Theorem ef4p 11635

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 8918	. 2 |- 4 = ( 3 + 1 )
3		3nn0 9132	. 2 |- 3 e. NN0
4		id 19	. 2 |- ( A e. CC -> A e. CC )
5		ax-1cn 7846	. . . 4 |- 1 e. CC
6		addcl 7878	. . . 4 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 + A ) e. CC )
7	5, 6	mpan 421	. . 3 |- ( A e. CC -> ( 1 + A ) e. CC )
8		sqcl 10516	. . . 4 |- ( A e. CC -> ( A ^ 2 ) e. CC )
9	8	halfcld 9101	. . 3 |- ( A e. CC -> ( ( A ^ 2 ) / 2 ) e. CC )
10	7, 9	addcld 7918	. 2 |- ( A e. CC -> ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) e. CC )
11		df-3 8917	. . 3 |- 3 = ( 2 + 1 )
12		2nn0 9131	. . 3 |- 2 e. NN0
13		df-2 8916	. . . 4 |- 2 = ( 1 + 1 )
14		1nn0 9130	. . . 4 |- 1 e. NN0
15	5	a1i 9	. . . 4 |- ( A e. CC -> 1 e. CC )
16		1e0p1 9363	. . . . 5 |- 1 = ( 0 + 1 )
17		0nn0 9129	. . . . 5 |- 0 e. NN0
18		0cnd 7892	. . . . 5 |- ( A e. CC -> 0 e. CC )
19	1	efval2 11606	. . . . . . . 8 |- ( A e. CC -> ( exp ` A ) = sum_ k e. NN0 ( F ` k ) )
20		nn0uz 9500	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
21	20	sumeq1i 11304	. . . . . . . 8 |- sum_ k e. NN0 ( F ` k ) = sum_ k e. ( ZZ>= ` 0 ) ( F ` k )
22	19, 21	eqtr2di 2216	. . . . . . 7 |- ( A e. CC -> sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) = ( exp ` A ) )
23	22	oveq2d 5858	. . . . . 6 |- ( A e. CC -> ( 0 + sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) ) = ( 0 + ( exp ` A ) ) )
24		efcl 11605	. . . . . . 7 |- ( A e. CC -> ( exp ` A ) e. CC )
25	24	addid2d 8048	. . . . . 6 |- ( A e. CC -> ( 0 + ( exp ` A ) ) = ( exp ` A ) )
26	23, 25	eqtr2d 2199	. . . . 5 |- ( A e. CC -> ( exp ` A ) = ( 0 + sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) ) )
27		eft0val 11634	. . . . . . 7 |- ( A e. CC -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = 1 )
28	27	oveq2d 5858	. . . . . 6 |- ( A e. CC -> ( 0 + ( ( A ^ 0 ) / ( ! ` 0 ) ) ) = ( 0 + 1 ) )
29		0p1e1 8971	. . . . . 6 |- ( 0 + 1 ) = 1
30	28, 29	eqtrdi 2215	. . . . 5 |- ( A e. CC -> ( 0 + ( ( A ^ 0 ) / ( ! ` 0 ) ) ) = 1 )
31	1, 16, 17, 4, 18, 26, 30	efsep 11632	. . . 4 |- ( A e. CC -> ( exp ` A ) = ( 1 + sum_ k e. ( ZZ>= ` 1 ) ( F ` k ) ) )
32		exp1 10461	. . . . . . 7 |- ( A e. CC -> ( A ^ 1 ) = A )
33		fac1 10642	. . . . . . . 8 |- ( ! ` 1 ) = 1
34	33	a1i 9	. . . . . . 7 |- ( A e. CC -> ( ! ` 1 ) = 1 )
35	32, 34	oveq12d 5860	. . . . . 6 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( A / 1 ) )
36		div1 8599	. . . . . 6 |- ( A e. CC -> ( A / 1 ) = A )
37	35, 36	eqtrd 2198	. . . . 5 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
38	37	oveq2d 5858	. . . 4 |- ( A e. CC -> ( 1 + ( ( A ^ 1 ) / ( ! ` 1 ) ) ) = ( 1 + A ) )
39	1, 13, 14, 4, 15, 31, 38	efsep 11632	. . 3 |- ( A e. CC -> ( exp ` A ) = ( ( 1 + A ) + sum_ k e. ( ZZ>= ` 2 ) ( F ` k ) ) )
40		fac2 10644	. . . . . 6 |- ( ! ` 2 ) = 2
41	40	oveq2i 5853	. . . . 5 |- ( ( A ^ 2 ) / ( ! ` 2 ) ) = ( ( A ^ 2 ) / 2 )
42	41	oveq2i 5853	. . . 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 11632	. 2 |- ( A e. CC -> ( exp ` A ) = ( ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) + sum_ k e. ( ZZ>= ` 3 ) ( F ` k ) ) )
45		fac3 10645	. . . . 5 |- ( ! ` 3 ) = 6
46	45	oveq2i 5853	. . . 4 |- ( ( A ^ 3 ) / ( ! ` 3 ) ) = ( ( A ^ 3 ) / 6 )
47	46	oveq2i 5853	. . 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 11632	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 1343   e. wcel 2136   |-> cmpt 4043   ` cfv 5188  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   + caddc 7756   / cdiv 8568   2c2 8908   3c3 8909   4c4 8910   6c6 8912   NN0cn0 9114   ZZ>=cuz 9466   ^cexp 10454   !cfa 10638   sum_csu 11294   expce 11583

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-5 8919  df-6 8920  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-ico 9830  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589

This theorem is referenced by:  efi4p  11658