Theorem mulexpzap 10761

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 9421	. . 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 10760	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
6	5	3expa 1206	. . . . 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 533	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC )
9		simplrl 535	. . . . . . 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 8128	. . . . . 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 534	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> A # 0 )
12		simplrr 536	. . . . . . 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 8766	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A x. B ) # 0 )
14		recn 8093	. . . . . . 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 9337	. . . . . . 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 10730	. . . . . 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 1251	. . . . 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 10730	. . . . . . . 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 1251	. . . . . . 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 10730	. . . . . . . 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 1251	. . . . . . 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 5985	. . . . . 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 10760	. . . . . . . . . 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 1250	. . . . . . . . 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 5983	. . . . . . . 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 9224	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
29	28	oveq1i 5977	. . . . . . . 8 |- ( ( 1 x. 1 ) / ( ( A ^ -u N ) x. ( B ^ -u N ) ) ) = ( 1 / ( ( A ^ -u N ) x. ( B ^ -u N ) ) )
30	27, 29	eqtr4di 2258	. . . . . . 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 10739	. . . . . . . . 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 9426	. . . . . . . . . 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 10753	. . . . . . . . 9 |- ( ( A e. CC /\ A # 0 /\ -u N e. ZZ ) -> ( A ^ -u N ) # 0 )
36	8, 11, 34, 35	syl3anc 1250	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) # 0 )
37		expcl 10739	. . . . . . . . 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 10753	. . . . . . . . 9 |- ( ( B e. CC /\ B # 0 /\ -u N e. ZZ ) -> ( B ^ -u N ) # 0 )
40	9, 12, 34, 39	syl3anc 1250	. . . . . . . 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 8053	. . . . . . . . 9 |- 1 e. CC
42		divmuldivap 8820	. . . . . . . . 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 1251	. . . . . . 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 2243	. . . . . 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 2243	. . . . 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 2243	. . . 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 799	. . 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 1197	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 710   /\ w3a 981   = wceq 1373   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   1c1 7961   x. cmul 7965   -ucneg 8279   # cap 8689   / cdiv 8780   NNcn 9071   NN0cn0 9330   ZZcz 9407   ^cexp 10720

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-seqfrec 10630  df-exp 10721

This theorem is referenced by:  exprecap  10762