Theorem efexp 11825

Description: The exponential of an integer power. Corollary 15-4.4 of [Gleason] p. 309, restricted to integers. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)

Assertion

Ref	Expression
efexp	|- ( ( A e. CC /\ N e. ZZ ) -> ( exp ` ( N x. A ) ) = ( ( exp ` A ) ^ N ) )

Proof of Theorem efexp

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

Step	Hyp	Ref	Expression
1		zcn 9322	. . . 4 |- ( N e. ZZ -> N e. CC )
2		mulcom 8001	. . . 4 |- ( ( A e. CC /\ N e. CC ) -> ( A x. N ) = ( N x. A ) )
3	1, 2	sylan2 286	. . 3 |- ( ( A e. CC /\ N e. ZZ ) -> ( A x. N ) = ( N x. A ) )
4	3	fveq2d 5558	. 2 |- ( ( A e. CC /\ N e. ZZ ) -> ( exp ` ( A x. N ) ) = ( exp ` ( N x. A ) ) )
5		oveq2 5926	. . . . . 6 |- ( j = 0 -> ( A x. j ) = ( A x. 0 ) )
6	5	fveq2d 5558	. . . . 5 |- ( j = 0 -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. 0 ) ) )
7		oveq2 5926	. . . . 5 |- ( j = 0 -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ 0 ) )
8	6, 7	eqeq12d 2208	. . . 4 |- ( j = 0 -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. 0 ) ) = ( ( exp ` A ) ^ 0 ) ) )
9		oveq2 5926	. . . . . 6 |- ( j = k -> ( A x. j ) = ( A x. k ) )
10	9	fveq2d 5558	. . . . 5 |- ( j = k -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. k ) ) )
11		oveq2 5926	. . . . 5 |- ( j = k -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ k ) )
12	10, 11	eqeq12d 2208	. . . 4 |- ( j = k -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) ) )
13		oveq2 5926	. . . . . 6 |- ( j = ( k + 1 ) -> ( A x. j ) = ( A x. ( k + 1 ) ) )
14	13	fveq2d 5558	. . . . 5 |- ( j = ( k + 1 ) -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. ( k + 1 ) ) ) )
15		oveq2 5926	. . . . 5 |- ( j = ( k + 1 ) -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ ( k + 1 ) ) )
16	14, 15	eqeq12d 2208	. . . 4 |- ( j = ( k + 1 ) -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. ( k + 1 ) ) ) = ( ( exp ` A ) ^ ( k + 1 ) ) ) )
17		oveq2 5926	. . . . . 6 |- ( j = -u k -> ( A x. j ) = ( A x. -u k ) )
18	17	fveq2d 5558	. . . . 5 |- ( j = -u k -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. -u k ) ) )
19		oveq2 5926	. . . . 5 |- ( j = -u k -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ -u k ) )
20	18, 19	eqeq12d 2208	. . . 4 |- ( j = -u k -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. -u k ) ) = ( ( exp ` A ) ^ -u k ) ) )
21		oveq2 5926	. . . . . 6 |- ( j = N -> ( A x. j ) = ( A x. N ) )
22	21	fveq2d 5558	. . . . 5 |- ( j = N -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. N ) ) )
23		oveq2 5926	. . . . 5 |- ( j = N -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ N ) )
24	22, 23	eqeq12d 2208	. . . 4 |- ( j = N -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. N ) ) = ( ( exp ` A ) ^ N ) ) )
25		ef0 11815	. . . . 5 |- ( exp ` 0 ) = 1
26		mul01 8408	. . . . . 6 |- ( A e. CC -> ( A x. 0 ) = 0 )
27	26	fveq2d 5558	. . . . 5 |- ( A e. CC -> ( exp ` ( A x. 0 ) ) = ( exp ` 0 ) )
28		efcl 11807	. . . . . 6 |- ( A e. CC -> ( exp ` A ) e. CC )
29	28	exp0d 10738	. . . . 5 |- ( A e. CC -> ( ( exp ` A ) ^ 0 ) = 1 )
30	25, 27, 29	3eqtr4a 2252	. . . 4 |- ( A e. CC -> ( exp ` ( A x. 0 ) ) = ( ( exp ` A ) ^ 0 ) )
31		oveq1 5925	. . . . . . 7 |- ( ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) -> ( ( exp ` ( A x. k ) ) x. ( exp ` A ) ) = ( ( ( exp ` A ) ^ k ) x. ( exp ` A ) ) )
32	31	adantl 277	. . . . . 6 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) ) -> ( ( exp ` ( A x. k ) ) x. ( exp ` A ) ) = ( ( ( exp ` A ) ^ k ) x. ( exp ` A ) ) )
33		nn0cn 9250	. . . . . . . . . 10 |- ( k e. NN0 -> k e. CC )
34		ax-1cn 7965	. . . . . . . . . . . 12 |- 1 e. CC
35		adddi 8004	. . . . . . . . . . . 12 |- ( ( A e. CC /\ k e. CC /\ 1 e. CC ) -> ( A x. ( k + 1 ) ) = ( ( A x. k ) + ( A x. 1 ) ) )
36	34, 35	mp3an3 1337	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. CC ) -> ( A x. ( k + 1 ) ) = ( ( A x. k ) + ( A x. 1 ) ) )
37		mulrid 8016	. . . . . . . . . . . . 13 |- ( A e. CC -> ( A x. 1 ) = A )
38	37	adantr 276	. . . . . . . . . . . 12 |- ( ( A e. CC /\ k e. CC ) -> ( A x. 1 ) = A )
39	38	oveq2d 5934	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. CC ) -> ( ( A x. k ) + ( A x. 1 ) ) = ( ( A x. k ) + A ) )
40	36, 39	eqtrd 2226	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. CC ) -> ( A x. ( k + 1 ) ) = ( ( A x. k ) + A ) )
41	33, 40	sylan2 286	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( A x. ( k + 1 ) ) = ( ( A x. k ) + A ) )
42	41	fveq2d 5558	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( exp ` ( A x. ( k + 1 ) ) ) = ( exp ` ( ( A x. k ) + A ) ) )
43		mulcl 7999	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. CC ) -> ( A x. k ) e. CC )
44	33, 43	sylan2 286	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( A x. k ) e. CC )
45		simpl 109	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> A e. CC )
46		efadd 11818	. . . . . . . . 9 |- ( ( ( A x. k ) e. CC /\ A e. CC ) -> ( exp ` ( ( A x. k ) + A ) ) = ( ( exp ` ( A x. k ) ) x. ( exp ` A ) ) )
47	44, 45, 46	syl2anc 411	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( exp ` ( ( A x. k ) + A ) ) = ( ( exp ` ( A x. k ) ) x. ( exp ` A ) ) )
48	42, 47	eqtrd 2226	. . . . . . 7 |- ( ( A e. CC /\ k e. NN0 ) -> ( exp ` ( A x. ( k + 1 ) ) ) = ( ( exp ` ( A x. k ) ) x. ( exp ` A ) ) )
49	48	adantr 276	. . . . . 6 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) ) -> ( exp ` ( A x. ( k + 1 ) ) ) = ( ( exp ` ( A x. k ) ) x. ( exp ` A ) ) )
50		expp1 10617	. . . . . . . 8 |- ( ( ( exp ` A ) e. CC /\ k e. NN0 ) -> ( ( exp ` A ) ^ ( k + 1 ) ) = ( ( ( exp ` A ) ^ k ) x. ( exp ` A ) ) )
51	28, 50	sylan 283	. . . . . . 7 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( exp ` A ) ^ ( k + 1 ) ) = ( ( ( exp ` A ) ^ k ) x. ( exp ` A ) ) )
52	51	adantr 276	. . . . . 6 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) ) -> ( ( exp ` A ) ^ ( k + 1 ) ) = ( ( ( exp ` A ) ^ k ) x. ( exp ` A ) ) )
53	32, 49, 52	3eqtr4d 2236	. . . . 5 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) ) -> ( exp ` ( A x. ( k + 1 ) ) ) = ( ( exp ` A ) ^ ( k + 1 ) ) )
54	53	exp31 364	. . . 4 |- ( A e. CC -> ( k e. NN0 -> ( ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) -> ( exp ` ( A x. ( k + 1 ) ) ) = ( ( exp ` A ) ^ ( k + 1 ) ) ) ) )
55		oveq2 5926	. . . . . 6 |- ( ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) -> ( 1 / ( exp ` ( A x. k ) ) ) = ( 1 / ( ( exp ` A ) ^ k ) ) )
56		nncn 8990	. . . . . . . . . 10 |- ( k e. NN -> k e. CC )
57		mulneg2 8415	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. CC ) -> ( A x. -u k ) = -u ( A x. k ) )
58	56, 57	sylan2 286	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN ) -> ( A x. -u k ) = -u ( A x. k ) )
59	58	fveq2d 5558	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN ) -> ( exp ` ( A x. -u k ) ) = ( exp ` -u ( A x. k ) ) )
60	56, 43	sylan2 286	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN ) -> ( A x. k ) e. CC )
61		efneg 11822	. . . . . . . . 9 |- ( ( A x. k ) e. CC -> ( exp ` -u ( A x. k ) ) = ( 1 / ( exp ` ( A x. k ) ) ) )
62	60, 61	syl 14	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN ) -> ( exp ` -u ( A x. k ) ) = ( 1 / ( exp ` ( A x. k ) ) ) )
63	59, 62	eqtrd 2226	. . . . . . 7 |- ( ( A e. CC /\ k e. NN ) -> ( exp ` ( A x. -u k ) ) = ( 1 / ( exp ` ( A x. k ) ) ) )
64		efap0 11820	. . . . . . . 8 |- ( A e. CC -> ( exp ` A ) # 0 )
65		nnnn0 9247	. . . . . . . 8 |- ( k e. NN -> k e. NN0 )
66		expnegap0 10618	. . . . . . . 8 |- ( ( ( exp ` A ) e. CC /\ ( exp ` A ) # 0 /\ k e. NN0 ) -> ( ( exp ` A ) ^ -u k ) = ( 1 / ( ( exp ` A ) ^ k ) ) )
67	28, 64, 65, 66	syl2an3an 1309	. . . . . . 7 |- ( ( A e. CC /\ k e. NN ) -> ( ( exp ` A ) ^ -u k ) = ( 1 / ( ( exp ` A ) ^ k ) ) )
68	63, 67	eqeq12d 2208	. . . . . 6 |- ( ( A e. CC /\ k e. NN ) -> ( ( exp ` ( A x. -u k ) ) = ( ( exp ` A ) ^ -u k ) <-> ( 1 / ( exp ` ( A x. k ) ) ) = ( 1 / ( ( exp ` A ) ^ k ) ) ) )
69	55, 68	imbitrrid 156	. . . . 5 |- ( ( A e. CC /\ k e. NN ) -> ( ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) -> ( exp ` ( A x. -u k ) ) = ( ( exp ` A ) ^ -u k ) ) )
70	69	ex 115	. . . 4 |- ( A e. CC -> ( k e. NN -> ( ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) -> ( exp ` ( A x. -u k ) ) = ( ( exp ` A ) ^ -u k ) ) ) )
71	8, 12, 16, 20, 24, 30, 54, 70	zindd 9435	. . 3 |- ( A e. CC -> ( N e. ZZ -> ( exp ` ( A x. N ) ) = ( ( exp ` A ) ^ N ) ) )
72	71	imp 124	. 2 |- ( ( A e. CC /\ N e. ZZ ) -> ( exp ` ( A x. N ) ) = ( ( exp ` A ) ^ N ) )
73	4, 72	eqtr3d 2228	1 |- ( ( A e. CC /\ N e. ZZ ) -> ( exp ` ( N x. A ) ) = ( ( exp ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   -ucneg 8191   # cap 8600   / cdiv 8691   NNcn 8982   NN0cn0 9240   ZZcz 9317   ^cexp 10609   expce 11785

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-disj 4007  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-ico 9960  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-exp 10610  df-fac 10797  df-bc 10819  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-sumdc 11497  df-ef 11791

This theorem is referenced by:  efzval  11826  efgt0  11827  tanval3ap  11857  demoivre  11916  ef2kpi  14941  reexplog  15006  relogexp  15007