Theorem efexp 12368

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 9582	. . . 4 |- ( N e. ZZ -> N e. CC )
2		mulcom 8256	. . . 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 5674	. 2 |- ( ( A e. CC /\ N e. ZZ ) -> ( exp ` ( A x. N ) ) = ( exp ` ( N x. A ) ) )
5		oveq2 6058	. . . . . 6 |- ( j = 0 -> ( A x. j ) = ( A x. 0 ) )
6	5	fveq2d 5674	. . . . 5 |- ( j = 0 -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. 0 ) ) )
7		oveq2 6058	. . . . 5 |- ( j = 0 -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ 0 ) )
8	6, 7	eqeq12d 2247	. . . 4 |- ( j = 0 -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. 0 ) ) = ( ( exp ` A ) ^ 0 ) ) )
9		oveq2 6058	. . . . . 6 |- ( j = k -> ( A x. j ) = ( A x. k ) )
10	9	fveq2d 5674	. . . . 5 |- ( j = k -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. k ) ) )
11		oveq2 6058	. . . . 5 |- ( j = k -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ k ) )
12	10, 11	eqeq12d 2247	. . . 4 |- ( j = k -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) ) )
13		oveq2 6058	. . . . . 6 |- ( j = ( k + 1 ) -> ( A x. j ) = ( A x. ( k + 1 ) ) )
14	13	fveq2d 5674	. . . . 5 |- ( j = ( k + 1 ) -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. ( k + 1 ) ) ) )
15		oveq2 6058	. . . . 5 |- ( j = ( k + 1 ) -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ ( k + 1 ) ) )
16	14, 15	eqeq12d 2247	. . . 4 |- ( j = ( k + 1 ) -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. ( k + 1 ) ) ) = ( ( exp ` A ) ^ ( k + 1 ) ) ) )
17		oveq2 6058	. . . . . 6 |- ( j = -u k -> ( A x. j ) = ( A x. -u k ) )
18	17	fveq2d 5674	. . . . 5 |- ( j = -u k -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. -u k ) ) )
19		oveq2 6058	. . . . 5 |- ( j = -u k -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ -u k ) )
20	18, 19	eqeq12d 2247	. . . 4 |- ( j = -u k -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. -u k ) ) = ( ( exp ` A ) ^ -u k ) ) )
21		oveq2 6058	. . . . . 6 |- ( j = N -> ( A x. j ) = ( A x. N ) )
22	21	fveq2d 5674	. . . . 5 |- ( j = N -> ( exp ` ( A x. j ) ) = ( exp ` ( A x. N ) ) )
23		oveq2 6058	. . . . 5 |- ( j = N -> ( ( exp ` A ) ^ j ) = ( ( exp ` A ) ^ N ) )
24	22, 23	eqeq12d 2247	. . . 4 |- ( j = N -> ( ( exp ` ( A x. j ) ) = ( ( exp ` A ) ^ j ) <-> ( exp ` ( A x. N ) ) = ( ( exp ` A ) ^ N ) ) )
25		ef0 12358	. . . . 5 |- ( exp ` 0 ) = 1
26		mul01 8662	. . . . . 6 |- ( A e. CC -> ( A x. 0 ) = 0 )
27	26	fveq2d 5674	. . . . 5 |- ( A e. CC -> ( exp ` ( A x. 0 ) ) = ( exp ` 0 ) )
28		efcl 12350	. . . . . 6 |- ( A e. CC -> ( exp ` A ) e. CC )
29	28	exp0d 11029	. . . . 5 |- ( A e. CC -> ( ( exp ` A ) ^ 0 ) = 1 )
30	25, 27, 29	3eqtr4a 2291	. . . 4 |- ( A e. CC -> ( exp ` ( A x. 0 ) ) = ( ( exp ` A ) ^ 0 ) )
31		oveq1 6057	. . . . . . 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 9506	. . . . . . . . . 10 |- ( k e. NN0 -> k e. CC )
34		ax-1cn 8220	. . . . . . . . . . . 12 |- 1 e. CC
35		adddi 8259	. . . . . . . . . . . 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 8271	. . . . . . . . . . . . 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 6066	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. CC ) -> ( ( A x. k ) + ( A x. 1 ) ) = ( ( A x. k ) + A ) )
40	36, 39	eqtrd 2265	. . . . . . . . . 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 5674	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( exp ` ( A x. ( k + 1 ) ) ) = ( exp ` ( ( A x. k ) + A ) ) )
43		mulcl 8254	. . . . . . . . . 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 12361	. . . . . . . . 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 2265	. . . . . . 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 10908	. . . . . . . 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 2275	. . . . 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 6058	. . . . . 6 |- ( ( exp ` ( A x. k ) ) = ( ( exp ` A ) ^ k ) -> ( 1 / ( exp ` ( A x. k ) ) ) = ( 1 / ( ( exp ` A ) ^ k ) ) )
56		nncn 9245	. . . . . . . . . 10 |- ( k e. NN -> k e. CC )
57		mulneg2 8669	. . . . . . . . . 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 5674	. . . . . . . 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 12365	. . . . . . . . 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 2265	. . . . . . 7 |- ( ( A e. CC /\ k e. NN ) -> ( exp ` ( A x. -u k ) ) = ( 1 / ( exp ` ( A x. k ) ) ) )
64		efap0 12363	. . . . . . . 8 |- ( A e. CC -> ( exp ` A ) # 0 )
65		nnnn0 9503	. . . . . . . 8 |- ( k e. NN -> k e. NN0 )
66		expnegap0 10909	. . . . . . . 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 2247	. . . . . 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 9696	. . 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 2267	1 |- ( ( A e. CC /\ N e. ZZ ) -> ( exp ` ( N x. A ) ) = ( ( exp ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   CCcc 8125   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   -ucneg 8445   # cap 8855   / cdiv 8946   NNcn 9237   NN0cn0 9496   ZZcz 9577   ^cexp 10900   expce 12328

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-disj 4086  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-sup 7275  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-ico 10227  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901  df-fac 11088  df-bc 11110  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-sumdc 12039  df-ef 12334

This theorem is referenced by:  efzval  12369  efgt0  12370  tanval3ap  12400  demoivre  12459  ef2kpi  15671  reexplog  15736  relogexp  15737