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Theorem ef4p 12393
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 9806 . 2  |-  4  =  ( 3  +  1 )
3 3nn0 9983 . 2  |-  3  e.  NN0
4 id 19 . 2  |-  ( A  e.  CC  ->  A  e.  CC )
5 ax-1cn 8795 . . . 4  |-  1  e.  CC
6 addcl 8819 . . . 4  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  A
)  e.  CC )
75, 6mpan 651 . . 3  |-  ( A  e.  CC  ->  (
1  +  A )  e.  CC )
8 sqcl 11166 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
98halfcld 9956 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 2 )  /  2 )  e.  CC )
107, 9addcld 8854 . 2  |-  ( A  e.  CC  ->  (
( 1  +  A
)  +  ( ( A ^ 2 )  /  2 ) )  e.  CC )
11 df-3 9805 . . 3  |-  3  =  ( 2  +  1 )
12 2nn0 9982 . . 3  |-  2  e.  NN0
13 df-2 9804 . . . 4  |-  2  =  ( 1  +  1 )
14 1nn0 9981 . . . 4  |-  1  e.  NN0
155a1i 10 . . . 4  |-  ( A  e.  CC  ->  1  e.  CC )
16 1e0p1 10152 . . . . 5  |-  1  =  ( 0  +  1 )
17 0nn0 9980 . . . . 5  |-  0  e.  NN0
18 0cn 8831 . . . . . 6  |-  0  e.  CC
1918a1i 10 . . . . 5  |-  ( A  e.  CC  ->  0  e.  CC )
201efval2 12365 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  A )  = 
sum_ k  e.  NN0  ( F `  k ) )
21 nn0uz 10262 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
2221sumeq1i 12171 . . . . . . . 8  |-  sum_ k  e.  NN0  ( F `  k )  =  sum_ k  e.  ( ZZ>= ` 
0 ) ( F `
 k )
2320, 22syl6req 2332 . . . . . . 7  |-  ( A  e.  CC  ->  sum_ k  e.  ( ZZ>= `  0 )
( F `  k
)  =  ( exp `  A ) )
2423oveq2d 5874 . . . . . 6  |-  ( A  e.  CC  ->  (
0  +  sum_ k  e.  ( ZZ>= `  0 )
( F `  k
) )  =  ( 0  +  ( exp `  A ) ) )
25 efcl 12364 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2625addid2d 9013 . . . . . 6  |-  ( A  e.  CC  ->  (
0  +  ( exp `  A ) )  =  ( exp `  A
) )
2724, 26eqtr2d 2316 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( 0  +  sum_ k  e.  ( ZZ>= ` 
0 ) ( F `
 k ) ) )
28 eft0val 12392 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
2928oveq2d 5874 . . . . . 6  |-  ( A  e.  CC  ->  (
0  +  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )  =  ( 0  +  1 ) )
30 0p1e1 9839 . . . . . 6  |-  ( 0  +  1 )  =  1
3129, 30syl6eq 2331 . . . . 5  |-  ( A  e.  CC  ->  (
0  +  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )  =  1 )
321, 16, 17, 4, 19, 27, 31efsep 12390 . . . 4  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( 1  +  sum_ k  e.  ( ZZ>= ` 
1 ) ( F `
 k ) ) )
33 exp1 11109 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
34 fac1 11292 . . . . . . . 8  |-  ( ! `
 1 )  =  1
3534a1i 10 . . . . . . 7  |-  ( A  e.  CC  ->  ( ! `  1 )  =  1 )
3633, 35oveq12d 5876 . . . . . 6  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  ( A  / 
1 ) )
37 div1 9453 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
3836, 37eqtrd 2315 . . . . 5  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
3938oveq2d 5874 . . . 4  |-  ( A  e.  CC  ->  (
1  +  ( ( A ^ 1 )  /  ( ! ` 
1 ) ) )  =  ( 1  +  A ) )
401, 13, 14, 4, 15, 32, 39efsep 12390 . . 3  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( ( 1  +  A )  +  sum_ k  e.  ( ZZ>= ` 
2 ) ( F `
 k ) ) )
41 fac2 11294 . . . . . 6  |-  ( ! `
 2 )  =  2
4241oveq2i 5869 . . . . 5  |-  ( ( A ^ 2 )  /  ( ! ` 
2 ) )  =  ( ( A ^
2 )  /  2
)
4342oveq2i 5869 . . . 4  |-  ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
( ! `  2
) ) )  =  ( ( 1  +  A )  +  ( ( A ^ 2 )  /  2 ) )
4443a1i 10 . . 3  |-  ( A  e.  CC  ->  (
( 1  +  A
)  +  ( ( A ^ 2 )  /  ( ! ` 
2 ) ) )  =  ( ( 1  +  A )  +  ( ( A ^
2 )  /  2
) ) )
451, 11, 12, 4, 7, 40, 44efsep 12390 . 2  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( ( ( 1  +  A )  +  ( ( A ^
2 )  /  2
) )  +  sum_ k  e.  ( ZZ>= ` 
3 ) ( F `
 k ) ) )
46 fac3 11295 . . . . 5  |-  ( ! `
 3 )  =  6
4746oveq2i 5869 . . . 4  |-  ( ( A ^ 3 )  /  ( ! ` 
3 ) )  =  ( ( A ^
3 )  /  6
)
4847oveq2i 5869 . . 3  |-  ( ( ( 1  +  A
)  +  ( ( A ^ 2 )  /  2 ) )  +  ( ( A ^ 3 )  / 
( ! `  3
) ) )  =  ( ( ( 1  +  A )  +  ( ( A ^
2 )  /  2
) )  +  ( ( A ^ 3 )  /  6 ) )
4948a1i 10 . 2  |-  ( A  e.  CC  ->  (
( ( 1  +  A )  +  ( ( A ^ 2 )  /  2 ) )  +  ( ( A ^ 3 )  /  ( ! ` 
3 ) ) )  =  ( ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
2 ) )  +  ( ( A ^
3 )  /  6
) ) )
501, 2, 3, 4, 10, 45, 49efsep 12390 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 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    / cdiv 9423   2c2 9795   3c3 9796   4c4 9797   6c6 9799   NN0cn0 9965   ZZ>=cuz 10230   ^cexp 11104   !cfa 11288   sum_csu 12158   expce 12343
This theorem is referenced by:  efi4p  12417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349
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