Theorem efi4p 8700

Description: Separate out the first four terms of the infinite series expansion of the exponential function of a pure imaginary number. (Contributed by Paul Chapman, 19-Jan-2008.)

Hypothesis

Ref	Expression
efit4pt.1	|- F = {<.j, y>. | (j e. NN0 /\ y = (((_i x. A)^j) / (!` j)))}

Assertion

Ref	Expression
efi4p	|- (A e. RR -> (exp` (_i x. A)) = (((1 - ((A^2) / 2)) + (_i x. (A - ((A^3) / 6)))) + sum_k e. (ZZ>=` 4)(F` k)))

Distinct variable groups:   A,j,k,y   k,F

Proof of Theorem efi4p

Step	Hyp	Ref	Expression
1		mulcl 6456	. . . 4 |- ((_i e. CC /\ A e. CC) -> (_i x. A) e. CC)
2		axicn 6423	. . . 4 |- _i e. CC
3		recn 6466	. . . 4 |- (A e. RR -> A e. CC)
4	1, 2, 3	sylancr 526	. . 3 |- (A e. RR -> (_i x. A) e. CC)
5		efit4pt.1	. . . 4 |- F = {<.j, y>. | (j e. NN0 /\ y = (((_i x. A)^j) / (!` j)))}
6	5	ef4p 8665	. . 3 |- ((_i x. A) e. CC -> (exp` (_i x. A)) = ((((1 + (_i x. A)) + (((_i x. A)^2) / 2)) + (((_i x. A)^3) / 6)) + sum_k e. (ZZ>=` 4)(F` k)))
7	4, 6	syl 12	. 2 |- (A e. RR -> (exp` (_i x. A)) = ((((1 + (_i x. A)) + (((_i x. A)^2) / 2)) + (((_i x. A)^3) / 6)) + sum_k e. (ZZ>=` 4)(F` k)))
8		addcl 6454	. . . . . 6 |- ((1 e. CC /\ (_i x. A) e. CC) -> (1 + (_i x. A)) e. CC)
9		ax1cn 6422	. . . . . 6 |- 1 e. CC
10	8, 9, 4	sylancr 526	. . . . 5 |- (A e. RR -> (1 + (_i x. A)) e. CC)
11		sqcl 7856	. . . . . . 7 |- ((_i x. A) e. CC -> ((_i x. A)^2) e. CC)
12	4, 11	syl 12	. . . . . 6 |- (A e. RR -> ((_i x. A)^2) e. CC)
13		2cn 7164	. . . . . . 7 |- 2 e. CC
14		2ne0 7174	. . . . . . 7 |- 2 =/= 0
15		divcl 6901	. . . . . . 7 |- ((((_i x. A)^2) e. CC /\ 2 e. CC /\ 2 =/= 0) -> (((_i x. A)^2) / 2) e. CC)
16	13, 14, 15	mp3an23 1183	. . . . . 6 |- (((_i x. A)^2) e. CC -> (((_i x. A)^2) / 2) e. CC)
17	12, 16	syl 12	. . . . 5 |- (A e. RR -> (((_i x. A)^2) / 2) e. CC)
18		expcl 7824	. . . . . . 7 |- (((_i x. A) e. CC /\ 3 e. NN0) -> ((_i x. A)^3) e. CC)
19		3nn 7184	. . . . . . . 8 |- 3 e. NN
20	19	nnnn0i 7316	. . . . . . 7 |- 3 e. NN0
21	18, 4, 20	sylancl 525	. . . . . 6 |- (A e. RR -> ((_i x. A)^3) e. CC)
22		6re 7168	. . . . . . . 8 |- 6 e. RR
23	22	recni 6467	. . . . . . 7 |- 6 e. CC
24		6pos 7178	. . . . . . . 8 |- 0 < 6
25	22, 24	gt0ne0ii 6799	. . . . . . 7 |- 6 =/= 0
26		divcl 6901	. . . . . . 7 |- ((((_i x. A)^3) e. CC /\ 6 e. CC /\ 6 =/= 0) -> (((_i x. A)^3) / 6) e. CC)
27	23, 25, 26	mp3an23 1183	. . . . . 6 |- (((_i x. A)^3) e. CC -> (((_i x. A)^3) / 6) e. CC)
28	21, 27	syl 12	. . . . 5 |- (A e. RR -> (((_i x. A)^3) / 6) e. CC)
29		addass 6460	. . . . 5 |- (((1 + (_i x. A)) e. CC /\ (((_i x. A)^2) / 2) e. CC /\ (((_i x. A)^3) / 6) e. CC) -> (((1 + (_i x. A)) + (((_i x. A)^2) / 2)) + (((_i x. A)^3) / 6)) = ((1 + (_i x. A)) + ((((_i x. A)^2) / 2) + (((_i x. A)^3) / 6))))
30	10, 17, 28, 29	syl111anc 1100	. . . 4 |- (A e. RR -> (((1 + (_i x. A)) + (((_i x. A)^2) / 2)) + (((_i x. A)^3) / 6)) = ((1 + (_i x. A)) + ((((_i x. A)^2) / 2) + (((_i x. A)^3) / 6))))
31	9	a1i 8	. . . . 5 |- (A e. RR -> 1 e. CC)
32		add4 6491	. . . . 5 |- (((1 e. CC /\ (_i x. A) e. CC) /\ ((((_i x. A)^2) / 2) e. CC /\ (((_i x. A)^3) / 6) e. CC)) -> ((1 + (_i x. A)) + ((((_i x. A)^2) / 2) + (((_i x. A)^3) / 6))) = ((1 + (((_i x. A)^2) / 2)) + ((_i x. A) + (((_i x. A)^3) / 6))))
33	31, 4, 17, 28, 32	syl22anc 1101	. . . 4 |- (A e. RR -> ((1 + (_i x. A)) + ((((_i x. A)^2) / 2) + (((_i x. A)^3) / 6))) = ((1 + (((_i x. A)^2) / 2)) + ((_i x. A) + (((_i x. A)^3) / 6))))
34		2nn0 7324	. . . . . . . . . . . 12 |- 2 e. NN0
35		mulexp 7836	. . . . . . . . . . . 12 |- ((_i e. CC /\ A e. CC /\ 2 e. NN0) -> ((_i x. A)^2) = ((_i^2) x. (A^2)))
36	2, 34, 35	mp3an13 1182	. . . . . . . . . . 11 |- (A e. CC -> ((_i x. A)^2) = ((_i^2) x. (A^2)))
37	3, 36	syl 12	. . . . . . . . . 10 |- (A e. RR -> ((_i x. A)^2) = ((_i^2) x. (A^2)))
38		i2 7982	. . . . . . . . . . . 12 |- (_i^2) = -u1
39	38	opreq1i 4892	. . . . . . . . . . 11 |- ((_i^2) x. (A^2)) = (-u1 x. (A^2))
40	39	a1i 8	. . . . . . . . . 10 |- (A e. RR -> ((_i^2) x. (A^2)) = (-u1 x. (A^2)))
41		sqcl 7856	. . . . . . . . . . . 12 |- (A e. CC -> (A^2) e. CC)
42	3, 41	syl 12	. . . . . . . . . . 11 |- (A e. RR -> (A^2) e. CC)
43		mulm1 6638	. . . . . . . . . . 11 |- ((A^2) e. CC -> (-u1 x. (A^2)) = -u(A^2))
44	42, 43	syl 12	. . . . . . . . . 10 |- (A e. RR -> (-u1 x. (A^2)) = -u(A^2))
45	37, 40, 44	3eqtrd 1929	. . . . . . . . 9 |- (A e. RR -> ((_i x. A)^2) = -u(A^2))
46	45	opreq1d 4897	. . . . . . . 8 |- (A e. RR -> (((_i x. A)^2) / 2) = (-u(A^2) / 2))
47		divneg 6950	. . . . . . . . . 10 |- (((A^2) e. CC /\ 2 e. CC /\ 2 =/= 0) -> -u((A^2) / 2) = (-u(A^2) / 2))
48	13, 14, 47	mp3an23 1183	. . . . . . . . 9 |- ((A^2) e. CC -> -u((A^2) / 2) = (-u(A^2) / 2))
49	42, 48	syl 12	. . . . . . . 8 |- (A e. RR -> -u((A^2) / 2) = (-u(A^2) / 2))
50	46, 49	eqtr4d 1928	. . . . . . 7 |- (A e. RR -> (((_i x. A)^2) / 2) = -u((A^2) / 2))
51	50	opreq2d 4898	. . . . . 6 |- (A e. RR -> (1 + (((_i x. A)^2) / 2)) = (1 + -u((A^2) / 2)))
52		negsub 6540	. . . . . . 7 |- ((1 e. CC /\ ((A^2) / 2) e. CC) -> (1 + -u((A^2) / 2)) = (1 - ((A^2) / 2)))
53		divcl 6901	. . . . . . . . 9 |- (((A^2) e. CC /\ 2 e. CC /\ 2 =/= 0) -> ((A^2) / 2) e. CC)
54	13, 14, 53	mp3an23 1183	. . . . . . . 8 |- ((A^2) e. CC -> ((A^2) / 2) e. CC)
55	42, 54	syl 12	. . . . . . 7 |- (A e. RR -> ((A^2) / 2) e. CC)
56	52, 9, 55	sylancr 526	. . . . . 6 |- (A e. RR -> (1 + -u((A^2) / 2)) = (1 - ((A^2) / 2)))
57	51, 56	eqtrd 1925	. . . . 5 |- (A e. RR -> (1 + (((_i x. A)^2) / 2)) = (1 - ((A^2) / 2)))
58		mulexp 7836	. . . . . . . . . . . 12 |- ((_i e. CC /\ A e. CC /\ 3 e. NN0) -> ((_i x. A)^3) = ((_i^3) x. (A^3)))
59	2, 20, 58	mp3an13 1182	. . . . . . . . . . 11 |- (A e. CC -> ((_i x. A)^3) = ((_i^3) x. (A^3)))
60	3, 59	syl 12	. . . . . . . . . 10 |- (A e. RR -> ((_i x. A)^3) = ((_i^3) x. (A^3)))
61		i3 7983	. . . . . . . . . . . 12 |- (_i^3) = -u_i
62	61	opreq1i 4892	. . . . . . . . . . 11 |- ((_i^3) x. (A^3)) = (-u_i x. (A^3))
63	62	a1i 8	. . . . . . . . . 10 |- (A e. RR -> ((_i^3) x. (A^3)) = (-u_i x. (A^3)))
64	60, 63	eqtrd 1925	. . . . . . . . 9 |- (A e. RR -> ((_i x. A)^3) = (-u_i x. (A^3)))
65	64	opreq1d 4897	. . . . . . . 8 |- (A e. RR -> (((_i x. A)^3) / 6) = ((-u_i x. (A^3)) / 6))
66		expcl 7824	. . . . . . . . . 10 |- ((A e. CC /\ 3 e. NN0) -> (A^3) e. CC)
67	66, 3, 20	sylancl 525	. . . . . . . . 9 |- (A e. RR -> (A^3) e. CC)
68	2	negcli 6526	. . . . . . . . . 10 |- -u_i e. CC
69	23, 25	pm3.2i 307	. . . . . . . . . 10 |- (6 e. CC /\ 6 =/= 0)
70		divass 6924	. . . . . . . . . 10 |- ((-u_i e. CC /\ (A^3) e. CC /\ (6 e. CC /\ 6 =/= 0)) -> ((-u_i x. (A^3)) / 6) = (-u_i x. ((A^3) / 6)))
71	68, 69, 70	mp3an13 1182	. . . . . . . . 9 |- ((A^3) e. CC -> ((-u_i x. (A^3)) / 6) = (-u_i x. ((A^3) / 6)))
72	67, 71	syl 12	. . . . . . . 8 |- (A e. RR -> ((-u_i x. (A^3)) / 6) = (-u_i x. ((A^3) / 6)))
73		mulneg12 6617	. . . . . . . . 9 |- ((_i e. CC /\ ((A^3) / 6) e. CC) -> (-u_i x. ((A^3) / 6)) = (_i x. -u((A^3) / 6)))
74		divcl 6901	. . . . . . . . . . 11 |- (((A^3) e. CC /\ 6 e. CC /\ 6 =/= 0) -> ((A^3) / 6) e. CC)
75	23, 25, 74	mp3an23 1183	. . . . . . . . . 10 |- ((A^3) e. CC -> ((A^3) / 6) e. CC)
76	67, 75	syl 12	. . . . . . . . 9 |- (A e. RR -> ((A^3) / 6) e. CC)
77	73, 2, 76	sylancr 526	. . . . . . . 8 |- (A e. RR -> (-u_i x. ((A^3) / 6)) = (_i x. -u((A^3) / 6)))
78	65, 72, 77	3eqtrd 1929	. . . . . . 7 |- (A e. RR -> (((_i x. A)^3) / 6) = (_i x. -u((A^3) / 6)))
79	78	opreq2d 4898	. . . . . 6 |- (A e. RR -> ((_i x. A) + (((_i x. A)^3) / 6)) = ((_i x. A) + (_i x. -u((A^3) / 6))))
80		negcl 6525	. . . . . . . 8 |- (((A^3) / 6) e. CC -> -u((A^3) / 6) e. CC)
81	76, 80	syl 12	. . . . . . 7 |- (A e. RR -> -u((A^3) / 6) e. CC)
82		adddi 6462	. . . . . . . 8 |- ((_i e. CC /\ A e. CC /\ -u((A^3) / 6) e. CC) -> (_i x. (A + -u((A^3) / 6))) = ((_i x. A) + (_i x. -u((A^3) / 6))))
83	2, 82	mp3an1 1178	. . . . . . 7 |- ((A e. CC /\ -u((A^3) / 6) e. CC) -> (_i x. (A + -u((A^3) / 6))) = ((_i x. A) + (_i x. -u((A^3) / 6))))
84	3, 81, 83	syl11anc 524	. . . . . 6 |- (A e. RR -> (_i x. (A + -u((A^3) / 6))) = ((_i x. A) + (_i x. -u((A^3) / 6))))
85		negsub 6540	. . . . . . . 8 |- ((A e. CC /\ ((A^3) / 6) e. CC) -> (A + -u((A^3) / 6)) = (A - ((A^3) / 6)))
86	3, 76, 85	syl11anc 524	. . . . . . 7 |- (A e. RR -> (A + -u((A^3) / 6)) = (A - ((A^3) / 6)))
87	86	opreq2d 4898	. . . . . 6 |- (A e. RR -> (_i x. (A + -u((A^3) / 6))) = (_i x. (A - ((A^3) / 6))))
88	79, 84, 87	3eqtr2d 1932	. . . . 5 |- (A e. RR -> ((_i x. A) + (((_i x. A)^3) / 6)) = (_i x. (A - ((A^3) / 6))))
89	57, 88	opreq12d 4900	. . . 4 |- (A e. RR -> ((1 + (((_i x. A)^2) / 2)) + ((_i x. A) + (((_i x. A)^3) / 6))) = ((1 - ((A^2) / 2)) + (_i x. (A - ((A^3) / 6)))))
90	30, 33, 89	3eqtrd 1929	. . 3 |- (A e. RR -> (((1 + (_i x. A)) + (((_i x. A)^2) / 2)) + (((_i x. A)^3) / 6)) = ((1 - ((A^2) / 2)) + (_i x. (A - ((A^3) / 6)))))
91	90	opreq1d 4897	. 2 |- (A e. RR -> ((((1 + (_i x. A)) + (((_i x. A)^2) / 2)) + (((_i x. A)^3) / 6)) + sum_k e. (ZZ>=` 4)(F` k)) = (((1 - ((A^2) / 2)) + (_i x. (A - ((A^3) / 6)))) + sum_k e. (ZZ>=` 4)(F` k)))
92	7, 91	eqtrd 1925	1 |- (A e. RR -> (exp` (_i x. A)) = (((1 - ((A^2) / 2)) + (_i x. (A - ((A^3) / 6)))) + sum_k e. (ZZ>=` 4)(F` k)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  {copab 3395  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447  NN0cn0 6450  2c2 7145  3c3 7146  4c4 7147  6c6 7149  ZZ>=cuz 7586  ^cexp 7811  !cfa 8183  sum_csu 8239  expce 8555

This theorem is referenced by:  resin4p 8701  recos4p 8702

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-5 7157  df-6 7158  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

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