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Theorem expdivap 10527
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 8605 . . . . 5 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
213expb 1199 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
323adant3 1012 . . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
43oveq1d 5868 . 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( A / B ) ^ N ) = ( ( A x. ( 1 / B ) ) ^ N ) )
5 recclap 8596 . . 3 |- ( ( B e. CC /\ B # 0 ) -> ( 1 / B ) e. CC )
6 mulexp 10515 . . 3 |- ( ( A e. CC /\ ( 1 / B ) e. CC /\ N e. NN0 ) -> ( ( A x. ( 1 / B ) ) ^ N ) = ( ( A ^ N ) x. ( ( 1 / B ) ^ N ) ) )
75, 6syl3an2 1267 . 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 1018 . . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> B e. CC )
9 simp2r 1019 . . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> B # 0 )
10 nn0z 9232 . . . . . 6 |- ( N e. NN0 -> N e. ZZ )
11103ad2ant3 1015 . . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> N e. ZZ )
12 exprecap 10517 . . . . 5 |- ( ( B e. CC /\ B # 0 /\ N e. ZZ ) -> ( ( 1 / B ) ^ N ) = ( 1 / ( B ^ N ) ) )
138, 9, 11, 12syl3anc 1233 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( 1 / B ) ^ N ) = ( 1 / ( B ^ N ) ) )
1413oveq2d 5869 . . 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 10494 . . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
16153adant2 1011 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( A ^ N ) e. CC )
17 expcl 10494 . . . . . 6 |- ( ( B e. CC /\ N e. NN0 ) -> ( B ^ N ) e. CC )
1817adantlr 474 . . . . 5 |- ( ( ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( B ^ N ) e. CC )
19183adant1 1010 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( B ^ N ) e. CC )
20 expap0i 10508 . . . . 5 |- ( ( B e. CC /\ B # 0 /\ N e. ZZ ) -> ( B ^ N ) # 0 )
218, 9, 11, 20syl3anc 1233 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( B ^ N ) # 0 )
2216, 19, 21divrecapd 8710 . . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( A ^ N ) / ( B ^ N ) ) = ( ( A ^ N ) x. ( 1 / ( B ^ N ) ) ) )
2314, 22eqtr4d 2206 . 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 2207 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 973   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   x. cmul 7779   # cap 8500   / cdiv 8589   NN0cn0 9135   ZZcz 9212   ^cexp 10475
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402  df-exp 10476
This theorem is referenced by:  expdivapd  10623
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