Theorem efi4p 12428

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 8238	. . . 4 |- _i e. CC
2		mulcl 8270	. . . 4 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
3	1, 2	mpan 424	. . 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 12405	. . 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 8236	. . . . . 6 |- 1 e. CC
8		addcl 8268	. . . . . 6 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC )
9	7, 3, 8	sylancr 414	. . . . 5 |- ( A e. CC -> ( 1 + ( _i x. A ) ) e. CC )
10	3	sqcld 11058	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) e. CC )
11	10	halfcld 9500	. . . . 5 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) e. CC )
12		3nn0 9531	. . . . . . 7 |- 3 e. NN0
13		expcl 10943	. . . . . . 7 |- ( ( ( _i x. A ) e. CC /\ 3 e. NN0 ) -> ( ( _i x. A ) ^ 3 ) e. CC )
14	3, 12, 13	sylancl 413	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) e. CC )
15		6cn 9336	. . . . . . 7 |- 6 e. CC
16		6re 9335	. . . . . . . 8 |- 6 e. RR
17		6pos 9355	. . . . . . . 8 |- 0 < 6
18	16, 17	gt0ap0ii 8919	. . . . . . 7 |- 6 # 0
19		divclap 8969	. . . . . . 7 |- ( ( ( ( _i x. A ) ^ 3 ) e. CC /\ 6 e. CC /\ 6 # 0 ) -> ( ( ( _i x. A ) ^ 3 ) / 6 ) e. CC )
20	15, 18, 19	mp3an23 1366	. . . . . 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 8312	. . . 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 8458	. . . 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 9530	. . . . . . . . . . 11 |- 2 e. NN0
26		mulexp 10964	. . . . . . . . . . 11 |- ( ( _i e. CC /\ A e. CC /\ 2 e. NN0 ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
27	1, 25, 26	mp3an13 1365	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
28		i2 11026	. . . . . . . . . . . 12 |- ( _i ^ 2 ) = -u 1
29	28	oveq1i 6068	. . . . . . . . . . 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 10986	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 2 ) e. CC )
32	31	mulm1d 8700	. . . . . . . . . 10 |- ( A e. CC -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
33	27, 30, 32	3eqtrd 2271	. . . . . . . . 9 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
34	33	oveq1d 6073	. . . . . . . 8 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
35		2cn 9325	. . . . . . . . . 10 |- 2 e. CC
36		2ap0 9347	. . . . . . . . . 10 |- 2 # 0
37		divnegap 8997	. . . . . . . . . 10 |- ( ( ( A ^ 2 ) e. CC /\ 2 e. CC /\ 2 # 0 ) -> -u ( ( A ^ 2 ) / 2 ) = ( -u ( A ^ 2 ) / 2 ) )
38	35, 36, 37	mp3an23 1366	. . . . . . . . 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 2270	. . . . . . 7 |- ( A e. CC -> ( ( ( _i x. A ) ^ 2 ) / 2 ) = -u ( ( A ^ 2 ) / 2 ) )
41	40	oveq2d 6074	. . . . . 6 |- ( A e. CC -> ( 1 + ( ( ( _i x. A ) ^ 2 ) / 2 ) ) = ( 1 + -u ( ( A ^ 2 ) / 2 ) ) )
42	31	halfcld 9500	. . . . . . 7 |- ( A e. CC -> ( ( A ^ 2 ) / 2 ) e. CC )
43		negsub 8537	. . . . . . 7 |- ( ( 1 e. CC /\ ( ( A ^ 2 ) / 2 ) e. CC ) -> ( 1 + -u ( ( A ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
44	7, 42, 43	sylancr 414	. . . . . 6 |- ( A e. CC -> ( 1 + -u ( ( A ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
45	41, 44	eqtrd 2267	. . . . 5 |- ( A e. CC -> ( 1 + ( ( ( _i x. A ) ^ 2 ) / 2 ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
46		mulexp 10964	. . . . . . . . . . 11 |- ( ( _i e. CC /\ A e. CC /\ 3 e. NN0 ) -> ( ( _i x. A ) ^ 3 ) = ( ( _i ^ 3 ) x. ( A ^ 3 ) ) )
47	1, 12, 46	mp3an13 1365	. . . . . . . . . 10 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) = ( ( _i ^ 3 ) x. ( A ^ 3 ) ) )
48		i3 11027	. . . . . . . . . . 11 |- ( _i ^ 3 ) = -u _i
49	48	oveq1i 6068	. . . . . . . . . 10 |- ( ( _i ^ 3 ) x. ( A ^ 3 ) ) = ( -u _i x. ( A ^ 3 ) )
50	47, 49	eqtrdi 2283	. . . . . . . . 9 |- ( A e. CC -> ( ( _i x. A ) ^ 3 ) = ( -u _i x. ( A ^ 3 ) ) )
51	50	oveq1d 6073	. . . . . . . 8 |- ( A e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) = ( ( -u _i x. ( A ^ 3 ) ) / 6 ) )
52		expcl 10943	. . . . . . . . . 10 |- ( ( A e. CC /\ 3 e. NN0 ) -> ( A ^ 3 ) e. CC )
53	12, 52	mpan2 425	. . . . . . . . 9 |- ( A e. CC -> ( A ^ 3 ) e. CC )
54		negicn 8490	. . . . . . . . . 10 |- -u _i e. CC
55	15, 18	pm3.2i 272	. . . . . . . . . 10 |- ( 6 e. CC /\ 6 # 0 )
56		divassap 8981	. . . . . . . . . 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 1365	. . . . . . . . 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 8969	. . . . . . . . . . 11 |- ( ( ( A ^ 3 ) e. CC /\ 6 e. CC /\ 6 # 0 ) -> ( ( A ^ 3 ) / 6 ) e. CC )
60	15, 18, 59	mp3an23 1366	. . . . . . . . . 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 8687	. . . . . . . . 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 414	. . . . . . . 8 |- ( A e. CC -> ( -u _i x. ( ( A ^ 3 ) / 6 ) ) = ( _i x. -u ( ( A ^ 3 ) / 6 ) ) )
64	51, 58, 63	3eqtrd 2271	. . . . . . 7 |- ( A e. CC -> ( ( ( _i x. A ) ^ 3 ) / 6 ) = ( _i x. -u ( ( A ^ 3 ) / 6 ) ) )
65	64	oveq2d 6074	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) + ( ( ( _i x. A ) ^ 3 ) / 6 ) ) = ( ( _i x. A ) + ( _i x. -u ( ( A ^ 3 ) / 6 ) ) ) )
66	61	negcld 8587	. . . . . . 7 |- ( A e. CC -> -u ( ( A ^ 3 ) / 6 ) e. CC )
67		adddi 8275	. . . . . . . 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 1361	. . . . . . 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 421	. . . . . 6 |- ( A e. CC -> ( _i x. ( A + -u ( ( A ^ 3 ) / 6 ) ) ) = ( ( _i x. A ) + ( _i x. -u ( ( A ^ 3 ) / 6 ) ) ) )
70		negsub 8537	. . . . . . . 8 |- ( ( A e. CC /\ ( ( A ^ 3 ) / 6 ) e. CC ) -> ( A + -u ( ( A ^ 3 ) / 6 ) ) = ( A - ( ( A ^ 3 ) / 6 ) ) )
71	61, 70	mpdan 421	. . . . . . 7 |- ( A e. CC -> ( A + -u ( ( A ^ 3 ) / 6 ) ) = ( A - ( ( A ^ 3 ) / 6 ) ) )
72	71	oveq2d 6074	. . . . . 6 |- ( A e. CC -> ( _i x. ( A + -u ( ( A ^ 3 ) / 6 ) ) ) = ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) )
73	65, 69, 72	3eqtr2d 2273	. . . . 5 |- ( A e. CC -> ( ( _i x. A ) + ( ( ( _i x. A ) ^ 3 ) / 6 ) ) = ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) )
74	45, 73	oveq12d 6076	. . . 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 2271	. . 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 6073	. 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 2267	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 104   = wceq 1398   e. wcel 2205   class class class wbr 4114   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   _ici 8145   + caddc 8146   x. cmul 8148   - cmin 8460   -ucneg 8461   # cap 8872   / cdiv 8963   2c2 9305   3c3 9306   4c4 9307   6c6 9309   NN0cn0 9513   ZZ>=cuz 9871   ^cexp 10924   !cfa 11112   sum_csu 12063   expce 12353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-ico 10246  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-fac 11113  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064  df-ef 12359

This theorem is referenced by:  resin4p  12429  recos4p  12430