Theorem demoivre 8752

Description: De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. (Contributed by Steve Rodriguez, 10-Nov-2006.) Warning: The HTML proof page is 0.6 megabyte in size.

Assertion

Ref	Expression
demoivre	|- ((A e. CC /\ N e. NN0) -> (((cos` A) + (_i x. (sin` A)))^N) = ((cos` (N x. A)) + (_i x. (sin` (N x. A)))))

Proof of Theorem demoivre

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . 5 |- (x = 0 -> (((cos` A) + (_i x. (sin` A)))^x) = (((cos` A) + (_i x. (sin` A)))^0))
2		opreq1 4889	. . . . . . 7 |- (x = 0 -> (x x. A) = (0 x. A))
3	2	fveq2d 4685	. . . . . 6 |- (x = 0 -> (cos` (x x. A)) = (cos` (0 x. A)))
4	2	fveq2d 4685	. . . . . . 7 |- (x = 0 -> (sin` (x x. A)) = (sin` (0 x. A)))
5	4	opreq2d 4898	. . . . . 6 |- (x = 0 -> (_i x. (sin` (x x. A))) = (_i x. (sin` (0 x. A))))
6	3, 5	opreq12d 4900	. . . . 5 |- (x = 0 -> ((cos` (x x. A)) + (_i x. (sin` (x x. A)))) = ((cos` (0 x. A)) + (_i x. (sin` (0 x. A)))))
7	1, 6	eqeq12d 1899	. . . 4 |- (x = 0 -> ((((cos` A) + (_i x. (sin` A)))^x) = ((cos` (x x. A)) + (_i x. (sin` (x x. A)))) <-> (((cos` A) + (_i x. (sin` A)))^0) = ((cos` (0 x. A)) + (_i x. (sin` (0 x. A))))))
8	7	imbi2d 674	. . 3 |- (x = 0 -> ((A e. CC -> (((cos` A) + (_i x. (sin` A)))^x) = ((cos` (x x. A)) + (_i x. (sin` (x x. A))))) <-> (A e. CC -> (((cos` A) + (_i x. (sin` A)))^0) = ((cos` (0 x. A)) + (_i x. (sin` (0 x. A)))))))
9		opreq2 4890	. . . . 5 |- (x = k -> (((cos` A) + (_i x. (sin` A)))^x) = (((cos` A) + (_i x. (sin` A)))^k))
10		opreq1 4889	. . . . . . 7 |- (x = k -> (x x. A) = (k x. A))
11	10	fveq2d 4685	. . . . . 6 |- (x = k -> (cos` (x x. A)) = (cos` (k x. A)))
12	10	fveq2d 4685	. . . . . . 7 |- (x = k -> (sin` (x x. A)) = (sin` (k x. A)))
13	12	opreq2d 4898	. . . . . 6 |- (x = k -> (_i x. (sin` (x x. A))) = (_i x. (sin` (k x. A))))
14	11, 13	opreq12d 4900	. . . . 5 |- (x = k -> ((cos` (x x. A)) + (_i x. (sin` (x x. A)))) = ((cos` (k x. A)) + (_i x. (sin` (k x. A)))))
15	9, 14	eqeq12d 1899	. . . 4 |- (x = k -> ((((cos` A) + (_i x. (sin` A)))^x) = ((cos` (x x. A)) + (_i x. (sin` (x x. A)))) <-> (((cos` A) + (_i x. (sin` A)))^k) = ((cos` (k x. A)) + (_i x. (sin` (k x. A))))))
16	15	imbi2d 674	. . 3 |- (x = k -> ((A e. CC -> (((cos` A) + (_i x. (sin` A)))^x) = ((cos` (x x. A)) + (_i x. (sin` (x x. A))))) <-> (A e. CC -> (((cos` A) + (_i x. (sin` A)))^k) = ((cos` (k x. A)) + (_i x. (sin` (k x. A)))))))
17		opreq2 4890	. . . . 5 |- (x = (k + 1) -> (((cos` A) + (_i x. (sin` A)))^x) = (((cos` A) + (_i x. (sin` A)))^(k + 1)))
18		opreq1 4889	. . . . . . 7 |- (x = (k + 1) -> (x x. A) = ((k + 1) x. A))
19	18	fveq2d 4685	. . . . . 6 |- (x = (k + 1) -> (cos` (x x. A)) = (cos` ((k + 1) x. A)))
20	18	fveq2d 4685	. . . . . . 7 |- (x = (k + 1) -> (sin` (x x. A)) = (sin` ((k + 1) x. A)))
21	20	opreq2d 4898	. . . . . 6 |- (x = (k + 1) -> (_i x. (sin` (x x. A))) = (_i x. (sin` ((k + 1) x. A))))
22	19, 21	opreq12d 4900	. . . . 5 |- (x = (k + 1) -> ((cos` (x x. A)) + (_i x. (sin` (x x. A)))) = ((cos` ((k + 1) x. A)) + (_i x. (sin` ((k + 1) x. A)))))
23	17, 22	eqeq12d 1899	. . . 4 |- (x = (k + 1) -> ((((cos` A) + (_i x. (sin` A)))^x) = ((cos` (x x. A)) + (_i x. (sin` (x x. A)))) <-> (((cos` A) + (_i x. (sin` A)))^(k + 1)) = ((cos` ((k + 1) x. A)) + (_i x. (sin` ((k + 1) x. A))))))
24	23	imbi2d 674	. . 3 |- (x = (k + 1) -> ((A e. CC -> (((cos` A) + (_i x. (sin` A)))^x) = ((cos` (x x. A)) + (_i x. (sin` (x x. A))))) <-> (A e. CC -> (((cos` A) + (_i x. (sin` A)))^(k + 1)) = ((cos` ((k + 1) x. A)) + (_i x. (sin` ((k + 1) x. A)))))))
25		opreq2 4890	. . . . 5 |- (x = N -> (((cos` A) + (_i x. (sin` A)))^x) = (((cos` A) + (_i x. (sin` A)))^N))
26		opreq1 4889	. . . . . . 7 |- (x = N -> (x x. A) = (N x. A))
27	26	fveq2d 4685	. . . . . 6 |- (x = N -> (cos` (x x. A)) = (cos` (N x. A)))
28	26	fveq2d 4685	. . . . . . 7 |- (x = N -> (sin` (x x. A)) = (sin` (N x. A)))
29	28	opreq2d 4898	. . . . . 6 |- (x = N -> (_i x. (sin` (x x. A))) = (_i x. (sin` (N x. A))))
30	27, 29	opreq12d 4900	. . . . 5 |- (x = N -> ((cos` (x x. A)) + (_i x. (sin` (x x. A)))) = ((cos` (N x. A)) + (_i x. (sin` (N x. A)))))
31	25, 30	eqeq12d 1899	. . . 4 |- (x = N -> ((((cos` A) + (_i x. (sin` A)))^x) = ((cos` (x x. A)) + (_i x. (sin` (x x. A)))) <-> (((cos` A) + (_i x. (sin` A)))^N) = ((cos` (N x. A)) + (_i x. (sin` (N x. A))))))
32	31	imbi2d 674	. . 3 |- (x = N -> ((A e. CC -> (((cos` A) + (_i x. (sin` A)))^x) = ((cos` (x x. A)) + (_i x. (sin` (x x. A))))) <-> (A e. CC -> (((cos` A) + (_i x. (sin` A)))^N) = ((cos` (N x. A)) + (_i x. (sin` (N x. A)))))))
33		coscl 8697	. . . . . 6 |- (A e. CC -> (cos` A) e. CC)
34		mulcl 6456	. . . . . . 7 |- ((_i e. CC /\ (sin` A) e. CC) -> (_i x. (sin` A)) e. CC)
35		axicn 6423	. . . . . . 7 |- _i e. CC
36		sincl 8696	. . . . . . 7 |- (A e. CC -> (sin` A) e. CC)
37	34, 35, 36	sylancr 526	. . . . . 6 |- (A e. CC -> (_i x. (sin` A)) e. CC)
38		addcl 6454	. . . . . 6 |- (((cos` A) e. CC /\ (_i x. (sin` A)) e. CC) -> ((cos` A) + (_i x. (sin` A))) e. CC)
39	33, 37, 38	syl11anc 524	. . . . 5 |- (A e. CC -> ((cos` A) + (_i x. (sin` A))) e. CC)
40		exp0 7814	. . . . 5 |- (((cos` A) + (_i x. (sin` A))) e. CC -> (((cos` A) + (_i x. (sin` A)))^0) = 1)
41	39, 40	syl 12	. . . 4 |- (A e. CC -> (((cos` A) + (_i x. (sin` A)))^0) = 1)
42		mul02 6607	. . . . . . . 8 |- (A e. CC -> (0 x. A) = 0)
43	42	fveq2d 4685	. . . . . . 7 |- (A e. CC -> (cos` (0 x. A)) = (cos` 0))
44		cos0 8711	. . . . . . 7 |- (cos` 0) = 1
45	43, 44	syl6eq 1944	. . . . . 6 |- (A e. CC -> (cos` (0 x. A)) = 1)
46	42	fveq2d 4685	. . . . . . . . 9 |- (A e. CC -> (sin` (0 x. A)) = (sin` 0))
47		sin0 8709	. . . . . . . . 9 |- (sin` 0) = 0
48	46, 47	syl6eq 1944	. . . . . . . 8 |- (A e. CC -> (sin` (0 x. A)) = 0)
49	48	opreq2d 4898	. . . . . . 7 |- (A e. CC -> (_i x. (sin` (0 x. A))) = (_i x. 0))
50	35	mul01i 6594	. . . . . . 7 |- (_i x. 0) = 0
51	49, 50	syl6eq 1944	. . . . . 6 |- (A e. CC -> (_i x. (sin` (0 x. A))) = 0)
52	45, 51	opreq12d 4900	. . . . 5 |- (A e. CC -> ((cos` (0 x. A)) + (_i x. (sin` (0 x. A)))) = (1 + 0))
53		ax1cn 6422	. . . . . 6 |- 1 e. CC
54	53	addid1i 6483	. . . . 5 |- (1 + 0) = 1
55	52, 54	syl6eq 1944	. . . 4 |- (A e. CC -> ((cos` (0 x. A)) + (_i x. (sin` (0 x. A)))) = 1)
56	41, 55	eqtr4d 1928	. . 3 |- (A e. CC -> (((cos` A) + (_i x. (sin` A)))^0) = ((cos` (0 x. A)) + (_i x. (sin` (0 x. A)))))
57		expp1 7817	. . . . . . . . 9 |- ((((cos` A) + (_i x. (sin` A))) e. CC /\ k e. NN0) -> (((cos` A) + (_i x. (sin` A)))^(k + 1)) = ((((cos` A) + (_i x. (sin` A)))^k) x. ((cos` A) + (_i x. (sin` A)))))
58	57, 39	sylan 497	. . . . . . . 8 |- ((A e. CC /\ k e. NN0) -> (((cos` A) + (_i x. (sin` A)))^(k + 1)) = ((((cos` A) + (_i x. (sin` A)))^k) x. ((cos` A) + (_i x. (sin` A)))))
59	58	ancoms 484	. . . . . . 7 |- ((k e. NN0 /\ A e. CC) -> (((cos` A) + (_i x. (sin` A)))^(k + 1)) = ((((cos` A) + (_i x. (sin` A)))^k) x. ((cos` A) + (_i x. (sin` A)))))
60	59	adantr 425	. . . . . 6 |- (((k e. NN0 /\ A e. CC) /\ (((cos` A) + (_i x. (sin` A)))^k) = ((cos` (k x. A)) + (_i x. (sin` (k x. A))))) -> (((cos` A) + (_i x. (sin` A)))^(k + 1)) = ((((cos` A) + (_i x. (sin` A)))^k) x. ((cos` A) + (_i x. (sin` A)))))
61		opreq1 4889	. . . . . . 7 |- ((((cos` A) + (_i x. (sin` A)))^k) = ((cos` (k x. A)) + (_i x. (sin` (k x. A)))) -> ((((cos` A) + (_i x. (sin` A)))^k) x. ((cos` A) + (_i x. (sin` A)))) = (((cos` (k x. A)) + (_i x. (sin` (k x. A)))) x. ((cos` A) + (_i x. (sin` A)))))
62	61	adantl 424	. . . . . 6 |- (((k e. NN0 /\ A e. CC) /\ (((cos` A) + (_i x. (sin` A)))^k) = ((cos` (k x. A)) + (_i x. (sin` (k x. A))))) -> ((((cos` A) + (_i x. (sin` A)))^k) x. ((cos` A) + (_i x. (sin` A)))) = (((cos` (k x. A)) + (_i x. (sin` (k x. A)))) x. ((cos` A) + (_i x. (sin` A)))))
63		mulcl 6456	. . . . . . . . . . . . 13 |- ((k e. CC /\ A e. CC) -> (k x. A) e. CC)
64		nn0cn 7318	. . . . . . . . . . . . 13 |- (k e. NN0 -> k e. CC)
65	63, 64	sylan 497	. . . . . . . . . . . 12 |- ((k e. NN0 /\ A e. CC) -> (k x. A) e. CC)
66		sinadd 8718	. . . . . . . . . . . 12 |- (((k x. A) e. CC /\ A e. CC) -> (sin` ((k x. A) + A)) = (((sin` (k x. A)) x. (cos` A)) + ((cos` (k x. A)) x. (sin` A))))
67	65, 66	sylancom 531	. . . . . . . . . . 11 |- ((k e. NN0 /\ A e. CC) -> (sin` ((k x. A) + A)) = (((sin` (k x. A)) x. (cos` A)) + ((cos` (k x. A)) x. (sin` A))))
68	33	adantl 424	. . . . . . . . . . . . 13 |- ((k e. NN0 /\ A e. CC) -> (cos` A) e. CC)
69		sincl 8696	. . . . . . . . . . . . . 14 |- ((k x. A) e. CC -> (sin` (k x. A)) e. CC)
70	65, 69	syl 12	. . . . . . . . . . . . 13 |- ((k e. NN0 /\ A e. CC) -> (sin` (k x. A)) e. CC)
71		mulcom 6459	. . . . . . . . . . . . 13 |- (((cos` A) e. CC /\ (sin` (k x. A)) e. CC) -> ((cos` A) x. (sin` (k x. A))) = ((sin` (k x. A)) x. (cos` A)))
72	68, 70, 71	syl11anc 524	. . . . . . . . . . . 12 |- ((k e. NN0 /\ A e. CC) -> ((cos` A) x. (sin` (k x. A))) = ((sin` (k x. A)) x. (cos` A)))
73	72	opreq1d 4897	. . . . . . . . . . 11 |- ((k e. NN0 /\ A e. CC) -> (((cos` A) x. (sin` (k x. A))) + ((cos` (k x. A)) x. (sin` A))) = (((sin` (k x. A)) x. (cos` A)) + ((cos` (k x. A)) x. (sin` A))))
74		mulcl 6456	. . . . . . . . . . . . 13 |- (((cos` A) e. CC /\ (sin` (k x. A)) e. CC) -> ((cos` A) x. (sin` (k x. A))) e. CC)
75	68, 70, 74	syl11anc 524	. . . . . . . . . . . 12 |- ((k e. NN0 /\ A e. CC) -> ((cos` A) x. (sin` (k x. A))) e. CC)
76		coscl 8697	. . . . . . . . . . . . . 14 |- ((k x. A) e. CC -> (cos` (k x. A)) e. CC)
77	65, 76	syl 12	. . . . . . . . . . . . 13 |- ((k e. NN0 /\ A e. CC) -> (cos` (k x. A)) e. CC)
78	36	adantl 424	. . . . . . . . . . . . 13 |- ((k e. NN0 /\ A e. CC) -> (sin` A) e. CC)
79		mulcl 6456	. . . . . . . . . . . . 13 |- (((cos` (k x. A)) e. CC /\ (sin` A) e. CC) -> ((cos` (k x. A)) x. (sin` A)) e. CC)
80	77, 78, 79	syl11anc 524	. . . . . . . . . . . 12 |- ((k e. NN0 /\ A e. CC) -> ((cos` (k x. A)) x. (sin` A)) e. CC)
81		addcom 6458	. . . . . . . . . . . 12 |- ((((cos` A) x. (sin` (k x. A))) e. CC /\ ((cos` (k x. A)) x. (sin` A)) e. CC) -> (((cos` A) x. (sin` (k x. A))) + ((cos` (k x. A)) x. (sin` A))) = (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))
82	75, 80, 81	syl11anc 524	. . . . . . . . . . 11 |- ((k e. NN0 /\ A e. CC) -> (((cos` A) x. (sin` (k x. A))) + ((cos` (k x. A)) x. (sin` A))) = (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))
83	67, 73, 82	3eqtr2d 1932	. . . . . . . . . 10 |- ((k e. NN0 /\ A e. CC) -> (sin` ((k x. A) + A)) = (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))
84	83	opreq2d 4898	. . . . . . . . 9 |- ((k e. NN0 /\ A e. CC) -> (_i x. (sin` ((k x. A) + A))) = (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A))))))
85	84	opreq2d 4898	. . . . . . . 8 |- ((k e. NN0 /\ A e. CC) -> ((cos` ((k x. A) + A)) + (_i x. (sin` ((k x. A) + A)))) = ((cos` ((k x. A) + A)) + (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))))
86		adddir 6472	. . . . . . . . . . . . 13 |- ((k e. CC /\ 1 e. CC /\ A e. CC) -> ((k + 1) x. A) = ((k x. A) + (1 x. A)))
87		mulid2 6578	. . . . . . . . . . . . . . 15 |- (A e. CC -> (1 x. A) = A)
88	87	opreq2d 4898	. . . . . . . . . . . . . 14 |- (A e. CC -> ((k x. A) + (1 x. A)) = ((k x. A) + A))
89	88	3ad2ant3 899	. . . . . . . . . . . . 13 |- ((k e. CC /\ 1 e. CC /\ A e. CC) -> ((k x. A) + (1 x. A)) = ((k x. A) + A))
90	86, 89	eqtrd 1925	. . . . . . . . . . . 12 |- ((k e. CC /\ 1 e. CC /\ A e. CC) -> ((k + 1) x. A) = ((k x. A) + A))
91	90, 64	syl3an1 1130	. . . . . . . . . . 11 |- ((k e. NN0 /\ 1 e. CC /\ A e. CC) -> ((k + 1) x. A) = ((k x. A) + A))
92	53, 91	mp3an2 1179	. . . . . . . . . 10 |- ((k e. NN0 /\ A e. CC) -> ((k + 1) x. A) = ((k x. A) + A))
93	92	fveq2d 4685	. . . . . . . . 9 |- ((k e. NN0 /\ A e. CC) -> (cos` ((k + 1) x. A)) = (cos` ((k x. A) + A)))
94	92	fveq2d 4685	. . . . . . . . . 10 |- ((k e. NN0 /\ A e. CC) -> (sin` ((k + 1) x. A)) = (sin` ((k x. A) + A)))
95	94	opreq2d 4898	. . . . . . . . 9 |- ((k e. NN0 /\ A e. CC) -> (_i x. (sin` ((k + 1) x. A))) = (_i x. (sin` ((k x. A) + A))))
96	93, 95	opreq12d 4900	. . . . . . . 8 |- ((k e. NN0 /\ A e. CC) -> ((cos` ((k + 1) x. A)) + (_i x. (sin` ((k + 1) x. A)))) = ((cos` ((k x. A) + A)) + (_i x. (sin` ((k x. A) + A)))))
97		mulcl 6456	. . . . . . . . . . . . . 14 |- ((_i e. CC /\ (sin` (k x. A)) e. CC) -> (_i x. (sin` (k x. A))) e. CC)
98	35, 97	mpan 759	. . . . . . . . . . . . 13 |- ((sin` (k x. A)) e. CC -> (_i x. (sin` (k x. A))) e. CC)
99	65, 69, 98	3syl 24	. . . . . . . . . . . 12 |- ((k e. NN0 /\ A e. CC) -> (_i x. (sin` (k x. A))) e. CC)
100	33, 37	jca 310	. . . . . . . . . . . . 13 |- (A e. CC -> ((cos` A) e. CC /\ (_i x. (sin` A)) e. CC))
101	100	adantl 424	. . . . . . . . . . . 12 |- ((k e. NN0 /\ A e. CC) -> ((cos` A) e. CC /\ (_i x. (sin` A)) e. CC))
102		muladd 6582	. . . . . . . . . . . 12 |- ((((cos` (k x. A)) e. CC /\ (_i x. (sin` (k x. A))) e. CC) /\ ((cos` A) e. CC /\ (_i x. (sin` A)) e. CC)) -> (((cos` (k x. A)) + (_i x. (sin` (k x. A)))) x. ((cos` A) + (_i x. (sin` A)))) = ((((cos` (k x. A)) x. (cos` A)) + ((_i x. (sin` A)) x. (_i x. (sin` (k x. A))))) + (((cos` (k x. A)) x. (_i x. (sin` A))) + ((cos` A) x. (_i x. (sin` (k x. A)))))))
103	77, 99, 101, 102	syl21anc 1099	. . . . . . . . . . 11 |- ((k e. NN0 /\ A e. CC) -> (((cos` (k x. A)) + (_i x. (sin` (k x. A)))) x. ((cos` A) + (_i x. (sin` A)))) = ((((cos` (k x. A)) x. (cos` A)) + ((_i x. (sin` A)) x. (_i x. (sin` (k x. A))))) + (((cos` (k x. A)) x. (_i x. (sin` A))) + ((cos` A) x. (_i x. (sin` (k x. A)))))))
104	78, 35	jctil 316	. . . . . . . . . . . . . 14 |- ((k e. NN0 /\ A e. CC) -> (_i e. CC /\ (sin` A) e. CC))
105	70, 35	jctil 316	. . . . . . . . . . . . . 14 |- ((k e. NN0 /\ A e. CC) -> (_i e. CC /\ (sin` (k x. A)) e. CC))
106		mul4 6581	. . . . . . . . . . . . . . 15 |- (((_i e. CC /\ (sin` A) e. CC) /\ (_i e. CC /\ (sin` (k x. A)) e. CC)) -> ((_i x. (sin` A)) x. (_i x. (sin` (k x. A)))) = ((_i x. _i) x. ((sin` A) x. (sin` (k x. A)))))
107		ixi 6872	. . . . . . . . . . . . . . . 16 |- (_i x. _i) = -u1
108	107	opreq1i 4892	. . . . . . . . . . . . . . 15 |- ((_i x. _i) x. ((sin` A) x. (sin` (k x. A)))) = (-u1 x. ((sin` A) x. (sin` (k x. A))))
109	106, 108	syl6eq 1944	. . . . . . . . . . . . . 14 |- (((_i e. CC /\ (sin` A) e. CC) /\ (_i e. CC /\ (sin` (k x. A)) e. CC)) -> ((_i x. (sin` A)) x. (_i x. (sin` (k x. A)))) = (-u1 x. ((sin` A) x. (sin` (k x. A)))))
110	104, 105, 109	syl11anc 524	. . . . . . . . . . . . 13 |- ((k e. NN0 /\ A e. CC) -> ((_i x. (sin` A)) x. (_i x. (sin` (k x. A)))) = (-u1 x. ((sin` A) x. (sin` (k x. A)))))
111	110	opreq2d 4898	. . . . . . . . . . . 12 |- ((k e. NN0 /\ A e. CC) -> (((cos` (k x. A)) x. (cos` A)) + ((_i x. (sin` A)) x. (_i x. (sin` (k x. A))))) = (((cos` (k x. A)) x. (cos` A)) + (-u1 x. ((sin` A) x. (sin` (k x. A))))))
112	111	opreq1d 4897	. . . . . . . . . . 11 |- ((k e. NN0 /\ A e. CC) -> ((((cos` (k x. A)) x. (cos` A)) + ((_i x. (sin` A)) x. (_i x. (sin` (k x. A))))) + (((cos` (k x. A)) x. (_i x. (sin` A))) + ((cos` A) x. (_i x. (sin` (k x. A)))))) = ((((cos` (k x. A)) x. (cos` A)) + (-u1 x. ((sin` A) x. (sin` (k x. A))))) + (((cos` (k x. A)) x. (_i x. (sin` A))) + ((cos` A) x. (_i x. (sin` (k x. A)))))))
113		mul12 6579	. . . . . . . . . . . . . . . 16 |- (((cos` (k x. A)) e. CC /\ _i e. CC /\ (sin` A) e. CC) -> ((cos` (k x. A)) x. (_i x. (sin` A))) = (_i x. ((cos` (k x. A)) x. (sin` A))))
114	35, 113	mp3an2 1179	. . . . . . . . . . . . . . 15 |- (((cos` (k x. A)) e. CC /\ (sin` A) e. CC) -> ((cos` (k x. A)) x. (_i x. (sin` A))) = (_i x. ((cos` (k x. A)) x. (sin` A))))
115	77, 78, 114	syl11anc 524	. . . . . . . . . . . . . 14 |- ((k e. NN0 /\ A e. CC) -> ((cos` (k x. A)) x. (_i x. (sin` A))) = (_i x. ((cos` (k x. A)) x. (sin` A))))
116		mul12 6579	. . . . . . . . . . . . . . . 16 |- (((cos` A) e. CC /\ _i e. CC /\ (sin` (k x. A)) e. CC) -> ((cos` A) x. (_i x. (sin` (k x. A)))) = (_i x. ((cos` A) x. (sin` (k x. A)))))
117	35, 116	mp3an2 1179	. . . . . . . . . . . . . . 15 |- (((cos` A) e. CC /\ (sin` (k x. A)) e. CC) -> ((cos` A) x. (_i x. (sin` (k x. A)))) = (_i x. ((cos` A) x. (sin` (k x. A)))))
118	68, 70, 117	syl11anc 524	. . . . . . . . . . . . . 14 |- ((k e. NN0 /\ A e. CC) -> ((cos` A) x. (_i x. (sin` (k x. A)))) = (_i x. ((cos` A) x. (sin` (k x. A)))))
119	115, 118	opreq12d 4900	. . . . . . . . . . . . 13 |- ((k e. NN0 /\ A e. CC) -> (((cos` (k x. A)) x. (_i x. (sin` A))) + ((cos` A) x. (_i x. (sin` (k x. A))))) = ((_i x. ((cos` (k x. A)) x. (sin` A))) + (_i x. ((cos` A) x. (sin` (k x. A))))))
120		adddi 6462	. . . . . . . . . . . . . . 15 |- ((_i e. CC /\ ((cos` (k x. A)) x. (sin` A)) e. CC /\ ((cos` A) x. (sin` (k x. A))) e. CC) -> (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A))))) = ((_i x. ((cos` (k x. A)) x. (sin` A))) + (_i x. ((cos` A) x. (sin` (k x. A))))))
121	35, 120	mp3an1 1178	. . . . . . . . . . . . . 14 |- ((((cos` (k x. A)) x. (sin` A)) e. CC /\ ((cos` A) x. (sin` (k x. A))) e. CC) -> (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A))))) = ((_i x. ((cos` (k x. A)) x. (sin` A))) + (_i x. ((cos` A) x. (sin` (k x. A))))))
122	80, 75, 121	syl11anc 524	. . . . . . . . . . . . 13 |- ((k e. NN0 /\ A e. CC) -> (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A))))) = ((_i x. ((cos` (k x. A)) x. (sin` A))) + (_i x. ((cos` A) x. (sin` (k x. A))))))
123	119, 122	eqtr4d 1928	. . . . . . . . . . . 12 |- ((k e. NN0 /\ A e. CC) -> (((cos` (k x. A)) x. (_i x. (sin` A))) + ((cos` A) x. (_i x. (sin` (k x. A))))) = (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A))))))
124	123	opreq2d 4898	. . . . . . . . . . 11 |- ((k e. NN0 /\ A e. CC) -> ((((cos` (k x. A)) x. (cos` A)) + (-u1 x. ((sin` A) x. (sin` (k x. A))))) + (((cos` (k x. A)) x. (_i x. (sin` A))) + ((cos` A) x. (_i x. (sin` (k x. A)))))) = ((((cos` (k x. A)) x. (cos` A)) + (-u1 x. ((sin` A) x. (sin` (k x. A))))) + (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))))
125	103, 112, 124	3eqtrd 1929	. . . . . . . . . 10 |- ((k e. NN0 /\ A e. CC) -> (((cos` (k x. A)) + (_i x. (sin` (k x. A)))) x. ((cos` A) + (_i x. (sin` A)))) = ((((cos` (k x. A)) x. (cos` A)) + (-u1 x. ((sin` A) x. (sin` (k x. A))))) + (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))))
126		mulcl 6456	. . . . . . . . . . . . . 14 |- (((sin` A) e. CC /\ (sin` (k x. A)) e. CC) -> ((sin` A) x. (sin` (k x. A))) e. CC)
127	78, 70, 126	syl11anc 524	. . . . . . . . . . . . 13 |- ((k e. NN0 /\ A e. CC) -> ((sin` A) x. (sin` (k x. A))) e. CC)
128		mulm1 6638	. . . . . . . . . . . . 13 |- (((sin` A) x. (sin` (k x. A))) e. CC -> (-u1 x. ((sin` A) x. (sin` (k x. A)))) = -u((sin` A) x. (sin` (k x. A))))
129	127, 128	syl 12	. . . . . . . . . . . 12 |- ((k e. NN0 /\ A e. CC) -> (-u1 x. ((sin` A) x. (sin` (k x. A)))) = -u((sin` A) x. (sin` (k x. A))))
130	129	opreq2d 4898	. . . . . . . . . . 11 |- ((k e. NN0 /\ A e. CC) -> (((cos` (k x. A)) x. (cos` A)) + (-u1 x. ((sin` A) x. (sin` (k x. A))))) = (((cos` (k x. A)) x. (cos` A)) + -u((sin` A) x. (sin` (k x. A)))))
131	130	opreq1d 4897	. . . . . . . . . 10 |- ((k e. NN0 /\ A e. CC) -> ((((cos` (k x. A)) x. (cos` A)) + (-u1 x. ((sin` A) x. (sin` (k x. A))))) + (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))) = ((((cos` (k x. A)) x. (cos` A)) + -u((sin` A) x. (sin` (k x. A)))) + (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))))
132		mulcl 6456	. . . . . . . . . . . . 13 |- (((cos` (k x. A)) e. CC /\ (cos` A) e. CC) -> ((cos` (k x. A)) x. (cos` A)) e. CC)
133	77, 68, 132	syl11anc 524	. . . . . . . . . . . 12 |- ((k e. NN0 /\ A e. CC) -> ((cos` (k x. A)) x. (cos` A)) e. CC)
134		negsub 6540	. . . . . . . . . . . 12 |- ((((cos` (k x. A)) x. (cos` A)) e. CC /\ ((sin` A) x. (sin` (k x. A))) e. CC) -> (((cos` (k x. A)) x. (cos` A)) + -u((sin` A) x. (sin` (k x. A)))) = (((cos` (k x. A)) x. (cos` A)) - ((sin` A) x. (sin` (k x. A)))))
135	133, 127, 134	syl11anc 524	. . . . . . . . . . 11 |- ((k e. NN0 /\ A e. CC) -> (((cos` (k x. A)) x. (cos` A)) + -u((sin` A) x. (sin` (k x. A)))) = (((cos` (k x. A)) x. (cos` A)) - ((sin` A) x. (sin` (k x. A)))))
136	135	opreq1d 4897	. . . . . . . . . 10 |- ((k e. NN0 /\ A e. CC) -> ((((cos` (k x. A)) x. (cos` A)) + -u((sin` A) x. (sin` (k x. A)))) + (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))) = ((((cos` (k x. A)) x. (cos` A)) - ((sin` A) x. (sin` (k x. A)))) + (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))))
137	125, 131, 136	3eqtrd 1929	. . . . . . . . 9 |- ((k e. NN0 /\ A e. CC) -> (((cos` (k x. A)) + (_i x. (sin` (k x. A)))) x. ((cos` A) + (_i x. (sin` A)))) = ((((cos` (k x. A)) x. (cos` A)) - ((sin` A) x. (sin` (k x. A)))) + (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))))
138		cosadd 8719	. . . . . . . . . . . 12 |- (((k x. A) e. CC /\ A e. CC) -> (cos` ((k x. A) + A)) = (((cos` (k x. A)) x. (cos` A)) - ((sin` (k x. A)) x. (sin` A))))
139	65, 138	sylancom 531	. . . . . . . . . . 11 |- ((k e. NN0 /\ A e. CC) -> (cos` ((k x. A) + A)) = (((cos` (k x. A)) x. (cos` A)) - ((sin` (k x. A)) x. (sin` A))))
140		mulcom 6459	. . . . . . . . . . . . 13 |- (((sin` (k x. A)) e. CC /\ (sin` A) e. CC) -> ((sin` (k x. A)) x. (sin` A)) = ((sin` A) x. (sin` (k x. A))))
141	70, 78, 140	syl11anc 524	. . . . . . . . . . . 12 |- ((k e. NN0 /\ A e. CC) -> ((sin` (k x. A)) x. (sin` A)) = ((sin` A) x. (sin` (k x. A))))
142	141	opreq2d 4898	. . . . . . . . . . 11 |- ((k e. NN0 /\ A e. CC) -> (((cos` (k x. A)) x. (cos` A)) - ((sin` (k x. A)) x. (sin` A))) = (((cos` (k x. A)) x. (cos` A)) - ((sin` A) x. (sin` (k x. A)))))
143	139, 142	eqtrd 1925	. . . . . . . . . 10 |- ((k e. NN0 /\ A e. CC) -> (cos` ((k x. A) + A)) = (((cos` (k x. A)) x. (cos` A)) - ((sin` A) x. (sin` (k x. A)))))
144	143	opreq1d 4897	. . . . . . . . 9 |- ((k e. NN0 /\ A e. CC) -> ((cos` ((k x. A) + A)) + (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))) = ((((cos` (k x. A)) x. (cos` A)) - ((sin` A) x. (sin` (k x. A)))) + (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))))
145	137, 144	eqtr4d 1928	. . . . . . . 8 |- ((k e. NN0 /\ A e. CC) -> (((cos` (k x. A)) + (_i x. (sin` (k x. A)))) x. ((cos` A) + (_i x. (sin` A)))) = ((cos` ((k x. A) + A)) + (_i x. (((cos` (k x. A)) x. (sin` A)) + ((cos` A) x. (sin` (k x. A)))))))
146	85, 96, 145	3eqtr4rd 1939	. . . . . . 7 |- ((k e. NN0 /\ A e. CC) -> (((cos` (k x. A)) + (_i x. (sin` (k x. A)))) x. ((cos` A) + (_i x. (sin` A)))) = ((cos` ((k + 1) x. A)) + (_i x. (sin` ((k + 1) x. A)))))
147	146	adantr 425	. . . . . 6 |- (((k e. NN0 /\ A e. CC) /\ (((cos` A) + (_i x. (sin` A)))^k) = ((cos` (k x. A)) + (_i x. (sin` (k x. A))))) -> (((cos` (k x. A)) + (_i x. (sin` (k x. A)))) x. ((cos` A) + (_i x. (sin` A)))) = ((cos` ((k + 1) x. A)) + (_i x. (sin` ((k + 1) x. A)))))
148	60, 62, 147	3eqtrd 1929	. . . . 5 |- (((k e. NN0 /\ A e. CC) /\ (((cos` A) + (_i x. (sin` A)))^k) = ((cos` (k x. A)) + (_i x. (sin` (k x. A))))) -> (((cos` A) + (_i x. (sin` A)))^(k + 1)) = ((cos` ((k + 1) x. A)) + (_i x. (sin` ((k + 1) x. A)))))
149	148	exp31 407	. . . 4 |- (k e. NN0 -> (A e. CC -> ((((cos` A) + (_i x. (sin` A)))^k) = ((cos` (k x. A)) + (_i x. (sin` (k x. A)))) -> (((cos` A) + (_i x. (sin` A)))^(k + 1)) = ((cos` ((k + 1) x. A)) + (_i x. (sin` ((k + 1) x. A)))))))
150	149	a2d 16	. . 3 |- (k e. NN0 -> ((A e. CC -> (((cos` A) + (_i x. (sin` A)))^k) = ((cos` (k x. A)) + (_i x. (sin` (k x. A))))) -> (A e. CC -> (((cos` A) + (_i x. (sin` A)))^(k + 1)) = ((cos` ((k + 1) x. A)) + (_i x. (sin` ((k + 1) x. A)))))))
151	8, 16, 24, 32, 56, 150	nn0ind 7424	. 2 |- (N e. NN0 -> (A e. CC -> (((cos` A) + (_i x. (sin` A)))^N) = ((cos` (N x. A)) + (_i x. (sin` (N x. A))))))
152	151	impcom 378	1 |- ((A e. CC /\ N e. NN0) -> (((cos` A) + (_i x. (sin` A)))^N) = ((cos` (N x. A)) + (_i x. (sin` (N x. A)))))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446  NN0cn0 6450  ^cexp 7811  sincsin 8557  cosccos 8558

This theorem is referenced by:  sinperlem1 10035

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-fac 8184  df-bc 8209  df-clim 8235  df-sum 8240  df-ef 8560  df-sin 8562  df-cos 8563

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