Theorem ef4p 11437

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 8805	. 2 |- 4 = ( 3 + 1 )
3		3nn0 9019	. 2 |- 3 e. NN0
4		id 19	. 2 |- ( A e. CC -> A e. CC )
5		ax-1cn 7737	. . . 4 |- 1 e. CC
6		addcl 7769	. . . 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 10385	. . . 4 |- ( A e. CC -> ( A ^ 2 ) e. CC )
9	8	halfcld 8988	. . 3 |- ( A e. CC -> ( ( A ^ 2 ) / 2 ) e. CC )
10	7, 9	addcld 7809	. 2 |- ( A e. CC -> ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) e. CC )
11		df-3 8804	. . 3 |- 3 = ( 2 + 1 )
12		2nn0 9018	. . 3 |- 2 e. NN0
13		df-2 8803	. . . 4 |- 2 = ( 1 + 1 )
14		1nn0 9017	. . . 4 |- 1 e. NN0
15	5	a1i 9	. . . 4 |- ( A e. CC -> 1 e. CC )
16		1e0p1 9247	. . . . 5 |- 1 = ( 0 + 1 )
17		0nn0 9016	. . . . 5 |- 0 e. NN0
18		0cnd 7783	. . . . 5 |- ( A e. CC -> 0 e. CC )
19	1	efval2 11408	. . . . . . . 8 |- ( A e. CC -> ( exp ` A ) = sum_ k e. NN0 ( F ` k ) )
20		nn0uz 9384	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
21	20	sumeq1i 11164	. . . . . . . 8 |- sum_ k e. NN0 ( F ` k ) = sum_ k e. ( ZZ>= ` 0 ) ( F ` k )
22	19, 21	eqtr2di 2190	. . . . . . 7 |- ( A e. CC -> sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) = ( exp ` A ) )
23	22	oveq2d 5798	. . . . . 6 |- ( A e. CC -> ( 0 + sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) ) = ( 0 + ( exp ` A ) ) )
24		efcl 11407	. . . . . . 7 |- ( A e. CC -> ( exp ` A ) e. CC )
25	24	addid2d 7936	. . . . . 6 |- ( A e. CC -> ( 0 + ( exp ` A ) ) = ( exp ` A ) )
26	23, 25	eqtr2d 2174	. . . . 5 |- ( A e. CC -> ( exp ` A ) = ( 0 + sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) ) )
27		eft0val 11436	. . . . . . 7 |- ( A e. CC -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = 1 )
28	27	oveq2d 5798	. . . . . 6 |- ( A e. CC -> ( 0 + ( ( A ^ 0 ) / ( ! ` 0 ) ) ) = ( 0 + 1 ) )
29		0p1e1 8858	. . . . . 6 |- ( 0 + 1 ) = 1
30	28, 29	eqtrdi 2189	. . . . 5 |- ( A e. CC -> ( 0 + ( ( A ^ 0 ) / ( ! ` 0 ) ) ) = 1 )
31	1, 16, 17, 4, 18, 26, 30	efsep 11434	. . . 4 |- ( A e. CC -> ( exp ` A ) = ( 1 + sum_ k e. ( ZZ>= ` 1 ) ( F ` k ) ) )
32		exp1 10330	. . . . . . 7 |- ( A e. CC -> ( A ^ 1 ) = A )
33		fac1 10507	. . . . . . . 8 |- ( ! ` 1 ) = 1
34	33	a1i 9	. . . . . . 7 |- ( A e. CC -> ( ! ` 1 ) = 1 )
35	32, 34	oveq12d 5800	. . . . . 6 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( A / 1 ) )
36		div1 8487	. . . . . 6 |- ( A e. CC -> ( A / 1 ) = A )
37	35, 36	eqtrd 2173	. . . . 5 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
38	37	oveq2d 5798	. . . 4 |- ( A e. CC -> ( 1 + ( ( A ^ 1 ) / ( ! ` 1 ) ) ) = ( 1 + A ) )
39	1, 13, 14, 4, 15, 31, 38	efsep 11434	. . 3 |- ( A e. CC -> ( exp ` A ) = ( ( 1 + A ) + sum_ k e. ( ZZ>= ` 2 ) ( F ` k ) ) )
40		fac2 10509	. . . . . 6 |- ( ! ` 2 ) = 2
41	40	oveq2i 5793	. . . . 5 |- ( ( A ^ 2 ) / ( ! ` 2 ) ) = ( ( A ^ 2 ) / 2 )
42	41	oveq2i 5793	. . . 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 11434	. 2 |- ( A e. CC -> ( exp ` A ) = ( ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) + sum_ k e. ( ZZ>= ` 3 ) ( F ` k ) ) )
45		fac3 10510	. . . . 5 |- ( ! ` 3 ) = 6
46	45	oveq2i 5793	. . . 4 |- ( ( A ^ 3 ) / ( ! ` 3 ) ) = ( ( A ^ 3 ) / 6 )
47	46	oveq2i 5793	. . 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 11434	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 1332   e. wcel 1481   |-> cmpt 3997   ` cfv 5131  (class class class)co 5782   CCcc 7642   0cc0 7644   1c1 7645   + caddc 7647   / cdiv 8456   2c2 8795   3c3 8796   4c4 8797   6c6 8799   NN0cn0 9001   ZZ>=cuz 9350   ^cexp 10323   !cfa 10503   sum_csu 11154   expce 11385

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-5 8806  df-6 8807  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-ico 9707  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-fac 10504  df-ihash 10554  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155  df-ef 11391

This theorem is referenced by:  efi4p  11460