Theorem ef4p 11657

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, 29-Apr-2014.)

Hypothesis

Ref	Expression
ef4p.1	|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )

Assertion

Ref	Expression
ef4p	|- ( A e. CC -> ( exp ` A ) = ( ( ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) + ( ( A ^ 3 ) / 6 ) ) + sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) )

Distinct variable groups:   k, n, A   k, F

Allowed substitution hint:   F( n)

Proof of Theorem ef4p

Step	Hyp	Ref	Expression
1		ef4p.1	. 2 |- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
2		df-4 8939	. 2 |- 4 = ( 3 + 1 )
3		3nn0 9153	. 2 |- 3 e. NN0
4		id 19	. 2 |- ( A e. CC -> A e. CC )
5		ax-1cn 7867	. . . 4 |- 1 e. CC
6		addcl 7899	. . . 4 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 + A ) e. CC )
7	5, 6	mpan 422	. . 3 |- ( A e. CC -> ( 1 + A ) e. CC )
8		sqcl 10537	. . . 4 |- ( A e. CC -> ( A ^ 2 ) e. CC )
9	8	halfcld 9122	. . 3 |- ( A e. CC -> ( ( A ^ 2 ) / 2 ) e. CC )
10	7, 9	addcld 7939	. 2 |- ( A e. CC -> ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) e. CC )
11		df-3 8938	. . 3 |- 3 = ( 2 + 1 )
12		2nn0 9152	. . 3 |- 2 e. NN0
13		df-2 8937	. . . 4 |- 2 = ( 1 + 1 )
14		1nn0 9151	. . . 4 |- 1 e. NN0
15	5	a1i 9	. . . 4 |- ( A e. CC -> 1 e. CC )
16		1e0p1 9384	. . . . 5 |- 1 = ( 0 + 1 )
17		0nn0 9150	. . . . 5 |- 0 e. NN0
18		0cnd 7913	. . . . 5 |- ( A e. CC -> 0 e. CC )
19	1	efval2 11628	. . . . . . . 8 |- ( A e. CC -> ( exp ` A ) = sum_ k e. NN0 ( F ` k ) )
20		nn0uz 9521	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
21	20	sumeq1i 11326	. . . . . . . 8 |- sum_ k e. NN0 ( F ` k ) = sum_ k e. ( ZZ>= ` 0 ) ( F ` k )
22	19, 21	eqtr2di 2220	. . . . . . 7 |- ( A e. CC -> sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) = ( exp ` A ) )
23	22	oveq2d 5869	. . . . . 6 |- ( A e. CC -> ( 0 + sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) ) = ( 0 + ( exp ` A ) ) )
24		efcl 11627	. . . . . . 7 |- ( A e. CC -> ( exp ` A ) e. CC )
25	24	addid2d 8069	. . . . . 6 |- ( A e. CC -> ( 0 + ( exp ` A ) ) = ( exp ` A ) )
26	23, 25	eqtr2d 2204	. . . . 5 |- ( A e. CC -> ( exp ` A ) = ( 0 + sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) ) )
27		eft0val 11656	. . . . . . 7 |- ( A e. CC -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = 1 )
28	27	oveq2d 5869	. . . . . 6 |- ( A e. CC -> ( 0 + ( ( A ^ 0 ) / ( ! ` 0 ) ) ) = ( 0 + 1 ) )
29		0p1e1 8992	. . . . . 6 |- ( 0 + 1 ) = 1
30	28, 29	eqtrdi 2219	. . . . 5 |- ( A e. CC -> ( 0 + ( ( A ^ 0 ) / ( ! ` 0 ) ) ) = 1 )
31	1, 16, 17, 4, 18, 26, 30	efsep 11654	. . . 4 |- ( A e. CC -> ( exp ` A ) = ( 1 + sum_ k e. ( ZZ>= ` 1 ) ( F ` k ) ) )
32		exp1 10482	. . . . . . 7 |- ( A e. CC -> ( A ^ 1 ) = A )
33		fac1 10663	. . . . . . . 8 |- ( ! ` 1 ) = 1
34	33	a1i 9	. . . . . . 7 |- ( A e. CC -> ( ! ` 1 ) = 1 )
35	32, 34	oveq12d 5871	. . . . . 6 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( A / 1 ) )
36		div1 8620	. . . . . 6 |- ( A e. CC -> ( A / 1 ) = A )
37	35, 36	eqtrd 2203	. . . . 5 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
38	37	oveq2d 5869	. . . 4 |- ( A e. CC -> ( 1 + ( ( A ^ 1 ) / ( ! ` 1 ) ) ) = ( 1 + A ) )
39	1, 13, 14, 4, 15, 31, 38	efsep 11654	. . 3 |- ( A e. CC -> ( exp ` A ) = ( ( 1 + A ) + sum_ k e. ( ZZ>= ` 2 ) ( F ` k ) ) )
40		fac2 10665	. . . . . 6 |- ( ! ` 2 ) = 2
41	40	oveq2i 5864	. . . . 5 |- ( ( A ^ 2 ) / ( ! ` 2 ) ) = ( ( A ^ 2 ) / 2 )
42	41	oveq2i 5864	. . . 4 |- ( ( 1 + A ) + ( ( A ^ 2 ) / ( ! ` 2 ) ) ) = ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) )
43	42	a1i 9	. . 3 |- ( A e. CC -> ( ( 1 + A ) + ( ( A ^ 2 ) / ( ! ` 2 ) ) ) = ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) )
44	1, 11, 12, 4, 7, 39, 43	efsep 11654	. 2 |- ( A e. CC -> ( exp ` A ) = ( ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) + sum_ k e. ( ZZ>= ` 3 ) ( F ` k ) ) )
45		fac3 10666	. . . . 5 |- ( ! ` 3 ) = 6
46	45	oveq2i 5864	. . . 4 |- ( ( A ^ 3 ) / ( ! ` 3 ) ) = ( ( A ^ 3 ) / 6 )
47	46	oveq2i 5864	. . 3 |- ( ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) + ( ( A ^ 3 ) / ( ! ` 3 ) ) ) = ( ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) + ( ( A ^ 3 ) / 6 ) )
48	47	a1i 9	. 2 |- ( A e. CC -> ( ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) + ( ( A ^ 3 ) / ( ! ` 3 ) ) ) = ( ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) + ( ( A ^ 3 ) / 6 ) ) )
49	1, 2, 3, 4, 10, 44, 48	efsep 11654	1 |- ( A e. CC -> ( exp ` A ) = ( ( ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) + ( ( A ^ 3 ) / 6 ) ) + sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   |-> cmpt 4050   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   / cdiv 8589   2c2 8929   3c3 8930   4c4 8931   6c6 8933   NN0cn0 9135   ZZ>=cuz 9487   ^cexp 10475   !cfa 10659   sum_csu 11316   expce 11605

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-ico 9851  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611

This theorem is referenced by:  efi4p  11680