Theorem cjexp 10902

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 5883	. . . . . 6 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
2	1	fveq2d 5520	. . . . 5 |- ( j = 0 -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ 0 ) ) )
3		oveq2 5883	. . . . 5 |- ( j = 0 -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ 0 ) )
4	2, 3	eqeq12d 2192	. . . 4 |- ( j = 0 -> ( ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) <-> ( * ` ( A ^ 0 ) ) = ( ( * ` A ) ^ 0 ) ) )
5	4	imbi2d 230	. . 3 |- ( j = 0 -> ( ( A e. CC -> ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) ) <-> ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( ( * ` A ) ^ 0 ) ) ) )
6		oveq2 5883	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
7	6	fveq2d 5520	. . . . 5 |- ( j = k -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ k ) ) )
8		oveq2 5883	. . . . 5 |- ( j = k -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ k ) )
9	7, 8	eqeq12d 2192	. . . 4 |- ( j = k -> ( ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) <-> ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) )
10	9	imbi2d 230	. . 3 |- ( j = k -> ( ( A e. CC -> ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) ) <-> ( A e. CC -> ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) ) )
11		oveq2 5883	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
12	11	fveq2d 5520	. . . . 5 |- ( j = ( k + 1 ) -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ ( k + 1 ) ) ) )
13		oveq2 5883	. . . . 5 |- ( j = ( k + 1 ) -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2192	. . . 4 |- ( j = ( k + 1 ) -> ( ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) <-> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) ) )
15	14	imbi2d 230	. . 3 |- ( j = ( k + 1 ) -> ( ( A e. CC -> ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) ) <-> ( A e. CC -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) ) ) )
16		oveq2 5883	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
17	16	fveq2d 5520	. . . . 5 |- ( j = N -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ N ) ) )
18		oveq2 5883	. . . . 5 |- ( j = N -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ N ) )
19	17, 18	eqeq12d 2192	. . . 4 |- ( j = N -> ( ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) <-> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) ) )
20	19	imbi2d 230	. . 3 |- ( j = N -> ( ( A e. CC -> ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) ) <-> ( A e. CC -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) ) ) )
21		exp0 10524	. . . . 5 |- ( A e. CC -> ( A ^ 0 ) = 1 )
22	21	fveq2d 5520	. . . 4 |- ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( * ` 1 ) )
23		cjcl 10857	. . . . 5 |- ( A e. CC -> ( * ` A ) e. CC )
24		exp0 10524	. . . . . 6 |- ( ( * ` A ) e. CC -> ( ( * ` A ) ^ 0 ) = 1 )
25		1re 7956	. . . . . . 7 |- 1 e. RR
26		cjre 10891	. . . . . . 7 |- ( 1 e. RR -> ( * ` 1 ) = 1 )
27	25, 26	ax-mp 5	. . . . . 6 |- ( * ` 1 ) = 1
28	24, 27	eqtr4di 2228	. . . . 5 |- ( ( * ` A ) e. CC -> ( ( * ` A ) ^ 0 ) = ( * ` 1 ) )
29	23, 28	syl 14	. . . 4 |- ( A e. CC -> ( ( * ` A ) ^ 0 ) = ( * ` 1 ) )
30	22, 29	eqtr4d 2213	. . 3 |- ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( ( * ` A ) ^ 0 ) )
31		expp1 10527	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
32	31	fveq2d 5520	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( * ` ( ( A ^ k ) x. A ) ) )
33		expcl 10538	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
34		simpl 109	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> A e. CC )
35		cjmul 10894	. . . . . . . . . 10 |- ( ( ( A ^ k ) e. CC /\ A e. CC ) -> ( * ` ( ( A ^ k ) x. A ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
36	33, 34, 35	syl2anc 411	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( ( A ^ k ) x. A ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
37	32, 36	eqtrd 2210	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
38	37	adantr 276	. . . . . . 7 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
39		oveq1 5882	. . . . . . . 8 |- ( ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) -> ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
40		expp1 10527	. . . . . . . . . 10 |- ( ( ( * ` A ) e. CC /\ k e. NN0 ) -> ( ( * ` A ) ^ ( k + 1 ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
41	23, 40	sylan 283	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( * ` A ) ^ ( k + 1 ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
42	41	eqcomd 2183	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
43	39, 42	sylan9eqr 2232	. . . . . . 7 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
44	38, 43	eqtrd 2210	. . . . . 6 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
45	44	exp31 364	. . . . 5 |- ( A e. CC -> ( k e. NN0 -> ( ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) ) ) )
46	45	com12 30	. . . 4 |- ( k e. NN0 -> ( A e. CC -> ( ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) ) ) )
47	46	a2d 26	. . 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 9367	. 2 |- ( N e. NN0 -> ( A e. CC -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) ) )
49	48	impcom 125	1 |- ( ( A e. CC /\ N e. NN0 ) -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   ` cfv 5217  (class class class)co 5875   CCcc 7809   RRcr 7810   0cc0 7811   1c1 7812   + caddc 7814   x. cmul 7816   NN0cn0 9176   ^cexp 10519   *ccj 10848

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 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929

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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-n0 9177  df-z 9254  df-uz 9529  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853

This theorem is referenced by:  cjexpd  10967  efcj  11681