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Theorem expdivap 10506
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 8584 . . . . 5 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
213expb 1194 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
323adant3 1007 . . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
43oveq1d 5857 . 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( A / B ) ^ N ) = ( ( A x. ( 1 / B ) ) ^ N ) )
5 recclap 8575 . . 3 |- ( ( B e. CC /\ B # 0 ) -> ( 1 / B ) e. CC )
6 mulexp 10494 . . 3 |- ( ( A e. CC /\ ( 1 / B ) e. CC /\ N e. NN0 ) -> ( ( A x. ( 1 / B ) ) ^ N ) = ( ( A ^ N ) x. ( ( 1 / B ) ^ N ) ) )
75, 6syl3an2 1262 . 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 1013 . . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> B e. CC )
9 simp2r 1014 . . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> B # 0 )
10 nn0z 9211 . . . . . 6 |- ( N e. NN0 -> N e. ZZ )
11103ad2ant3 1010 . . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> N e. ZZ )
12 exprecap 10496 . . . . 5 |- ( ( B e. CC /\ B # 0 /\ N e. ZZ ) -> ( ( 1 / B ) ^ N ) = ( 1 / ( B ^ N ) ) )
138, 9, 11, 12syl3anc 1228 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( 1 / B ) ^ N ) = ( 1 / ( B ^ N ) ) )
1413oveq2d 5858 . . 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 10473 . . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
16153adant2 1006 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( A ^ N ) e. CC )
17 expcl 10473 . . . . . 6 |- ( ( B e. CC /\ N e. NN0 ) -> ( B ^ N ) e. CC )
1817adantlr 469 . . . . 5 |- ( ( ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( B ^ N ) e. CC )
19183adant1 1005 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( B ^ N ) e. CC )
20 expap0i 10487 . . . . 5 |- ( ( B e. CC /\ B # 0 /\ N e. ZZ ) -> ( B ^ N ) # 0 )
218, 9, 11, 20syl3anc 1228 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( B ^ N ) # 0 )
2216, 19, 21divrecapd 8689 . . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( A ^ N ) / ( B ^ N ) ) = ( ( A ^ N ) x. ( 1 / ( B ^ N ) ) ) )
2314, 22eqtr4d 2201 . 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 2202 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 103   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   x. cmul 7758   # cap 8479   / cdiv 8568   NN0cn0 9114   ZZcz 9191   ^cexp 10454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-seqfrec 10381  df-exp 10455
This theorem is referenced by:  expdivapd  10602
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