Theorem mulexpzap 10333

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 9068	. . 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 10332	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
6	5	3expa 1181	. . . . 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 522	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC )
9		simplrl 524	. . . . . . 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 7786	. . . . . 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 523	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> A # 0 )
12		simplrr 525	. . . . . . 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 8419	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A x. B ) # 0 )
14		recn 7753	. . . . . . 7 |- ( N e. RR -> N e. CC )
15	14	ad2antrl 481	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
16		nnnn0 8984	. . . . . . 7 |- ( -u N e. NN -> -u N e. NN0 )
17	16	ad2antll 482	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN0 )
18		expineg2 10302	. . . . . 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 1217	. . . . 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 10302	. . . . . . . 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 1217	. . . . . . 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 10302	. . . . . . . 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 1217	. . . . . . 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 5792	. . . . . 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 10332	. . . . . . . . . 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 1216	. . . . . . . . 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 5790	. . . . . . . 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 8872	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
29	28	oveq1i 5784	. . . . . . . 8 |- ( ( 1 x. 1 ) / ( ( A ^ -u N ) x. ( B ^ -u N ) ) ) = ( 1 / ( ( A ^ -u N ) x. ( B ^ -u N ) ) )
30	27, 29	syl6eqr 2190	. . . . . . 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 10311	. . . . . . . . 9 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( A ^ -u N ) e. CC )
32	8, 17, 31	syl2anc 408	. . . . . . . 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 9073	. . . . . . . . . 10 |- ( -u N e. NN -> -u N e. ZZ )
34	33	ad2antll 482	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
35		expap0i 10325	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
36	8, 11, 34, 35	syl3anc 1216	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) # 0 )
37		expcl 10311	. . . . . . . . 9 |- ( ( B e. CC /\ -u N e. NN0 ) -> ( B ^ -u N ) e. CC )
38	9, 17, 37	syl2anc 408	. . . . . . . 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 10325	. . . . . . . . 9 |- ( ( B e. CC /\ B # 0 /\ -u N e. ZZ ) -> ( B ^ -u N ) # 0 )
40	9, 12, 34, 39	syl3anc 1216	. . . . . . . 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 7713	. . . . . . . . 9 |- 1 e. CC
42		divmuldivap 8472	. . . . . . . . 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 432	. . . . . . . 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 1217	. . . . . . 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 2175	. . . . . 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 2175	. . . . 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 2175	. . . 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 786	. . 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 1176	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 697   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   1c1 7621   x. cmul 7625   -ucneg 7934   # cap 8343   / cdiv 8432   NNcn 8720   NN0cn0 8977   ZZcz 9054   ^cexp 10292

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-seqfrec 10219  df-exp 10293

This theorem is referenced by:  exprecap  10334