Theorem mulexp 10964

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 6066	. . . . . 6 |- ( j = 0 -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ 0 ) )
2		oveq2 6066	. . . . . . 7 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
3		oveq2 6066	. . . . . . 7 |- ( j = 0 -> ( B ^ j ) = ( B ^ 0 ) )
4	2, 3	oveq12d 6076	. . . . . 6 |- ( j = 0 -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
5	1, 4	eqeq12d 2249	. . . . 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 6066	. . . . . 6 |- ( j = k -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ k ) )
8		oveq2 6066	. . . . . . 7 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
9		oveq2 6066	. . . . . . 7 |- ( j = k -> ( B ^ j ) = ( B ^ k ) )
10	8, 9	oveq12d 6076	. . . . . 6 |- ( j = k -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ k ) x. ( B ^ k ) ) )
11	7, 10	eqeq12d 2249	. . . . 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 6066	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ ( k + 1 ) ) )
14		oveq2 6066	. . . . . . 7 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
15		oveq2 6066	. . . . . . 7 |- ( j = ( k + 1 ) -> ( B ^ j ) = ( B ^ ( k + 1 ) ) )
16	14, 15	oveq12d 6076	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) )
17	13, 16	eqeq12d 2249	. . . . 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 6066	. . . . . 6 |- ( j = N -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ N ) )
20		oveq2 6066	. . . . . . 7 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
21		oveq2 6066	. . . . . . 7 |- ( j = N -> ( B ^ j ) = ( B ^ N ) )
22	20, 21	oveq12d 6076	. . . . . 6 |- ( j = N -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ N ) x. ( B ^ N ) ) )
23	19, 22	eqeq12d 2249	. . . . 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 8270	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
26		exp0 10929	. . . . . 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 10929	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
29		exp0 10929	. . . . . . 7 |- ( B e. CC -> ( B ^ 0 ) = 1 )
30	28, 29	oveqan12d 6077	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = ( 1 x. 1 ) )
31		1t1e1 9407	. . . . . 6 |- ( 1 x. 1 ) = 1
32	30, 31	eqtrdi 2283	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = 1 )
33	27, 32	eqtr4d 2270	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ 0 ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
34		expp1 10932	. . . . . . . . . 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 6065	. . . . . . . . 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 10943	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
39		expcl 10943	. . . . . . . . . . . . 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 597	. . . . . . . . . . 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 8421	. . . . . . . . . . 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 10932	. . . . . . . . . . . 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 10932	. . . . . . . . . . . 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 6076	. . . . . . . . . 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 2270	. . . . . . . . 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 2289	. . . . . . . 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 2267	. . . . . . 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 9710	. . 3 |- ( N e. NN0 -> ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) )
57	56	expdcom 1488	. 2 |- ( A e. CC -> ( B e. CC -> ( N e. NN0 -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) ) )
58	57	3imp 1220	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 1005   = wceq 1398   e. wcel 2205  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   NN0cn0 9513   ^cexp 10924

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-seqfrec 10834  df-exp 10925

This theorem is referenced by:  mulexpzap  10965  expdivap  10976  expubnd  10982  sqmul  10987  mulexpd  11075  efi4p  12428