Theorem cjexp 10870

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 5873	. . . . . 6 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
2	1	fveq2d 5511	. . . . 5 |- ( j = 0 -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ 0 ) ) )
3		oveq2 5873	. . . . 5 |- ( j = 0 -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ 0 ) )
4	2, 3	eqeq12d 2190	. . . 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 5873	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
7	6	fveq2d 5511	. . . . 5 |- ( j = k -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ k ) ) )
8		oveq2 5873	. . . . 5 |- ( j = k -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ k ) )
9	7, 8	eqeq12d 2190	. . . 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 5873	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
12	11	fveq2d 5511	. . . . 5 |- ( j = ( k + 1 ) -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ ( k + 1 ) ) ) )
13		oveq2 5873	. . . . 5 |- ( j = ( k + 1 ) -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2190	. . . 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 5873	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
17	16	fveq2d 5511	. . . . 5 |- ( j = N -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ N ) ) )
18		oveq2 5873	. . . . 5 |- ( j = N -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ N ) )
19	17, 18	eqeq12d 2190	. . . 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 10494	. . . . 5 |- ( A e. CC -> ( A ^ 0 ) = 1 )
22	21	fveq2d 5511	. . . 4 |- ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( * ` 1 ) )
23		cjcl 10825	. . . . 5 |- ( A e. CC -> ( * ` A ) e. CC )
24		exp0 10494	. . . . . 6 |- ( ( * ` A ) e. CC -> ( ( * ` A ) ^ 0 ) = 1 )
25		1re 7931	. . . . . . 7 |- 1 e. RR
26		cjre 10859	. . . . . . 7 |- ( 1 e. RR -> ( * ` 1 ) = 1 )
27	25, 26	ax-mp 5	. . . . . 6 |- ( * ` 1 ) = 1
28	24, 27	eqtr4di 2226	. . . . 5 |- ( ( * ` A ) e. CC -> ( ( * ` A ) ^ 0 ) = ( * ` 1 ) )
29	23, 28	syl 14	. . . 4 |- ( A e. CC -> ( ( * ` A ) ^ 0 ) = ( * ` 1 ) )
30	22, 29	eqtr4d 2211	. . 3 |- ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( ( * ` A ) ^ 0 ) )
31		expp1 10497	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
32	31	fveq2d 5511	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( * ` ( ( A ^ k ) x. A ) ) )
33		expcl 10508	. . . . . . . . . 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 10862	. . . . . . . . . 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 2208	. . . . . . . 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 5872	. . . . . . . 8 |- ( ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) -> ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
40		expp1 10497	. . . . . . . . . 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 2181	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
43	39, 42	sylan9eqr 2230	. . . . . . 7 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
44	38, 43	eqtrd 2208	. . . . . 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 9340	. 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 2146   ` cfv 5208  (class class class)co 5865   CCcc 7784   RRcr 7785   0cc0 7786   1c1 7787   + caddc 7789   x. cmul 7791   NN0cn0 9149   ^cexp 10489   *ccj 10816

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904

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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-n0 9150  df-z 9227  df-uz 9502  df-seqfrec 10416  df-exp 10490  df-cj 10819  df-re 10820  df-im 10821

This theorem is referenced by:  cjexpd  10935  efcj  11649