Theorem cjexp 10835

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 5850	. . . . . 6 |- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) )
2	1	fveq2d 5490	. . . . 5 |- ( j = 0 -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ 0 ) ) )
3		oveq2 5850	. . . . 5 |- ( j = 0 -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ 0 ) )
4	2, 3	eqeq12d 2180	. . . 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 5850	. . . . . 6 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
7	6	fveq2d 5490	. . . . 5 |- ( j = k -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ k ) ) )
8		oveq2 5850	. . . . 5 |- ( j = k -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ k ) )
9	7, 8	eqeq12d 2180	. . . 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 5850	. . . . . 6 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
12	11	fveq2d 5490	. . . . 5 |- ( j = ( k + 1 ) -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ ( k + 1 ) ) ) )
13		oveq2 5850	. . . . 5 |- ( j = ( k + 1 ) -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2180	. . . 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 5850	. . . . . 6 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
17	16	fveq2d 5490	. . . . 5 |- ( j = N -> ( * ` ( A ^ j ) ) = ( * ` ( A ^ N ) ) )
18		oveq2 5850	. . . . 5 |- ( j = N -> ( ( * ` A ) ^ j ) = ( ( * ` A ) ^ N ) )
19	17, 18	eqeq12d 2180	. . . 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 10459	. . . . 5 |- ( A e. CC -> ( A ^ 0 ) = 1 )
22	21	fveq2d 5490	. . . 4 |- ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( * ` 1 ) )
23		cjcl 10790	. . . . 5 |- ( A e. CC -> ( * ` A ) e. CC )
24		exp0 10459	. . . . . 6 |- ( ( * ` A ) e. CC -> ( ( * ` A ) ^ 0 ) = 1 )
25		1re 7898	. . . . . . 7 |- 1 e. RR
26		cjre 10824	. . . . . . 7 |- ( 1 e. RR -> ( * ` 1 ) = 1 )
27	25, 26	ax-mp 5	. . . . . 6 |- ( * ` 1 ) = 1
28	24, 27	eqtr4di 2217	. . . . 5 |- ( ( * ` A ) e. CC -> ( ( * ` A ) ^ 0 ) = ( * ` 1 ) )
29	23, 28	syl 14	. . . 4 |- ( A e. CC -> ( ( * ` A ) ^ 0 ) = ( * ` 1 ) )
30	22, 29	eqtr4d 2201	. . 3 |- ( A e. CC -> ( * ` ( A ^ 0 ) ) = ( ( * ` A ) ^ 0 ) )
31		expp1 10462	. . . . . . . . . 10 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
32	31	fveq2d 5490	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( * ` ( ( A ^ k ) x. A ) ) )
33		expcl 10473	. . . . . . . . . 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 10827	. . . . . . . . . 10 |- ( ( ( A ^ k ) e. CC /\ A e. CC ) -> ( * ` ( ( A ^ k ) x. A ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
36	33, 34, 35	syl2anc 409	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( ( A ^ k ) x. A ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
37	32, 36	eqtrd 2198	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
38	37	adantr 274	. . . . . . 7 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) )
39		oveq1 5849	. . . . . . . 8 |- ( ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) -> ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
40		expp1 10462	. . . . . . . . . 10 |- ( ( ( * ` A ) e. CC /\ k e. NN0 ) -> ( ( * ` A ) ^ ( k + 1 ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
41	23, 40	sylan 281	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( * ` A ) ^ ( k + 1 ) ) = ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) )
42	41	eqcomd 2171	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( ( * ` A ) ^ k ) x. ( * ` A ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
43	39, 42	sylan9eqr 2221	. . . . . . 7 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( ( * ` ( A ^ k ) ) x. ( * ` A ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
44	38, 43	eqtrd 2198	. . . . . 6 |- ( ( ( A e. CC /\ k e. NN0 ) /\ ( * ` ( A ^ k ) ) = ( ( * ` A ) ^ k ) ) -> ( * ` ( A ^ ( k + 1 ) ) ) = ( ( * ` A ) ^ ( k + 1 ) ) )
45	44	exp31 362	. . . . 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 9305	. 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 1343   e. wcel 2136   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   NN0cn0 9114   ^cexp 10454   *ccj 10781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192  df-uz 9467  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786

This theorem is referenced by:  cjexpd  10900  efcj  11614