Theorem expmul 10676

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 5930	. . . . . . 7 |- ( j = 0 -> ( M x. j ) = ( M x. 0 ) )
2	1	oveq2d 5938	. . . . . 6 |- ( j = 0 -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. 0 ) ) )
3		oveq2 5930	. . . . . 6 |- ( j = 0 -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ 0 ) )
4	2, 3	eqeq12d 2211	. . . . 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 5930	. . . . . . 7 |- ( j = k -> ( M x. j ) = ( M x. k ) )
7	6	oveq2d 5938	. . . . . 6 |- ( j = k -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. k ) ) )
8		oveq2 5930	. . . . . 6 |- ( j = k -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ k ) )
9	7, 8	eqeq12d 2211	. . . . 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 5930	. . . . . . 7 |- ( j = ( k + 1 ) -> ( M x. j ) = ( M x. ( k + 1 ) ) )
12	11	oveq2d 5938	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. ( k + 1 ) ) ) )
13		oveq2 5930	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2211	. . . . 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 5930	. . . . . . 7 |- ( j = N -> ( M x. j ) = ( M x. N ) )
17	16	oveq2d 5938	. . . . . 6 |- ( j = N -> ( A ^ ( M x. j ) ) = ( A ^ ( M x. N ) ) )
18		oveq2 5930	. . . . . 6 |- ( j = N -> ( ( A ^ M ) ^ j ) = ( ( A ^ M ) ^ N ) )
19	17, 18	eqeq12d 2211	. . . . 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 9259	. . . . . . . 8 |- ( M e. NN0 -> M e. CC )
22	21	mul01d 8419	. . . . . . 7 |- ( M e. NN0 -> ( M x. 0 ) = 0 )
23	22	oveq2d 5938	. . . . . 6 |- ( M e. NN0 -> ( A ^ ( M x. 0 ) ) = ( A ^ 0 ) )
24		exp0 10635	. . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
25	23, 24	sylan9eqr 2251	. . . . 5 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. 0 ) ) = 1 )
26		expcl 10649	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC )
27		exp0 10635	. . . . . 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 2232	. . . 4 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M x. 0 ) ) = ( ( A ^ M ) ^ 0 ) )
30		oveq1 5929	. . . . . . 7 |- ( ( A ^ ( M x. k ) ) = ( ( A ^ M ) ^ k ) -> ( ( A ^ ( M x. k ) ) x. ( A ^ M ) ) = ( ( ( A ^ M ) ^ k ) x. ( A ^ M ) ) )
31		nn0cn 9259	. . . . . . . . . . . 12 |- ( k e. NN0 -> k e. CC )
32		ax-1cn 7972	. . . . . . . . . . . . . 14 |- 1 e. CC
33		adddi 8011	. . . . . . . . . . . . . 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 8023	. . . . . . . . . . . . . . 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 5938	. . . . . . . . . . . . 13 |- ( ( M e. CC /\ k e. CC ) -> ( ( M x. k ) + ( M x. 1 ) ) = ( ( M x. k ) + M ) )
38	34, 37	eqtrd 2229	. . . . . . . . . . . 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 5938	. . . . . . . . 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 9285	. . . . . . . . . . 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 10673	. . . . . . . . . 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 2229	. . . . . . . 8 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( M x. ( k + 1 ) ) ) = ( ( A ^ ( M x. k ) ) x. ( A ^ M ) ) )
49		expp1 10638	. . . . . . . . 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 2211	. . . . . . 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 9440	. . 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 2167  (class class class)co 5922   CCcc 7877   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   NN0cn0 9249   ^cexp 10630

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-seqfrec 10540  df-exp 10631

This theorem is referenced by:  expmulzap  10677  expnass  10737  expmuld  10768