Theorem ef4p 12405

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 9315	. 2 |- 4 = ( 3 + 1 )
3		3nn0 9531	. 2 |- 3 e. NN0
4		id 19	. 2 |- ( A e. CC -> A e. CC )
5		ax-1cn 8236	. . . 4 |- 1 e. CC
6		addcl 8268	. . . 4 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 + A ) e. CC )
7	5, 6	mpan 424	. . 3 |- ( A e. CC -> ( 1 + A ) e. CC )
8		sqcl 10986	. . . 4 |- ( A e. CC -> ( A ^ 2 ) e. CC )
9	8	halfcld 9500	. . 3 |- ( A e. CC -> ( ( A ^ 2 ) / 2 ) e. CC )
10	7, 9	addcld 8309	. 2 |- ( A e. CC -> ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) e. CC )
11		df-3 9314	. . 3 |- 3 = ( 2 + 1 )
12		2nn0 9530	. . 3 |- 2 e. NN0
13		df-2 9313	. . . 4 |- 2 = ( 1 + 1 )
14		1nn0 9529	. . . 4 |- 1 e. NN0
15	5	a1i 9	. . . 4 |- ( A e. CC -> 1 e. CC )
16		1e0p1 9768	. . . . 5 |- 1 = ( 0 + 1 )
17		0nn0 9528	. . . . 5 |- 0 e. NN0
18		0cnd 8283	. . . . 5 |- ( A e. CC -> 0 e. CC )
19	1	efval2 12376	. . . . . . . 8 |- ( A e. CC -> ( exp ` A ) = sum_ k e. NN0 ( F ` k ) )
20		nn0uz 9907	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
21	20	sumeq1i 12073	. . . . . . . 8 |- sum_ k e. NN0 ( F ` k ) = sum_ k e. ( ZZ>= ` 0 ) ( F ` k )
22	19, 21	eqtr2di 2284	. . . . . . 7 |- ( A e. CC -> sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) = ( exp ` A ) )
23	22	oveq2d 6074	. . . . . 6 |- ( A e. CC -> ( 0 + sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) ) = ( 0 + ( exp ` A ) ) )
24		efcl 12375	. . . . . . 7 |- ( A e. CC -> ( exp ` A ) e. CC )
25	24	addlidd 8439	. . . . . 6 |- ( A e. CC -> ( 0 + ( exp ` A ) ) = ( exp ` A ) )
26	23, 25	eqtr2d 2268	. . . . 5 |- ( A e. CC -> ( exp ` A ) = ( 0 + sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) ) )
27		eft0val 12404	. . . . . . 7 |- ( A e. CC -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = 1 )
28	27	oveq2d 6074	. . . . . 6 |- ( A e. CC -> ( 0 + ( ( A ^ 0 ) / ( ! ` 0 ) ) ) = ( 0 + 1 ) )
29		0p1e1 9368	. . . . . 6 |- ( 0 + 1 ) = 1
30	28, 29	eqtrdi 2283	. . . . 5 |- ( A e. CC -> ( 0 + ( ( A ^ 0 ) / ( ! ` 0 ) ) ) = 1 )
31	1, 16, 17, 4, 18, 26, 30	efsep 12402	. . . 4 |- ( A e. CC -> ( exp ` A ) = ( 1 + sum_ k e. ( ZZ>= ` 1 ) ( F ` k ) ) )
32		exp1 10931	. . . . . . 7 |- ( A e. CC -> ( A ^ 1 ) = A )
33		fac1 11116	. . . . . . . 8 |- ( ! ` 1 ) = 1
34	33	a1i 9	. . . . . . 7 |- ( A e. CC -> ( ! ` 1 ) = 1 )
35	32, 34	oveq12d 6076	. . . . . 6 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( A / 1 ) )
36		div1 8994	. . . . . 6 |- ( A e. CC -> ( A / 1 ) = A )
37	35, 36	eqtrd 2267	. . . . 5 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
38	37	oveq2d 6074	. . . 4 |- ( A e. CC -> ( 1 + ( ( A ^ 1 ) / ( ! ` 1 ) ) ) = ( 1 + A ) )
39	1, 13, 14, 4, 15, 31, 38	efsep 12402	. . 3 |- ( A e. CC -> ( exp ` A ) = ( ( 1 + A ) + sum_ k e. ( ZZ>= ` 2 ) ( F ` k ) ) )
40		fac2 11118	. . . . . 6 |- ( ! ` 2 ) = 2
41	40	oveq2i 6069	. . . . 5 |- ( ( A ^ 2 ) / ( ! ` 2 ) ) = ( ( A ^ 2 ) / 2 )
42	41	oveq2i 6069	. . . 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 12402	. 2 |- ( A e. CC -> ( exp ` A ) = ( ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) + sum_ k e. ( ZZ>= ` 3 ) ( F ` k ) ) )
45		fac3 11119	. . . . 5 |- ( ! ` 3 ) = 6
46	45	oveq2i 6069	. . . 4 |- ( ( A ^ 3 ) / ( ! ` 3 ) ) = ( ( A ^ 3 ) / 6 )
47	46	oveq2i 6069	. . 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 12402	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 1398   e. wcel 2205   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   / 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:  efi4p  12428