Theorem cjexp 9493

Description: Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)

Assertion

Ref	Expression
cjexp	|- ( ( A e. CC /\ N e. NN0 ) -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) )

Proof of Theorem cjexp

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5520	. . . . . 6 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
2	1	fveq2d 5182	. . . . 5 |- ( j = 0 -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ 0 ) ) )
3		oveq2 5520	. . . . 5 |- ( j = 0 -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ 0 ) )
4	2, 3	eqeq12d 2054	. . . 4 |- ( j = 0 -> ( ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) <-> ( * ` ( A ^ 0 ) ) = ( ( * ` A ) ^ 0 ) ) )
5	4	imbi2d 219	. . 3 |- ( j = 0 -> ( ( A e. CC -> ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) ) <-> ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( ( * ` A ) ^ 0 ) ) ) )
6		oveq2 5520	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
7	6	fveq2d 5182	. . . . 5 |- ( j = k -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ k ) ) )
8		oveq2 5520	. . . . 5 |- ( j = k -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ k ) )
9	7, 8	eqeq12d 2054	. . . 4 |- ( j = k -> ( ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) <-> ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) )
10	9	imbi2d 219	. . 3 |- ( j = k -> ( ( A e. CC -> ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) ) <-> ( A e. CC -> ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) ) )
11		oveq2 5520	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
12	11	fveq2d 5182	. . . . 5 |- ( j = ( k + 1 ) -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ ( k + 1 ) ) ) )
13		oveq2 5520	. . . . 5 |- ( j = ( k + 1 ) -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2054	. . . 4 |- ( j = ( k + 1 ) -> ( ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) <-> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) ) )
15	14	imbi2d 219	. . 3 |- ( j = ( k + 1 ) -> ( ( A e. CC -> ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) ) <-> ( A e. CC -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) ) ) )
16		oveq2 5520	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
17	16	fveq2d 5182	. . . . 5 |- ( j = N -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ N ) ) )
18		oveq2 5520	. . . . 5 |- ( j = N -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ N ) )
19	17, 18	eqeq12d 2054	. . . 4 |- ( j = N -> ( ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) <-> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) ) )
20	19	imbi2d 219	. . 3 |- ( j = N -> ( ( A e. CC -> ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) ) <-> ( A e. CC -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) ) ) )
21		exp0 9259	. . . . 5 |- ( A e. CC -> ( A ^ 0 ) = 1 )
22	21	fveq2d 5182	. . . 4 |- ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( * ` 1 ) )
23		cjcl 9448	. . . . 5 |- ( A e. CC -> ( * ` A ) e. CC )
24		exp0 9259	. . . . . 6 |- ( ( * ` A ) e. CC -> ( ( * ` A ) ^ 0 ) = 1 )
25		1re 7026	. . . . . . 7 |- 1 e. RR
26		cjre 9482	. . . . . . 7 |- ( 1 e. RR -> ( * ` 1 ) = 1 )
27	25, 26	ax-mp 7	. . . . . 6 |- ( * ` 1 ) = 1
28	24, 27	syl6eqr 2090	. . . . 5 |- ( ( * ` A ) e. CC -> ( ( * ` A ) ^ 0 ) = ( * ` 1 ) )
29	23, 28	syl 14	. . . 4 |- ( A e. CC -> ( ( * ` A ) ^ 0 ) = ( * ` 1 ) )
30	22, 29	eqtr4d 2075	. . 3 |- ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( ( * ` A ) ^ 0 ) )
31		expp1 9262	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
32	31	fveq2d 5182	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( * ` ( ( A ^ k ) x. A ) ) )
33		expcl 9273	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
34		simpl 102	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> A e. CC )
35		cjmul 9485	. . . . . . . . . 10 |- ( ( ( A ^ k ) e. CC /\ A e. CC ) -> ( * ` ( ( A ^ k ) x. A ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
36	33, 34, 35	syl2anc 391	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( ( A ^ k ) x. A ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
37	32, 36	eqtrd 2072	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
38	37	adantr 261	. . . . . . 7 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
39		oveq1 5519	. . . . . . . 8 |- ( ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) -> ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
40		expp1 9262	. . . . . . . . . 10 |- ( ( ( * ` A ) e. CC /\ k e. NN0 ) -> ( ( * ` A ) ^ ( k + 1 ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
41	23, 40	sylan 267	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( * ` A ) ^ ( k + 1 ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
42	41	eqcomd 2045	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
43	39, 42	sylan9eqr 2094	. . . . . . 7 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
44	38, 43	eqtrd 2072	. . . . . 6 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
45	44	exp31 346	. . . . 5 |- ( A e. CC -> ( k e. NN0 -> ( ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) ) ) )
46	45	com12 27	. . . 4 |- ( k e. NN0 -> ( A e. CC -> ( ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) ) ) )
47	46	a2d 23	. . 3 |- ( k e. NN0 -> ( ( A e. CC -> ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( A e. CC -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) ) ) )
48	5, 10, 15, 20, 30, 47	nn0ind 8352	. 2 |- ( N e. NN0 -> ( A e. CC -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) ) )
49	48	impcom 116	1 |- ( ( A e. CC /\ N e. NN0 ) -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   ` cfv 4902  (class class class)co 5512   CCcc 6887   RRcr 6888   0cc0 6889   1c1 6890   + caddc 6892   x. cmul 6894   NN0cn0 8181   ^cexp 9254   *ccj 9439

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 7002

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-div 7652  df-inn 7915  df-2 7973  df-n0 8182  df-z 8246  df-uz 8474  df-iseq 9212  df-iexp 9255  df-cj 9442  df-re 9443  df-im 9444

This theorem is referenced by:  cjexpd  9558