Theorem expmul 10599

Description: Product of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.)

Assertion

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

Proof of Theorem expmul

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

Step	Hyp	Ref	Expression
1		oveq2 5905	. . . . . . 7 |- ( j = 0 -> ( M x. j ) = ( M x. 0 ) )
2	1	oveq2d 5913	. . . . . 6 |- ( j = 0 -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. 0 ) ) )
3		oveq2 5905	. . . . . 6 |- ( j = 0 -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ 0 ) )
4	2, 3	eqeq12d 2204	. . . . 5 |- ( j = 0 -> ( ( A ^ ( M x. j ) ) = ( ( A ^ M ) ^ j ) <-> ( A ^ ( M x. 0 ) ) = ( ( A ^ M ) ^ 0 ) ) )
5	4	imbi2d 230	. . . 4 |- ( j = 0 -> ( ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. j ) ) = ( ( A ^ M ) ^ j ) ) <-> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. 0 ) ) = ( ( A ^ M ) ^ 0 ) ) ) )
6		oveq2 5905	. . . . . . 7 |- ( j = k -> ( M x. j ) = ( M x. k ) )
7	6	oveq2d 5913	. . . . . 6 |- ( j = k -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. k ) ) )
8		oveq2 5905	. . . . . 6 |- ( j = k -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ k ) )
9	7, 8	eqeq12d 2204	. . . . 5 |- ( j = k -> ( ( A ^ ( M x. j ) ) = ( ( A ^ M ) ^ j ) <-> ( A ^ ( M x. k ) ) = ( ( A ^ M ) ^ k ) ) )
10	9	imbi2d 230	. . . 4 |- ( j = k -> ( ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. j ) ) = ( ( A ^ M ) ^ j ) ) <-> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. k ) ) = ( ( A ^ M ) ^ k ) ) ) )
11		oveq2 5905	. . . . . . 7 |- ( j = ( k + 1 ) -> ( M x. j ) = ( M x. ( k + 1 ) ) )
12	11	oveq2d 5913	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. ( k + 1 ) ) ) )
13		oveq2 5905	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2204	. . . . 5 |- ( j = ( k + 1 ) -> ( ( A ^ ( M x. j ) ) = ( ( A ^ M ) ^ j ) <-> ( A ^ ( M x. ( k + 1 ) ) ) = ( ( A ^ M ) ^ ( k + 1 ) ) ) )
15	14	imbi2d 230	. . . 4 |- ( j = ( k + 1 ) -> ( ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. j ) ) = ( ( A ^ M ) ^ j ) ) <-> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. ( k + 1 ) ) ) = ( ( A ^ M ) ^ ( k + 1 ) ) ) ) )
16		oveq2 5905	. . . . . . 7 |- ( j = N -> ( M x. j ) = ( M x. N ) )
17	16	oveq2d 5913	. . . . . 6 |- ( j = N -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. N ) ) )
18		oveq2 5905	. . . . . 6 |- ( j = N -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ N ) )
19	17, 18	eqeq12d 2204	. . . . 5 |- ( j = N -> ( ( A ^ ( M x. j ) ) = ( ( A ^ M ) ^ j ) <-> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) )
20	19	imbi2d 230	. . . 4 |- ( j = N -> ( ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. j ) ) = ( ( A ^ M ) ^ j ) ) <-> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) ) )
21		nn0cn 9217	. . . . . . . 8 |- ( M e. NN0 -> M e. CC )
22	21	mul01d 8381	. . . . . . 7 |- ( M e. NN0 -> ( M x. 0 ) = 0 )
23	22	oveq2d 5913	. . . . . 6 |- ( M e. NN0 -> ( A ^ ( M x. 0 ) ) = ( A ^ 0 ) )
24		exp0 10558	. . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
25	23, 24	sylan9eqr 2244	. . . . 5 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. 0 ) ) = 1 )
26		expcl 10572	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC )
27		exp0 10558	. . . . . 6 |- ( ( A ^ M ) e. CC -> ( ( A ^ M ) ^ 0 ) = 1 )
28	26, 27	syl 14	. . . . 5 |- ( ( A e. CC /\ M e. NN0 ) -> ( ( A ^ M ) ^ 0 ) = 1 )
29	25, 28	eqtr4d 2225	. . . 4 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. 0 ) ) = ( ( A ^ M ) ^ 0 ) )
30		oveq1 5904	. . . . . . 7 |- ( ( A ^ ( M x. k ) ) = ( ( A ^ M ) ^ k ) -> ( ( A ^ ( M x. k ) ) x. ( A ^ M ) ) = ( ( ( A ^ M ) ^ k ) x. ( A ^ M ) ) )
31		nn0cn 9217	. . . . . . . . . . . 12 |- ( k e. NN0 -> k e. CC )
32		ax-1cn 7935	. . . . . . . . . . . . . 14 |- 1 e. CC
33		adddi 7974	. . . . . . . . . . . . . 14 |- ( ( M e. CC /\ k e. CC /\ 1 e. CC ) -> ( M x. ( k + 1 ) ) = ( ( M x. k ) + ( M x. 1 ) ) )
34	32, 33	mp3an3 1337	. . . . . . . . . . . . 13 |- ( ( M e. CC /\ k e. CC ) -> ( M x. ( k + 1 ) ) = ( ( M x. k ) + ( M x. 1 ) ) )
35		mulrid 7985	. . . . . . . . . . . . . . 15 |- ( M e. CC -> ( M x. 1 ) = M )
36	35	adantr 276	. . . . . . . . . . . . . 14 |- ( ( M e. CC /\ k e. CC ) -> ( M x. 1 ) = M )
37	36	oveq2d 5913	. . . . . . . . . . . . 13 |- ( ( M e. CC /\ k e. CC ) -> ( ( M x. k ) + ( M x. 1 ) ) = ( ( M x. k ) + M ) )
38	34, 37	eqtrd 2222	. . . . . . . . . . . 12 |- ( ( M e. CC /\ k e. CC ) -> ( M x. ( k + 1 ) ) = ( ( M x. k ) + M ) )
39	21, 31, 38	syl2an 289	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ k e. NN0 ) -> ( M x. ( k + 1 ) ) = ( ( M x. k ) + M ) )
40	39	adantll 476	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( M x. ( k + 1 ) ) = ( ( M x. k ) + M ) )
41	40	oveq2d 5913	. . . . . . . . 9 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( M x. ( k + 1 ) ) ) = ( A ^ ( ( M x. k ) + M ) ) )
42		simpll 527	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> A e. CC )
43		nn0mulcl 9243	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ k e. NN0 ) -> ( M x. k ) e. NN0 )
44	43	adantll 476	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( M x. k ) e. NN0 )
45		simplr 528	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> M e. NN0 )
46		expadd 10596	. . . . . . . . . 10 |- ( ( A e. CC /\ ( M x. k ) e. NN0 /\ M e. NN0 ) -> ( A ^ ( ( M x. k ) + M ) ) = ( ( A ^ ( M x. k ) ) x. ( A ^ M ) ) )
47	42, 44, 45, 46	syl3anc 1249	. . . . . . . . 9 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( ( M x. k ) + M ) ) = ( ( A ^ ( M x. k ) ) x. ( A ^ M ) ) )
48	41, 47	eqtrd 2222	. . . . . . . 8 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( M x. ( k + 1 ) ) ) = ( ( A ^ ( M x. k ) ) x. ( A ^ M ) ) )
49		expp1 10561	. . . . . . . . 9 |- ( ( ( A ^ M ) e. CC /\ k e. NN0 ) -> ( ( A ^ M ) ^ ( k + 1 ) ) = ( ( ( A ^ M ) ^ k ) x. ( A ^ M ) ) )
50	26, 49	sylan 283	. . . . . . . 8 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( A ^ M ) ^ ( k + 1 ) ) = ( ( ( A ^ M ) ^ k ) x. ( A ^ M ) ) )
51	48, 50	eqeq12d 2204	. . . . . . 7 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( A ^ ( M x. ( k + 1 ) ) ) = ( ( A ^ M ) ^ ( k + 1 ) ) <-> ( ( A ^ ( M x. k ) ) x. ( A ^ M ) ) = ( ( ( A ^ M ) ^ k ) x. ( A ^ M ) ) ) )
52	30, 51	imbitrrid 156	. . . . . 6 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( A ^ ( M x. k ) ) = ( ( A ^ M ) ^ k ) -> ( A ^ ( M x. ( k + 1 ) ) ) = ( ( A ^ M ) ^ ( k + 1 ) ) ) )
53	52	expcom 116	. . . . 5 |- ( k e. NN0 -> ( ( A e. CC /\ M e. NN0 ) -> ( ( A ^ ( M x. k ) ) = ( ( A ^ M ) ^ k ) -> ( A ^ ( M x. ( k + 1 ) ) ) = ( ( A ^ M ) ^ ( k + 1 ) ) ) ) )
54	53	a2d 26	. . . 4 |- ( k e. NN0 -> ( ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. k ) ) = ( ( A ^ M ) ^ k ) ) -> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. ( k + 1 ) ) ) = ( ( A ^ M ) ^ ( k + 1 ) ) ) ) )
55	5, 10, 15, 20, 29, 54	nn0ind 9398	. . 3 |- ( N e. NN0 -> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) )
56	55	expdcom 1453	. 2 |- ( A e. CC -> ( M e. NN0 -> ( N e. NN0 -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) ) )
57	56	3imp 1195	1 |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160  (class class class)co 5897   CCcc 7840   0cc0 7842   1c1 7843   + caddc 7845   x. cmul 7847   NN0cn0 9207   ^cexp 10553

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960

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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-n0 9208  df-z 9285  df-uz 9560  df-seqfrec 10479  df-exp 10554

This theorem is referenced by:  expmulzap  10600  expnass  10660  expmuld  10691