Theorem mulexp 10440

Description: Positive 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 5826	. . . . . 6 |- ( j = 0 -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ 0 ) )
2		oveq2 5826	. . . . . . 7 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
3		oveq2 5826	. . . . . . 7 |- ( j = 0 -> ( B ^ j ) = ( B ^ 0 ) )
4	2, 3	oveq12d 5836	. . . . . 6 |- ( j = 0 -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
5	1, 4	eqeq12d 2172	. . . . 5 |- ( j = 0 -> ( ( ( A x. B ) ^ j ) = ( ( A ^ j ) x. ( B ^ j ) ) <-> ( ( A x. B ) ^ 0 ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) ) )
6	5	imbi2d 229	. . . 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 5826	. . . . . 6 |- ( j = k -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ k ) )
8		oveq2 5826	. . . . . . 7 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
9		oveq2 5826	. . . . . . 7 |- ( j = k -> ( B ^ j ) = ( B ^ k ) )
10	8, 9	oveq12d 5836	. . . . . 6 |- ( j = k -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ k ) x. ( B ^ k ) ) )
11	7, 10	eqeq12d 2172	. . . . 5 |- ( j = k -> ( ( ( A x. B ) ^ j ) = ( ( A ^ j ) x. ( B ^ j ) ) <-> ( ( A x. B ) ^ k ) = ( ( A ^ k ) x. ( B ^ k ) ) ) )
12	11	imbi2d 229	. . . 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 5826	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ ( k + 1 ) ) )
14		oveq2 5826	. . . . . . 7 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
15		oveq2 5826	. . . . . . 7 |- ( j = ( k + 1 ) -> ( B ^ j ) = ( B ^ ( k + 1 ) ) )
16	14, 15	oveq12d 5836	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ ( k + 1 ) ) x. ( B ^ ( k + 1 ) ) ) )
17	13, 16	eqeq12d 2172	. . . . 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 229	. . . 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 5826	. . . . . 6 |- ( j = N -> ( ( A x. B ) ^ j ) = ( ( A x. B ) ^ N ) )
20		oveq2 5826	. . . . . . 7 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
21		oveq2 5826	. . . . . . 7 |- ( j = N -> ( B ^ j ) = ( B ^ N ) )
22	20, 21	oveq12d 5836	. . . . . 6 |- ( j = N -> ( ( A ^ j ) x. ( B ^ j ) ) = ( ( A ^ N ) x. ( B ^ N ) ) )
23	19, 22	eqeq12d 2172	. . . . 5 |- ( j = N -> ( ( ( A x. B ) ^ j ) = ( ( A ^ j ) x. ( B ^ j ) ) <-> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) )
24	23	imbi2d 229	. . . 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 7842	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
26		exp0 10405	. . . . . 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 10405	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
29		exp0 10405	. . . . . . 7 |- ( B e. CC -> ( B ^ 0 ) = 1 )
30	28, 29	oveqan12d 5837	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = ( 1 x. 1 ) )
31		1t1e1 8968	. . . . . 6 |- ( 1 x. 1 ) = 1
32	30, 31	eqtrdi 2206	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 0 ) x. ( B ^ 0 ) ) = 1 )
33	27, 32	eqtr4d 2193	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ 0 ) = ( ( A ^ 0 ) x. ( B ^ 0 ) ) )
34		expp1 10408	. . . . . . . . . 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 281	. . . . . . . . 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 274	. . . . . . . 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 5825	. . . . . . . . 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 10419	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
39		expcl 10419	. . . . . . . . . . . . 13 |- ( ( B e. CC /\ k e. NN0 ) -> ( B ^ k ) e. CC )
40	38, 39	anim12i 336	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( B e. CC /\ k e. NN0 ) ) -> ( ( A ^ k ) e. CC /\ ( B ^ k ) e. CC ) )
41	40	anandirs 583	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( ( A ^ k ) e. CC /\ ( B ^ k ) e. CC ) )
42		simpl 108	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( A e. CC /\ B e. CC ) )
43		mul4 7990	. . . . . . . . . . 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 409	. . . . . . . . . 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 10408	. . . . . . . . . . . 12 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
46	45	adantlr 469	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
47		expp1 10408	. . . . . . . . . . . 12 |- ( ( B e. CC /\ k e. NN0 ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
48	47	adantll 468	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ B e. CC ) /\ k e. NN0 ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
49	46, 48	oveq12d 5836	. . . . . . . . . 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 2193	. . . . . . . . 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 2212	. . . . . . . 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 2190	. . . . . . 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 362	. . . . . 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 9261	. . 3 |- ( N e. NN0 -> ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) )
57	56	expdcom 1422	. 2 |- ( A e. CC -> ( B e. CC -> ( N e. NN0 -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) ) )
58	57	3imp 1176	1 |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1335   e. wcel 2128  (class class class)co 5818   CCcc 7713   0cc0 7715   1c1 7716   + caddc 7718   x. cmul 7720   NN0cn0 9073   ^cexp 10400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-frec 6332  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-n0 9074  df-z 9151  df-uz 9423  df-seqfrec 10327  df-exp 10401

This theorem is referenced by:  mulexpzap  10441  expdivap  10452  expubnd  10458  sqmul  10463  mulexpd  10548  efi4p  11596