Theorem efper 10101

Description: The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.)

Assertion

Ref	Expression
efper	|- ((A e. CC /\ K e. ZZ) -> (exp` (A + ((_i x. (2 x. pi)) x. K))) = (exp` A))

Proof of Theorem efper

Step	Hyp	Ref	Expression
1		efadd 8629	. . 3 |- ((A e. CC /\ ((_i x. (2 x. pi)) x. K) e. CC) -> (exp` (A + ((_i x. (2 x. pi)) x. K))) = ((exp` A) x. (exp` ((_i x. (2 x. pi)) x. K))))
2		zcn 7349	. . . 4 |- (K e. ZZ -> K e. CC)
3		axicn 6423	. . . . . 6 |- _i e. CC
4		2re 7163	. . . . . . . 8 |- 2 e. RR
5		pire 10026	. . . . . . . 8 |- pi e. RR
6	4, 5	remulcli 6488	. . . . . . 7 |- (2 x. pi) e. RR
7	6	recni 6467	. . . . . 6 |- (2 x. pi) e. CC
8	3, 7	mulcli 6474	. . . . 5 |- (_i x. (2 x. pi)) e. CC
9		mulcl 6456	. . . . 5 |- (((_i x. (2 x. pi)) e. CC /\ K e. CC) -> ((_i x. (2 x. pi)) x. K) e. CC)
10	8, 9	mpan 759	. . . 4 |- (K e. CC -> ((_i x. (2 x. pi)) x. K) e. CC)
11	2, 10	syl 12	. . 3 |- (K e. ZZ -> ((_i x. (2 x. pi)) x. K) e. CC)
12	1, 11	sylan2 500	. 2 |- ((A e. CC /\ K e. ZZ) -> (exp` (A + ((_i x. (2 x. pi)) x. K))) = ((exp` A) x. (exp` ((_i x. (2 x. pi)) x. K))))
13		mulcl 6456	. . . . . . 7 |- ((K e. CC /\ (2 x. pi) e. CC) -> (K x. (2 x. pi)) e. CC)
14	7, 13	mpan2 760	. . . . . 6 |- (K e. CC -> (K x. (2 x. pi)) e. CC)
15		efival 8712	. . . . . 6 |- ((K x. (2 x. pi)) e. CC -> (exp` (_i x. (K x. (2 x. pi)))) = ((cos` (K x. (2 x. pi))) + (_i x. (sin` (K x. (2 x. pi))))))
16	2, 14, 15	3syl 24	. . . . 5 |- (K e. ZZ -> (exp` (_i x. (K x. (2 x. pi)))) = ((cos` (K x. (2 x. pi))) + (_i x. (sin` (K x. (2 x. pi))))))
17		mul12 6579	. . . . . . . . 9 |- ((_i e. CC /\ K e. CC /\ (2 x. pi) e. CC) -> (_i x. (K x. (2 x. pi))) = (K x. (_i x. (2 x. pi))))
18	3, 7, 17	mp3an13 1182	. . . . . . . 8 |- (K e. CC -> (_i x. (K x. (2 x. pi))) = (K x. (_i x. (2 x. pi))))
19		mulcom 6459	. . . . . . . . 9 |- ((K e. CC /\ (_i x. (2 x. pi)) e. CC) -> (K x. (_i x. (2 x. pi))) = ((_i x. (2 x. pi)) x. K))
20	8, 19	mpan2 760	. . . . . . . 8 |- (K e. CC -> (K x. (_i x. (2 x. pi))) = ((_i x. (2 x. pi)) x. K))
21	18, 20	eqtrd 1925	. . . . . . 7 |- (K e. CC -> (_i x. (K x. (2 x. pi))) = ((_i x. (2 x. pi)) x. K))
22	2, 21	syl 12	. . . . . 6 |- (K e. ZZ -> (_i x. (K x. (2 x. pi))) = ((_i x. (2 x. pi)) x. K))
23	22	fveq2d 4685	. . . . 5 |- (K e. ZZ -> (exp` (_i x. (K x. (2 x. pi)))) = (exp` ((_i x. (2 x. pi)) x. K)))
24		cos2kpi 10038	. . . . . . 7 |- (K e. ZZ -> (cos` (K x. (2 x. pi))) = 1)
25		sin2kpi 10037	. . . . . . . . 9 |- (K e. ZZ -> (sin` (K x. (2 x. pi))) = 0)
26	25	opreq2d 4898	. . . . . . . 8 |- (K e. ZZ -> (_i x. (sin` (K x. (2 x. pi)))) = (_i x. 0))
27	3	mul01i 6594	. . . . . . . 8 |- (_i x. 0) = 0
28	26, 27	syl6eq 1944	. . . . . . 7 |- (K e. ZZ -> (_i x. (sin` (K x. (2 x. pi)))) = 0)
29	24, 28	opreq12d 4900	. . . . . 6 |- (K e. ZZ -> ((cos` (K x. (2 x. pi))) + (_i x. (sin` (K x. (2 x. pi))))) = (1 + 0))
30		ax1cn 6422	. . . . . . 7 |- 1 e. CC
31	30	addid1i 6483	. . . . . 6 |- (1 + 0) = 1
32	29, 31	syl6eq 1944	. . . . 5 |- (K e. ZZ -> ((cos` (K x. (2 x. pi))) + (_i x. (sin` (K x. (2 x. pi))))) = 1)
33	16, 23, 32	3eqtr3d 1934	. . . 4 |- (K e. ZZ -> (exp` ((_i x. (2 x. pi)) x. K)) = 1)
34	33	opreq2d 4898	. . 3 |- (K e. ZZ -> ((exp` A) x. (exp` ((_i x. (2 x. pi)) x. K))) = ((exp` A) x. 1))
35		efcl 8574	. . . 4 |- (A e. CC -> (exp` A) e. CC)
36		mulid1 6464	. . . 4 |- ((exp` A) e. CC -> ((exp` A) x. 1) = (exp` A))
37	35, 36	syl 12	. . 3 |- (A e. CC -> ((exp` A) x. 1) = (exp` A))
38	34, 37	sylan9eqr 1951	. 2 |- ((A e. CC /\ K e. ZZ) -> ((exp` A) x. (exp` ((_i x. (2 x. pi)) x. K))) = (exp` A))
39	12, 38	eqtrd 1925	1 |- ((A e. CC /\ K e. ZZ) -> (exp` (A + ((_i x. (2 x. pi)) x. K))) = (exp` A))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391  ZZcz 6451  2c2 7145  expce 8555  sincsin 8557  cosccos 8558  picpi 8559

This theorem is referenced by:  pilog 10122

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-5 7157  df-6 7158  df-7 7159  df-8 7160  df-9 7161  df-rp 7232  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-ioc 7529  df-ico 7530  df-icc 7531  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-fac 8184  df-bc 8209  df-clim 8235  df-sum 8240  df-cncf 8525  df-ef 8560  df-sin 8562  df-cos 8563  df-pi 8564  df-top 8861  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200

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