Theorem efexp 11674

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 9247	. . . 4 |- ( N e. ZZ -> N e. CC )
2		mulcom 7931	. . . 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 5515	. 2 |- ( ( A e. CC /\ N e. ZZ ) -> ( exp ` ( A x. N ) ) = ( exp ` ( N x. A ) ) )
5		oveq2 5877	. . . . . 6 |- ( j = 0 -> ( A x. j ) = ( A x. 0 ) )
6	5	fveq2d 5515	. . . . 5 |- ( j = 0 -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. 0 ) ) )
7		oveq2 5877	. . . . 5 |- ( j = 0 -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ 0 ) )
8	6, 7	eqeq12d 2192	. . . 4 |- ( j = 0 -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. 0 ) ) = ( ( exp ` A ) ^ 0 ) ) )
9		oveq2 5877	. . . . . 6 |- ( j = k -> ( A x. j ) = ( A x. k ) )
10	9	fveq2d 5515	. . . . 5 |- ( j = k -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. k ) ) )
11		oveq2 5877	. . . . 5 |- ( j = k -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ k ) )
12	10, 11	eqeq12d 2192	. . . 4 |- ( j = k -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) ) )
13		oveq2 5877	. . . . . 6 |- ( j = ( k + 1 ) -> ( A x. j ) = ( A x. ( k + 1 ) ) )
14	13	fveq2d 5515	. . . . 5 |- ( j = ( k + 1 ) -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. ( k + 1 ) ) ) )
15		oveq2 5877	. . . . 5 |- ( j = ( k + 1 ) -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ ( k + 1 ) ) )
16	14, 15	eqeq12d 2192	. . . 4 |- ( j = ( k + 1 ) -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. ( k + 1 ) ) ) = ( ( exp ` A ) ^ ( k + 1 ) ) ) )
17		oveq2 5877	. . . . . 6 |- ( j = -u k -> ( A x. j ) = ( A x. -u k ) )
18	17	fveq2d 5515	. . . . 5 |- ( j = -u k -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. -u k ) ) )
19		oveq2 5877	. . . . 5 |- ( j = -u k -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ -u k ) )
20	18, 19	eqeq12d 2192	. . . 4 |- ( j = -u k -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. -u k ) ) = ( ( exp ` A ) ^ -u k ) ) )
21		oveq2 5877	. . . . . 6 |- ( j = N -> ( A x. j ) = ( A x. N ) )
22	21	fveq2d 5515	. . . . 5 |- ( j = N -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. N ) ) )
23		oveq2 5877	. . . . 5 |- ( j = N -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ N ) )
24	22, 23	eqeq12d 2192	. . . 4 |- ( j = N -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. N ) ) = ( ( exp ` A ) ^ N ) ) )
25		ef0 11664	. . . . 5 |- ( exp ` 0 ) = 1
26		mul01 8336	. . . . . 6 |- ( A e. CC -> ( A x. 0 ) = 0 )
27	26	fveq2d 5515	. . . . 5 |- ( A e. CC -> ( exp ` ( A x. 0 ) ) = ( exp ` 0 ) )
28		efcl 11656	. . . . . 6 |- ( A e. CC -> ( exp ` A ) e. CC )
29	28	exp0d 10633	. . . . 5 |- ( A e. CC -> ( ( exp ` A ) ^ 0 ) = 1 )
30	25, 27, 29	3eqtr4a 2236	. . . 4 |- ( A e. CC -> ( exp ` ( A x. 0 ) ) = ( ( exp ` A ) ^ 0 ) )
31		oveq1 5876	. . . . . . 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 9175	. . . . . . . . . 10 |- ( k e. NN0 -> k e. CC )
34		ax-1cn 7895	. . . . . . . . . . . 12 |- 1 e. CC
35		adddi 7934	. . . . . . . . . . . 12 |- ( ( A e. CC /\ k e. CC /\ 1 e. CC ) -> ( A x. ( k + 1 ) ) = ( ( A x. k ) + ( A x. 1 ) ) )
36	34, 35	mp3an3 1326	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. CC ) -> ( A x. ( k + 1 ) ) = ( ( A x. k ) + ( A x. 1 ) ) )
37		mulid1 7945	. . . . . . . . . . . . 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 5885	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. CC ) -> ( ( A x. k ) + ( A x. 1 ) ) = ( ( A x. k ) + A ) )
40	36, 39	eqtrd 2210	. . . . . . . . . 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 5515	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( exp ` ( A x. ( k + 1 ) ) ) = ( exp ` ( ( A x. k ) + A ) ) )
43		mulcl 7929	. . . . . . . . . 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 11667	. . . . . . . . 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 2210	. . . . . . 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 10513	. . . . . . . 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 2220	. . . . 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 5877	. . . . . 6 |- ( ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) -> ( 1 / ( exp ` ( A x. k ) ) ) = ( 1 / ( ( exp ` A ) ^ k ) ) )
56		nncn 8916	. . . . . . . . . 10 |- ( k e. NN -> k e. CC )
57		mulneg2 8343	. . . . . . . . . 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 5515	. . . . . . . 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 11671	. . . . . . . . 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 2210	. . . . . . 7 |- ( ( A e. CC /\ k e. NN ) -> ( exp ` ( A x. -u k ) ) = ( 1 / ( exp ` ( A x. k ) ) ) )
64		efap0 11669	. . . . . . . 8 |- ( A e. CC -> ( exp ` A ) # 0 )
65		nnnn0 9172	. . . . . . . 8 |- ( k e. NN -> k e. NN0 )
66		expnegap0 10514	. . . . . . . 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 1298	. . . . . . 7 |- ( ( A e. CC /\ k e. NN ) -> ( ( exp ` A ) ^ -u k ) = ( 1 / ( ( exp ` A ) ^ k ) ) )
68	63, 67	eqeq12d 2192	. . . . . 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	syl5ibr 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 9360	. . 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 2212	1 |- ( ( A e. CC /\ N e. ZZ ) -> ( exp ` ( N x. A ) ) = ( ( exp ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   -ucneg 8119   # cap 8528   / cdiv 8618   NNcn 8908   NN0cn0 9165   ZZcz 9242   ^cexp 10505   expce 11634

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-disj 3978  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-ico 9881  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-fac 10690  df-bc 10712  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346  df-ef 11640

This theorem is referenced by:  efzval  11675  efgt0  11676  tanval3ap  11706  demoivre  11764  ef2kpi  13894  reexplog  13959  relogexp  13960