Theorem efi4p 13413

Description: Separate out the first four terms of the infinite series expansion of the exponential function of an imaginary number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Hypothesis

Ref	Expression
efi4p.1	|- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )

Assertion

Ref	Expression
efi4p	|- ( A e. CC -> ( 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, k, n   k, F

Allowed substitution hint:   F( n)

Proof of Theorem efi4p

Step	Hyp	Ref	Expression
1		ax-icn 9333	. . . 4 |- _i e. CC
2		mulcl 9358	. . . 4 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
3	1, 2	mpan 670	. . 3 |- ( A e. CC -> ( _i x. A ) e. CC )
4		efi4p.1	. . . 4 |- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )
5	4	ef4p 13389	. . 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 ) ) )
6	3, 5	syl 16	. 2 |- ( 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		ax-1cn 9332	. . . . . 6 |- 1 e. CC
8		addcl 9356	. . . . . 6 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC )
9	7, 3, 8	sylancr 663	. . . . 5 |- ( A e. CC -> ( 1 + ( _i x. A ) ) e. CC )
10		sqcl 11920	. . . . . . 7 |- ( ( _i x. A ) e. CC -> ( ( _i x. A ) ^ 2 ) e. CC )
11	3, 10	syl 16	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) e. CC )
12	11	halfcld 10561	. . . . 5 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) e. CC )
13		3nn0 10589	. . . . . . 7 |- 3 e. NN0
14		expcl 11875	. . . . . . 7 |- ( ( ( _i x. A ) e. CC /\ 3 e. NN0 ) -> ( ( _i x. A ) ^ 3 ) e. CC )
15	3, 13, 14	sylancl 662	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) e. CC )
16		6cn 10395	. . . . . . 7 |- 6 e. CC
17		6re 10394	. . . . . . . 8 |- 6 e. RR
18		6pos 10412	. . . . . . . 8 |- 0 < 6
19	17, 18	gt0ne0ii 9868	. . . . . . 7 |- 6 =/= 0
20		divcl 9992	. . . . . . 7 |- ( ( ( ( _i x. A ) ^ 3 ) e. CC /\ 6 e. CC /\ 6 =/= 0 ) -> ( ( ( _i x. A ) ^ 3 ) / 6 ) e. CC )
21	16, 19, 20	mp3an23 1306	. . . . . 6 |- ( ( ( _i x. A ) ^ 3 ) e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) e. CC )
22	15, 21	syl 16	. . . . 5 |- ( A e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) e. CC )
23	9, 12, 22	addassd 9400	. . . 4 |- ( A 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 ) ) ) )
24	7	a1i 11	. . . . 5 |- ( A e. CC -> 1 e. CC )
25	24, 3, 12, 22	add4d 9585	. . . 4 |- ( A 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 ) ) ) )
26		2nn0 10588	. . . . . . . . . . 11 |- 2 e. NN0
27		mulexp 11895	. . . . . . . . . . 11 |- ( ( _i e. CC /\ A e. CC /\ 2 e. NN0 ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
28	1, 26, 27	mp3an13 1305	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
29		i2 11958	. . . . . . . . . . . 12 |- ( _i ^ 2 ) = -u 1
30	29	oveq1i 6096	. . . . . . . . . . 11 |- ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) )
31	30	a1i 11	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) ) )
32		sqcl 11920	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 2 ) e. CC )
33	32	mulm1d 9788	. . . . . . . . . 10 |- ( A e. CC -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
34	28, 31, 33	3eqtrd 2474	. . . . . . . . 9 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
35	34	oveq1d 6101	. . . . . . . 8 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
36		2cn 10384	. . . . . . . . . 10 |- 2 e. CC
37		2ne0 10406	. . . . . . . . . 10 |- 2 =/= 0
38		divneg 10018	. . . . . . . . . 10 |- ( ( ( A ^ 2 ) e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> -u ( ( A ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
39	36, 37, 38	mp3an23 1306	. . . . . . . . 9 |- ( ( A ^ 2 ) e. CC -> -u ( ( A ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
40	32, 39	syl 16	. . . . . . . 8 |- ( A e. CC -> -u ( ( A ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
41	35, 40	eqtr4d 2473	. . . . . . 7 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) = -u ( ( A ^ 2 ) / 2 ) )
42	41	oveq2d 6102	. . . . . 6 |- ( A e. CC -> ( 1 + ( ( ( _i x. A ) ^ 2 ) / 2 ) ) = ( 1 + -u ( ( A ^ 2 ) / 2 ) ) )
43	32	halfcld 10561	. . . . . . 7 |- ( A e. CC -> ( ( A ^ 2 ) / 2 ) e. CC )
44		negsub 9649	. . . . . . 7 |- ( ( 1 e. CC /\ ( ( A ^ 2 ) / 2 ) e. CC ) -> ( 1 + -u ( ( A ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
45	7, 43, 44	sylancr 663	. . . . . 6 |- ( A e. CC -> ( 1 + -u ( ( A ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
46	42, 45	eqtrd 2470	. . . . 5 |- ( A e. CC -> ( 1 + ( ( ( _i x. A ) ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
47		mulexp 11895	. . . . . . . . . . 11 |- ( ( _i e. CC /\ A e. CC /\ 3 e. NN0 ) -> ( ( _i x. A ) ^ 3 ) = ( ( _i ^ 3 ) x. ( A ^ 3 ) ) )
48	1, 13, 47	mp3an13 1305	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) = ( ( _i ^ 3 ) x. ( A ^ 3 ) ) )
49		i3 11959	. . . . . . . . . . 11 |- ( _i ^ 3 ) = -u _i
50	49	oveq1i 6096	. . . . . . . . . 10 |- ( ( _i ^ 3 ) x. ( A ^ 3 ) ) = ( -u _i x. ( A ^ 3 ) )
51	48, 50	syl6eq 2486	. . . . . . . . 9 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) = ( -u _i x. ( A ^ 3 ) ) )
52	51	oveq1d 6101	. . . . . . . 8 |- ( A e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) = ( ( -u _i x. ( A ^ 3 ) ) / 6 ) )
53		expcl 11875	. . . . . . . . . 10 |- ( ( A e. CC /\ 3 e. NN0 ) -> ( A ^ 3 ) e. CC )
54	13, 53	mpan2 671	. . . . . . . . 9 |- ( A e. CC -> ( A ^ 3 ) e. CC )
55		negicn 9603	. . . . . . . . . 10 |- -u _i e. CC
56	16, 19	pm3.2i 455	. . . . . . . . . 10 |- ( 6 e. CC /\ 6 =/= 0 )
57		divass 10004	. . . . . . . . . 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 ) ) )
58	55, 56, 57	mp3an13 1305	. . . . . . . . 9 |- ( ( A ^ 3 ) e. CC -> ( ( -u _i x. ( A ^ 3 ) ) / 6 ) = ( -u _i x. ( ( A ^ 3 ) / 6 ) ) )
59	54, 58	syl 16	. . . . . . . 8 |- ( A e. CC -> ( ( -u _i x. ( A ^ 3 ) ) / 6 ) = ( -u _i x. ( ( A ^ 3 ) / 6 ) ) )
60		divcl 9992	. . . . . . . . . . 11 |- ( ( ( A ^ 3 ) e. CC /\ 6 e. CC /\ 6 =/= 0 ) -> ( ( A ^ 3 ) / 6 ) e. CC )
61	16, 19, 60	mp3an23 1306	. . . . . . . . . 10 |- ( ( A ^ 3 ) e. CC -> ( ( A ^ 3 ) / 6 ) e. CC )
62	54, 61	syl 16	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^ 3 ) / 6 ) e. CC )
63		mulneg12 9775	. . . . . . . . 9 |- ( ( _i e. CC /\ ( ( A ^ 3 ) / 6 ) e. CC ) -> ( -u _i x. ( ( A ^ 3 ) / 6 ) ) = ( _i x. -u ( ( A ^ 3 ) / 6 ) ) )
64	1, 62, 63	sylancr 663	. . . . . . . 8 |- ( A e. CC -> ( -u _i x. ( ( A ^ 3 ) / 6 ) ) = ( _i x. -u ( ( A ^ 3 ) / 6 ) ) )
65	52, 59, 64	3eqtrd 2474	. . . . . . 7 |- ( A e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) = ( _i x. -u ( ( A ^ 3 ) / 6 ) ) )
66	65	oveq2d 6102	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) + ( ( ( _i x. A ) ^ 3 ) / 6 ) ) = ( ( _i x. A ) + ( _i x. -u ( ( A ^ 3 ) / 6 ) ) ) )
67	62	negcld 9698	. . . . . . 7 |- ( A e. CC -> -u ( ( A ^ 3 ) / 6 ) e. CC )
68		adddi 9363	. . . . . . . 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 ) ) ) )
69	1, 68	mp3an1 1301	. . . . . . 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 ) ) ) )
70	67, 69	mpdan 668	. . . . . 6 |- ( A e. CC -> ( _i x. ( A + -u ( ( A ^ 3 ) / 6 ) ) ) = ( ( _i x. A ) + ( _i x. -u ( ( A ^ 3 ) / 6 ) ) ) )
71		negsub 9649	. . . . . . . 8 |- ( ( A e. CC /\ ( ( A ^ 3 ) / 6 ) e. CC ) -> ( A + -u ( ( A ^ 3 ) / 6 ) ) = ( A - ( ( A ^ 3 ) / 6 ) ) )
72	62, 71	mpdan 668	. . . . . . 7 |- ( A e. CC -> ( A + -u ( ( A ^ 3 ) / 6 ) ) = ( A - ( ( A ^ 3 ) / 6 ) ) )
73	72	oveq2d 6102	. . . . . 6 |- ( A e. CC -> ( _i x. ( A + -u ( ( A ^ 3 ) / 6 ) ) ) = ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) )
74	66, 70, 73	3eqtr2d 2476	. . . . 5 |- ( A e. CC -> ( ( _i x. A ) + ( ( ( _i x. A ) ^ 3 ) / 6 ) ) = ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) )
75	46, 74	oveq12d 6104	. . . 4 |- ( A e. CC -> ( ( 1 + ( ( ( _i x. A ) ^ 2 ) / 2 ) ) + ( ( _i x. A ) + ( ( ( _i x. A ) ^ 3 ) / 6 ) ) ) = ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) )
76	23, 25, 75	3eqtrd 2474	. . 3 |- ( A e. CC -> ( ( ( 1 + ( _i x. A ) ) + ( ( ( _i x. A ) ^ 2 ) / 2 ) ) + ( ( ( _i x. A ) ^ 3 ) / 6 ) ) = ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) )
77	76	oveq1d 6101	. 2 |- ( A e. CC -> ( ( ( ( 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 ) ) )
78	6, 77	eqtrd 2470	1 |- ( A e. CC -> ( exp ` ( _i x. A ) ) = ( ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) + sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2601   e. cmpt 4345   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274   1c1 9275   _ici 9276   + caddc 9277   x. cmul 9279   - cmin 9587   -ucneg 9588   / cdiv 9985   2c2 10363   3c3 10364   4c4 10365   6c6 10367   NN0cn0 10571   ZZ>=cuz 10853   ^cexp 11857   !cfa 12043   sum_csu 13155   expce 13339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-ico 11298  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-fac 12044  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345

This theorem is referenced by:  resin4p  13414  recos4p  13415