Theorem efi4p 11069

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 7501	. . . 4 |- _i e. CC
2		mulcl 7530	. . . 4 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
3	1, 2	mpan 416	. . 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 11045	. . 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 7499	. . . . . 6 |- 1 e. CC
8		addcl 7528	. . . . . 6 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC )
9	7, 3, 8	sylancr 406	. . . . 5 |- ( A e. CC -> ( 1 + ( _i x. A ) ) e. CC )
10	3	sqcld 10145	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) e. CC )
11	10	halfcld 8721	. . . . 5 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) e. CC )
12		3nn0 8752	. . . . . . 7 |- 3 e. NN0
13		expcl 10034	. . . . . . 7 |- ( ( ( _i x. A ) e. CC /\ 3 e. NN0 ) -> ( ( _i x. A ) ^ 3 ) e. CC )
14	3, 12, 13	sylancl 405	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) e. CC )
15		6cn 8565	. . . . . . 7 |- 6 e. CC
16		6re 8564	. . . . . . . 8 |- 6 e. RR
17		6pos 8584	. . . . . . . 8 |- 0 < 6
18	16, 17	gt0ap0ii 8165	. . . . . . 7 |- 6 # 0
19		divclap 8206	. . . . . . 7 |- ( ( ( ( _i x. A ) ^ 3 ) e. CC /\ 6 e. CC /\ 6 # 0 ) -> ( ( ( _i x. A ) ^ 3 ) / 6 ) e. CC )
20	15, 18, 19	mp3an23 1266	. . . . . 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 7571	. . . 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 7712	. . . 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 8751	. . . . . . . . . . 11 |- 2 e. NN0
26		mulexp 10055	. . . . . . . . . . 11 |- ( ( _i e. CC /\ A e. CC /\ 2 e. NN0 ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
27	1, 25, 26	mp3an13 1265	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
28		i2 10116	. . . . . . . . . . . 12 |- ( _i ^ 2 ) = -u 1
29	28	oveq1i 5676	. . . . . . . . . . 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 10077	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 2 ) e. CC )
32	31	mulm1d 7949	. . . . . . . . . 10 |- ( A e. CC -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
33	27, 30, 32	3eqtrd 2125	. . . . . . . . 9 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
34	33	oveq1d 5681	. . . . . . . 8 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
35		2cn 8554	. . . . . . . . . 10 |- 2 e. CC
36		2ap0 8576	. . . . . . . . . 10 |- 2 # 0
37		divnegap 8234	. . . . . . . . . 10 |- ( ( ( A ^ 2 ) e. CC /\ 2 e. CC /\ 2 # 0 ) -> -u ( ( A ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
38	35, 36, 37	mp3an23 1266	. . . . . . . . 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 2124	. . . . . . 7 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) = -u ( ( A ^ 2 ) / 2 ) )
41	40	oveq2d 5682	. . . . . 6 |- ( A e. CC -> ( 1 + ( ( ( _i x. A ) ^ 2 ) / 2 ) ) = ( 1 + -u ( ( A ^ 2 ) / 2 ) ) )
42	31	halfcld 8721	. . . . . . 7 |- ( A e. CC -> ( ( A ^ 2 ) / 2 ) e. CC )
43		negsub 7791	. . . . . . 7 |- ( ( 1 e. CC /\ ( ( A ^ 2 ) / 2 ) e. CC ) -> ( 1 + -u ( ( A ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
44	7, 42, 43	sylancr 406	. . . . . 6 |- ( A e. CC -> ( 1 + -u ( ( A ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
45	41, 44	eqtrd 2121	. . . . 5 |- ( A e. CC -> ( 1 + ( ( ( _i x. A ) ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
46		mulexp 10055	. . . . . . . . . . 11 |- ( ( _i e. CC /\ A e. CC /\ 3 e. NN0 ) -> ( ( _i x. A ) ^ 3 ) = ( ( _i ^ 3 ) x. ( A ^ 3 ) ) )
47	1, 12, 46	mp3an13 1265	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) = ( ( _i ^ 3 ) x. ( A ^ 3 ) ) )
48		i3 10117	. . . . . . . . . . 11 |- ( _i ^ 3 ) = -u _i
49	48	oveq1i 5676	. . . . . . . . . 10 |- ( ( _i ^ 3 ) x. ( A ^ 3 ) ) = ( -u _i x. ( A ^ 3 ) )
50	47, 49	syl6eq 2137	. . . . . . . . 9 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) = ( -u _i x. ( A ^ 3 ) ) )
51	50	oveq1d 5681	. . . . . . . 8 |- ( A e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) = ( ( -u _i x. ( A ^ 3 ) ) / 6 ) )
52		expcl 10034	. . . . . . . . . 10 |- ( ( A e. CC /\ 3 e. NN0 ) -> ( A ^ 3 ) e. CC )
53	12, 52	mpan2 417	. . . . . . . . 9 |- ( A e. CC -> ( A ^ 3 ) e. CC )
54		negicn 7744	. . . . . . . . . 10 |- -u _i e. CC
55	15, 18	pm3.2i 267	. . . . . . . . . 10 |- ( 6 e. CC /\ 6 # 0 )
56		divassap 8218	. . . . . . . . . 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 1265	. . . . . . . . 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 8206	. . . . . . . . . . 11 |- ( ( ( A ^ 3 ) e. CC /\ 6 e. CC /\ 6 # 0 ) -> ( ( A ^ 3 ) / 6 ) e. CC )
60	15, 18, 59	mp3an23 1266	. . . . . . . . . 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 7936	. . . . . . . . 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 406	. . . . . . . 8 |- ( A e. CC -> ( -u _i x. ( ( A ^ 3 ) / 6 ) ) = ( _i x. -u ( ( A ^ 3 ) / 6 ) ) )
64	51, 58, 63	3eqtrd 2125	. . . . . . 7 |- ( A e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) = ( _i x. -u ( ( A ^ 3 ) / 6 ) ) )
65	64	oveq2d 5682	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) + ( ( ( _i x. A ) ^ 3 ) / 6 ) ) = ( ( _i x. A ) + ( _i x. -u ( ( A ^ 3 ) / 6 ) ) ) )
66	61	negcld 7841	. . . . . . 7 |- ( A e. CC -> -u ( ( A ^ 3 ) / 6 ) e. CC )
67		adddi 7535	. . . . . . . 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 1261	. . . . . . 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 413	. . . . . 6 |- ( A e. CC -> ( _i x. ( A + -u ( ( A ^ 3 ) / 6 ) ) ) = ( ( _i x. A ) + ( _i x. -u ( ( A ^ 3 ) / 6 ) ) ) )
70		negsub 7791	. . . . . . . 8 |- ( ( A e. CC /\ ( ( A ^ 3 ) / 6 ) e. CC ) -> ( A + -u ( ( A ^ 3 ) / 6 ) ) = ( A - ( ( A ^ 3 ) / 6 ) ) )
71	61, 70	mpdan 413	. . . . . . 7 |- ( A e. CC -> ( A + -u ( ( A ^ 3 ) / 6 ) ) = ( A - ( ( A ^ 3 ) / 6 ) ) )
72	71	oveq2d 5682	. . . . . 6 |- ( A e. CC -> ( _i x. ( A + -u ( ( A ^ 3 ) / 6 ) ) ) = ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) )
73	65, 69, 72	3eqtr2d 2127	. . . . 5 |- ( A e. CC -> ( ( _i x. A ) + ( ( ( _i x. A ) ^ 3 ) / 6 ) ) = ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) )
74	45, 73	oveq12d 5684	. . . 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 2125	. . 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 5681	. 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 2121	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 1290   e. wcel 1439   class class class wbr 3851   |-> cmpt 3905   ` cfv 5028  (class class class)co 5666   CCcc 7409   0cc0 7411   1c1 7412   _ici 7413   + caddc 7414   x. cmul 7416   - cmin 7714   -ucneg 7715   # cap 8119   / cdiv 8200   2c2 8534   3c3 8535   4c4 8536   6c6 8538   NN0cn0 8734   ZZ>=cuz 9080   ^cexp 10015   !cfa 10194   sum_csu 10803   expce 10993

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525  ax-caucvg 7526

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-3 8543  df-4 8544  df-5 8545  df-6 8546  df-n0 8735  df-z 8812  df-uz 9081  df-q 9166  df-rp 9196  df-ico 9373  df-fz 9486  df-fzo 9615  df-iseq 9914  df-seq3 9915  df-exp 10016  df-fac 10195  df-ihash 10245  df-cj 10337  df-re 10338  df-im 10339  df-rsqrt 10492  df-abs 10493  df-clim 10728  df-isum 10804  df-ef 10999

This theorem is referenced by:  resin4p  11070  recos4p  11071