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Theorem expdivap 9694
Description: Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
Assertion
Ref Expression
expdivap  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( ( A  /  B ) ^ N
)  =  ( ( A ^ N )  /  ( B ^ N ) ) )

Proof of Theorem expdivap
StepHypRef Expression
1 divrecap 7913 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  =  ( A  x.  (
1  /  B ) ) )
213expb 1140 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( A  /  B )  =  ( A  x.  ( 1  /  B ) ) )
323adant3 959 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( A  /  B
)  =  ( A  x.  ( 1  /  B ) ) )
43oveq1d 5579 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( ( A  /  B ) ^ N
)  =  ( ( A  x.  ( 1  /  B ) ) ^ N ) )
5 recclap 7904 . . 3  |-  ( ( B  e.  CC  /\  B #  0 )  ->  (
1  /  B )  e.  CC )
6 mulexp 9682 . . 3  |-  ( ( A  e.  CC  /\  ( 1  /  B
)  e.  CC  /\  N  e.  NN0 )  -> 
( ( A  x.  ( 1  /  B
) ) ^ N
)  =  ( ( A ^ N )  x.  ( ( 1  /  B ) ^ N ) ) )
75, 6syl3an2 1204 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( ( A  x.  ( 1  /  B
) ) ^ N
)  =  ( ( A ^ N )  x.  ( ( 1  /  B ) ^ N ) ) )
8 simp2l 965 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  ->  B  e.  CC )
9 simp2r 966 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  ->  B #  0 )
10 nn0z 8522 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  ZZ )
11103ad2ant3 962 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  ->  N  e.  ZZ )
12 exprecap 9684 . . . . 5  |-  ( ( B  e.  CC  /\  B #  0  /\  N  e.  ZZ )  ->  (
( 1  /  B
) ^ N )  =  ( 1  / 
( B ^ N
) ) )
138, 9, 11, 12syl3anc 1170 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( ( 1  /  B ) ^ N
)  =  ( 1  /  ( B ^ N ) ) )
1413oveq2d 5580 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( ( A ^ N )  x.  (
( 1  /  B
) ^ N ) )  =  ( ( A ^ N )  x.  ( 1  / 
( B ^ N
) ) ) )
15 expcl 9661 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  CC )
16153adant2 958 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  CC )
17 expcl 9661 . . . . . 6  |-  ( ( B  e.  CC  /\  N  e.  NN0 )  -> 
( B ^ N
)  e.  CC )
1817adantlr 461 . . . . 5  |-  ( ( ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( B ^ N
)  e.  CC )
19183adant1 957 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( B ^ N
)  e.  CC )
20 expap0i 9675 . . . . 5  |-  ( ( B  e.  CC  /\  B #  0  /\  N  e.  ZZ )  ->  ( B ^ N ) #  0 )
218, 9, 11, 20syl3anc 1170 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( B ^ N
) #  0 )
2216, 19, 21divrecapd 8017 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( ( A ^ N )  /  ( B ^ N ) )  =  ( ( A ^ N )  x.  ( 1  /  ( B ^ N ) ) ) )
2314, 22eqtr4d 2118 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( ( A ^ N )  x.  (
( 1  /  B
) ^ N ) )  =  ( ( A ^ N )  /  ( B ^ N ) ) )
244, 7, 233eqtrd 2119 1  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  NN0 )  -> 
( ( A  /  B ) ^ N
)  =  ( ( A ^ N )  /  ( B ^ N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3805  (class class class)co 5564   CCcc 7111   0cc0 7113   1c1 7114    x. cmul 7118   # cap 7818    / cdiv 7897   NN0cn0 8425   ZZcz 8502   ^cexp 9642
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-mulrcl 7207  ax-addcom 7208  ax-mulcom 7209  ax-addass 7210  ax-mulass 7211  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-1rid 7215  ax-0id 7216  ax-rnegex 7217  ax-precex 7218  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-apti 7223  ax-pre-ltadd 7224  ax-pre-mulgt0 7225  ax-pre-mulext 7226
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-reap 7812  df-ap 7819  df-div 7898  df-inn 8177  df-n0 8426  df-z 8503  df-uz 8771  df-iseq 9592  df-iexp 9643
This theorem is referenced by:  expdivapd  9786
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