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Theorem expsub 11386
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 10273 . . 3  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
2 expaddz 11383 . . 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 635 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ ( M  +  -u N ) )  =  ( ( A ^ M )  x.  ( A ^ -u N ) ) )
4 zcn 10247 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  CC )
5 zcn 10247 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  CC )
6 negsub 9309 . . . . 5  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  +  -u N )  =  ( M  -  N ) )
74, 5, 6syl2an 464 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  -u N )  =  ( M  -  N ) )
87adantl 453 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  +  -u N )  =  ( M  -  N ) )
98oveq2d 6060 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ ( M  +  -u N ) )  =  ( A ^ ( M  -  N ) ) )
10 expnegz 11373 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
11103expa 1153 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  N  e.  ZZ )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
1211adantrl 697 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
1312oveq2d 6060 . . 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 11365 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  M  e.  ZZ )  ->  ( A ^ M )  e.  CC )
15143expa 1153 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  M  e.  ZZ )  ->  ( A ^ M )  e.  CC )
1615adantrr 698 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ M
)  e.  CC )
17 expclz 11365 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  CC )
18173expa 1153 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  CC )
1918adantrl 697 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ N
)  e.  CC )
20 expne0i 11371 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  =/=  0 )
21203expa 1153 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  N  e.  ZZ )  ->  ( A ^ N )  =/=  0
)
2221adantrl 697 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( A ^ N
)  =/=  0 )
2316, 19, 22divrecd 9753 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( A ^ M )  /  ( A ^ N ) )  =  ( ( A ^ M )  x.  ( 1  /  ( A ^ N ) ) ) )
2413, 23eqtr4d 2443 . 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 2448 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 359    = wceq 1649    e. wcel 1721    =/= wne 2571  (class class class)co 6044   CCcc 8948   0cc0 8950   1c1 8951    + caddc 8953    x. cmul 8955    - cmin 9251   -ucneg 9252    / cdiv 9637   ZZcz 10242   ^cexp 11341
This theorem is referenced by:  expm1  11388  ltexp2a  11390  leexp2a  11394  iexpcyc  11444  expmulnbnd  11470  expsubd  11493  aaliou3lem8  20219  m1expaddsub  27293  psgnuni  27294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-n0 10182  df-z 10243  df-uz 10449  df-seq 11283  df-exp 11342
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