Theorem expmul 10655

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 5926	. . . . . . 7 |- ( j = 0 -> ( M x. j ) = ( M x. 0 ) )
2	1	oveq2d 5934	. . . . . 6 |- ( j = 0 -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. 0 ) ) )
3		oveq2 5926	. . . . . 6 |- ( j = 0 -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ 0 ) )
4	2, 3	eqeq12d 2208	. . . . 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 5926	. . . . . . 7 |- ( j = k -> ( M x. j ) = ( M x. k ) )
7	6	oveq2d 5934	. . . . . 6 |- ( j = k -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. k ) ) )
8		oveq2 5926	. . . . . 6 |- ( j = k -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ k ) )
9	7, 8	eqeq12d 2208	. . . . 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 5926	. . . . . . 7 |- ( j = ( k + 1 ) -> ( M x. j ) = ( M x. ( k + 1 ) ) )
12	11	oveq2d 5934	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. ( k + 1 ) ) ) )
13		oveq2 5926	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2208	. . . . 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 5926	. . . . . . 7 |- ( j = N -> ( M x. j ) = ( M x. N ) )
17	16	oveq2d 5934	. . . . . 6 |- ( j = N -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. N ) ) )
18		oveq2 5926	. . . . . 6 |- ( j = N -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ N ) )
19	17, 18	eqeq12d 2208	. . . . 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 9250	. . . . . . . 8 |- ( M e. NN0 -> M e. CC )
22	21	mul01d 8412	. . . . . . 7 |- ( M e. NN0 -> ( M x. 0 ) = 0 )
23	22	oveq2d 5934	. . . . . 6 |- ( M e. NN0 -> ( A ^ ( M x. 0 ) ) = ( A ^ 0 ) )
24		exp0 10614	. . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
25	23, 24	sylan9eqr 2248	. . . . 5 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. 0 ) ) = 1 )
26		expcl 10628	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC )
27		exp0 10614	. . . . . 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 2229	. . . 4 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. 0 ) ) = ( ( A ^ M ) ^ 0 ) )
30		oveq1 5925	. . . . . . 7 |- ( ( A ^ ( M x. k ) ) = ( ( A ^ M ) ^ k ) -> ( ( A ^ ( M x. k ) ) x. ( A ^ M ) ) = ( ( ( A ^ M ) ^ k ) x. ( A ^ M ) ) )
31		nn0cn 9250	. . . . . . . . . . . 12 |- ( k e. NN0 -> k e. CC )
32		ax-1cn 7965	. . . . . . . . . . . . . 14 |- 1 e. CC
33		adddi 8004	. . . . . . . . . . . . . 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 8016	. . . . . . . . . . . . . . 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 5934	. . . . . . . . . . . . 13 |- ( ( M e. CC /\ k e. CC ) -> ( ( M x. k ) + ( M x. 1 ) ) = ( ( M x. k ) + M ) )
38	34, 37	eqtrd 2226	. . . . . . . . . . . 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 5934	. . . . . . . . 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 9276	. . . . . . . . . . 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 10652	. . . . . . . . . 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 2226	. . . . . . . 8 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( M x. ( k + 1 ) ) ) = ( ( A ^ ( M x. k ) ) x. ( A ^ M ) ) )
49		expp1 10617	. . . . . . . . 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 2208	. . . . . . 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 9431	. . 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 2164  (class class class)co 5918   CCcc 7870   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   NN0cn0 9240   ^cexp 10609

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-seqfrec 10519  df-exp 10610

This theorem is referenced by:  expmulzap  10656  expnass  10716  expmuld  10747