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Theorem ef4p 12440
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 9851 . 2  |-  4  =  ( 3  +  1 )
3 3nn0 10030 . 2  |-  3  e.  NN0
4 id 19 . 2  |-  ( A  e.  CC  ->  A  e.  CC )
5 ax-1cn 8840 . . . 4  |-  1  e.  CC
6 addcl 8864 . . . 4  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  A
)  e.  CC )
75, 6mpan 651 . . 3  |-  ( A  e.  CC  ->  (
1  +  A )  e.  CC )
8 sqcl 11213 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
98halfcld 10003 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 2 )  /  2 )  e.  CC )
107, 9addcld 8899 . 2  |-  ( A  e.  CC  ->  (
( 1  +  A
)  +  ( ( A ^ 2 )  /  2 ) )  e.  CC )
11 df-3 9850 . . 3  |-  3  =  ( 2  +  1 )
12 2nn0 10029 . . 3  |-  2  e.  NN0
13 df-2 9849 . . . 4  |-  2  =  ( 1  +  1 )
14 1nn0 10028 . . . 4  |-  1  e.  NN0
155a1i 10 . . . 4  |-  ( A  e.  CC  ->  1  e.  CC )
16 1e0p1 10199 . . . . 5  |-  1  =  ( 0  +  1 )
17 0nn0 10027 . . . . 5  |-  0  e.  NN0
18 0cn 8876 . . . . . 6  |-  0  e.  CC
1918a1i 10 . . . . 5  |-  ( A  e.  CC  ->  0  e.  CC )
201efval2 12412 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  A )  = 
sum_ k  e.  NN0  ( F `  k ) )
21 nn0uz 10309 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
2221sumeq1i 12218 . . . . . . . 8  |-  sum_ k  e.  NN0  ( F `  k )  =  sum_ k  e.  ( ZZ>= ` 
0 ) ( F `
 k )
2320, 22syl6req 2365 . . . . . . 7  |-  ( A  e.  CC  ->  sum_ k  e.  ( ZZ>= `  0 )
( F `  k
)  =  ( exp `  A ) )
2423oveq2d 5916 . . . . . 6  |-  ( A  e.  CC  ->  (
0  +  sum_ k  e.  ( ZZ>= `  0 )
( F `  k
) )  =  ( 0  +  ( exp `  A ) ) )
25 efcl 12411 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2625addid2d 9058 . . . . . 6  |-  ( A  e.  CC  ->  (
0  +  ( exp `  A ) )  =  ( exp `  A
) )
2724, 26eqtr2d 2349 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( 0  +  sum_ k  e.  ( ZZ>= ` 
0 ) ( F `
 k ) ) )
28 eft0val 12439 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
2928oveq2d 5916 . . . . . 6  |-  ( A  e.  CC  ->  (
0  +  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )  =  ( 0  +  1 ) )
30 0p1e1 9884 . . . . . 6  |-  ( 0  +  1 )  =  1
3129, 30syl6eq 2364 . . . . 5  |-  ( A  e.  CC  ->  (
0  +  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )  =  1 )
321, 16, 17, 4, 19, 27, 31efsep 12437 . . . 4  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( 1  +  sum_ k  e.  ( ZZ>= ` 
1 ) ( F `
 k ) ) )
33 exp1 11156 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
34 fac1 11339 . . . . . . . 8  |-  ( ! `
 1 )  =  1
3534a1i 10 . . . . . . 7  |-  ( A  e.  CC  ->  ( ! `  1 )  =  1 )
3633, 35oveq12d 5918 . . . . . 6  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  ( A  / 
1 ) )
37 div1 9498 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
3836, 37eqtrd 2348 . . . . 5  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
3938oveq2d 5916 . . . 4  |-  ( A  e.  CC  ->  (
1  +  ( ( A ^ 1 )  /  ( ! ` 
1 ) ) )  =  ( 1  +  A ) )
401, 13, 14, 4, 15, 32, 39efsep 12437 . . 3  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( ( 1  +  A )  +  sum_ k  e.  ( ZZ>= ` 
2 ) ( F `
 k ) ) )
41 fac2 11341 . . . . . 6  |-  ( ! `
 2 )  =  2
4241oveq2i 5911 . . . . 5  |-  ( ( A ^ 2 )  /  ( ! ` 
2 ) )  =  ( ( A ^
2 )  /  2
)
4342oveq2i 5911 . . . 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 12437 . 2  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( ( ( 1  +  A )  +  ( ( A ^
2 )  /  2
) )  +  sum_ k  e.  ( ZZ>= ` 
3 ) ( F `
 k ) ) )
46 fac3 11342 . . . . 5  |-  ( ! `
 3 )  =  6
4746oveq2i 5911 . . . 4  |-  ( ( A ^ 3 )  /  ( ! ` 
3 ) )  =  ( ( A ^
3 )  /  6
)
4847oveq2i 5911 . . 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 12437 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 1633    e. wcel 1701    e. cmpt 4114   ` cfv 5292  (class class class)co 5900   CCcc 8780   0cc0 8782   1c1 8783    + caddc 8785    / cdiv 9468   2c2 9840   3c3 9841   4c4 9842   6c6 9844   NN0cn0 10012   ZZ>=cuz 10277   ^cexp 11151   !cfa 11335   sum_csu 12205   expce 12390
This theorem is referenced by:  efi4p  12464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-pm 6818  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-oi 7270  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-ico 10709  df-fz 10830  df-fzo 10918  df-fl 10972  df-seq 11094  df-exp 11152  df-fac 11336  df-hash 11385  df-shft 11609  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-limsup 11992  df-clim 12009  df-rlim 12010  df-sum 12206  df-ef 12396
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