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Theorem ef4p 13860
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 10617 . 2  |-  4  =  ( 3  +  1 )
3 3nn0 10834 . 2  |-  3  e.  NN0
4 id 22 . 2  |-  ( A  e.  CC  ->  A  e.  CC )
5 ax-1cn 9567 . . . 4  |-  1  e.  CC
6 addcl 9591 . . . 4  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  A
)  e.  CC )
75, 6mpan 670 . . 3  |-  ( A  e.  CC  ->  (
1  +  A )  e.  CC )
8 sqcl 12233 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
98halfcld 10804 . . 3  |-  ( A  e.  CC  ->  (
( A ^ 2 )  /  2 )  e.  CC )
107, 9addcld 9632 . 2  |-  ( A  e.  CC  ->  (
( 1  +  A
)  +  ( ( A ^ 2 )  /  2 ) )  e.  CC )
11 df-3 10616 . . 3  |-  3  =  ( 2  +  1 )
12 2nn0 10833 . . 3  |-  2  e.  NN0
13 df-2 10615 . . . 4  |-  2  =  ( 1  +  1 )
14 1nn0 10832 . . . 4  |-  1  e.  NN0
155a1i 11 . . . 4  |-  ( A  e.  CC  ->  1  e.  CC )
16 1e0p1 11028 . . . . 5  |-  1  =  ( 0  +  1 )
17 0nn0 10831 . . . . 5  |-  0  e.  NN0
18 0cnd 9606 . . . . 5  |-  ( A  e.  CC  ->  0  e.  CC )
191efval2 13831 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  A )  = 
sum_ k  e.  NN0  ( F `  k ) )
20 nn0uz 11140 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
2120sumeq1i 13532 . . . . . . . 8  |-  sum_ k  e.  NN0  ( F `  k )  =  sum_ k  e.  ( ZZ>= ` 
0 ) ( F `
 k )
2219, 21syl6req 2515 . . . . . . 7  |-  ( A  e.  CC  ->  sum_ k  e.  ( ZZ>= `  0 )
( F `  k
)  =  ( exp `  A ) )
2322oveq2d 6312 . . . . . 6  |-  ( A  e.  CC  ->  (
0  +  sum_ k  e.  ( ZZ>= `  0 )
( F `  k
) )  =  ( 0  +  ( exp `  A ) ) )
24 efcl 13830 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2524addid2d 9798 . . . . . 6  |-  ( A  e.  CC  ->  (
0  +  ( exp `  A ) )  =  ( exp `  A
) )
2623, 25eqtr2d 2499 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( 0  +  sum_ k  e.  ( ZZ>= ` 
0 ) ( F `
 k ) ) )
27 eft0val 13859 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
2827oveq2d 6312 . . . . . 6  |-  ( A  e.  CC  ->  (
0  +  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )  =  ( 0  +  1 ) )
29 0p1e1 10668 . . . . . 6  |-  ( 0  +  1 )  =  1
3028, 29syl6eq 2514 . . . . 5  |-  ( A  e.  CC  ->  (
0  +  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )  =  1 )
311, 16, 17, 4, 18, 26, 30efsep 13857 . . . 4  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( 1  +  sum_ k  e.  ( ZZ>= ` 
1 ) ( F `
 k ) ) )
32 exp1 12175 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
33 fac1 12360 . . . . . . . 8  |-  ( ! `
 1 )  =  1
3433a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  ( ! `  1 )  =  1 )
3532, 34oveq12d 6314 . . . . . 6  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  ( A  / 
1 ) )
36 div1 10257 . . . . . 6  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
3735, 36eqtrd 2498 . . . . 5  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
3837oveq2d 6312 . . . 4  |-  ( A  e.  CC  ->  (
1  +  ( ( A ^ 1 )  /  ( ! ` 
1 ) ) )  =  ( 1  +  A ) )
391, 13, 14, 4, 15, 31, 38efsep 13857 . . 3  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( ( 1  +  A )  +  sum_ k  e.  ( ZZ>= ` 
2 ) ( F `
 k ) ) )
40 fac2 12362 . . . . . 6  |-  ( ! `
 2 )  =  2
4140oveq2i 6307 . . . . 5  |-  ( ( A ^ 2 )  /  ( ! ` 
2 ) )  =  ( ( A ^
2 )  /  2
)
4241oveq2i 6307 . . . 4  |-  ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
( ! `  2
) ) )  =  ( ( 1  +  A )  +  ( ( A ^ 2 )  /  2 ) )
4342a1i 11 . . 3  |-  ( A  e.  CC  ->  (
( 1  +  A
)  +  ( ( A ^ 2 )  /  ( ! ` 
2 ) ) )  =  ( ( 1  +  A )  +  ( ( A ^
2 )  /  2
) ) )
441, 11, 12, 4, 7, 39, 43efsep 13857 . 2  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( ( ( 1  +  A )  +  ( ( A ^
2 )  /  2
) )  +  sum_ k  e.  ( ZZ>= ` 
3 ) ( F `
 k ) ) )
45 fac3 12363 . . . . 5  |-  ( ! `
 3 )  =  6
4645oveq2i 6307 . . . 4  |-  ( ( A ^ 3 )  /  ( ! ` 
3 ) )  =  ( ( A ^
3 )  /  6
)
4746oveq2i 6307 . . 3  |-  ( ( ( 1  +  A
)  +  ( ( A ^ 2 )  /  2 ) )  +  ( ( A ^ 3 )  / 
( ! `  3
) ) )  =  ( ( ( 1  +  A )  +  ( ( A ^
2 )  /  2
) )  +  ( ( A ^ 3 )  /  6 ) )
4847a1i 11 . 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 13857 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 setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    / cdiv 10227   2c2 10606   3c3 10607   4c4 10608   6c6 10610   NN0cn0 10816   ZZ>=cuz 11106   ^cexp 12169   !cfa 12356   sum_csu 13520   expce 13809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-ico 11560  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-fac 12357  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815
This theorem is referenced by:  efi4p  13884
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