Theorem mulexpzap 10495

Description: Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)

Assertion

Ref	Expression
mulexpzap	|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ N e. ZZ ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )

Proof of Theorem mulexpzap

Step	Hyp	Ref	Expression
1		elznn0nn 9205	. . 3 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2		simpl 108	. . . . . 6 |- ( ( A e. CC /\ A # 0 ) -> A e. CC )
3		simpl 108	. . . . . 6 |- ( ( B e. CC /\ B # 0 ) -> B e. CC )
4	2, 3	anim12i 336	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( A e. CC /\ B e. CC ) )
5		mulexp 10494	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
6	5	3expa 1193	. . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
7	4, 6	sylan 281	. . . 4 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
8		simplll 523	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC )
9		simplrl 525	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> B e. CC )
10	8, 9	mulcld 7919	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A x. B ) e. CC )
11		simpllr 524	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> A # 0 )
12		simplrr 526	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> B # 0 )
13	8, 9, 11, 12	mulap0d 8555	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A x. B ) # 0 )
14		recn 7886	. . . . . . 7 |- ( N e. RR -> N e. CC )
15	14	ad2antrl 482	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
16		nnnn0 9121	. . . . . . 7 |- ( -u N e. NN -> -u N e. NN0 )
17	16	ad2antll 483	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN0 )
18		expineg2 10464	. . . . . 6 |- ( ( ( ( A x. B ) e. CC /\ ( A x. B ) # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) -> ( ( A x. B ) ^ N ) = ( 1 / ( ( A x. B ) ^ -u N ) ) )
19	10, 13, 15, 17, 18	syl22anc 1229	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( A x. B ) ^ N ) = ( 1 / ( ( A x. B ) ^ -u N ) ) )
20		expineg2 10464	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
21	8, 11, 15, 17, 20	syl22anc 1229	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
22		expineg2 10464	. . . . . . . 8 |- ( ( ( B e. CC /\ B # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) -> ( B ^ N ) = ( 1 / ( B ^ -u N ) ) )
23	9, 12, 15, 17, 22	syl22anc 1229	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( B ^ N ) = ( 1 / ( B ^ -u N ) ) )
24	21, 23	oveq12d 5860	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( A ^ N ) x. ( B ^ N ) ) = ( ( 1 / ( A ^ -u N ) ) x. ( 1 / ( B ^ -u N ) ) ) )
25		mulexp 10494	. . . . . . . . . 10 |- ( ( A e. CC /\ B e. CC /\ -u N e. NN0 ) -> ( ( A x. B ) ^ -u N ) = ( ( A ^ -u N ) x. ( B ^ -u N ) ) )
26	8, 9, 17, 25	syl3anc 1228	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( A x. B ) ^ -u N ) = ( ( A ^ -u N ) x. ( B ^ -u N ) ) )
27	26	oveq2d 5858	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( 1 / ( ( A x. B ) ^ -u N ) ) = ( 1 / ( ( A ^ -u N ) x. ( B ^ -u N ) ) ) )
28		1t1e1 9009	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
29	28	oveq1i 5852	. . . . . . . 8 |- ( ( 1 x. 1 ) / ( ( A ^ -u N ) x. ( B ^ -u N ) ) ) = ( 1 / ( ( A ^ -u N ) x. ( B ^ -u N ) ) )
30	27, 29	eqtr4di 2217	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( 1 / ( ( A x. B ) ^ -u N ) ) = ( ( 1 x. 1 ) / ( ( A ^ -u N ) x. ( B ^ -u N ) ) ) )
31		expcl 10473	. . . . . . . . 9 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( A ^ -u N ) e. CC )
32	8, 17, 31	syl2anc 409	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) e. CC )
33		nnz 9210	. . . . . . . . . 10 |- ( -u N e. NN -> -u N e. ZZ )
34	33	ad2antll 483	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
35		expap0i 10487	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
36	8, 11, 34, 35	syl3anc 1228	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) # 0 )
37		expcl 10473	. . . . . . . . 9 |- ( ( B e. CC /\ -u N e. NN0 ) -> ( B ^ -u N ) e. CC )
38	9, 17, 37	syl2anc 409	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( B ^ -u N ) e. CC )
39		expap0i 10487	. . . . . . . . 9 |- ( ( B e. CC /\ B # 0 /\ -u N e. ZZ ) -> ( B ^ -u N ) # 0 )
40	9, 12, 34, 39	syl3anc 1228	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( B ^ -u N ) # 0 )
41		ax-1cn 7846	. . . . . . . . 9 |- 1 e. CC
42		divmuldivap 8608	. . . . . . . . 9 |- ( ( ( 1 e. CC /\ 1 e. CC ) /\ ( ( ( A ^ -u N ) e. CC /\ ( A ^ -u N ) # 0 ) /\ ( ( B ^ -u N ) e. CC /\ ( B ^ -u N ) # 0 ) ) ) -> ( ( 1 / ( A ^ -u N ) ) x. ( 1 / ( B ^ -u N ) ) ) = ( ( 1 x. 1 ) / ( ( A ^ -u N ) x. ( B ^ -u N ) ) ) )
43	41, 41, 42	mpanl12 433	. . . . . . . 8 |- ( ( ( ( A ^ -u N ) e. CC /\ ( A ^ -u N ) # 0 ) /\ ( ( B ^ -u N ) e. CC /\ ( B ^ -u N ) # 0 ) ) -> ( ( 1 / ( A ^ -u N ) ) x. ( 1 / ( B ^ -u N ) ) ) = ( ( 1 x. 1 ) / ( ( A ^ -u N ) x. ( B ^ -u N ) ) ) )
44	32, 36, 38, 40, 43	syl22anc 1229	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( 1 / ( A ^ -u N ) ) x. ( 1 / ( B ^ -u N ) ) ) = ( ( 1 x. 1 ) / ( ( A ^ -u N ) x. ( B ^ -u N ) ) ) )
45	30, 44	eqtr4d 2201	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( 1 / ( ( A x. B ) ^ -u N ) ) = ( ( 1 / ( A ^ -u N ) ) x. ( 1 / ( B ^ -u N ) ) ) )
46	24, 45	eqtr4d 2201	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( A ^ N ) x. ( B ^ N ) ) = ( 1 / ( ( A x. B ) ^ -u N ) ) )
47	19, 46	eqtr4d 2201	. . . 4 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
48	7, 47	jaodan 787	. . 3 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
49	1, 48	sylan2b 285	. 2 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ N e. ZZ ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
50	49	3impa 1184	1 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ N e. ZZ ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   1c1 7754   x. cmul 7758   -ucneg 8070   # cap 8479   / cdiv 8568   NNcn 8857   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:  exprecap  10496