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Theorem expdivap 10807
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 8831 . . . . 5 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
213expb 1228 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
323adant3 1041 . . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
43oveq1d 6015 . 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( A / B ) ^ N ) = ( ( A x. ( 1 / B ) ) ^ N ) )
5 recclap 8822 . . 3 |- ( ( B e. CC /\ B # 0 ) -> ( 1 / B ) e. CC )
6 mulexp 10795 . . 3 |- ( ( A e. CC /\ ( 1 / B ) e. CC /\ N e. NN0 ) -> ( ( A x. ( 1 / B ) ) ^ N ) = ( ( A ^ N ) x. ( ( 1 / B ) ^ N ) ) )
75, 6syl3an2 1305 . 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 1047 . . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> B e. CC )
9 simp2r 1048 . . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> B # 0 )
10 nn0z 9462 . . . . . 6 |- ( N e. NN0 -> N e. ZZ )
11103ad2ant3 1044 . . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> N e. ZZ )
12 exprecap 10797 . . . . 5 |- ( ( B e. CC /\ B # 0 /\ N e. ZZ ) -> ( ( 1 / B ) ^ N ) = ( 1 / ( B ^ N ) ) )
138, 9, 11, 12syl3anc 1271 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( 1 / B ) ^ N ) = ( 1 / ( B ^ N ) ) )
1413oveq2d 6016 . . 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 10774 . . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
16153adant2 1040 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( A ^ N ) e. CC )
17 expcl 10774 . . . . . 6 |- ( ( B e. CC /\ N e. NN0 ) -> ( B ^ N ) e. CC )
1817adantlr 477 . . . . 5 |- ( ( ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( B ^ N ) e. CC )
19183adant1 1039 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( B ^ N ) e. CC )
20 expap0i 10788 . . . . 5 |- ( ( B e. CC /\ B # 0 /\ N e. ZZ ) -> ( B ^ N ) # 0 )
218, 9, 11, 20syl3anc 1271 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( B ^ N ) # 0 )
2216, 19, 21divrecapd 8936 . . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( A ^ N ) / ( B ^ N ) ) = ( ( A ^ N ) x. ( 1 / ( B ^ N ) ) ) )
2314, 22eqtr4d 2265 . 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 2266 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 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4082  (class class class)co 6000   CCcc 7993   0cc0 7995   1c1 7996   x. cmul 8000   # cap 8724   / cdiv 8815   NN0cn0 9365   ZZcz 9442   ^cexp 10755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-seqfrec 10665  df-exp 10756
This theorem is referenced by:  expdivapd  10904
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