Theorem sinaddi 8716

Description: Sine addition formula for complex arguments. Equation 14 of [Gleason] p. 310.

Hypotheses

Ref	Expression
sinadd.1	|- A e. CC
sinadd.2	|- B e. CC

Assertion

Ref	Expression
sinaddi	|- (sin` (A + B)) = (((sin` A) x. (cos` B)) + ((cos` A) x. (sin` B)))

Proof of Theorem sinaddi

Step	Hyp	Ref	Expression
1		sinadd.1	. . . 4 |- A e. CC
2		sinadd.2	. . . 4 |- B e. CC
3	1, 2	addcli 6473	. . 3 |- (A + B) e. CC
4		sinval 8694	. . 3 |- ((A + B) e. CC -> (sin` (A + B)) = (((exp` (_i x. (A + B))) - (exp` (-u_i x. (A + B)))) / (2 x. _i)))
5	3, 4	ax-mp 7	. 2 |- (sin` (A + B)) = (((exp` (_i x. (A + B))) - (exp` (-u_i x. (A + B)))) / (2 x. _i))
6		coscl 8697	. . . . . . . 8 |- (A e. CC -> (cos` A) e. CC)
7	1, 6	ax-mp 7	. . . . . . 7 |- (cos` A) e. CC
8		coscl 8697	. . . . . . . 8 |- (B e. CC -> (cos` B) e. CC)
9	2, 8	ax-mp 7	. . . . . . 7 |- (cos` B) e. CC
10	7, 9	mulcli 6474	. . . . . 6 |- ((cos` A) x. (cos` B)) e. CC
11		axicn 6423	. . . . . . . 8 |- _i e. CC
12		sincl 8696	. . . . . . . . 9 |- (B e. CC -> (sin` B) e. CC)
13	2, 12	ax-mp 7	. . . . . . . 8 |- (sin` B) e. CC
14	11, 13	mulcli 6474	. . . . . . 7 |- (_i x. (sin` B)) e. CC
15		sincl 8696	. . . . . . . . 9 |- (A e. CC -> (sin` A) e. CC)
16	1, 15	ax-mp 7	. . . . . . . 8 |- (sin` A) e. CC
17	11, 16	mulcli 6474	. . . . . . 7 |- (_i x. (sin` A)) e. CC
18	14, 17	mulcli 6474	. . . . . 6 |- ((_i x. (sin` B)) x. (_i x. (sin` A))) e. CC
19	10, 18	addcli 6473	. . . . 5 |- (((cos` A) x. (cos` B)) + ((_i x. (sin` B)) x. (_i x. (sin` A)))) e. CC
20	7, 14	mulcli 6474	. . . . . 6 |- ((cos` A) x. (_i x. (sin` B))) e. CC
21	9, 17	mulcli 6474	. . . . . 6 |- ((cos` B) x. (_i x. (sin` A))) e. CC
22	20, 21	addcli 6473	. . . . 5 |- (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))) e. CC
23	19, 22, 22	pnncani 6649	. . . 4 |- (((((cos` A) x. (cos` B)) + ((_i x. (sin` B)) x. (_i x. (sin` A)))) + (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))) - ((((cos` A) x. (cos` B)) + ((_i x. (sin` B)) x. (_i x. (sin` A)))) - (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))))) = ((((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))) + (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))))
24	11, 1, 2	adddii 6479	. . . . . . 7 |- (_i x. (A + B)) = ((_i x. A) + (_i x. B))
25	24	fveq2i 4684	. . . . . 6 |- (exp` (_i x. (A + B))) = (exp` ((_i x. A) + (_i x. B)))
26	11, 1	mulcli 6474	. . . . . . 7 |- (_i x. A) e. CC
27	11, 2	mulcli 6474	. . . . . . 7 |- (_i x. B) e. CC
28	26, 27	efaddi 8628	. . . . . 6 |- (exp` ((_i x. A) + (_i x. B))) = ((exp` (_i x. A)) x. (exp` (_i x. B)))
29		efival 8712	. . . . . . . . 9 |- (A e. CC -> (exp` (_i x. A)) = ((cos` A) + (_i x. (sin` A))))
30	1, 29	ax-mp 7	. . . . . . . 8 |- (exp` (_i x. A)) = ((cos` A) + (_i x. (sin` A)))
31		efival 8712	. . . . . . . . 9 |- (B e. CC -> (exp` (_i x. B)) = ((cos` B) + (_i x. (sin` B))))
32	2, 31	ax-mp 7	. . . . . . . 8 |- (exp` (_i x. B)) = ((cos` B) + (_i x. (sin` B)))
33	30, 32	opreq12i 4894	. . . . . . 7 |- ((exp` (_i x. A)) x. (exp` (_i x. B))) = (((cos` A) + (_i x. (sin` A))) x. ((cos` B) + (_i x. (sin` B))))
34	7, 17, 9, 14	muladdi 6589	. . . . . . 7 |- (((cos` A) + (_i x. (sin` A))) x. ((cos` B) + (_i x. (sin` B)))) = ((((cos` A) x. (cos` B)) + ((_i x. (sin` B)) x. (_i x. (sin` A)))) + (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))))
35	33, 34	eqtri 1908	. . . . . 6 |- ((exp` (_i x. A)) x. (exp` (_i x. B))) = ((((cos` A) x. (cos` B)) + ((_i x. (sin` B)) x. (_i x. (sin` A)))) + (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))))
36	25, 28, 35	3eqtri 1912	. . . . 5 |- (exp` (_i x. (A + B))) = ((((cos` A) x. (cos` B)) + ((_i x. (sin` B)) x. (_i x. (sin` A)))) + (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))))
37	11	negcli 6526	. . . . . . . 8 |- -u_i e. CC
38	37, 1, 2	adddii 6479	. . . . . . 7 |- (-u_i x. (A + B)) = ((-u_i x. A) + (-u_i x. B))
39	38	fveq2i 4684	. . . . . 6 |- (exp` (-u_i x. (A + B))) = (exp` ((-u_i x. A) + (-u_i x. B)))
40	37, 1	mulcli 6474	. . . . . . 7 |- (-u_i x. A) e. CC
41	37, 2	mulcli 6474	. . . . . . 7 |- (-u_i x. B) e. CC
42	40, 41	efaddi 8628	. . . . . 6 |- (exp` ((-u_i x. A) + (-u_i x. B))) = ((exp` (-u_i x. A)) x. (exp` (-u_i x. B)))
43		efmival 8713	. . . . . . . . 9 |- (A e. CC -> (exp` (-u_i x. A)) = ((cos` A) - (_i x. (sin` A))))
44	1, 43	ax-mp 7	. . . . . . . 8 |- (exp` (-u_i x. A)) = ((cos` A) - (_i x. (sin` A)))
45		efmival 8713	. . . . . . . . 9 |- (B e. CC -> (exp` (-u_i x. B)) = ((cos` B) - (_i x. (sin` B))))
46	2, 45	ax-mp 7	. . . . . . . 8 |- (exp` (-u_i x. B)) = ((cos` B) - (_i x. (sin` B)))
47	44, 46	opreq12i 4894	. . . . . . 7 |- ((exp` (-u_i x. A)) x. (exp` (-u_i x. B))) = (((cos` A) - (_i x. (sin` A))) x. ((cos` B) - (_i x. (sin` B))))
48	7, 17	pm3.2i 307	. . . . . . . 8 |- ((cos` A) e. CC /\ (_i x. (sin` A)) e. CC)
49	9, 14	pm3.2i 307	. . . . . . . 8 |- ((cos` B) e. CC /\ (_i x. (sin` B)) e. CC)
50		mulsub 6644	. . . . . . . 8 |- ((((cos` A) e. CC /\ (_i x. (sin` A)) e. CC) /\ ((cos` B) e. CC /\ (_i x. (sin` B)) e. CC)) -> (((cos` A) - (_i x. (sin` A))) x. ((cos` B) - (_i x. (sin` B)))) = ((((cos` A) x. (cos` B)) + ((_i x. (sin` B)) x. (_i x. (sin` A)))) - (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))))
51	48, 49, 50	mp2an 761	. . . . . . 7 |- (((cos` A) - (_i x. (sin` A))) x. ((cos` B) - (_i x. (sin` B)))) = ((((cos` A) x. (cos` B)) + ((_i x. (sin` B)) x. (_i x. (sin` A)))) - (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))))
52	47, 51	eqtri 1908	. . . . . 6 |- ((exp` (-u_i x. A)) x. (exp` (-u_i x. B))) = ((((cos` A) x. (cos` B)) + ((_i x. (sin` B)) x. (_i x. (sin` A)))) - (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))))
53	39, 42, 52	3eqtri 1912	. . . . 5 |- (exp` (-u_i x. (A + B))) = ((((cos` A) x. (cos` B)) + ((_i x. (sin` B)) x. (_i x. (sin` A)))) - (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))))
54	36, 53	opreq12i 4894	. . . 4 |- ((exp` (_i x. (A + B))) - (exp` (-u_i x. (A + B)))) = (((((cos` A) x. (cos` B)) + ((_i x. (sin` B)) x. (_i x. (sin` A)))) + (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))) - ((((cos` A) x. (cos` B)) + ((_i x. (sin` B)) x. (_i x. (sin` A)))) - (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))))
55	22	2timesi 7187	. . . 4 |- (2 x. (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))) = ((((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))) + (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))))
56	23, 54, 55	3eqtr4i 1921	. . 3 |- ((exp` (_i x. (A + B))) - (exp` (-u_i x. (A + B)))) = (2 x. (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))))
57	56	opreq1i 4892	. 2 |- (((exp` (_i x. (A + B))) - (exp` (-u_i x. (A + B)))) / (2 x. _i)) = ((2 x. (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))) / (2 x. _i))
58		2cn 7164	. . . . . 6 |- 2 e. CC
59	7, 13	mulcli 6474	. . . . . . 7 |- ((cos` A) x. (sin` B)) e. CC
60	16, 9	mulcli 6474	. . . . . . 7 |- ((sin` A) x. (cos` B)) e. CC
61	59, 60	addcli 6473	. . . . . 6 |- (((cos` A) x. (sin` B)) + ((sin` A) x. (cos` B))) e. CC
62	58, 11, 61	mulassi 6478	. . . . 5 |- ((2 x. _i) x. (((cos` A) x. (sin` B)) + ((sin` A) x. (cos` B)))) = (2 x. (_i x. (((cos` A) x. (sin` B)) + ((sin` A) x. (cos` B)))))
63	11, 59, 60	adddii 6479	. . . . . . 7 |- (_i x. (((cos` A) x. (sin` B)) + ((sin` A) x. (cos` B)))) = ((_i x. ((cos` A) x. (sin` B))) + (_i x. ((sin` A) x. (cos` B))))
64	11, 7, 13	mul12i 6585	. . . . . . . 8 |- (_i x. ((cos` A) x. (sin` B))) = ((cos` A) x. (_i x. (sin` B)))
65	16, 9	mulcomi 6476	. . . . . . . . . 10 |- ((sin` A) x. (cos` B)) = ((cos` B) x. (sin` A))
66	65	opreq2i 4893	. . . . . . . . 9 |- (_i x. ((sin` A) x. (cos` B))) = (_i x. ((cos` B) x. (sin` A)))
67	11, 9, 16	mul12i 6585	. . . . . . . . 9 |- (_i x. ((cos` B) x. (sin` A))) = ((cos` B) x. (_i x. (sin` A)))
68	66, 67	eqtri 1908	. . . . . . . 8 |- (_i x. ((sin` A) x. (cos` B))) = ((cos` B) x. (_i x. (sin` A)))
69	64, 68	opreq12i 4894	. . . . . . 7 |- ((_i x. ((cos` A) x. (sin` B))) + (_i x. ((sin` A) x. (cos` B)))) = (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))
70	63, 69	eqtri 1908	. . . . . 6 |- (_i x. (((cos` A) x. (sin` B)) + ((sin` A) x. (cos` B)))) = (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))
71	70	opreq2i 4893	. . . . 5 |- (2 x. (_i x. (((cos` A) x. (sin` B)) + ((sin` A) x. (cos` B))))) = (2 x. (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))))
72	62, 71	eqtri 1908	. . . 4 |- ((2 x. _i) x. (((cos` A) x. (sin` B)) + ((sin` A) x. (cos` B)))) = (2 x. (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A)))))
73	58, 22	mulcli 6474	. . . . 5 |- (2 x. (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))) e. CC
74	58, 11	mulcli 6474	. . . . 5 |- (2 x. _i) e. CC
75		2ne0 7174	. . . . . 6 |- 2 =/= 0
76		ine0 6597	. . . . . 6 |- _i =/= 0
77	58, 11, 75, 76	mulne0i 6888	. . . . 5 |- (2 x. _i) =/= 0
78	73, 74, 61, 77	divmuli 6894	. . . 4 |- (((2 x. (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))) / (2 x. _i)) = (((cos` A) x. (sin` B)) + ((sin` A) x. (cos` B))) <-> ((2 x. _i) x. (((cos` A) x. (sin` B)) + ((sin` A) x. (cos` B)))) = (2 x. (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))))
79	72, 78	mpbir 207	. . 3 |- ((2 x. (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))) / (2 x. _i)) = (((cos` A) x. (sin` B)) + ((sin` A) x. (cos` B)))
80	60, 59	addcomi 6475	. . 3 |- (((sin` A) x. (cos` B)) + ((cos` A) x. (sin` B))) = (((cos` A) x. (sin` B)) + ((sin` A) x. (cos` B)))
81	79, 80	eqtr4i 1911	. 2 |- ((2 x. (((cos` A) x. (_i x. (sin` B))) + ((cos` B) x. (_i x. (sin` A))))) / (2 x. _i)) = (((sin` A) x. (cos` B)) + ((cos` A) x. (sin` B)))
82	5, 57, 81	3eqtri 1912	1 |- (sin` (A + B)) = (((sin` A) x. (cos` B)) + ((cos` A) x. (sin` B)))

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  CCcc 6384  _ici 6388   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447  2c2 7145  expce 8555  sincsin 8557  cosccos 8558

This theorem is referenced by:  sinadd 8718

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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