Theorem mulexpzap 10364

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 9092	. . 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 10363	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
6	5	3expa 1182	. . . . 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 7810	. . . . . 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 8443	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A x. B ) # 0 )
14		recn 7777	. . . . . . 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 9008	. . . . . . 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 10333	. . . . . 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 1218	. . . . 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 10333	. . . . . . . 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 1218	. . . . . . 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 10333	. . . . . . . 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 1218	. . . . . . 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 5800	. . . . . 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 10363	. . . . . . . . . 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 1217	. . . . . . . . 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 5798	. . . . . . . 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 8896	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
29	28	oveq1i 5792	. . . . . . . 8 |- ( ( 1 x. 1 ) / ( ( A ^ -u N ) x. ( B ^ -u N ) ) ) = ( 1 / ( ( A ^ -u N ) x. ( B ^ -u N ) ) )
30	27, 29	eqtr4di 2191	. . . . . . 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 10342	. . . . . . . . 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 9097	. . . . . . . . . 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 10356	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
36	8, 11, 34, 35	syl3anc 1217	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) # 0 )
37		expcl 10342	. . . . . . . . 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 10356	. . . . . . . . 9 |- ( ( B e. CC /\ B # 0 /\ -u N e. ZZ ) -> ( B ^ -u N ) # 0 )
40	9, 12, 34, 39	syl3anc 1217	. . . . . . . 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 7737	. . . . . . . . 9 |- 1 e. CC
42		divmuldivap 8496	. . . . . . . . 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 1218	. . . . . . 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 2176	. . . . . 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 2176	. . . . 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 2176	. . . 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 1177	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 963   = wceq 1332   e. wcel 1481   class class class wbr 3937  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645   x. cmul 7649   -ucneg 7958   # cap 8367   / cdiv 8456   NNcn 8744   NN0cn0 9001   ZZcz 9078   ^cexp 10323

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-seqfrec 10250  df-exp 10324

This theorem is referenced by:  exprecap  10365