Theorem mulexp 10687

Description: Nonnegative integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 13-Feb-2005.)

Assertion

Ref	Expression
mulexp	|- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )

Proof of Theorem mulexp

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5933	. . . . . 6 |- ( j = 0 -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ 0 ) )
2		oveq2 5933	. . . . . . 7 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
3		oveq2 5933	. . . . . . 7 |- ( j = 0 -> ( B ^ j ) = ( B ^ 0 ) )
4	2, 3	oveq12d 5943	. . . . . 6 |- ( j = 0 -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
5	1, 4	eqeq12d 2211	. . . . 5 |- ( j = 0 -> ( ( ( A x. B ) ^ j ) = ( ( A ^ j ) x. ( B ^ j ) ) <-> ( ( A x. B ) ^ 0 ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) )
6	5	imbi2d 230	. . . 4 |- ( j = 0 -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ j ) = ( ( A ^ j ) x. ( B ^ j ) ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ 0 ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) ) )
7		oveq2 5933	. . . . . 6 |- ( j = k -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ k ) )
8		oveq2 5933	. . . . . . 7 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
9		oveq2 5933	. . . . . . 7 |- ( j = k -> ( B ^ j ) = ( B ^ k ) )
10	8, 9	oveq12d 5943	. . . . . 6 |- ( j = k -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ k ) x. ( B ^ k ) ) )
11	7, 10	eqeq12d 2211	. . . . 5 |- ( j = k -> ( ( ( A x. B ) ^ j ) = ( ( A ^ j ) x. ( B ^ j ) ) <-> ( ( A x. B ) ^ k ) = ( ( A ^ k ) x. ( B ^ k ) ) ) )
12	11	imbi2d 230	. . . 4 |- ( j = k -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ j ) = ( ( A ^ j ) x. ( B ^ j ) ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ k ) = ( ( A ^ k ) x. ( B ^ k ) ) ) ) )
13		oveq2 5933	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ ( k + 1 ) ) )
14		oveq2 5933	. . . . . . 7 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
15		oveq2 5933	. . . . . . 7 |- ( j = ( k + 1 ) -> ( B ^ j ) = ( B ^ ( k + 1 ) ) )
16	14, 15	oveq12d 5943	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) )
17	13, 16	eqeq12d 2211	. . . . 5 |- ( j = ( k + 1 ) -> ( ( ( A x. B ) ^ j ) = ( ( A ^ j ) x. ( B ^ j ) ) <-> ( ( A x. B ) ^ ( k + 1 ) ) = ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) ) )
18	17	imbi2d 230	. . . 4 |- ( j = ( k + 1 ) -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ j ) = ( ( A ^ j ) x. ( B ^ j ) ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ ( k + 1 ) ) = ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) ) ) )
19		oveq2 5933	. . . . . 6 |- ( j = N -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ N ) )
20		oveq2 5933	. . . . . . 7 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
21		oveq2 5933	. . . . . . 7 |- ( j = N -> ( B ^ j ) = ( B ^ N ) )
22	20, 21	oveq12d 5943	. . . . . 6 |- ( j = N -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ N ) x. ( B ^ N ) ) )
23	19, 22	eqeq12d 2211	. . . . 5 |- ( j = N -> ( ( ( A x. B ) ^ j ) = ( ( A ^ j ) x. ( B ^ j ) ) <-> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) )
24	23	imbi2d 230	. . . 4 |- ( j = N -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ j ) = ( ( A ^ j ) x. ( B ^ j ) ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) ) )
25		mulcl 8023	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
26		exp0 10652	. . . . . 6 |- ( ( A x. B ) e. CC -> ( ( A x. B ) ^ 0 ) = 1 )
27	25, 26	syl 14	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ 0 ) = 1 )
28		exp0 10652	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
29		exp0 10652	. . . . . . 7 |- ( B e. CC -> ( B ^ 0 ) = 1 )
30	28, 29	oveqan12d 5944	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = ( 1 x. 1 ) )
31		1t1e1 9160	. . . . . 6 |- ( 1 x. 1 ) = 1
32	30, 31	eqtrdi 2245	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = 1 )
33	27, 32	eqtr4d 2232	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ 0 ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
34		expp1 10655	. . . . . . . . . 10 |- ( ( ( A x. B ) e. CC /\ k e. NN0 ) -> ( ( A x. B ) ^ ( k + 1 ) ) = ( ( ( A x. B ) ^ k ) x. ( A x. B ) ) )
35	25, 34	sylan 283	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( ( A x. B ) ^ ( k + 1 ) ) = ( ( ( A x. B ) ^ k ) x. ( A x. B ) ) )
36	35	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ ( ( A x. B ) ^ k ) = ( ( A ^ k ) x. ( B ^ k ) ) ) -> ( ( A x. B ) ^ ( k + 1 ) ) = ( ( ( A x. B ) ^ k ) x. ( A x. B ) ) )
37		oveq1 5932	. . . . . . . . 9 |- ( ( ( A x. B ) ^ k ) = ( ( A ^ k ) x. ( B ^ k ) ) -> ( ( ( A x. B ) ^ k ) x. ( A x. B ) ) = ( ( ( A ^ k ) x. ( B ^ k ) ) x. ( A x. B ) ) )
38		expcl 10666	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
39		expcl 10666	. . . . . . . . . . . . 13 |- ( ( B e. CC /\ k e. NN0 ) -> ( B ^ k ) e. CC )
40	38, 39	anim12i 338	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( B e. CC /\ k e. NN0 ) ) -> ( ( A ^ k ) e. CC /\ ( B ^ k ) e. CC ) )
41	40	anandirs 593	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( ( A ^ k ) e. CC /\ ( B ^ k ) e. CC ) )
42		simpl 109	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( A e. CC /\ B e. CC ) )
43		mul4 8175	. . . . . . . . . . 11 |- ( ( ( ( A ^ k ) e. CC /\ ( B ^ k ) e. CC ) /\ ( A e. CC /\ B e. CC ) ) -> ( ( ( A ^ k ) x. ( B ^ k ) ) x. ( A x. B ) ) = ( ( ( A ^ k ) x. A ) x. ( ( B ^ k ) x. B ) ) )
44	41, 42, 43	syl2anc 411	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( ( ( A ^ k ) x. ( B ^ k ) ) x. ( A x. B ) ) = ( ( ( A ^ k ) x. A ) x. ( ( B ^ k ) x. B ) ) )
45		expp1 10655	. . . . . . . . . . . 12 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
46	45	adantlr 477	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
47		expp1 10655	. . . . . . . . . . . 12 |- ( ( B e. CC /\ k e. NN0 ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
48	47	adantll 476	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
49	46, 48	oveq12d 5943	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) = ( ( ( A ^ k ) x. A ) x. ( ( B ^ k ) x. B ) ) )
50	44, 49	eqtr4d 2232	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( ( ( A ^ k ) x. ( B ^ k ) ) x. ( A x. B ) ) = ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) )
51	37, 50	sylan9eqr 2251	. . . . . . . 8 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ ( ( A x. B ) ^ k ) = ( ( A ^ k ) x. ( B ^ k ) ) ) -> ( ( ( A x. B ) ^ k ) x. ( A x. B ) ) = ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) )
52	36, 51	eqtrd 2229	. . . . . . 7 |- ( ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) /\ ( ( A x. B ) ^ k ) = ( ( A ^ k ) x. ( B ^ k ) ) ) -> ( ( A x. B ) ^ ( k + 1 ) ) = ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) )
53	52	exp31 364	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( k e. NN0 -> ( ( ( A x. B ) ^ k ) = ( ( A ^ k ) x. ( B ^ k ) ) -> ( ( A x. B ) ^ ( k + 1 ) ) = ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) ) ) )
54	53	com12 30	. . . . 5 |- ( k e. NN0 -> ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) ^ k ) = ( ( A ^ k ) x. ( B ^ k ) ) -> ( ( A x. B ) ^ ( k + 1 ) ) = ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) ) ) )
55	54	a2d 26	. . . 4 |- ( k e. NN0 -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ k ) = ( ( A ^ k ) x. ( B ^ k ) ) ) -> ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ ( k + 1 ) ) = ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) ) ) )
56	6, 12, 18, 24, 33, 55	nn0ind 9457	. . 3 |- ( N e. NN0 -> ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) )
57	56	expdcom 1453	. 2 |- ( A e. CC -> ( B e. CC -> ( N e. NN0 -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) ) )
58	57	3imp 1195	1 |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167  (class class class)co 5925   CCcc 7894   0cc0 7896   1c1 7897   + caddc 7899   x. cmul 7901   NN0cn0 9266   ^cexp 10647

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-seqfrec 10557  df-exp 10648

This theorem is referenced by:  mulexpzap  10688  expdivap  10699  expubnd  10705  sqmul  10710  mulexpd  10797  efi4p  11899