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Theorem expsubap 9991
Description: Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
Assertion
Ref Expression
expsubap  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  -  N ) )  =  ( ( A ^ M )  /  ( A ^ N ) ) )

Proof of Theorem expsubap
StepHypRef Expression
1 znegcl 8771 . . 3  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
2 expaddzap 9987 . . 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 397 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  +  -u N ) )  =  ( ( A ^ M )  x.  ( A ^ -u N ) ) )
4 zcn 8745 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  CC )
5 zcn 8745 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  CC )
6 negsub 7720 . . . . 5  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( M  +  -u N )  =  ( M  -  N ) )
74, 5, 6syl2an 283 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  -u N )  =  ( M  -  N ) )
87adantl 271 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  +  -u N )  =  ( M  -  N
) )
98oveq2d 5660 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  +  -u N ) )  =  ( A ^ ( M  -  N )
) )
10 expnegzap 9977 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ -u N )  =  ( 1  / 
( A ^ N
) ) )
11103expa 1143 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  ZZ )  ->  ( A ^ -u N
)  =  ( 1  /  ( A ^ N ) ) )
1211adantrl 462 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
1312oveq2d 5660 . . 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 expclzap 9968 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  M  e.  ZZ )  ->  ( A ^ M )  e.  CC )
15143expa 1143 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  M  e.  ZZ )  ->  ( A ^ M
)  e.  CC )
1615adantrr 463 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ M )  e.  CC )
17 expclzap 9968 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  CC )
18173expa 1143 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  ZZ )  ->  ( A ^ N
)  e.  CC )
1918adantrl 462 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ N )  e.  CC )
20 expap0i 9975 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ N ) #  0 )
21203expa 1143 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  N  e.  ZZ )  ->  ( A ^ N
) #  0 )
2221adantrl 462 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ N ) #  0 )
2316, 19, 22divrecapd 8250 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( A ^ M )  / 
( A ^ N
) )  =  ( ( A ^ M
)  x.  ( 1  /  ( A ^ N ) ) ) )
2413, 23eqtr4d 2123 . 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 2128 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 102    = wceq 1289    e. wcel 1438   class class class wbr 3843  (class class class)co 5644   CCcc 7338   0cc0 7340   1c1 7341    + caddc 7343    x. cmul 7345    - cmin 7643   -ucneg 7644   # cap 8048    / cdiv 8129   ZZcz 8740   ^cexp 9942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-mulrcl 7434  ax-addcom 7435  ax-mulcom 7436  ax-addass 7437  ax-mulass 7438  ax-distr 7439  ax-i2m1 7440  ax-0lt1 7441  ax-1rid 7442  ax-0id 7443  ax-rnegex 7444  ax-precex 7445  ax-cnre 7446  ax-pre-ltirr 7447  ax-pre-ltwlin 7448  ax-pre-lttrn 7449  ax-pre-apti 7450  ax-pre-ltadd 7451  ax-pre-mulgt0 7452  ax-pre-mulext 7453
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3392  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-po 4121  df-iso 4122  df-iord 4191  df-on 4193  df-ilim 4194  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-frec 6148  df-pnf 7514  df-mnf 7515  df-xr 7516  df-ltxr 7517  df-le 7518  df-sub 7645  df-neg 7646  df-reap 8042  df-ap 8049  df-div 8130  df-inn 8413  df-n0 8664  df-z 8741  df-uz 9010  df-iseq 9841  df-seq3 9842  df-exp 9943
This theorem is referenced by:  expm1ap  9993  ltexp2a  9995  leexp2a  9996  iexpcyc  10047  expsubapd  10085
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