Theorem efi4p 11424

Description: Separate out the first four terms of the infinite series expansion of the exponential function. (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 7715	. . . 4 |- _i e. CC
2		mulcl 7747	. . . 4 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
3	1, 2	mpan 420	. . 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 11400	. . 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 14	. 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 7713	. . . . . 6 |- 1 e. CC
8		addcl 7745	. . . . . 6 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC )
9	7, 3, 8	sylancr 410	. . . . 5 |- ( A e. CC -> ( 1 + ( _i x. A ) ) e. CC )
10	3	sqcld 10422	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) e. CC )
11	10	halfcld 8964	. . . . 5 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) e. CC )
12		3nn0 8995	. . . . . . 7 |- 3 e. NN0
13		expcl 10311	. . . . . . 7 |- ( ( ( _i x. A ) e. CC /\ 3 e. NN0 ) -> ( ( _i x. A ) ^ 3 ) e. CC )
14	3, 12, 13	sylancl 409	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) e. CC )
15		6cn 8802	. . . . . . 7 |- 6 e. CC
16		6re 8801	. . . . . . . 8 |- 6 e. RR
17		6pos 8821	. . . . . . . 8 |- 0 < 6
18	16, 17	gt0ap0ii 8390	. . . . . . 7 |- 6 # 0
19		divclap 8438	. . . . . . 7 |- ( ( ( ( _i x. A ) ^ 3 ) e. CC /\ 6 e. CC /\ 6 # 0 ) -> ( ( ( _i x. A ) ^ 3 ) / 6 ) e. CC )
20	15, 18, 19	mp3an23 1307	. . . . . 6 |- ( ( ( _i x. A ) ^ 3 ) e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) e. CC )
21	14, 20	syl 14	. . . . 5 |- ( A e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) e. CC )
22	9, 11, 21	addassd 7788	. . . 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 ) ) ) )
23	7	a1i 9	. . . . 5 |- ( A e. CC -> 1 e. CC )
24	23, 3, 11, 21	add4d 7931	. . . 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 ) ) ) )
25		2nn0 8994	. . . . . . . . . . 11 |- 2 e. NN0
26		mulexp 10332	. . . . . . . . . . 11 |- ( ( _i e. CC /\ A e. CC /\ 2 e. NN0 ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
27	1, 25, 26	mp3an13 1306	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
28		i2 10393	. . . . . . . . . . . 12 |- ( _i ^ 2 ) = -u 1
29	28	oveq1i 5784	. . . . . . . . . . 11 |- ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) )
30	29	a1i 9	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) ) )
31		sqcl 10354	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 2 ) e. CC )
32	31	mulm1d 8172	. . . . . . . . . 10 |- ( A e. CC -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
33	27, 30, 32	3eqtrd 2176	. . . . . . . . 9 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
34	33	oveq1d 5789	. . . . . . . 8 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
35		2cn 8791	. . . . . . . . . 10 |- 2 e. CC
36		2ap0 8813	. . . . . . . . . 10 |- 2 # 0
37		divnegap 8466	. . . . . . . . . 10 |- ( ( ( A ^ 2 ) e. CC /\ 2 e. CC /\ 2 # 0 ) -> -u ( ( A ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
38	35, 36, 37	mp3an23 1307	. . . . . . . . 9 |- ( ( A ^ 2 ) e. CC -> -u ( ( A ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
39	31, 38	syl 14	. . . . . . . 8 |- ( A e. CC -> -u ( ( A ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
40	34, 39	eqtr4d 2175	. . . . . . 7 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) = -u ( ( A ^ 2 ) / 2 ) )
41	40	oveq2d 5790	. . . . . 6 |- ( A e. CC -> ( 1 + ( ( ( _i x. A ) ^ 2 ) / 2 ) ) = ( 1 + -u ( ( A ^ 2 ) / 2 ) ) )
42	31	halfcld 8964	. . . . . . 7 |- ( A e. CC -> ( ( A ^ 2 ) / 2 ) e. CC )
43		negsub 8010	. . . . . . 7 |- ( ( 1 e. CC /\ ( ( A ^ 2 ) / 2 ) e. CC ) -> ( 1 + -u ( ( A ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
44	7, 42, 43	sylancr 410	. . . . . 6 |- ( A e. CC -> ( 1 + -u ( ( A ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
45	41, 44	eqtrd 2172	. . . . 5 |- ( A e. CC -> ( 1 + ( ( ( _i x. A ) ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
46		mulexp 10332	. . . . . . . . . . 11 |- ( ( _i e. CC /\ A e. CC /\ 3 e. NN0 ) -> ( ( _i x. A ) ^ 3 ) = ( ( _i ^ 3 ) x. ( A ^ 3 ) ) )
47	1, 12, 46	mp3an13 1306	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) = ( ( _i ^ 3 ) x. ( A ^ 3 ) ) )
48		i3 10394	. . . . . . . . . . 11 |- ( _i ^ 3 ) = -u _i
49	48	oveq1i 5784	. . . . . . . . . 10 |- ( ( _i ^ 3 ) x. ( A ^ 3 ) ) = ( -u _i x. ( A ^ 3 ) )
50	47, 49	syl6eq 2188	. . . . . . . . 9 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) = ( -u _i x. ( A ^ 3 ) ) )
51	50	oveq1d 5789	. . . . . . . 8 |- ( A e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) = ( ( -u _i x. ( A ^ 3 ) ) / 6 ) )
52		expcl 10311	. . . . . . . . . 10 |- ( ( A e. CC /\ 3 e. NN0 ) -> ( A ^ 3 ) e. CC )
53	12, 52	mpan2 421	. . . . . . . . 9 |- ( A e. CC -> ( A ^ 3 ) e. CC )
54		negicn 7963	. . . . . . . . . 10 |- -u _i e. CC
55	15, 18	pm3.2i 270	. . . . . . . . . 10 |- ( 6 e. CC /\ 6 # 0 )
56		divassap 8450	. . . . . . . . . 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 ) ) )
57	54, 55, 56	mp3an13 1306	. . . . . . . . 9 |- ( ( A ^ 3 ) e. CC -> ( ( -u _i x. ( A ^ 3 ) ) / 6 ) = ( -u _i x. ( ( A ^ 3 ) / 6 ) ) )
58	53, 57	syl 14	. . . . . . . 8 |- ( A e. CC -> ( ( -u _i x. ( A ^ 3 ) ) / 6 ) = ( -u _i x. ( ( A ^ 3 ) / 6 ) ) )
59		divclap 8438	. . . . . . . . . . 11 |- ( ( ( A ^ 3 ) e. CC /\ 6 e. CC /\ 6 # 0 ) -> ( ( A ^ 3 ) / 6 ) e. CC )
60	15, 18, 59	mp3an23 1307	. . . . . . . . . 10 |- ( ( A ^ 3 ) e. CC -> ( ( A ^ 3 ) / 6 ) e. CC )
61	53, 60	syl 14	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^ 3 ) / 6 ) e. CC )
62		mulneg12 8159	. . . . . . . . 9 |- ( ( _i e. CC /\ ( ( A ^ 3 ) / 6 ) e. CC ) -> ( -u _i x. ( ( A ^ 3 ) / 6 ) ) = ( _i x. -u ( ( A ^ 3 ) / 6 ) ) )
63	1, 61, 62	sylancr 410	. . . . . . . 8 |- ( A e. CC -> ( -u _i x. ( ( A ^ 3 ) / 6 ) ) = ( _i x. -u ( ( A ^ 3 ) / 6 ) ) )
64	51, 58, 63	3eqtrd 2176	. . . . . . 7 |- ( A e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) = ( _i x. -u ( ( A ^ 3 ) / 6 ) ) )
65	64	oveq2d 5790	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) + ( ( ( _i x. A ) ^ 3 ) / 6 ) ) = ( ( _i x. A ) + ( _i x. -u ( ( A ^ 3 ) / 6 ) ) ) )
66	61	negcld 8060	. . . . . . 7 |- ( A e. CC -> -u ( ( A ^ 3 ) / 6 ) e. CC )
67		adddi 7752	. . . . . . . 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 ) ) ) )
68	1, 67	mp3an1 1302	. . . . . . 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 ) ) ) )
69	66, 68	mpdan 417	. . . . . 6 |- ( A e. CC -> ( _i x. ( A + -u ( ( A ^ 3 ) / 6 ) ) ) = ( ( _i x. A ) + ( _i x. -u ( ( A ^ 3 ) / 6 ) ) ) )
70		negsub 8010	. . . . . . . 8 |- ( ( A e. CC /\ ( ( A ^ 3 ) / 6 ) e. CC ) -> ( A + -u ( ( A ^ 3 ) / 6 ) ) = ( A - ( ( A ^ 3 ) / 6 ) ) )
71	61, 70	mpdan 417	. . . . . . 7 |- ( A e. CC -> ( A + -u ( ( A ^ 3 ) / 6 ) ) = ( A - ( ( A ^ 3 ) / 6 ) ) )
72	71	oveq2d 5790	. . . . . 6 |- ( A e. CC -> ( _i x. ( A + -u ( ( A ^ 3 ) / 6 ) ) ) = ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) )
73	65, 69, 72	3eqtr2d 2178	. . . . 5 |- ( A e. CC -> ( ( _i x. A ) + ( ( ( _i x. A ) ^ 3 ) / 6 ) ) = ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) )
74	45, 73	oveq12d 5792	. . . 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 ) ) ) ) )
75	22, 24, 74	3eqtrd 2176	. . 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 ) ) ) ) )
76	75	oveq1d 5789	. 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 ) ) )
77	6, 76	eqtrd 2172	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 103   = wceq 1331   e. wcel 1480   class class class wbr 3929   |-> cmpt 3989   ` cfv 5123  (class class class)co 5774   CCcc 7618   0cc0 7620   1c1 7621   _ici 7622   + caddc 7623   x. cmul 7625   - cmin 7933   -ucneg 7934   # cap 8343   / cdiv 8432   2c2 8771   3c3 8772   4c4 8773   6c6 8775   NN0cn0 8977   ZZ>=cuz 9326   ^cexp 10292   !cfa 10471   sum_csu 11122   expce 11348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-5 8782  df-6 8783  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-ico 9677  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-fac 10472  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123  df-ef 11354

This theorem is referenced by:  resin4p  11425  recos4p  11426