Theorem efi4p 13732

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 9550	. . . 4 |- _i e. CC
2		mulcl 9575	. . . 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 13708	. . 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 9549	. . . . . 6 |- 1 e. CC
8		addcl 9573	. . . . . 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 12197	. . . . . . 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 10782	. . . . 5 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) e. CC )
13		3nn0 10812	. . . . . . 7 |- 3 e. NN0
14		expcl 12151	. . . . . . 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 10616	. . . . . . 7 |- 6 e. CC
17		6re 10615	. . . . . . . 8 |- 6 e. RR
18		6pos 10633	. . . . . . . 8 |- 0 < 6
19	17, 18	gt0ne0ii 10088	. . . . . . 7 |- 6 =/= 0
20		divcl 10212	. . . . . . 7 |- ( ( ( ( _i x. A ) ^ 3 ) e. CC /\ 6 e. CC /\ 6 =/= 0 ) -> ( ( ( _i x. A ) ^ 3 ) / 6 ) e. CC )
21	16, 19, 20	mp3an23 1316	. . . . . 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 9617	. . . 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 9802	. . . 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 10811	. . . . . . . . . . 11 |- 2 e. NN0
27		mulexp 12172	. . . . . . . . . . 11 |- ( ( _i e. CC /\ A e. CC /\ 2 e. NN0 ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
28	1, 26, 27	mp3an13 1315	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
29		i2 12235	. . . . . . . . . . . 12 |- ( _i ^ 2 ) = -u 1
30	29	oveq1i 6293	. . . . . . . . . . 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 12197	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 2 ) e. CC )
33	32	mulm1d 10007	. . . . . . . . . 10 |- ( A e. CC -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
34	28, 31, 33	3eqtrd 2512	. . . . . . . . 9 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
35	34	oveq1d 6298	. . . . . . . 8 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
36		2cn 10605	. . . . . . . . . 10 |- 2 e. CC
37		2ne0 10627	. . . . . . . . . 10 |- 2 =/= 0
38		divneg 10238	. . . . . . . . . 10 |- ( ( ( A ^ 2 ) e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> -u ( ( A ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
39	36, 37, 38	mp3an23 1316	. . . . . . . . 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 2511	. . . . . . 7 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) = -u ( ( A ^ 2 ) / 2 ) )
42	41	oveq2d 6299	. . . . . 6 |- ( A e. CC -> ( 1 + ( ( ( _i x. A ) ^ 2 ) / 2 ) ) = ( 1 + -u ( ( A ^ 2 ) / 2 ) ) )
43	32	halfcld 10782	. . . . . . 7 |- ( A e. CC -> ( ( A ^ 2 ) / 2 ) e. CC )
44		negsub 9866	. . . . . . 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 2508	. . . . 5 |- ( A e. CC -> ( 1 + ( ( ( _i x. A ) ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
47		mulexp 12172	. . . . . . . . . . 11 |- ( ( _i e. CC /\ A e. CC /\ 3 e. NN0 ) -> ( ( _i x. A ) ^ 3 ) = ( ( _i ^ 3 ) x. ( A ^ 3 ) ) )
48	1, 13, 47	mp3an13 1315	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) = ( ( _i ^ 3 ) x. ( A ^ 3 ) ) )
49		i3 12236	. . . . . . . . . . 11 |- ( _i ^ 3 ) = -u _i
50	49	oveq1i 6293	. . . . . . . . . 10 |- ( ( _i ^ 3 ) x. ( A ^ 3 ) ) = ( -u _i x. ( A ^ 3 ) )
51	48, 50	syl6eq 2524	. . . . . . . . 9 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) = ( -u _i x. ( A ^ 3 ) ) )
52	51	oveq1d 6298	. . . . . . . 8 |- ( A e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) = ( ( -u _i x. ( A ^ 3 ) ) / 6 ) )
53		expcl 12151	. . . . . . . . . 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 9820	. . . . . . . . . 10 |- -u _i e. CC
56	16, 19	pm3.2i 455	. . . . . . . . . 10 |- ( 6 e. CC /\ 6 =/= 0 )
57		divass 10224	. . . . . . . . . 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 1315	. . . . . . . . 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 10212	. . . . . . . . . . 11 |- ( ( ( A ^ 3 ) e. CC /\ 6 e. CC /\ 6 =/= 0 ) -> ( ( A ^ 3 ) / 6 ) e. CC )
61	16, 19, 60	mp3an23 1316	. . . . . . . . . 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 9994	. . . . . . . . 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 2512	. . . . . . 7 |- ( A e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) = ( _i x. -u ( ( A ^ 3 ) / 6 ) ) )
66	65	oveq2d 6299	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) + ( ( ( _i x. A ) ^ 3 ) / 6 ) ) = ( ( _i x. A ) + ( _i x. -u ( ( A ^ 3 ) / 6 ) ) ) )
67	62	negcld 9916	. . . . . . 7 |- ( A e. CC -> -u ( ( A ^ 3 ) / 6 ) e. CC )
68		adddi 9580	. . . . . . . 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 1311	. . . . . . 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 9866	. . . . . . . 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 6299	. . . . . 6 |- ( A e. CC -> ( _i x. ( A + -u ( ( A ^ 3 ) / 6 ) ) ) = ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) )
74	66, 70, 73	3eqtr2d 2514	. . . . 5 |- ( A e. CC -> ( ( _i x. A ) + ( ( ( _i x. A ) ^ 3 ) / 6 ) ) = ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) )
75	46, 74	oveq12d 6301	. . . 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 2512	. . 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 6298	. 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 2508	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 1379   e. wcel 1767   =/= wne 2662   |-> cmpt 4505   ` cfv 5587  (class class class)co 6283   CCcc 9489   0cc0 9491   1c1 9492   _ici 9493   + caddc 9494   x. cmul 9496   - cmin 9804   -ucneg 9805   / cdiv 10205   2c2 10584   3c3 10585   4c4 10586   6c6 10588   NN0cn0 10794   ZZ>=cuz 11081   ^cexp 12133   !cfa 12320   sum_csu 13470   expce 13658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-oi 7934  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-n0 10795  df-z 10864  df-uz 11082  df-rp 11220  df-ico 11534  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-fac 12321  df-hash 12373  df-shft 12862  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-limsup 13256  df-clim 13273  df-rlim 13274  df-sum 13471  df-ef 13664

This theorem is referenced by:  resin4p  13733  recos4p  13734