Theorem expadd 10366

Description: Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)

Assertion

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

Proof of Theorem expadd

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

Step	Hyp	Ref	Expression
1		oveq2 5790	. . . . . . 7 |- ( j = 0 -> ( M + j ) = ( M + 0 ) )
2	1	oveq2d 5798	. . . . . 6 |- ( j = 0 -> ( A ^ ( M + j ) ) = ( A ^ ( M + 0 ) ) )
3		oveq2 5790	. . . . . . 7 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
4	3	oveq2d 5798	. . . . . 6 |- ( j = 0 -> ( ( A ^ M ) x. ( A ^ j ) ) = ( ( A ^ M ) x. ( A ^ 0 ) ) )
5	2, 4	eqeq12d 2155	. . . . 5 |- ( j = 0 -> ( ( A ^ ( M + j ) ) = ( ( A ^ M ) x. ( A ^ j ) ) <-> ( A ^ ( M + 0 ) ) = ( ( A ^ M ) x. ( A ^ 0 ) ) ) )
6	5	imbi2d 229	. . . 4 |- ( j = 0 -> ( ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + j ) ) = ( ( A ^ M ) x. ( A ^ j ) ) ) <-> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + 0 ) ) = ( ( A ^ M ) x. ( A ^ 0 ) ) ) ) )
7		oveq2 5790	. . . . . . 7 |- ( j = k -> ( M + j ) = ( M + k ) )
8	7	oveq2d 5798	. . . . . 6 |- ( j = k -> ( A ^ ( M + j ) ) = ( A ^ ( M + k ) ) )
9		oveq2 5790	. . . . . . 7 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
10	9	oveq2d 5798	. . . . . 6 |- ( j = k -> ( ( A ^ M ) x. ( A ^ j ) ) = ( ( A ^ M ) x. ( A ^ k ) ) )
11	8, 10	eqeq12d 2155	. . . . 5 |- ( j = k -> ( ( A ^ ( M + j ) ) = ( ( A ^ M ) x. ( A ^ j ) ) <-> ( A ^ ( M + k ) ) = ( ( A ^ M ) x. ( A ^ k ) ) ) )
12	11	imbi2d 229	. . . 4 |- ( j = k -> ( ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + j ) ) = ( ( A ^ M ) x. ( A ^ j ) ) ) <-> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + k ) ) = ( ( A ^ M ) x. ( A ^ k ) ) ) ) )
13		oveq2 5790	. . . . . . 7 |- ( j = ( k + 1 ) -> ( M + j ) = ( M + ( k + 1 ) ) )
14	13	oveq2d 5798	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ ( M + j ) ) = ( A ^ ( M + ( k + 1 ) ) ) )
15		oveq2 5790	. . . . . . 7 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
16	15	oveq2d 5798	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A ^ M ) x. ( A ^ j ) ) = ( ( A ^ M ) x. ( A ^ ( k + 1 ) ) ) )
17	14, 16	eqeq12d 2155	. . . . 5 |- ( j = ( k + 1 ) -> ( ( A ^ ( M + j ) ) = ( ( A ^ M ) x. ( A ^ j ) ) <-> ( A ^ ( M + ( k + 1 ) ) ) = ( ( A ^ M ) x. ( A ^ ( k + 1 ) ) ) ) )
18	17	imbi2d 229	. . . 4 |- ( j = ( k + 1 ) -> ( ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + j ) ) = ( ( A ^ M ) x. ( A ^ j ) ) ) <-> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + ( k + 1 ) ) ) = ( ( A ^ M ) x. ( A ^ ( k + 1 ) ) ) ) ) )
19		oveq2 5790	. . . . . . 7 |- ( j = N -> ( M + j ) = ( M + N ) )
20	19	oveq2d 5798	. . . . . 6 |- ( j = N -> ( A ^ ( M + j ) ) = ( A ^ ( M + N ) ) )
21		oveq2 5790	. . . . . . 7 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
22	21	oveq2d 5798	. . . . . 6 |- ( j = N -> ( ( A ^ M ) x. ( A ^ j ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
23	20, 22	eqeq12d 2155	. . . . 5 |- ( j = N -> ( ( A ^ ( M + j ) ) = ( ( A ^ M ) x. ( A ^ j ) ) <-> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
24	23	imbi2d 229	. . . 4 |- ( j = N -> ( ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + j ) ) = ( ( A ^ M ) x. ( A ^ j ) ) ) <-> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) ) )
25		nn0cn 9011	. . . . . . . . 9 |- ( M e. NN0 -> M e. CC )
26	25	addid1d 7935	. . . . . . . 8 |- ( M e. NN0 -> ( M + 0 ) = M )
27	26	adantl 275	. . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> ( M + 0 ) = M )
28	27	oveq2d 5798	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + 0 ) ) = ( A ^ M ) )
29		expcl 10342	. . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC )
30	29	mulid1d 7807	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( ( A ^ M ) x. 1 ) = ( A ^ M ) )
31	28, 30	eqtr4d 2176	. . . . 5 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + 0 ) ) = ( ( A ^ M ) x. 1 ) )
32		exp0 10328	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
33	32	adantr 274	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ 0 ) = 1 )
34	33	oveq2d 5798	. . . . 5 |- ( ( A e. CC /\ M e. NN0 ) -> ( ( A ^ M ) x. ( A ^ 0 ) ) = ( ( A ^ M ) x. 1 ) )
35	31, 34	eqtr4d 2176	. . . 4 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + 0 ) ) = ( ( A ^ M ) x. ( A ^ 0 ) ) )
36		oveq1 5789	. . . . . . 7 |- ( ( A ^ ( M + k ) ) = ( ( A ^ M ) x. ( A ^ k ) ) -> ( ( A ^ ( M + k ) ) x. A ) = ( ( ( A ^ M ) x. ( A ^ k ) ) x. A ) )
37		nn0cn 9011	. . . . . . . . . . . 12 |- ( k e. NN0 -> k e. CC )
38		ax-1cn 7737	. . . . . . . . . . . . 13 |- 1 e. CC
39		addass 7774	. . . . . . . . . . . . 13 |- ( ( M e. CC /\ k e. CC /\ 1 e. CC ) -> ( ( M + k ) + 1 ) = ( M + ( k + 1 ) ) )
40	38, 39	mp3an3 1305	. . . . . . . . . . . 12 |- ( ( M e. CC /\ k e. CC ) -> ( ( M + k ) + 1 ) = ( M + ( k + 1 ) ) )
41	25, 37, 40	syl2an 287	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ k e. NN0 ) -> ( ( M + k ) + 1 ) = ( M + ( k + 1 ) ) )
42	41	adantll 468	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( M + k ) + 1 ) = ( M + ( k + 1 ) ) )
43	42	oveq2d 5798	. . . . . . . . 9 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( ( M + k ) + 1 ) ) = ( A ^ ( M + ( k + 1 ) ) ) )
44		simpll 519	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> A e. CC )
45		nn0addcl 9036	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ k e. NN0 ) -> ( M + k ) e. NN0 )
46	45	adantll 468	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( M + k ) e. NN0 )
47		expp1 10331	. . . . . . . . . 10 |- ( ( A e. CC /\ ( M + k ) e. NN0 ) -> ( A ^ ( ( M + k ) + 1 ) ) = ( ( A ^ ( M + k ) ) x. A ) )
48	44, 46, 47	syl2anc 409	. . . . . . . . 9 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( ( M + k ) + 1 ) ) = ( ( A ^ ( M + k ) ) x. A ) )
49	43, 48	eqtr3d 2175	. . . . . . . 8 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( M + ( k + 1 ) ) ) = ( ( A ^ ( M + k ) ) x. A ) )
50		expp1 10331	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
51	50	adantlr 469	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
52	51	oveq2d 5798	. . . . . . . . 9 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( A ^ M ) x. ( A ^ ( k + 1 ) ) ) = ( ( A ^ M ) x. ( ( A ^ k ) x. A ) ) )
53	29	adantr 274	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ M ) e. CC )
54		expcl 10342	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
55	54	adantlr 469	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ k ) e. CC )
56	53, 55, 44	mulassd 7813	. . . . . . . . 9 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( ( A ^ M ) x. ( A ^ k ) ) x. A ) = ( ( A ^ M ) x. ( ( A ^ k ) x. A ) ) )
57	52, 56	eqtr4d 2176	. . . . . . . 8 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( A ^ M ) x. ( A ^ ( k + 1 ) ) ) = ( ( ( A ^ M ) x. ( A ^ k ) ) x. A ) )
58	49, 57	eqeq12d 2155	. . . . . . 7 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( A ^ ( M + ( k + 1 ) ) ) = ( ( A ^ M ) x. ( A ^ ( k + 1 ) ) ) <-> ( ( A ^ ( M + k ) ) x. A ) = ( ( ( A ^ M ) x. ( A ^ k ) ) x. A ) ) )
59	36, 58	syl5ibr 155	. . . . . 6 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( A ^ ( M + k ) ) = ( ( A ^ M ) x. ( A ^ k ) ) -> ( A ^ ( M + ( k + 1 ) ) ) = ( ( A ^ M ) x. ( A ^ ( k + 1 ) ) ) ) )
60	59	expcom 115	. . . . 5 |- ( k e. NN0 -> ( ( A e. CC /\ M e. NN0 ) -> ( ( A ^ ( M + k ) ) = ( ( A ^ M ) x. ( A ^ k ) ) -> ( A ^ ( M + ( k + 1 ) ) ) = ( ( A ^ M ) x. ( A ^ ( k + 1 ) ) ) ) ) )
61	60	a2d 26	. . . 4 |- ( k e. NN0 -> ( ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + k ) ) = ( ( A ^ M ) x. ( A ^ k ) ) ) -> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + ( k + 1 ) ) ) = ( ( A ^ M ) x. ( A ^ ( k + 1 ) ) ) ) ) )
62	6, 12, 18, 24, 35, 61	nn0ind 9189	. . 3 |- ( N e. NN0 -> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
63	62	expdcom 1419	. 2 |- ( A e. CC -> ( M e. NN0 -> ( N e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) ) )
64	63	3imp 1176	1 |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1332   e. wcel 1481  (class class class)co 5782   CCcc 7642   0cc0 7644   1c1 7645   + caddc 7647   x. cmul 7649   NN0cn0 9001   ^cexp 10323

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-seqfrec 10250  df-exp 10324

This theorem is referenced by:  expaddzaplem  10367  expaddzap  10368  expmul  10369  i4  10426  expaddd  10457  ef01bndlem  11499