Theorem efexp 12393

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 9599	. . . 4 |- ( N e. ZZ -> N e. CC )
2		mulcom 8272	. . . 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 5679	. 2 |- ( ( A e. CC /\ N e. ZZ ) -> ( exp ` ( A x. N ) ) = ( exp ` ( N x. A ) ) )
5		oveq2 6066	. . . . . 6 |- ( j = 0 -> ( A x. j ) = ( A x. 0 ) )
6	5	fveq2d 5679	. . . . 5 |- ( j = 0 -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. 0 ) ) )
7		oveq2 6066	. . . . 5 |- ( j = 0 -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ 0 ) )
8	6, 7	eqeq12d 2249	. . . 4 |- ( j = 0 -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. 0 ) ) = ( ( exp ` A ) ^ 0 ) ) )
9		oveq2 6066	. . . . . 6 |- ( j = k -> ( A x. j ) = ( A x. k ) )
10	9	fveq2d 5679	. . . . 5 |- ( j = k -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. k ) ) )
11		oveq2 6066	. . . . 5 |- ( j = k -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ k ) )
12	10, 11	eqeq12d 2249	. . . 4 |- ( j = k -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) ) )
13		oveq2 6066	. . . . . 6 |- ( j = ( k + 1 ) -> ( A x. j ) = ( A x. ( k + 1 ) ) )
14	13	fveq2d 5679	. . . . 5 |- ( j = ( k + 1 ) -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. ( k + 1 ) ) ) )
15		oveq2 6066	. . . . 5 |- ( j = ( k + 1 ) -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ ( k + 1 ) ) )
16	14, 15	eqeq12d 2249	. . . 4 |- ( j = ( k + 1 ) -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. ( k + 1 ) ) ) = ( ( exp ` A ) ^ ( k + 1 ) ) ) )
17		oveq2 6066	. . . . . 6 |- ( j = -u k -> ( A x. j ) = ( A x. -u k ) )
18	17	fveq2d 5679	. . . . 5 |- ( j = -u k -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. -u k ) ) )
19		oveq2 6066	. . . . 5 |- ( j = -u k -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ -u k ) )
20	18, 19	eqeq12d 2249	. . . 4 |- ( j = -u k -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. -u k ) ) = ( ( exp ` A ) ^ -u k ) ) )
21		oveq2 6066	. . . . . 6 |- ( j = N -> ( A x. j ) = ( A x. N ) )
22	21	fveq2d 5679	. . . . 5 |- ( j = N -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. N ) ) )
23		oveq2 6066	. . . . 5 |- ( j = N -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ N ) )
24	22, 23	eqeq12d 2249	. . . 4 |- ( j = N -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. N ) ) = ( ( exp ` A ) ^ N ) ) )
25		ef0 12383	. . . . 5 |- ( exp ` 0 ) = 1
26		mul01 8679	. . . . . 6 |- ( A e. CC -> ( A x. 0 ) = 0 )
27	26	fveq2d 5679	. . . . 5 |- ( A e. CC -> ( exp ` ( A x. 0 ) ) = ( exp ` 0 ) )
28		efcl 12375	. . . . . 6 |- ( A e. CC -> ( exp ` A ) e. CC )
29	28	exp0d 11054	. . . . 5 |- ( A e. CC -> ( ( exp ` A ) ^ 0 ) = 1 )
30	25, 27, 29	3eqtr4a 2293	. . . 4 |- ( A e. CC -> ( exp ` ( A x. 0 ) ) = ( ( exp ` A ) ^ 0 ) )
31		oveq1 6065	. . . . . . 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 9523	. . . . . . . . . 10 |- ( k e. NN0 -> k e. CC )
34		ax-1cn 8236	. . . . . . . . . . . 12 |- 1 e. CC
35		adddi 8275	. . . . . . . . . . . 12 |- ( ( A e. CC /\ k e. CC /\ 1 e. CC ) -> ( A x. ( k + 1 ) ) = ( ( A x. k ) + ( A x. 1 ) ) )
36	34, 35	mp3an3 1363	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. CC ) -> ( A x. ( k + 1 ) ) = ( ( A x. k ) + ( A x. 1 ) ) )
37		mulrid 8287	. . . . . . . . . . . . 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 6074	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. CC ) -> ( ( A x. k ) + ( A x. 1 ) ) = ( ( A x. k ) + A ) )
40	36, 39	eqtrd 2267	. . . . . . . . . 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 5679	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( exp ` ( A x. ( k + 1 ) ) ) = ( exp ` ( ( A x. k ) + A ) ) )
43		mulcl 8270	. . . . . . . . . 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 12386	. . . . . . . . 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 2267	. . . . . . 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 10932	. . . . . . . 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 2277	. . . . 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 6066	. . . . . 6 |- ( ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) -> ( 1 / ( exp ` ( A x. k ) ) ) = ( 1 / ( ( exp ` A ) ^ k ) ) )
56		nncn 9262	. . . . . . . . . 10 |- ( k e. NN -> k e. CC )
57		mulneg2 8686	. . . . . . . . . 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 5679	. . . . . . . 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 12390	. . . . . . . . 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 2267	. . . . . . 7 |- ( ( A e. CC /\ k e. NN ) -> ( exp ` ( A x. -u k ) ) = ( 1 / ( exp ` ( A x. k ) ) ) )
64		efap0 12388	. . . . . . . 8 |- ( A e. CC -> ( exp ` A ) # 0 )
65		nnnn0 9520	. . . . . . . 8 |- ( k e. NN -> k e. NN0 )
66		expnegap0 10933	. . . . . . . 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 1335	. . . . . . 7 |- ( ( A e. CC /\ k e. NN ) -> ( ( exp ` A ) ^ -u k ) = ( 1 / ( ( exp ` A ) ^ k ) ) )
68	63, 67	eqeq12d 2249	. . . . . 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 9714	. . 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 2269	1 |- ( ( A e. CC /\ N e. ZZ ) -> ( exp ` ( N x. A ) ) = ( ( exp ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   -ucneg 8461   # cap 8872   / cdiv 8963   NNcn 9254   NN0cn0 9513   ZZcz 9594   ^cexp 10924   expce 12353

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-disj 4091  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-ico 10246  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-fac 11113  df-bc 11135  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064  df-ef 12359

This theorem is referenced by:  efzval  12394  efgt0  12395  tanval3ap  12425  demoivre  12484  ef2kpi  15797  reexplog  15862  relogexp  15863