Theorem expadd 11018

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 6093	. . . . . . 7 |- ( j = 0 -> ( M + j ) = ( M + 0 ) )
2	1	oveq2d 6101	. . . . . 6 |- ( j = 0 -> ( A ^ ( M + j ) ) = ( A ^ ( M + 0 ) ) )
3		oveq2 6093	. . . . . . 7 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
4	3	oveq2d 6101	. . . . . 6 |- ( j = 0 -> ( ( A ^ M ) x. ( A ^ j ) ) = ( ( A ^ M ) x. ( A ^ 0 ) ) )
5	2, 4	eqeq12d 2253	. . . . 5 |- ( j = 0 -> ( ( A ^ ( M + j ) ) = ( ( A ^ M ) x. ( A ^ j ) ) <-> ( A ^ ( M + 0 ) ) = ( ( A ^ M ) x. ( A ^ 0 ) ) ) )
6	5	imbi2d 230	. . . 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 6093	. . . . . . 7 |- ( j = k -> ( M + j ) = ( M + k ) )
8	7	oveq2d 6101	. . . . . 6 |- ( j = k -> ( A ^ ( M + j ) ) = ( A ^ ( M + k ) ) )
9		oveq2 6093	. . . . . . 7 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
10	9	oveq2d 6101	. . . . . 6 |- ( j = k -> ( ( A ^ M ) x. ( A ^ j ) ) = ( ( A ^ M ) x. ( A ^ k ) ) )
11	8, 10	eqeq12d 2253	. . . . 5 |- ( j = k -> ( ( A ^ ( M + j ) ) = ( ( A ^ M ) x. ( A ^ j ) ) <-> ( A ^ ( M + k ) ) = ( ( A ^ M ) x. ( A ^ k ) ) ) )
12	11	imbi2d 230	. . . 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 6093	. . . . . . 7 |- ( j = ( k + 1 ) -> ( M + j ) = ( M + ( k + 1 ) ) )
14	13	oveq2d 6101	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ ( M + j ) ) = ( A ^ ( M + ( k + 1 ) ) ) )
15		oveq2 6093	. . . . . . 7 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
16	15	oveq2d 6101	. . . . . 6 |- ( j = ( k + 1 ) -> ( ( A ^ M ) x. ( A ^ j ) ) = ( ( A ^ M ) x. ( A ^ ( k + 1 ) ) ) )
17	14, 16	eqeq12d 2253	. . . . 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 230	. . . 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 6093	. . . . . . 7 |- ( j = N -> ( M + j ) = ( M + N ) )
20	19	oveq2d 6101	. . . . . 6 |- ( j = N -> ( A ^ ( M + j ) ) = ( A ^ ( M + N ) ) )
21		oveq2 6093	. . . . . . 7 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
22	21	oveq2d 6101	. . . . . 6 |- ( j = N -> ( ( A ^ M ) x. ( A ^ j ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
23	20, 22	eqeq12d 2253	. . . . 5 |- ( j = N -> ( ( A ^ ( M + j ) ) = ( ( A ^ M ) x. ( A ^ j ) ) <-> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
24	23	imbi2d 230	. . . 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 9573	. . . . . . . . 9 |- ( M e. NN0 -> M e. CC )
26	25	addridd 8475	. . . . . . . 8 |- ( M e. NN0 -> ( M + 0 ) = M )
27	26	adantl 277	. . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> ( M + 0 ) = M )
28	27	oveq2d 6101	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + 0 ) ) = ( A ^ M ) )
29		expcl 10994	. . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ M ) e. CC )
30	29	mulridd 8343	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( ( A ^ M ) x. 1 ) = ( A ^ M ) )
31	28, 30	eqtr4d 2274	. . . . 5 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + 0 ) ) = ( ( A ^ M ) x. 1 ) )
32		exp0 10980	. . . . . . 7 |- ( A e. CC -> ( A ^ 0 ) = 1 )
33	32	adantr 276	. . . . . 6 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ 0 ) = 1 )
34	33	oveq2d 6101	. . . . 5 |- ( ( A e. CC /\ M e. NN0 ) -> ( ( A ^ M ) x. ( A ^ 0 ) ) = ( ( A ^ M ) x. 1 ) )
35	31, 34	eqtr4d 2274	. . . 4 |- ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + 0 ) ) = ( ( A ^ M ) x. ( A ^ 0 ) ) )
36		oveq1 6092	. . . . . . 7 |- ( ( A ^ ( M + k ) ) = ( ( A ^ M ) x. ( A ^ k ) ) -> ( ( A ^ ( M + k ) ) x. A ) = ( ( ( A ^ M ) x. ( A ^ k ) ) x. A ) )
37		nn0cn 9573	. . . . . . . . . . . 12 |- ( k e. NN0 -> k e. CC )
38		ax-1cn 8272	. . . . . . . . . . . . 13 |- 1 e. CC
39		addass 8309	. . . . . . . . . . . . 13 |- ( ( M e. CC /\ k e. CC /\ 1 e. CC ) -> ( ( M + k ) + 1 ) = ( M + ( k + 1 ) ) )
40	38, 39	mp3an3 1367	. . . . . . . . . . . 12 |- ( ( M e. CC /\ k e. CC ) -> ( ( M + k ) + 1 ) = ( M + ( k + 1 ) ) )
41	25, 37, 40	syl2an 289	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ k e. NN0 ) -> ( ( M + k ) + 1 ) = ( M + ( k + 1 ) ) )
42	41	adantll 480	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( M + k ) + 1 ) = ( M + ( k + 1 ) ) )
43	42	oveq2d 6101	. . . . . . . . 9 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( ( M + k ) + 1 ) ) = ( A ^ ( M + ( k + 1 ) ) ) )
44		simpll 531	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> A e. CC )
45		nn0addcl 9598	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ k e. NN0 ) -> ( M + k ) e. NN0 )
46	45	adantll 480	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( M + k ) e. NN0 )
47		expp1 10983	. . . . . . . . . 10 |- ( ( A e. CC /\ ( M + k ) e. NN0 ) -> ( A ^ ( ( M + k ) + 1 ) ) = ( ( A ^ ( M + k ) ) x. A ) )
48	44, 46, 47	syl2anc 415	. . . . . . . . 9 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( ( M + k ) + 1 ) ) = ( ( A ^ ( M + k ) ) x. A ) )
49	43, 48	eqtr3d 2273	. . . . . . . 8 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( M + ( k + 1 ) ) ) = ( ( A ^ ( M + k ) ) x. A ) )
50		expp1 10983	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
51	50	adantlr 481	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
52	51	oveq2d 6101	. . . . . . . . 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 276	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ M ) e. CC )
54		expcl 10994	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
55	54	adantlr 481	. . . . . . . . . 10 |- ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( A ^ k ) e. CC )
56	53, 55, 44	mulassd 8349	. . . . . . . . 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 2274	. . . . . . . 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 2253	. . . . . . 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	imbitrrid 156	. . . . . 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 116	. . . . 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 9760	. . 3 |- ( N e. NN0 -> ( ( A e. CC /\ M e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) )
63	62	expdcom 1492	. 2 |- ( A e. CC -> ( M e. NN0 -> ( N e. NN0 -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) ) ) )
64	63	3imp 1224	1 |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   NN0cn0 9563   ^cexp 10975

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-seqfrec 10885  df-exp 10976

This theorem is used by:  expaddzaplem  11019  expaddzap  11020  expmul  11021  i4  11079  expaddd  11113  ef01bndlem  12523  modxai  13195  numexp2x  13204  2exp5  13211  2exp11  13215  log2ublem2  16084