Theorem mulexpzap 10840

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 9492	. . 3 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2		simpl 109	. . . . . 6 |- ( ( A e. CC /\ A # 0 ) -> A e. CC )
3		simpl 109	. . . . . 6 |- ( ( B e. CC /\ B # 0 ) -> B e. CC )
4	2, 3	anim12i 338	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( A e. CC /\ B e. CC ) )
5		mulexp 10839	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
6	5	3expa 1229	. . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
7	4, 6	sylan 283	. . . 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 535	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC )
9		simplrl 537	. . . . . . 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 8199	. . . . . 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 536	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> A # 0 )
12		simplrr 538	. . . . . . 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 8837	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A x. B ) # 0 )
14		recn 8164	. . . . . . 7 |- ( N e. RR -> N e. CC )
15	14	ad2antrl 490	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
16		nnnn0 9408	. . . . . . 7 |- ( -u N e. NN -> -u N e. NN0 )
17	16	ad2antll 491	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN0 )
18		expineg2 10809	. . . . . 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 1274	. . . . 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 10809	. . . . . . . 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 1274	. . . . . . 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 10809	. . . . . . . 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 1274	. . . . . . 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 6035	. . . . . 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 10839	. . . . . . . . . 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 1273	. . . . . . . . 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 6033	. . . . . . . 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 9295	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
29	28	oveq1i 6027	. . . . . . . 8 |- ( ( 1 x. 1 ) / ( ( A ^ -u N ) x. ( B ^ -u N ) ) ) = ( 1 / ( ( A ^ -u N ) x. ( B ^ -u N ) ) )
30	27, 29	eqtr4di 2282	. . . . . . 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 10818	. . . . . . . . 9 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( A ^ -u N ) e. CC )
32	8, 17, 31	syl2anc 411	. . . . . . . 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 9497	. . . . . . . . . 10 |- ( -u N e. NN -> -u N e. ZZ )
34	33	ad2antll 491	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
35		expap0i 10832	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
36	8, 11, 34, 35	syl3anc 1273	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) # 0 )
37		expcl 10818	. . . . . . . . 9 |- ( ( B e. CC /\ -u N e. NN0 ) -> ( B ^ -u N ) e. CC )
38	9, 17, 37	syl2anc 411	. . . . . . . 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 10832	. . . . . . . . 9 |- ( ( B e. CC /\ B # 0 /\ -u N e. ZZ ) -> ( B ^ -u N ) # 0 )
40	9, 12, 34, 39	syl3anc 1273	. . . . . . . 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 8124	. . . . . . . . 9 |- 1 e. CC
42		divmuldivap 8891	. . . . . . . . 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 436	. . . . . . . 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 1274	. . . . . . 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 2267	. . . . . 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 2267	. . . . 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 2267	. . . 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 804	. . 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 287	. 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 1220	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 104   \/ wo 715   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   1c1 8032   x. cmul 8036   -ucneg 8350   # cap 8760   / cdiv 8851   NNcn 9142   NN0cn0 9401   ZZcz 9478   ^cexp 10799

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-seqfrec 10709  df-exp 10800

This theorem is referenced by:  exprecap  10841