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Theorem ef4p 11038
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
StepHypRef Expression
1 ef4p.1 . 2  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
2 df-4 8537 . 2  |-  4  =  ( 3  +  1 )
3 3nn0 8745 . 2  |-  3  e.  NN0
4 id 19 . 2  |-  ( A  e.  CC  ->  A  e.  CC )
5 ax-1cn 7492 . . . 4  |-  1  e.  CC
6 addcl 7521 . . . 4  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  A
)  e.  CC )
75, 6mpan 416 . . 3  |-  ( A  e.  CC  ->  (
1  +  A )  e.  CC )
8 sqcl 10070 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
98halfcld 8714 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 2 )  /  2 )  e.  CC )
107, 9addcld 7561 . 2  |-  ( A  e.  CC  ->  (
( 1  +  A
)  +  ( ( A ^ 2 )  /  2 ) )  e.  CC )
11 df-3 8536 . . 3  |-  3  =  ( 2  +  1 )
12 2nn0 8744 . . 3  |-  2  e.  NN0
13 df-2 8535 . . . 4  |-  2  =  ( 1  +  1 )
14 1nn0 8743 . . . 4  |-  1  e.  NN0
155a1i 9 . . . 4  |-  ( A  e.  CC  ->  1  e.  CC )
16 1e0p1 8972 . . . . 5  |-  1  =  ( 0  +  1 )
17 0nn0 8742 . . . . 5  |-  0  e.  NN0
18 0cnd 7535 . . . . 5  |-  ( A  e.  CC  ->  0  e.  CC )
191efval2 11009 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  A )  = 
sum_ k  e.  NN0  ( F `  k ) )
20 nn0uz 9107 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
2120sumeq1i 10806 . . . . . . . 8  |-  sum_ k  e.  NN0  ( F `  k )  =  sum_ k  e.  ( ZZ>= ` 
0 ) ( F `
 k )
2219, 21syl6req 2138 . . . . . . 7  |-  ( A  e.  CC  ->  sum_ k  e.  ( ZZ>= `  0 )
( F `  k
)  =  ( exp `  A ) )
2322oveq2d 5682 . . . . . 6  |-  ( A  e.  CC  ->  (
0  +  sum_ k  e.  ( ZZ>= `  0 )
( F `  k
) )  =  ( 0  +  ( exp `  A ) ) )
24 efcl 11008 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2524addid2d 7686 . . . . . 6  |-  ( A  e.  CC  ->  (
0  +  ( exp `  A ) )  =  ( exp `  A
) )
2623, 25eqtr2d 2122 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( 0  +  sum_ k  e.  ( ZZ>= ` 
0 ) ( F `
 k ) ) )
27 eft0val 11037 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
2827oveq2d 5682 . . . . . 6  |-  ( A  e.  CC  ->  (
0  +  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )  =  ( 0  +  1 ) )
29 0p1e1 8590 . . . . . 6  |-  ( 0  +  1 )  =  1
3028, 29syl6eq 2137 . . . . 5  |-  ( A  e.  CC  ->  (
0  +  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )  =  1 )
311, 16, 17, 4, 18, 26, 30efsep 11035 . . . 4  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( 1  +  sum_ k  e.  ( ZZ>= ` 
1 ) ( F `
 k ) ) )
32 exp1 10015 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
33 fac1 10191 . . . . . . . 8  |-  ( ! `
 1 )  =  1
3433a1i 9 . . . . . . 7  |-  ( A  e.  CC  ->  ( ! `  1 )  =  1 )
3532, 34oveq12d 5684 . . . . . 6  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  ( A  / 
1 ) )
36 div1 8224 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
3735, 36eqtrd 2121 . . . . 5  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
3837oveq2d 5682 . . . 4  |-  ( A  e.  CC  ->  (
1  +  ( ( A ^ 1 )  /  ( ! ` 
1 ) ) )  =  ( 1  +  A ) )
391, 13, 14, 4, 15, 31, 38efsep 11035 . . 3  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( ( 1  +  A )  +  sum_ k  e.  ( ZZ>= ` 
2 ) ( F `
 k ) ) )
40 fac2 10193 . . . . . 6  |-  ( ! `
 2 )  =  2
4140oveq2i 5677 . . . . 5  |-  ( ( A ^ 2 )  /  ( ! ` 
2 ) )  =  ( ( A ^
2 )  /  2
)
4241oveq2i 5677 . . . 4  |-  ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
( ! `  2
) ) )  =  ( ( 1  +  A )  +  ( ( A ^ 2 )  /  2 ) )
4342a1i 9 . . 3  |-  ( A  e.  CC  ->  (
( 1  +  A
)  +  ( ( A ^ 2 )  /  ( ! ` 
2 ) ) )  =  ( ( 1  +  A )  +  ( ( A ^
2 )  /  2
) ) )
441, 11, 12, 4, 7, 39, 43efsep 11035 . 2  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( ( ( 1  +  A )  +  ( ( A ^
2 )  /  2
) )  +  sum_ k  e.  ( ZZ>= ` 
3 ) ( F `
 k ) ) )
45 fac3 10194 . . . . 5  |-  ( ! `
 3 )  =  6
4645oveq2i 5677 . . . 4  |-  ( ( A ^ 3 )  /  ( ! ` 
3 ) )  =  ( ( A ^
3 )  /  6
)
4746oveq2i 5677 . . 3  |-  ( ( ( 1  +  A
)  +  ( ( A ^ 2 )  /  2 ) )  +  ( ( A ^ 3 )  / 
( ! `  3
) ) )  =  ( ( ( 1  +  A )  +  ( ( A ^
2 )  /  2
) )  +  ( ( A ^ 3 )  /  6 ) )
4847a1i 9 . 2  |-  ( A  e.  CC  ->  (
( ( 1  +  A )  +  ( ( A ^ 2 )  /  2 ) )  +  ( ( A ^ 3 )  /  ( ! ` 
3 ) ) )  =  ( ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
2 ) )  +  ( ( A ^
3 )  /  6
) ) )
491, 2, 3, 4, 10, 44, 48efsep 11035 1  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( ( ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
2 ) )  +  ( ( A ^
3 )  /  6
) )  +  sum_ k  e.  ( ZZ>= ` 
4 ) ( F `
 k ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290    e. wcel 1439    |-> cmpt 3905   ` cfv 5028  (class class class)co 5666   CCcc 7402   0cc0 7404   1c1 7405    + caddc 7407    / cdiv 8193   2c2 8527   3c3 8528   4c4 8529   6c6 8531   NN0cn0 8727   ZZ>=cuz 9073   ^cexp 10008   !cfa 10187   sum_csu 10796   expce 10986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517  ax-arch 7518  ax-caucvg 7519
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113  df-div 8194  df-inn 8477  df-2 8535  df-3 8536  df-4 8537  df-5 8538  df-6 8539  df-n0 8728  df-z 8805  df-uz 9074  df-q 9159  df-rp 9189  df-ico 9366  df-fz 9479  df-fzo 9608  df-iseq 9907  df-seq3 9908  df-exp 10009  df-fac 10188  df-ihash 10238  df-cj 10330  df-re 10331  df-im 10332  df-rsqrt 10485  df-abs 10486  df-clim 10721  df-isum 10797  df-ef 10992
This theorem is referenced by:  efi4p  11062
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