Theorem iexpcyc 10866

Description: Taking _i to the K-th power is the same as using the K mod 4 -th power instead, by i4 10864. (Contributed by Mario Carneiro, 7-Jul-2014.)

Assertion

Ref	Expression
iexpcyc	|- ( K e. ZZ -> ( _i ^ ( K mod 4 ) ) = ( _i ^ K ) )

Proof of Theorem iexpcyc

Step	Hyp	Ref	Expression
1		zq 9821	. . . 4 |- ( K e. ZZ -> K e. QQ )
2		4z 9476	. . . . . 6 |- 4 e. ZZ
3		zq 9821	. . . . . 6 |- ( 4 e. ZZ -> 4 e. QQ )
4	2, 3	ax-mp 5	. . . . 5 |- 4 e. QQ
5		4pos 9207	. . . . 5 |- 0 < 4
6		modqval 10546	. . . . 5 |- ( ( K e. QQ /\ 4 e. QQ /\ 0 < 4 ) -> ( K mod 4 ) = ( K - ( 4 x. ( |_ ` ( K / 4 ) ) ) ) )
7	4, 5, 6	mp3an23 1363	. . . 4 |- ( K e. QQ -> ( K mod 4 ) = ( K - ( 4 x. ( |_ ` ( K / 4 ) ) ) ) )
8	1, 7	syl 14	. . 3 |- ( K e. ZZ -> ( K mod 4 ) = ( K - ( 4 x. ( |_ ` ( K / 4 ) ) ) ) )
9	8	oveq2d 6017	. 2 |- ( K e. ZZ -> ( _i ^ ( K mod 4 ) ) = ( _i ^ ( K - ( 4 x. ( |_ ` ( K / 4 ) ) ) ) ) )
10		4nn 9274	. . . . . . 7 |- 4 e. NN
11		znq 9819	. . . . . . 7 |- ( ( K e. ZZ /\ 4 e. NN ) -> ( K / 4 ) e. QQ )
12	10, 11	mpan2 425	. . . . . 6 |- ( K e. ZZ -> ( K / 4 ) e. QQ )
13	12	flqcld 10497	. . . . 5 |- ( K e. ZZ -> ( |_ ` ( K / 4 ) ) e. ZZ )
14		zmulcl 9500	. . . . 5 |- ( ( 4 e. ZZ /\ ( |_ ` ( K / 4 ) ) e. ZZ ) -> ( 4 x. ( |_ ` ( K / 4 ) ) ) e. ZZ )
15	2, 13, 14	sylancr 414	. . . 4 |- ( K e. ZZ -> ( 4 x. ( |_ ` ( K / 4 ) ) ) e. ZZ )
16		ax-icn 8094	. . . . 5 |- _i e. CC
17		iap0 9334	. . . . 5 |- _i # 0
18		expsubap 10809	. . . . 5 |- ( ( ( _i e. CC /\ _i # 0 ) /\ ( K e. ZZ /\ ( 4 x. ( |_ ` ( K / 4 ) ) ) e. ZZ ) ) -> ( _i ^ ( K - ( 4 x. ( |_ ` ( K / 4 ) ) ) ) ) = ( ( _i ^ K ) / ( _i ^ ( 4 x. ( |_ ` ( K / 4 ) ) ) ) ) )
19	16, 17, 18	mpanl12 436	. . . 4 |- ( ( K e. ZZ /\ ( 4 x. ( |_ ` ( K / 4 ) ) ) e. ZZ ) -> ( _i ^ ( K - ( 4 x. ( |_ ` ( K / 4 ) ) ) ) ) = ( ( _i ^ K ) / ( _i ^ ( 4 x. ( |_ ` ( K / 4 ) ) ) ) ) )
20	15, 19	mpdan 421	. . 3 |- ( K e. ZZ -> ( _i ^ ( K - ( 4 x. ( |_ ` ( K / 4 ) ) ) ) ) = ( ( _i ^ K ) / ( _i ^ ( 4 x. ( |_ ` ( K / 4 ) ) ) ) ) )
21		expmulzap 10807	. . . . . . . 8 |- ( ( ( _i e. CC /\ _i # 0 ) /\ ( 4 e. ZZ /\ ( |_ ` ( K / 4 ) ) e. ZZ ) ) -> ( _i ^ ( 4 x. ( |_ ` ( K / 4 ) ) ) ) = ( ( _i ^ 4 ) ^ ( |_ ` ( K / 4 ) ) ) )
22	16, 17, 21	mpanl12 436	. . . . . . 7 |- ( ( 4 e. ZZ /\ ( |_ ` ( K / 4 ) ) e. ZZ ) -> ( _i ^ ( 4 x. ( |_ ` ( K / 4 ) ) ) ) = ( ( _i ^ 4 ) ^ ( |_ ` ( K / 4 ) ) ) )
23	2, 13, 22	sylancr 414	. . . . . 6 |- ( K e. ZZ -> ( _i ^ ( 4 x. ( |_ ` ( K / 4 ) ) ) ) = ( ( _i ^ 4 ) ^ ( |_ ` ( K / 4 ) ) ) )
24		i4 10864	. . . . . . . 8 |- ( _i ^ 4 ) = 1
25	24	oveq1i 6011	. . . . . . 7 |- ( ( _i ^ 4 ) ^ ( |_ ` ( K / 4 ) ) ) = ( 1 ^ ( |_ ` ( K / 4 ) ) )
26		1exp 10790	. . . . . . . 8 |- ( ( |_ ` ( K / 4 ) ) e. ZZ -> ( 1 ^ ( |_ ` ( K / 4 ) ) ) = 1 )
27	13, 26	syl 14	. . . . . . 7 |- ( K e. ZZ -> ( 1 ^ ( |_ ` ( K / 4 ) ) ) = 1 )
28	25, 27	eqtrid 2274	. . . . . 6 |- ( K e. ZZ -> ( ( _i ^ 4 ) ^ ( |_ ` ( K / 4 ) ) ) = 1 )
29	23, 28	eqtrd 2262	. . . . 5 |- ( K e. ZZ -> ( _i ^ ( 4 x. ( |_ ` ( K / 4 ) ) ) ) = 1 )
30	29	oveq2d 6017	. . . 4 |- ( K e. ZZ -> ( ( _i ^ K ) / ( _i ^ ( 4 x. ( |_ ` ( K / 4 ) ) ) ) ) = ( ( _i ^ K ) / 1 ) )
31		expclzap 10786	. . . . . 6 |- ( ( _i e. CC /\ _i # 0 /\ K e. ZZ ) -> ( _i ^ K ) e. CC )
32	16, 17, 31	mp3an12 1361	. . . . 5 |- ( K e. ZZ -> ( _i ^ K ) e. CC )
33	32	div1d 8927	. . . 4 |- ( K e. ZZ -> ( ( _i ^ K ) / 1 ) = ( _i ^ K ) )
34	30, 33	eqtrd 2262	. . 3 |- ( K e. ZZ -> ( ( _i ^ K ) / ( _i ^ ( 4 x. ( |_ ` ( K / 4 ) ) ) ) ) = ( _i ^ K ) )
35	20, 34	eqtrd 2262	. 2 |- ( K e. ZZ -> ( _i ^ ( K - ( 4 x. ( |_ ` ( K / 4 ) ) ) ) ) = ( _i ^ K ) )
36	9, 35	eqtrd 2262	1 |- ( K e. ZZ -> ( _i ^ ( K mod 4 ) ) = ( _i ^ K ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   CCcc 7997   0cc0 7999   1c1 8000   _ici 8001   x. cmul 8004   < clt 8181   - cmin 8317   # cap 8728   / cdiv 8819   NNcn 9110   4c4 9163   ZZcz 9446   QQcq 9814   |_cfl 10488   mod cmo 10544   ^cexp 10760

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761

This theorem is referenced by: (None)