Theorem efival 8712

Description: The exponential function in terms of sine and cosine.

Assertion

Ref	Expression
efival	|- (A e. CC -> (exp` (_i x. A)) = ((cos` A) + (_i x. (sin` A))))

Proof of Theorem efival

Step	Hyp	Ref	Expression
1		axicn 6423	. . . . . 6 |- _i e. CC
2		mulcl 6456	. . . . . 6 |- ((_i e. CC /\ A e. CC) -> (_i x. A) e. CC)
3	1, 2	mpan 759	. . . . 5 |- (A e. CC -> (_i x. A) e. CC)
4		efcl 8574	. . . . 5 |- ((_i x. A) e. CC -> (exp` (_i x. A)) e. CC)
5	3, 4	syl 12	. . . 4 |- (A e. CC -> (exp` (_i x. A)) e. CC)
6	1	negcli 6526	. . . . . 6 |- -u_i e. CC
7		mulcl 6456	. . . . . 6 |- ((-u_i e. CC /\ A e. CC) -> (-u_i x. A) e. CC)
8	6, 7	mpan 759	. . . . 5 |- (A e. CC -> (-u_i x. A) e. CC)
9		efcl 8574	. . . . 5 |- ((-u_i x. A) e. CC -> (exp` (-u_i x. A)) e. CC)
10	8, 9	syl 12	. . . 4 |- (A e. CC -> (exp` (-u_i x. A)) e. CC)
11		addcl 6454	. . . 4 |- (((exp` (_i x. A)) e. CC /\ (exp` (-u_i x. A)) e. CC) -> ((exp` (_i x. A)) + (exp` (-u_i x. A))) e. CC)
12	5, 10, 11	syl11anc 524	. . 3 |- (A e. CC -> ((exp` (_i x. A)) + (exp` (-u_i x. A))) e. CC)
13		subcl 6524	. . . 4 |- (((exp` (_i x. A)) e. CC /\ (exp` (-u_i x. A)) e. CC) -> ((exp` (_i x. A)) - (exp` (-u_i x. A))) e. CC)
14	5, 10, 13	syl11anc 524	. . 3 |- (A e. CC -> ((exp` (_i x. A)) - (exp` (-u_i x. A))) e. CC)
15		2cn 7164	. . . . 5 |- 2 e. CC
16		2ne0 7174	. . . . 5 |- 2 =/= 0
17	15, 16	pm3.2i 307	. . . 4 |- (2 e. CC /\ 2 =/= 0)
18		divdir 6933	. . . 4 |- ((((exp` (_i x. A)) + (exp` (-u_i x. A))) e. CC /\ ((exp` (_i x. A)) - (exp` (-u_i x. A))) e. CC /\ (2 e. CC /\ 2 =/= 0)) -> ((((exp` (_i x. A)) + (exp` (-u_i x. A))) + ((exp` (_i x. A)) - (exp` (-u_i x. A)))) / 2) = ((((exp` (_i x. A)) + (exp` (-u_i x. A))) / 2) + (((exp` (_i x. A)) - (exp` (-u_i x. A))) / 2)))
19	17, 18	mp3an3 1180	. . 3 |- ((((exp` (_i x. A)) + (exp` (-u_i x. A))) e. CC /\ ((exp` (_i x. A)) - (exp` (-u_i x. A))) e. CC) -> ((((exp` (_i x. A)) + (exp` (-u_i x. A))) + ((exp` (_i x. A)) - (exp` (-u_i x. A)))) / 2) = ((((exp` (_i x. A)) + (exp` (-u_i x. A))) / 2) + (((exp` (_i x. A)) - (exp` (-u_i x. A))) / 2)))
20	12, 14, 19	syl11anc 524	. 2 |- (A e. CC -> ((((exp` (_i x. A)) + (exp` (-u_i x. A))) + ((exp` (_i x. A)) - (exp` (-u_i x. A)))) / 2) = ((((exp` (_i x. A)) + (exp` (-u_i x. A))) / 2) + (((exp` (_i x. A)) - (exp` (-u_i x. A))) / 2)))
21		pncan3 6534	. . . . . . 7 |- (((exp` (-u_i x. A)) e. CC /\ (exp` (_i x. A)) e. CC) -> ((exp` (-u_i x. A)) + ((exp` (_i x. A)) - (exp` (-u_i x. A)))) = (exp` (_i x. A)))
22	10, 5, 21	syl11anc 524	. . . . . 6 |- (A e. CC -> ((exp` (-u_i x. A)) + ((exp` (_i x. A)) - (exp` (-u_i x. A)))) = (exp` (_i x. A)))
23	22	opreq2d 4898	. . . . 5 |- (A e. CC -> ((exp` (_i x. A)) + ((exp` (-u_i x. A)) + ((exp` (_i x. A)) - (exp` (-u_i x. A))))) = ((exp` (_i x. A)) + (exp` (_i x. A))))
24		addass 6460	. . . . . 6 |- (((exp` (_i x. A)) e. CC /\ (exp` (-u_i x. A)) e. CC /\ ((exp` (_i x. A)) - (exp` (-u_i x. A))) e. CC) -> (((exp` (_i x. A)) + (exp` (-u_i x. A))) + ((exp` (_i x. A)) - (exp` (-u_i x. A)))) = ((exp` (_i x. A)) + ((exp` (-u_i x. A)) + ((exp` (_i x. A)) - (exp` (-u_i x. A))))))
25	5, 10, 14, 24	syl111anc 1100	. . . . 5 |- (A e. CC -> (((exp` (_i x. A)) + (exp` (-u_i x. A))) + ((exp` (_i x. A)) - (exp` (-u_i x. A)))) = ((exp` (_i x. A)) + ((exp` (-u_i x. A)) + ((exp` (_i x. A)) - (exp` (-u_i x. A))))))
26		2times 7188	. . . . . 6 |- ((exp` (_i x. A)) e. CC -> (2 x. (exp` (_i x. A))) = ((exp` (_i x. A)) + (exp` (_i x. A))))
27	5, 26	syl 12	. . . . 5 |- (A e. CC -> (2 x. (exp` (_i x. A))) = ((exp` (_i x. A)) + (exp` (_i x. A))))
28	23, 25, 27	3eqtr4d 1937	. . . 4 |- (A e. CC -> (((exp` (_i x. A)) + (exp` (-u_i x. A))) + ((exp` (_i x. A)) - (exp` (-u_i x. A)))) = (2 x. (exp` (_i x. A))))
29	28	opreq1d 4897	. . 3 |- (A e. CC -> ((((exp` (_i x. A)) + (exp` (-u_i x. A))) + ((exp` (_i x. A)) - (exp` (-u_i x. A)))) / 2) = ((2 x. (exp` (_i x. A))) / 2))
30		divcan3 6938	. . . . 5 |- (((exp` (_i x. A)) e. CC /\ 2 e. CC /\ 2 =/= 0) -> ((2 x. (exp` (_i x. A))) / 2) = (exp` (_i x. A)))
31	15, 16, 30	mp3an23 1183	. . . 4 |- ((exp` (_i x. A)) e. CC -> ((2 x. (exp` (_i x. A))) / 2) = (exp` (_i x. A)))
32	5, 31	syl 12	. . 3 |- (A e. CC -> ((2 x. (exp` (_i x. A))) / 2) = (exp` (_i x. A)))
33	29, 32	eqtr2d 1926	. 2 |- (A e. CC -> (exp` (_i x. A)) = ((((exp` (_i x. A)) + (exp` (-u_i x. A))) + ((exp` (_i x. A)) - (exp` (-u_i x. A)))) / 2))
34		cosval 8695	. . 3 |- (A e. CC -> (cos` A) = (((exp` (_i x. A)) + (exp` (-u_i x. A))) / 2))
35	15, 1	mulcli 6474	. . . . . . 7 |- (2 x. _i) e. CC
36		ine0 6597	. . . . . . . 8 |- _i =/= 0
37	15, 1, 16, 36	mulne0i 6888	. . . . . . 7 |- (2 x. _i) =/= 0
38	35, 37	pm3.2i 307	. . . . . 6 |- ((2 x. _i) e. CC /\ (2 x. _i) =/= 0)
39		div12 6927	. . . . . 6 |- ((_i e. CC /\ ((exp` (_i x. A)) - (exp` (-u_i x. A))) e. CC /\ ((2 x. _i) e. CC /\ (2 x. _i) =/= 0)) -> (_i x. (((exp` (_i x. A)) - (exp` (-u_i x. A))) / (2 x. _i))) = (((exp` (_i x. A)) - (exp` (-u_i x. A))) x. (_i / (2 x. _i))))
40	1, 38, 39	mp3an13 1182	. . . . 5 |- (((exp` (_i x. A)) - (exp` (-u_i x. A))) e. CC -> (_i x. (((exp` (_i x. A)) - (exp` (-u_i x. A))) / (2 x. _i))) = (((exp` (_i x. A)) - (exp` (-u_i x. A))) x. (_i / (2 x. _i))))
41	14, 40	syl 12	. . . 4 |- (A e. CC -> (_i x. (((exp` (_i x. A)) - (exp` (-u_i x. A))) / (2 x. _i))) = (((exp` (_i x. A)) - (exp` (-u_i x. A))) x. (_i / (2 x. _i))))
42		sinval 8694	. . . . 5 |- (A e. CC -> (sin` A) = (((exp` (_i x. A)) - (exp` (-u_i x. A))) / (2 x. _i)))
43	42	opreq2d 4898	. . . 4 |- (A e. CC -> (_i x. (sin` A)) = (_i x. (((exp` (_i x. A)) - (exp` (-u_i x. A))) / (2 x. _i))))
44		divrec 6922	. . . . . . 7 |- ((((exp` (_i x. A)) - (exp` (-u_i x. A))) e. CC /\ 2 e. CC /\ 2 =/= 0) -> (((exp` (_i x. A)) - (exp` (-u_i x. A))) / 2) = (((exp` (_i x. A)) - (exp` (-u_i x. A))) x. (1 / 2)))
45	15, 16, 44	mp3an23 1183	. . . . . 6 |- (((exp` (_i x. A)) - (exp` (-u_i x. A))) e. CC -> (((exp` (_i x. A)) - (exp` (-u_i x. A))) / 2) = (((exp` (_i x. A)) - (exp` (-u_i x. A))) x. (1 / 2)))
46	14, 45	syl 12	. . . . 5 |- (A e. CC -> (((exp` (_i x. A)) - (exp` (-u_i x. A))) / 2) = (((exp` (_i x. A)) - (exp` (-u_i x. A))) x. (1 / 2)))
47	1	mulid2i 6486	. . . . . . . 8 |- (1 x. _i) = _i
48	47	opreq1i 4892	. . . . . . 7 |- ((1 x. _i) / (2 x. _i)) = (_i / (2 x. _i))
49	1, 36	dividi 6946	. . . . . . . . 9 |- (_i / _i) = 1
50	49	opreq2i 4893	. . . . . . . 8 |- ((1 / 2) x. (_i / _i)) = ((1 / 2) x. 1)
51		ax1cn 6422	. . . . . . . . 9 |- 1 e. CC
52	51, 15, 1, 1, 16, 36	divmuldivi 6963	. . . . . . . 8 |- ((1 / 2) x. (_i / _i)) = ((1 x. _i) / (2 x. _i))
53	15, 16	reccli 6902	. . . . . . . . 9 |- (1 / 2) e. CC
54	53	mulid1i 6485	. . . . . . . 8 |- ((1 / 2) x. 1) = (1 / 2)
55	50, 52, 54	3eqtr3i 1918	. . . . . . 7 |- ((1 x. _i) / (2 x. _i)) = (1 / 2)
56	48, 55	eqtr3i 1910	. . . . . 6 |- (_i / (2 x. _i)) = (1 / 2)
57	56	opreq2i 4893	. . . . 5 |- (((exp` (_i x. A)) - (exp` (-u_i x. A))) x. (_i / (2 x. _i))) = (((exp` (_i x. A)) - (exp` (-u_i x. A))) x. (1 / 2))
58	46, 57	syl6eqr 1946	. . . 4 |- (A e. CC -> (((exp` (_i x. A)) - (exp` (-u_i x. A))) / 2) = (((exp` (_i x. A)) - (exp` (-u_i x. A))) x. (_i / (2 x. _i))))
59	41, 43, 58	3eqtr4d 1937	. . 3 |- (A e. CC -> (_i x. (sin` A)) = (((exp` (_i x. A)) - (exp` (-u_i x. A))) / 2))
60	34, 59	opreq12d 4900	. 2 |- (A e. CC -> ((cos` A) + (_i x. (sin` A))) = ((((exp` (_i x. A)) + (exp` (-u_i x. A))) / 2) + (((exp` (_i x. A)) - (exp` (-u_i x. A))) / 2)))
61	20, 33, 60	3eqtr4d 1937	1 |- (A e. CC -> (exp` (_i x. A)) = ((cos` A) + (_i x. (sin` A))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387  _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:  efmival 8713  efeul 8714  efieq 8715  sinaddi 8716  cosaddi 8717  absefi 8748  demoivreALT 8753  eulerid 10032  efimpi 10047  efifolem7 10082  efif1lem3 10086  efper 10101

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-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-clim 8235  df-sum 8240  df-ef 8560  df-sin 8562  df-cos 8563

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