221 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  M  e.  ZZ )  ->  ( A ^ M )  e.  CC )
15143expa 1151 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  ZZ )  ->  ( A ^ M )  e.  CC )
1615adantrr 697 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ M
)  e.  CC )
17 expclz 11221 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  CC )
18173expa 1151 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  CC )
1918adantrl 696 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ N
)  e.  CC )
20 expne0i 11227 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  =/=  0 )
21203expa 1151 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  N  e.  ZZ )  ->  ( A ^ N )  =/=  0
)
2221adantrl 696 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ N
)  =/=  0 )
2316, 19, 22divrecd 9629 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( A ^ M )  /  ( A ^ N ) )  =  ( ( A ^ M )  x.  ( 1  /  ( A ^ N ) ) ) )
2413, 23eqtr4d 2393 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( A ^ M )  x.  ( A ^ -u N ) )  =  ( ( A ^ M )  /  ( A ^ N ) ) )
253, 9, 243eqtr3d 2398 1  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ ( M  -  N )
)  =  ( ( A ^ M )  /  ( A ^ N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521  (class class class)co 5945   CCcc 8825   0cc0 8827   1c1 8828    + caddc 8830    x. cmul 8832    - cmin 9127   -ucneg 9128    / cdiv 9513   ZZcz 10116   ^cexp 11197
This theorem is referenced by:  expm1  11244  ltexp2a  11246  leexp2a  11250  iexpcyc  11300  expmulnbnd  11326  expsubd  11349  aaliou3lem8  19829  m1expaddsub  26744  psgnuni  26745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-n0 10058  df-z 10117  df-uz 10323  df-seq 11139  df-exp 11198
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