Theorem ef4p 11686

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 8969	. 2 |- 4 = ( 3 + 1 )
3		3nn0 9183	. 2 |- 3 e. NN0
4		id 19	. 2 |- ( A e. CC -> A e. CC )
5		ax-1cn 7895	. . . 4 |- 1 e. CC
6		addcl 7927	. . . 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 10567	. . . 4 |- ( A e. CC -> ( A ^ 2 ) e. CC )
9	8	halfcld 9152	. . 3 |- ( A e. CC -> ( ( A ^ 2 ) / 2 ) e. CC )
10	7, 9	addcld 7967	. 2 |- ( A e. CC -> ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) e. CC )
11		df-3 8968	. . 3 |- 3 = ( 2 + 1 )
12		2nn0 9182	. . 3 |- 2 e. NN0
13		df-2 8967	. . . 4 |- 2 = ( 1 + 1 )
14		1nn0 9181	. . . 4 |- 1 e. NN0
15	5	a1i 9	. . . 4 |- ( A e. CC -> 1 e. CC )
16		1e0p1 9414	. . . . 5 |- 1 = ( 0 + 1 )
17		0nn0 9180	. . . . 5 |- 0 e. NN0
18		0cnd 7941	. . . . 5 |- ( A e. CC -> 0 e. CC )
19	1	efval2 11657	. . . . . . . 8 |- ( A e. CC -> ( exp ` A ) = sum_ k e. NN0 ( F ` k ) )
20		nn0uz 9551	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
21	20	sumeq1i 11355	. . . . . . . 8 |- sum_ k e. NN0 ( F ` k ) = sum_ k e. ( ZZ>= ` 0 ) ( F ` k )
22	19, 21	eqtr2di 2227	. . . . . . 7 |- ( A e. CC -> sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) = ( exp ` A ) )
23	22	oveq2d 5885	. . . . . 6 |- ( A e. CC -> ( 0 + sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) ) = ( 0 + ( exp ` A ) ) )
24		efcl 11656	. . . . . . 7 |- ( A e. CC -> ( exp ` A ) e. CC )
25	24	addid2d 8097	. . . . . 6 |- ( A e. CC -> ( 0 + ( exp ` A ) ) = ( exp ` A ) )
26	23, 25	eqtr2d 2211	. . . . 5 |- ( A e. CC -> ( exp ` A ) = ( 0 + sum_ k e. ( ZZ>= ` 0 ) ( F ` k ) ) )
27		eft0val 11685	. . . . . . 7 |- ( A e. CC -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = 1 )
28	27	oveq2d 5885	. . . . . 6 |- ( A e. CC -> ( 0 + ( ( A ^ 0 ) / ( ! ` 0 ) ) ) = ( 0 + 1 ) )
29		0p1e1 9022	. . . . . 6 |- ( 0 + 1 ) = 1
30	28, 29	eqtrdi 2226	. . . . 5 |- ( A e. CC -> ( 0 + ( ( A ^ 0 ) / ( ! ` 0 ) ) ) = 1 )
31	1, 16, 17, 4, 18, 26, 30	efsep 11683	. . . 4 |- ( A e. CC -> ( exp ` A ) = ( 1 + sum_ k e. ( ZZ>= ` 1 ) ( F ` k ) ) )
32		exp1 10512	. . . . . . 7 |- ( A e. CC -> ( A ^ 1 ) = A )
33		fac1 10693	. . . . . . . 8 |- ( ! ` 1 ) = 1
34	33	a1i 9	. . . . . . 7 |- ( A e. CC -> ( ! ` 1 ) = 1 )
35	32, 34	oveq12d 5887	. . . . . 6 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( A / 1 ) )
36		div1 8649	. . . . . 6 |- ( A e. CC -> ( A / 1 ) = A )
37	35, 36	eqtrd 2210	. . . . 5 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
38	37	oveq2d 5885	. . . 4 |- ( A e. CC -> ( 1 + ( ( A ^ 1 ) / ( ! ` 1 ) ) ) = ( 1 + A ) )
39	1, 13, 14, 4, 15, 31, 38	efsep 11683	. . 3 |- ( A e. CC -> ( exp ` A ) = ( ( 1 + A ) + sum_ k e. ( ZZ>= ` 2 ) ( F ` k ) ) )
40		fac2 10695	. . . . . 6 |- ( ! ` 2 ) = 2
41	40	oveq2i 5880	. . . . 5 |- ( ( A ^ 2 ) / ( ! ` 2 ) ) = ( ( A ^ 2 ) / 2 )
42	41	oveq2i 5880	. . . 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 11683	. 2 |- ( A e. CC -> ( exp ` A ) = ( ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) + sum_ k e. ( ZZ>= ` 3 ) ( F ` k ) ) )
45		fac3 10696	. . . . 5 |- ( ! ` 3 ) = 6
46	45	oveq2i 5880	. . . 4 |- ( ( A ^ 3 ) / ( ! ` 3 ) ) = ( ( A ^ 3 ) / 6 )
47	46	oveq2i 5880	. . 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 11683	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 1353   e. wcel 2148   |-> cmpt 4061   ` cfv 5212  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803   + caddc 7805   / cdiv 8618   2c2 8959   3c3 8960   4c4 8961   6c6 8963   NN0cn0 9165   ZZ>=cuz 9517   ^cexp 10505   !cfa 10689   sum_csu 11345   expce 11634

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-ico 9881  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-fac 10690  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346  df-ef 11640

This theorem is referenced by:  efi4p  11709