Theorem cjexp 10458

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 5698	. . . . . 6 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
2	1	fveq2d 5344	. . . . 5 |- ( j = 0 -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ 0 ) ) )
3		oveq2 5698	. . . . 5 |- ( j = 0 -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ 0 ) )
4	2, 3	eqeq12d 2109	. . . 4 |- ( j = 0 -> ( ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) <-> ( * ` ( A ^ 0 ) ) = ( ( * ` A ) ^ 0 ) ) )
5	4	imbi2d 229	. . 3 |- ( j = 0 -> ( ( A e. CC -> ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) ) <-> ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( ( * ` A ) ^ 0 ) ) ) )
6		oveq2 5698	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
7	6	fveq2d 5344	. . . . 5 |- ( j = k -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ k ) ) )
8		oveq2 5698	. . . . 5 |- ( j = k -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ k ) )
9	7, 8	eqeq12d 2109	. . . 4 |- ( j = k -> ( ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) <-> ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) )
10	9	imbi2d 229	. . 3 |- ( j = k -> ( ( A e. CC -> ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) ) <-> ( A e. CC -> ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) ) )
11		oveq2 5698	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
12	11	fveq2d 5344	. . . . 5 |- ( j = ( k + 1 ) -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ ( k + 1 ) ) ) )
13		oveq2 5698	. . . . 5 |- ( j = ( k + 1 ) -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2109	. . . 4 |- ( j = ( k + 1 ) -> ( ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) <-> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) ) )
15	14	imbi2d 229	. . 3 |- ( j = ( k + 1 ) -> ( ( A e. CC -> ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) ) <-> ( A e. CC -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) ) ) )
16		oveq2 5698	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
17	16	fveq2d 5344	. . . . 5 |- ( j = N -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ N ) ) )
18		oveq2 5698	. . . . 5 |- ( j = N -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ N ) )
19	17, 18	eqeq12d 2109	. . . 4 |- ( j = N -> ( ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) <-> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) ) )
20	19	imbi2d 229	. . 3 |- ( j = N -> ( ( A e. CC -> ( * ` ( A ^ j ) ) = ( ( * ` A ) ^ j ) ) <-> ( A e. CC -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) ) ) )
21		exp0 10090	. . . . 5 |- ( A e. CC -> ( A ^ 0 ) = 1 )
22	21	fveq2d 5344	. . . 4 |- ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( * ` 1 ) )
23		cjcl 10413	. . . . 5 |- ( A e. CC -> ( * ` A ) e. CC )
24		exp0 10090	. . . . . 6 |- ( ( * ` A ) e. CC -> ( ( * ` A ) ^ 0 ) = 1 )
25		1re 7584	. . . . . . 7 |- 1 e. RR
26		cjre 10447	. . . . . . 7 |- ( 1 e. RR -> ( * ` 1 ) = 1 )
27	25, 26	ax-mp 7	. . . . . 6 |- ( * ` 1 ) = 1
28	24, 27	syl6eqr 2145	. . . . 5 |- ( ( * ` A ) e. CC -> ( ( * ` A ) ^ 0 ) = ( * ` 1 ) )
29	23, 28	syl 14	. . . 4 |- ( A e. CC -> ( ( * ` A ) ^ 0 ) = ( * ` 1 ) )
30	22, 29	eqtr4d 2130	. . 3 |- ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( ( * ` A ) ^ 0 ) )
31		expp1 10093	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
32	31	fveq2d 5344	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( * ` ( ( A ^ k ) x. A ) ) )
33		expcl 10104	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
34		simpl 108	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> A e. CC )
35		cjmul 10450	. . . . . . . . . 10 |- ( ( ( A ^ k ) e. CC /\ A e. CC ) -> ( * ` ( ( A ^ k ) x. A ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
36	33, 34, 35	syl2anc 404	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( ( A ^ k ) x. A ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
37	32, 36	eqtrd 2127	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
38	37	adantr 271	. . . . . . 7 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
39		oveq1 5697	. . . . . . . 8 |- ( ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) -> ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
40		expp1 10093	. . . . . . . . . 10 |- ( ( ( * ` A ) e. CC /\ k e. NN0 ) -> ( ( * ` A ) ^ ( k + 1 ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
41	23, 40	sylan 278	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( * ` A ) ^ ( k + 1 ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
42	41	eqcomd 2100	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
43	39, 42	sylan9eqr 2149	. . . . . . 7 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
44	38, 43	eqtrd 2127	. . . . . 6 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
45	44	exp31 357	. . . . 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 8959	. 2 |- ( N e. NN0 -> ( A e. CC -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) ) )
49	48	impcom 124	1 |- ( ( A e. CC /\ N e. NN0 ) -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1296   e. wcel 1445   ` cfv 5049  (class class class)co 5690   CCcc 7445   RRcr 7446   0cc0 7447   1c1 7448   + caddc 7450   x. cmul 7452   NN0cn0 8771   ^cexp 10085   *ccj 10404

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-2 8579  df-n0 8772  df-z 8849  df-uz 9119  df-iseq 10002  df-seq3 10003  df-exp 10086  df-cj 10407  df-re 10408  df-im 10409

This theorem is referenced by:  cjexpd  10523  efcj  11127