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Theorem expdivap 10602
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 8675 . . . . 5 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
213expb 1206 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
323adant3 1019 . . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
43oveq1d 5911 . 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( A / B ) ^ N ) = ( ( A x. ( 1 / B ) ) ^ N ) )
5 recclap 8666 . . 3 |- ( ( B e. CC /\ B # 0 ) -> ( 1 / B ) e. CC )
6 mulexp 10590 . . 3 |- ( ( A e. CC /\ ( 1 / B ) e. CC /\ N e. NN0 ) -> ( ( A x. ( 1 / B ) ) ^ N ) = ( ( A ^ N ) x. ( ( 1 / B ) ^ N ) ) )
75, 6syl3an2 1283 . 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 1025 . . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> B e. CC )
9 simp2r 1026 . . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> B # 0 )
10 nn0z 9303 . . . . . 6 |- ( N e. NN0 -> N e. ZZ )
11103ad2ant3 1022 . . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> N e. ZZ )
12 exprecap 10592 . . . . 5 |- ( ( B e. CC /\ B # 0 /\ N e. ZZ ) -> ( ( 1 / B ) ^ N ) = ( 1 / ( B ^ N ) ) )
138, 9, 11, 12syl3anc 1249 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( 1 / B ) ^ N ) = ( 1 / ( B ^ N ) ) )
1413oveq2d 5912 . . 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 10569 . . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
16153adant2 1018 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( A ^ N ) e. CC )
17 expcl 10569 . . . . . 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 1017 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( B ^ N ) e. CC )
20 expap0i 10583 . . . . 5 |- ( ( B e. CC /\ B # 0 /\ N e. ZZ ) -> ( B ^ N ) # 0 )
218, 9, 11, 20syl3anc 1249 . . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( B ^ N ) # 0 )
2216, 19, 21divrecapd 8780 . . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( A ^ N ) / ( B ^ N ) ) = ( ( A ^ N ) x. ( 1 / ( B ^ N ) ) ) )
2314, 22eqtr4d 2225 . 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 2226 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 980   = wceq 1364   e. wcel 2160   class class class wbr 4018  (class class class)co 5896   CCcc 7839   0cc0 7841   1c1 7842   x. cmul 7846   # cap 8568   / cdiv 8659   NN0cn0 9206   ZZcz 9283   ^cexp 10550
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-n0 9207  df-z 9284  df-uz 9559  df-seqfrec 10477  df-exp 10551
This theorem is referenced by:  expdivapd  10699
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