Theorem expmul 9521

Description: Product of exponents law for positive 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 5540	. . . . . . 7 |- ( j = 0 -> ( M x. j ) = ( M x. 0 ) )
2	1	oveq2d 5548	. . . . . 6 |- ( j = 0 -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. 0 ) ) )
3		oveq2 5540	. . . . . 6 |- ( j = 0 -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ 0 ) )
4	2, 3	eqeq12d 2095	. . . . 5 |- ( j = 0 -> ( ( A ^ ( M x. j ) ) = ( ( A ^ M ) ^ j ) <-> ( A ^ ( M x. 0 ) ) = ( ( A ^ M ) ^ 0 ) ) )
5	4	imbi2d 228	. . . 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 5540	. . . . . . 7 |- ( j = k -> ( M x. j ) = ( M x. k ) )
7	6	oveq2d 5548	. . . . . 6 |- ( j = k -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. k ) ) )
8		oveq2 5540	. . . . . 6 |- ( j = k -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ k ) )
9	7, 8	eqeq12d 2095	. . . . 5 |- ( j = k -> ( ( A ^ ( M x. j ) ) = ( ( A ^ M ) ^ j ) <-> ( A ^ ( M x. k ) ) = ( ( A ^ M ) ^ k ) ) )
10	9	imbi2d 228	. . . 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 5540	. . . . . . 7 |- ( j = ( k + 1 ) -> ( M x. j ) = ( M x. ( k + 1 ) ) )
12	11	oveq2d 5548	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. ( k + 1 ) ) ) )
13		oveq2 5540	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2095	. . . . 5 |- ( j = ( k + 1 ) -> ( ( A ^ ( M x. j ) ) = ( ( A ^ M ) ^ j ) <-> ( A ^ ( M x. ( k + 1 ) ) ) = ( ( A ^ M ) ^ ( k + 1 ) ) ) )
15	14	imbi2d 228	. . . 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 5540	. . . . . . 7 |- ( j = N -> ( M x. j ) = ( M x. N ) )
17	16	oveq2d 5548	. . . . . 6 |- ( j = N -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. N ) ) )
18		oveq2 5540	. . . . . 6 |- ( j = N -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ N ) )
19	17, 18	eqeq12d 2095	. . . . 5 |- ( j = N -> ( ( A ^ ( M x. j ) ) = ( ( A ^ M ) ^ j ) <-> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) )
20	19	imbi2d 228	. . . 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 8298	. . . . . . . 8 |- ( M e. NN0 -> M e. CC )
22	21	mul01d 7497	. . . . . . 7 |- ( M e. NN0 -> ( M x. 0 ) = 0 )
23	22	oveq2d 5548	. . . . . 6 |- ( M e. NN0 -> ( A ^ ( M x. 0 ) ) = ( A ^ 0 ) )
24		exp0 9480	. . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
25	23, 24	sylan9eqr 2135	. . . . 5 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. 0 ) ) = 1 )
26		expcl 9494	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC )
27		exp0 9480	. . . . . 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 2116	. . . 4 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. 0 ) ) = ( ( A ^ M ) ^ 0 ) )
30		oveq1 5539	. . . . . . 7 |- ( ( A ^ ( M x. k ) ) = ( ( A ^ M ) ^ k ) -> ( ( A ^ ( M x. k ) ) x. ( A ^ M ) ) = ( ( ( A ^ M ) ^ k ) x. ( A ^ M ) ) )
31		nn0cn 8298	. . . . . . . . . . . 12 |- ( k e. NN0 -> k e. CC )
32		ax-1cn 7069	. . . . . . . . . . . . . 14 |- 1 e. CC
33		adddi 7105	. . . . . . . . . . . . . 14 |- ( ( M e. CC /\ k e. CC /\ 1 e. CC ) -> ( M x. ( k + 1 ) ) = ( ( M x. k ) + ( M x. 1 ) ) )
34	32, 33	mp3an3 1257	. . . . . . . . . . . . 13 |- ( ( M e. CC /\ k e. CC ) -> ( M x. ( k + 1 ) ) = ( ( M x. k ) + ( M x. 1 ) ) )
35		mulid1 7116	. . . . . . . . . . . . . . 15 |- ( M e. CC -> ( M x. 1 ) = M )
36	35	adantr 270	. . . . . . . . . . . . . 14 |- ( ( M e. CC /\ k e. CC ) -> ( M x. 1 ) = M )
37	36	oveq2d 5548	. . . . . . . . . . . . 13 |- ( ( M e. CC /\ k e. CC ) -> ( ( M x. k ) + ( M x. 1 ) ) = ( ( M x. k ) + M ) )
38	34, 37	eqtrd 2113	. . . . . . . . . . . 12 |- ( ( M e. CC /\ k e. CC ) -> ( M x. ( k + 1 ) ) = ( ( M x. k ) + M ) )
39	21, 31, 38	syl2an 283	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ k e. NN0 ) -> ( M x. ( k + 1 ) ) = ( ( M x. k ) + M ) )
40	39	adantll 459	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( M x. ( k + 1 ) ) = ( ( M x. k ) + M ) )
41	40	oveq2d 5548	. . . . . . . . 9 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( M x. ( k + 1 ) ) ) = ( A ^ ( ( M x. k ) + M ) ) )
42		simpll 495	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> A e. CC )
43		nn0mulcl 8324	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ k e. NN0 ) -> ( M x. k ) e. NN0 )
44	43	adantll 459	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( M x. k ) e. NN0 )
45		simplr 496	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> M e. NN0 )
46		expadd 9518	. . . . . . . . . 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 1169	. . . . . . . . 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 2113	. . . . . . . 8 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( M x. ( k + 1 ) ) ) = ( ( A ^ ( M x. k ) ) x. ( A ^ M ) ) )
49		expp1 9483	. . . . . . . . 9 |- ( ( ( A ^ M ) e. CC /\ k e. NN0 ) -> ( ( A ^ M ) ^ ( k + 1 ) ) = ( ( ( A ^ M ) ^ k ) x. ( A ^ M ) ) )
50	26, 49	sylan 277	. . . . . . . 8 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( A ^ M ) ^ ( k + 1 ) ) = ( ( ( A ^ M ) ^ k ) x. ( A ^ M ) ) )
51	48, 50	eqeq12d 2095	. . . . . . 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	syl5ibr 154	. . . . . 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 114	. . . . 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 8461	. . 3 |- ( N e. NN0 -> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) )
56	55	expdcom 1371	. 2 |- ( A e. CC -> ( M e. NN0 -> ( N e. NN0 -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) ) )
57	56	3imp 1132	1 |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433  (class class class)co 5532   CCcc 6979   0cc0 6981   1c1 6982   + caddc 6984   x. cmul 6986   NN0cn0 8288   ^cexp 9475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432  df-iexp 9476

This theorem is referenced by:  expmulzap  9522  expnass  9580  expmuld  9608