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Theorem efi4p 11334
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, 30-Apr-2014.)
Hypothesis
Ref Expression
efi4p.1  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
Assertion
Ref Expression
efi4p  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( ( 1  -  ( ( A ^ 2 )  / 
2 ) )  +  ( _i  x.  ( A  -  ( ( A ^ 3 )  / 
6 ) ) ) )  +  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
) ) )
Distinct variable groups:    A, k, n   
k, F
Allowed substitution hint:    F( n)

Proof of Theorem efi4p
StepHypRef Expression
1 ax-icn 7679 . . . 4  |-  _i  e.  CC
2 mulcl 7711 . . . 4  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
31, 2mpan 418 . . 3  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
4 efi4p.1 . . . 4  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
54ef4p 11310 . . 3  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( ( ( 1  +  ( _i  x.  A ) )  +  ( ( ( _i  x.  A ) ^ 2 )  / 
2 ) )  +  ( ( ( _i  x.  A ) ^
3 )  /  6
) )  +  sum_ k  e.  ( ZZ>= ` 
4 ) ( F `
 k ) ) )
63, 5syl 14 . 2  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( ( ( 1  +  ( _i  x.  A ) )  +  ( ( ( _i  x.  A ) ^ 2 )  / 
2 ) )  +  ( ( ( _i  x.  A ) ^
3 )  /  6
) )  +  sum_ k  e.  ( ZZ>= ` 
4 ) ( F `
 k ) ) )
7 ax-1cn 7677 . . . . . 6  |-  1  e.  CC
8 addcl 7709 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
97, 3, 8sylancr 408 . . . . 5  |-  ( A  e.  CC  ->  (
1  +  ( _i  x.  A ) )  e.  CC )
103sqcld 10373 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  e.  CC )
1110halfcld 8918 . . . . 5  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  e.  CC )
12 3nn0 8949 . . . . . . 7  |-  3  e.  NN0
13 expcl 10262 . . . . . . 7  |-  ( ( ( _i  x.  A
)  e.  CC  /\  3  e.  NN0 )  -> 
( ( _i  x.  A ) ^ 3 )  e.  CC )
143, 12, 13sylancl 407 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 3 )  e.  CC )
15 6cn 8762 . . . . . . 7  |-  6  e.  CC
16 6re 8761 . . . . . . . 8  |-  6  e.  RR
17 6pos 8781 . . . . . . . 8  |-  0  <  6
1816, 17gt0ap0ii 8353 . . . . . . 7  |-  6 #  0
19 divclap 8401 . . . . . . 7  |-  ( ( ( ( _i  x.  A ) ^ 3 )  e.  CC  /\  6  e.  CC  /\  6 #  0 )  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  e.  CC )
2015, 18, 19mp3an23 1290 . . . . . 6  |-  ( ( ( _i  x.  A
) ^ 3 )  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  e.  CC )
2114, 20syl 14 . . . . 5  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  e.  CC )
229, 11, 21addassd 7752 . . . 4  |-  ( A  e.  CC  ->  (
( ( 1  +  ( _i  x.  A
) )  +  ( ( ( _i  x.  A ) ^ 2 )  /  2 ) )  +  ( ( ( _i  x.  A
) ^ 3 )  /  6 ) )  =  ( ( 1  +  ( _i  x.  A ) )  +  ( ( ( ( _i  x.  A ) ^ 2 )  / 
2 )  +  ( ( ( _i  x.  A ) ^ 3 )  /  6 ) ) ) )
237a1i 9 . . . . 5  |-  ( A  e.  CC  ->  1  e.  CC )
2423, 3, 11, 21add4d 7895 . . . 4  |-  ( A  e.  CC  ->  (
( 1  +  ( _i  x.  A ) )  +  ( ( ( ( _i  x.  A ) ^ 2 )  /  2 )  +  ( ( ( _i  x.  A ) ^ 3 )  / 
6 ) ) )  =  ( ( 1  +  ( ( ( _i  x.  A ) ^ 2 )  / 
2 ) )  +  ( ( _i  x.  A )  +  ( ( ( _i  x.  A ) ^ 3 )  /  6 ) ) ) )
25 2nn0 8948 . . . . . . . . . . 11  |-  2  e.  NN0
26 mulexp 10283 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC  /\  2  e.  NN0 )  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
271, 25, 26mp3an13 1289 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
28 i2 10344 . . . . . . . . . . . 12  |-  ( _i
^ 2 )  = 
-u 1
2928oveq1i 5750 . . . . . . . . . . 11  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
3029a1i 9 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i ^ 2 )  x.  ( A ^ 2 ) )  =  ( -u 1  x.  ( A ^ 2 ) ) )
31 sqcl 10305 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
3231mulm1d 8136 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( -u 1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
3327, 30, 323eqtrd 2152 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  -u ( A ^
2 ) )
3433oveq1d 5755 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
35 2cn 8751 . . . . . . . . . 10  |-  2  e.  CC
36 2ap0 8773 . . . . . . . . . 10  |-  2 #  0
37 divnegap 8429 . . . . . . . . . 10  |-  ( ( ( A ^ 2 )  e.  CC  /\  2  e.  CC  /\  2 #  0 )  ->  -u (
( A ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
3835, 36, 37mp3an23 1290 . . . . . . . . 9  |-  ( ( A ^ 2 )  e.  CC  ->  -u (
( A ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
3931, 38syl 14 . . . . . . . 8  |-  ( A  e.  CC  ->  -u (
( A ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
4034, 39eqtr4d 2151 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  =  -u ( ( A ^ 2 )  / 
2 ) )
4140oveq2d 5756 . . . . . 6  |-  ( A  e.  CC  ->  (
1  +  ( ( ( _i  x.  A
) ^ 2 )  /  2 ) )  =  ( 1  + 
-u ( ( A ^ 2 )  / 
2 ) ) )
4231halfcld 8918 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 2 )  /  2 )  e.  CC )
43 negsub 7974 . . . . . . 7  |-  ( ( 1  e.  CC  /\  ( ( A ^
2 )  /  2
)  e.  CC )  ->  ( 1  + 
-u ( ( A ^ 2 )  / 
2 ) )  =  ( 1  -  (
( A ^ 2 )  /  2 ) ) )
447, 42, 43sylancr 408 . . . . . 6  |-  ( A  e.  CC  ->  (
1  +  -u (
( A ^ 2 )  /  2 ) )  =  ( 1  -  ( ( A ^ 2 )  / 
2 ) ) )
4541, 44eqtrd 2148 . . . . 5  |-  ( A  e.  CC  ->  (
1  +  ( ( ( _i  x.  A
) ^ 2 )  /  2 ) )  =  ( 1  -  ( ( A ^
2 )  /  2
) ) )
46 mulexp 10283 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC  /\  3  e.  NN0 )  ->  (
( _i  x.  A
) ^ 3 )  =  ( ( _i
^ 3 )  x.  ( A ^ 3 ) ) )
471, 12, 46mp3an13 1289 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 3 )  =  ( ( _i
^ 3 )  x.  ( A ^ 3 ) ) )
48 i3 10345 . . . . . . . . . . 11  |-  ( _i
^ 3 )  = 
-u _i
4948oveq1i 5750 . . . . . . . . . 10  |-  ( ( _i ^ 3 )  x.  ( A ^
3 ) )  =  ( -u _i  x.  ( A ^ 3 ) )
5047, 49syl6eq 2164 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 3 )  =  ( -u _i  x.  ( A ^ 3 ) ) )
5150oveq1d 5755 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  =  ( ( -u _i  x.  ( A ^
3 ) )  / 
6 ) )
52 expcl 10262 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  CC )
5312, 52mpan2 419 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A ^ 3 )  e.  CC )
54 negicn 7927 . . . . . . . . . 10  |-  -u _i  e.  CC
5515, 18pm3.2i 268 . . . . . . . . . 10  |-  ( 6  e.  CC  /\  6 #  0 )
56 divassap 8413 . . . . . . . . . 10  |-  ( (
-u _i  e.  CC  /\  ( A ^ 3 )  e.  CC  /\  ( 6  e.  CC  /\  6 #  0 ) )  ->  ( ( -u _i  x.  ( A ^
3 ) )  / 
6 )  =  (
-u _i  x.  (
( A ^ 3 )  /  6 ) ) )
5754, 55, 56mp3an13 1289 . . . . . . . . 9  |-  ( ( A ^ 3 )  e.  CC  ->  (
( -u _i  x.  ( A ^ 3 ) )  /  6 )  =  ( -u _i  x.  ( ( A ^
3 )  /  6
) ) )
5853, 57syl 14 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( -u _i  x.  ( A ^ 3 ) )  /  6 )  =  ( -u _i  x.  ( ( A ^
3 )  /  6
) ) )
59 divclap 8401 . . . . . . . . . . 11  |-  ( ( ( A ^ 3 )  e.  CC  /\  6  e.  CC  /\  6 #  0 )  ->  (
( A ^ 3 )  /  6 )  e.  CC )
6015, 18, 59mp3an23 1290 . . . . . . . . . 10  |-  ( ( A ^ 3 )  e.  CC  ->  (
( A ^ 3 )  /  6 )  e.  CC )
6153, 60syl 14 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 3 )  /  6 )  e.  CC )
62 mulneg12 8123 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  ( ( A ^
3 )  /  6
)  e.  CC )  ->  ( -u _i  x.  ( ( A ^
3 )  /  6
) )  =  ( _i  x.  -u (
( A ^ 3 )  /  6 ) ) )
631, 61, 62sylancr 408 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u _i  x.  ( ( A ^ 3 )  /  6 ) )  =  ( _i  x.  -u ( ( A ^
3 )  /  6
) ) )
6451, 58, 633eqtrd 2152 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  =  ( _i  x.  -u ( ( A ^
3 )  /  6
) ) )
6564oveq2d 5756 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( ( ( _i  x.  A
) ^ 3 )  /  6 ) )  =  ( ( _i  x.  A )  +  ( _i  x.  -u (
( A ^ 3 )  /  6 ) ) ) )
6661negcld 8024 . . . . . . 7  |-  ( A  e.  CC  ->  -u (
( A ^ 3 )  /  6 )  e.  CC )
67 adddi 7716 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC  /\  -u (
( A ^ 3 )  /  6 )  e.  CC )  -> 
( _i  x.  ( A  +  -u ( ( A ^ 3 )  /  6 ) ) )  =  ( ( _i  x.  A )  +  ( _i  x.  -u ( ( A ^
3 )  /  6
) ) ) )
681, 67mp3an1 1285 . . . . . . 7  |-  ( ( A  e.  CC  /\  -u ( ( A ^
3 )  /  6
)  e.  CC )  ->  ( _i  x.  ( A  +  -u (
( A ^ 3 )  /  6 ) ) )  =  ( ( _i  x.  A
)  +  ( _i  x.  -u ( ( A ^ 3 )  / 
6 ) ) ) )
6966, 68mpdan 415 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( A  +  -u ( ( A ^ 3 )  / 
6 ) ) )  =  ( ( _i  x.  A )  +  ( _i  x.  -u (
( A ^ 3 )  /  6 ) ) ) )
70 negsub 7974 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( A ^
3 )  /  6
)  e.  CC )  ->  ( A  +  -u ( ( A ^
3 )  /  6
) )  =  ( A  -  ( ( A ^ 3 )  /  6 ) ) )
7161, 70mpdan 415 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  +  -u ( ( A ^ 3 )  /  6 ) )  =  ( A  -  ( ( A ^
3 )  /  6
) ) )
7271oveq2d 5756 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( A  +  -u ( ( A ^ 3 )  / 
6 ) ) )  =  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) )
7365, 69, 723eqtr2d 2154 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( ( ( _i  x.  A
) ^ 3 )  /  6 ) )  =  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) )
7445, 73oveq12d 5758 . . . 4  |-  ( A  e.  CC  ->  (
( 1  +  ( ( ( _i  x.  A ) ^ 2 )  /  2 ) )  +  ( ( _i  x.  A )  +  ( ( ( _i  x.  A ) ^ 3 )  / 
6 ) ) )  =  ( ( 1  -  ( ( A ^ 2 )  / 
2 ) )  +  ( _i  x.  ( A  -  ( ( A ^ 3 )  / 
6 ) ) ) ) )
7522, 24, 743eqtrd 2152 . . 3  |-  ( A  e.  CC  ->  (
( ( 1  +  ( _i  x.  A
) )  +  ( ( ( _i  x.  A ) ^ 2 )  /  2 ) )  +  ( ( ( _i  x.  A
) ^ 3 )  /  6 ) )  =  ( ( 1  -  ( ( A ^ 2 )  / 
2 ) )  +  ( _i  x.  ( A  -  ( ( A ^ 3 )  / 
6 ) ) ) ) )
7675oveq1d 5755 . 2  |-  ( A  e.  CC  ->  (
( ( ( 1  +  ( _i  x.  A ) )  +  ( ( ( _i  x.  A ) ^
2 )  /  2
) )  +  ( ( ( _i  x.  A ) ^ 3 )  /  6 ) )  +  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
) )  =  ( ( ( 1  -  ( ( A ^
2 )  /  2
) )  +  ( _i  x.  ( A  -  ( ( A ^ 3 )  / 
6 ) ) ) )  +  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
) ) )
776, 76eqtrd 2148 1  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( ( 1  -  ( ( A ^ 2 )  / 
2 ) )  +  ( _i  x.  ( A  -  ( ( A ^ 3 )  / 
6 ) ) ) )  +  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   class class class wbr 3897    |-> cmpt 3957   ` cfv 5091  (class class class)co 5740   CCcc 7582   0cc0 7584   1c1 7585   _ici 7586    + caddc 7587    x. cmul 7589    - cmin 7897   -ucneg 7898   # cap 8306    / cdiv 8395   2c2 8731   3c3 8732   4c4 8733   6c6 8735   NN0cn0 8931   ZZ>=cuz 9278   ^cexp 10243   !cfa 10422   sum_csu 11073   expce 11258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-5 8742  df-6 8743  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-ico 9628  df-fz 9742  df-fzo 9871  df-seqfrec 10170  df-exp 10244  df-fac 10423  df-ihash 10473  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-clim 10999  df-sumdc 11074  df-ef 11264
This theorem is referenced by:  resin4p  11335  recos4p  11336
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