Theorem cjexpt 6817

Description: Complex conjugate of natural number exponentiation.

Assertion

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

Proof of Theorem cjexpt

Step	Hyp	Ref	Expression
1		opreq2 3969	. . . . . 6 |- (j = 0 -> (A^j) = (A^0))
2	1	fveq2d 3728	. . . . 5 |- (j = 0 -> (*` (A^j)) = (*` (A^0)))
3		opreq2 3969	. . . . 5 |- (j = 0 -> ((*` A)^j) = ((*` A)^0))
4	2, 3	eqeq12d 1489	. . . 4 |- (j = 0 -> ((*` (A^j)) = ((*` A)^j) <-> (*` (A^0)) = ((*` A)^0)))
5	4	imbi2d 612	. . 3 |- (j = 0 -> ((A e. CC -> (*` (A^j)) = ((*` A)^j)) <-> (A e. CC -> (*` (A^0)) = ((*` A)^0))))
6		opreq2 3969	. . . . . 6 |- (j = k -> (A^j) = (A^k))
7	6	fveq2d 3728	. . . . 5 |- (j = k -> (*` (A^j)) = (*` (A^k)))
8		opreq2 3969	. . . . 5 |- (j = k -> ((*` A)^j) = ((*` A)^k))
9	7, 8	eqeq12d 1489	. . . 4 |- (j = k -> ((*` (A^j)) = ((*` A)^j) <-> (*` (A^k)) = ((*` A)^k)))
10	9	imbi2d 612	. . 3 |- (j = k -> ((A e. CC -> (*` (A^j)) = ((*` A)^j)) <-> (A e. CC -> (*` (A^k)) = ((*` A)^k))))
11		opreq2 3969	. . . . . 6 |- (j = (k + 1) -> (A^j) = (A^(k + 1)))
12	11	fveq2d 3728	. . . . 5 |- (j = (k + 1) -> (*` (A^j)) = (*` (A^(k + 1))))
13		opreq2 3969	. . . . 5 |- (j = (k + 1) -> ((*` A)^j) = ((*` A)^(k + 1)))
14	12, 13	eqeq12d 1489	. . . 4 |- (j = (k + 1) -> ((*` (A^j)) = ((*` A)^j) <-> (*` (A^(k + 1))) = ((*` A)^(k + 1))))
15	14	imbi2d 612	. . 3 |- (j = (k + 1) -> ((A e. CC -> (*` (A^j)) = ((*` A)^j)) <-> (A e. CC -> (*` (A^(k + 1))) = ((*` A)^(k + 1)))))
16		opreq2 3969	. . . . . 6 |- (j = N -> (A^j) = (A^N))
17	16	fveq2d 3728	. . . . 5 |- (j = N -> (*` (A^j)) = (*` (A^N)))
18		opreq2 3969	. . . . 5 |- (j = N -> ((*` A)^j) = ((*` A)^N))
19	17, 18	eqeq12d 1489	. . . 4 |- (j = N -> ((*` (A^j)) = ((*` A)^j) <-> (*` (A^N)) = ((*` A)^N)))
20	19	imbi2d 612	. . 3 |- (j = N -> ((A e. CC -> (*` (A^j)) = ((*` A)^j)) <-> (A e. CC -> (*` (A^N)) = ((*` A)^N))))
21		exp0t 6571	. . . . 5 |- (A e. CC -> (A^0) = 1)
22	21	fveq2d 3728	. . . 4 |- (A e. CC -> (*` (A^0)) = (*` 1))
23		cjclt 6764	. . . . 5 |- (A e. CC -> (*` A) e. CC)
24		exp0t 6571	. . . . . 6 |- ((*` A) e. CC -> ((*` A)^0) = 1)
25		1re 5435	. . . . . . 7 |- 1 e. RR
26		cjret 6810	. . . . . . 7 |- (1 e. RR -> (*` 1) = 1)
27	25, 26	ax-mp 7	. . . . . 6 |- (*` 1) = 1
28	24, 27	syl6eqr 1525	. . . . 5 |- ((*` A) e. CC -> ((*` A)^0) = (*` 1))
29	23, 28	syl 10	. . . 4 |- (A e. CC -> ((*` A)^0) = (*` 1))
30	22, 29	eqtr4d 1510	. . 3 |- (A e. CC -> (*` (A^0)) = ((*` A)^0))
31		expp1t 6574	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> (A^(k + 1)) = ((A^k) x. A))
32	31	fveq2d 3728	. . . . . . . . 9 |- ((A e. CC /\ k e. NN0) -> (*` (A^(k + 1))) = (*` ((A^k) x. A)))
33		cjmult 6813	. . . . . . . . . 10 |- (((A^k) e. CC /\ A e. CC) -> (*` ((A^k) x. A)) = ((*` (A^k)) x. (*` A)))
34		expclt 6581	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> (A^k) e. CC)
35		pm3.26 319	. . . . . . . . . 10 |- ((A e. CC /\ k e. NN0) -> A e. CC)
36	33, 34, 35	sylanc 471	. . . . . . . . 9 |- ((A e. CC /\ k e. NN0) -> (*` ((A^k) x. A)) = ((*` (A^k)) x. (*` A)))
37	32, 36	eqtrd 1507	. . . . . . . 8 |- ((A e. CC /\ k e. NN0) -> (*` (A^(k + 1))) = ((*` (A^k)) x. (*` A)))
38	37	adantr 389	. . . . . . 7 |- (((A e. CC /\ k e. NN0) /\ (*` (A^k)) = ((*` A)^k)) -> (*` (A^(k + 1))) = ((*` (A^k)) x. (*` A)))
39		opreq1 3968	. . . . . . . 8 |- ((*` (A^k)) = ((*` A)^k) -> ((*` (A^k)) x. (*` A)) = (((*` A)^k) x. (*` A)))
40		expp1t 6574	. . . . . . . . . 10 |- (((*` A) e. CC /\ k e. NN0) -> ((*` A)^(k + 1)) = (((*` A)^k) x. (*` A)))
41	40, 23	sylan 448	. . . . . . . . 9 |- ((A e. CC /\ k e. NN0) -> ((*` A)^(k + 1)) = (((*` A)^k) x. (*` A)))
42	41	eqcomd 1480	. . . . . . . 8 |- ((A e. CC /\ k e. NN0) -> (((*` A)^k) x. (*` A)) = ((*` A)^(k + 1)))
43	39, 42	sylan9eqr 1529	. . . . . . 7 |- (((A e. CC /\ k e. NN0) /\ (*` (A^k)) = ((*` A)^k)) -> ((*` (A^k)) x. (*` A)) = ((*` A)^(k + 1)))
44	38, 43	eqtrd 1507	. . . . . 6 |- (((A e. CC /\ k e. NN0) /\ (*` (A^k)) = ((*` A)^k)) -> (*` (A^(k + 1))) = ((*` A)^(k + 1)))
45	44	exp31 376	. . . . 5 |- (A e. CC -> (k e. NN0 -> ((*` (A^k)) = ((*` A)^k) -> (*` (A^(k + 1))) = ((*` A)^(k + 1)))))
46	45	com12 11	. . . 4 |- (k e. NN0 -> (A e. CC -> ((*` (A^k)) = ((*` A)^k) -> (*` (A^(k + 1))) = ((*` A)^(k + 1)))))
47	46	a2d 13	. . 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 6212	. 2 |- (N e. NN0 -> (A e. CC -> (*` (A^N)) = ((*` A)^N)))
49	48	impcom 351	1 |- ((A e. CC /\ N e. NN0) -> (*` (A^N)) = ((*` A)^N))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  ` cfv 3182  (class class class)co 3963  CCcc 5232  RRcr 5233  0cc0 5234  1c1 5235   + caddc 5237   x. cmul 5239  NN0cn0 5297  ^cexp 6568  *ccj 6749

This theorem is referenced by:  efcj 7336

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi