Theorem expadd 10470

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 5834	. . . . . . 7 |- ( j = 0 -> ( M + j ) = ( M + 0 ) )
2	1	oveq2d 5842	. . . . . 6 |- ( j = 0 -> ( A ^ ( M + j ) ) = ( A ^ ( M + 0 ) ) )
3		oveq2 5834	. . . . . . 7 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
4	3	oveq2d 5842	. . . . . 6 |- ( j = 0 -> ( ( A ^ M ) x. ( A ^ j ) ) = ( ( A ^ M ) x. ( A ^ 0 ) ) )
5	2, 4	eqeq12d 2172	. . . . 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 5834	. . . . . . 7 |- ( j = k -> ( M + j ) = ( M + k ) )
8	7	oveq2d 5842	. . . . . 6 |- ( j = k -> ( A ^ ( M + j ) ) = ( A ^ ( M + k ) ) )
9		oveq2 5834	. . . . . . 7 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
10	9	oveq2d 5842	. . . . . 6 |- ( j = k -> ( ( A ^ M ) x. ( A ^ j ) ) = ( ( A ^ M ) x. ( A ^ k ) ) )
11	8, 10	eqeq12d 2172	. . . . 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 5834	. . . . . . 7 |- ( j = ( k + 1 ) -> ( M + j ) = ( M + ( k + 1 ) ) )
14	13	oveq2d 5842	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ ( M + j ) ) = ( A ^ ( M + ( k + 1 ) ) ) )
15		oveq2 5834	. . . . . . 7 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
16	15	oveq2d 5842	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A ^ M ) x. ( A ^ j ) ) = ( ( A ^ M ) x. ( A ^ ( k + 1 ) ) ) )
17	14, 16	eqeq12d 2172	. . . . 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 5834	. . . . . . 7 |- ( j = N -> ( M + j ) = ( M + N ) )
20	19	oveq2d 5842	. . . . . 6 |- ( j = N -> ( A ^ ( M + j ) ) = ( A ^ ( M + N ) ) )
21		oveq2 5834	. . . . . . 7 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
22	21	oveq2d 5842	. . . . . 6 |- ( j = N -> ( ( A ^ M ) x. ( A ^ j ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
23	20, 22	eqeq12d 2172	. . . . 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 9105	. . . . . . . . 9 |- ( M e. NN0 -> M e. CC )
26	25	addid1d 8028	. . . . . . . 8 |- ( M e. NN0 -> ( M + 0 ) = M )
27	26	adantl 275	. . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> ( M + 0 ) = M )
28	27	oveq2d 5842	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + 0 ) ) = ( A ^ M ) )
29		expcl 10446	. . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC )
30	29	mulid1d 7897	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( ( A ^ M ) x. 1 ) = ( A ^ M ) )
31	28, 30	eqtr4d 2193	. . . . 5 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + 0 ) ) = ( ( A ^ M ) x. 1 ) )
32		exp0 10432	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
33	32	adantr 274	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ 0 ) = 1 )
34	33	oveq2d 5842	. . . . 5 |- ( ( A e. CC /\ M e. NN0 ) -> ( ( A ^ M ) x. ( A ^ 0 ) ) = ( ( A ^ M ) x. 1 ) )
35	31, 34	eqtr4d 2193	. . . 4 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + 0 ) ) = ( ( A ^ M ) x. ( A ^ 0 ) ) )
36		oveq1 5833	. . . . . . 7 |- ( ( A ^ ( M + k ) ) = ( ( A ^ M ) x. ( A ^ k ) ) -> ( ( A ^ ( M + k ) ) x. A ) = ( ( ( A ^ M ) x. ( A ^ k ) ) x. A ) )
37		nn0cn 9105	. . . . . . . . . . . 12 |- ( k e. NN0 -> k e. CC )
38		ax-1cn 7827	. . . . . . . . . . . . 13 |- 1 e. CC
39		addass 7864	. . . . . . . . . . . . 13 |- ( ( M e. CC /\ k e. CC /\ 1 e. CC ) -> ( ( M + k ) + 1 ) = ( M + ( k + 1 ) ) )
40	38, 39	mp3an3 1308	. . . . . . . . . . . 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 5842	. . . . . . . . 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 9130	. . . . . . . . . . 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 10435	. . . . . . . . . 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 2192	. . . . . . . 8 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( M + ( k + 1 ) ) ) = ( ( A ^ ( M + k ) ) x. A ) )
50		expp1 10435	. . . . . . . . . . 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 5842	. . . . . . . . 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 10446	. . . . . . . . . . 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 7903	. . . . . . . . 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 2193	. . . . . . . 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 2172	. . . . . . 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 9283	. . 3 |- ( N e. NN0 -> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
63	62	expdcom 1422	. 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 1335   e. wcel 2128  (class class class)co 5826   CCcc 7732   0cc0 7734   1c1 7735   + caddc 7737   x. cmul 7739   NN0cn0 9095   ^cexp 10427

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-frec 6340  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550  df-inn 8839  df-n0 9096  df-z 9173  df-uz 9445  df-seqfrec 10354  df-exp 10428

This theorem is referenced by:  expaddzaplem  10471  expaddzap  10472  expmul  10473  i4  10530  expaddd  10562  ef01bndlem  11664