Theorem mulexp 10998

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 6087	. . . . . 6 |- ( j = 0 -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ 0 ) )
2		oveq2 6087	. . . . . . 7 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
3		oveq2 6087	. . . . . . 7 |- ( j = 0 -> ( B ^ j ) = ( B ^ 0 ) )
4	2, 3	oveq12d 6097	. . . . . 6 |- ( j = 0 -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
5	1, 4	eqeq12d 2253	. . . . 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 6087	. . . . . 6 |- ( j = k -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ k ) )
8		oveq2 6087	. . . . . . 7 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
9		oveq2 6087	. . . . . . 7 |- ( j = k -> ( B ^ j ) = ( B ^ k ) )
10	8, 9	oveq12d 6097	. . . . . 6 |- ( j = k -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ k ) x. ( B ^ k ) ) )
11	7, 10	eqeq12d 2253	. . . . 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 6087	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ ( k + 1 ) ) )
14		oveq2 6087	. . . . . . 7 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
15		oveq2 6087	. . . . . . 7 |- ( j = ( k + 1 ) -> ( B ^ j ) = ( B ^ ( k + 1 ) ) )
16	14, 15	oveq12d 6097	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) )
17	13, 16	eqeq12d 2253	. . . . 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 6087	. . . . . 6 |- ( j = N -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ N ) )
20		oveq2 6087	. . . . . . 7 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
21		oveq2 6087	. . . . . . 7 |- ( j = N -> ( B ^ j ) = ( B ^ N ) )
22	20, 21	oveq12d 6097	. . . . . 6 |- ( j = N -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ N ) x. ( B ^ N ) ) )
23	19, 22	eqeq12d 2253	. . . . 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 8300	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
26		exp0 10963	. . . . . 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 10963	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
29		exp0 10963	. . . . . . 7 |- ( B e. CC -> ( B ^ 0 ) = 1 )
30	28, 29	oveqan12d 6098	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = ( 1 x. 1 ) )
31		1t1e1 9440	. . . . . 6 |- ( 1 x. 1 ) = 1
32	30, 31	eqtrdi 2287	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = 1 )
33	27, 32	eqtr4d 2274	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ 0 ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
34		expp1 10966	. . . . . . . . . 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 6086	. . . . . . . . 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 10977	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
39		expcl 10977	. . . . . . . . . . . . 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 601	. . . . . . . . . . 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 8452	. . . . . . . . . . 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 415	. . . . . . . . . 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 10966	. . . . . . . . . . . 12 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
46	45	adantlr 481	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
47		expp1 10966	. . . . . . . . . . . 12 |- ( ( B e. CC /\ k e. NN0 ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
48	47	adantll 480	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
49	46, 48	oveq12d 6097	. . . . . . . . . 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 2274	. . . . . . . . 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 2293	. . . . . . . 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 2271	. . . . . . 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 9743	. . 3 |- ( N e. NN0 -> ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) )
57	56	expdcom 1492	. 2 |- ( A e. CC -> ( B e. CC -> ( N e. NN0 -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) ) )
58	57	3imp 1224	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 1009   = wceq 1402   e. wcel 2209  (class class class)co 6079   CCcc 8171   0cc0 8173   1c1 8174   + caddc 8176   x. cmul 8178   NN0cn0 9546   ^cexp 10958

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-seqfrec 10868  df-exp 10959

This theorem is referenced by:  mulexpzap  10999  expdivap  11010  expubnd  11016  sqmul  11021  mulexpd  11109  efi4p  12467