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Theorem efi4p 13413
Description: Separate out the first four terms of the infinite series expansion of the exponential function of an imaginary number. (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 9333 . . . 4  |-  _i  e.  CC
2 mulcl 9358 . . . 4  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
31, 2mpan 670 . . 3  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
4 efi4p.1 . . . 4  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
54ef4p 13389 . . 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 16 . 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 9332 . . . . . 6  |-  1  e.  CC
8 addcl 9356 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
97, 3, 8sylancr 663 . . . . 5  |-  ( A  e.  CC  ->  (
1  +  ( _i  x.  A ) )  e.  CC )
10 sqcl 11920 . . . . . . 7  |-  ( ( _i  x.  A )  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  e.  CC )
113, 10syl 16 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  e.  CC )
1211halfcld 10561 . . . . 5  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  e.  CC )
13 3nn0 10589 . . . . . . 7  |-  3  e.  NN0
14 expcl 11875 . . . . . . 7  |-  ( ( ( _i  x.  A
)  e.  CC  /\  3  e.  NN0 )  -> 
( ( _i  x.  A ) ^ 3 )  e.  CC )
153, 13, 14sylancl 662 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 3 )  e.  CC )
16 6cn 10395 . . . . . . 7  |-  6  e.  CC
17 6re 10394 . . . . . . . 8  |-  6  e.  RR
18 6pos 10412 . . . . . . . 8  |-  0  <  6
1917, 18gt0ne0ii 9868 . . . . . . 7  |-  6  =/=  0
20 divcl 9992 . . . . . . 7  |-  ( ( ( ( _i  x.  A ) ^ 3 )  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  e.  CC )
2116, 19, 20mp3an23 1306 . . . . . 6  |-  ( ( ( _i  x.  A
) ^ 3 )  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  e.  CC )
2215, 21syl 16 . . . . 5  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  e.  CC )
239, 12, 22addassd 9400 . . . 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 ) ) ) )
247a1i 11 . . . . 5  |-  ( A  e.  CC  ->  1  e.  CC )
2524, 3, 12, 22add4d 9585 . . . 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 ) ) ) )
26 2nn0 10588 . . . . . . . . . . 11  |-  2  e.  NN0
27 mulexp 11895 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC  /\  2  e.  NN0 )  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
281, 26, 27mp3an13 1305 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
29 i2 11958 . . . . . . . . . . . 12  |-  ( _i
^ 2 )  = 
-u 1
3029oveq1i 6096 . . . . . . . . . . 11  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
3130a1i 11 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i ^ 2 )  x.  ( A ^ 2 ) )  =  ( -u 1  x.  ( A ^ 2 ) ) )
32 sqcl 11920 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
3332mulm1d 9788 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( -u 1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
3428, 31, 333eqtrd 2474 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  -u ( A ^
2 ) )
3534oveq1d 6101 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
36 2cn 10384 . . . . . . . . . 10  |-  2  e.  CC
37 2ne0 10406 . . . . . . . . . 10  |-  2  =/=  0
38 divneg 10018 . . . . . . . . . 10  |-  ( ( ( A ^ 2 )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
( A ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
3936, 37, 38mp3an23 1306 . . . . . . . . 9  |-  ( ( A ^ 2 )  e.  CC  ->  -u (
( A ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
4032, 39syl 16 . . . . . . . 8  |-  ( A  e.  CC  ->  -u (
( A ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
4135, 40eqtr4d 2473 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  =  -u ( ( A ^ 2 )  / 
2 ) )
4241oveq2d 6102 . . . . . 6  |-  ( A  e.  CC  ->  (
1  +  ( ( ( _i  x.  A
) ^ 2 )  /  2 ) )  =  ( 1  + 
-u ( ( A ^ 2 )  / 
2 ) ) )
4332halfcld 10561 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 2 )  /  2 )  e.  CC )
44 negsub 9649 . . . . . . 7  |-  ( ( 1  e.  CC  /\  ( ( A ^
2 )  /  2
)  e.  CC )  ->  ( 1  + 
-u ( ( A ^ 2 )  / 
2 ) )  =  ( 1  -  (
( A ^ 2 )  /  2 ) ) )
457, 43, 44sylancr 663 . . . . . 6  |-  ( A  e.  CC  ->  (
1  +  -u (
( A ^ 2 )  /  2 ) )  =  ( 1  -  ( ( A ^ 2 )  / 
2 ) ) )
4642, 45eqtrd 2470 . . . . 5  |-  ( A  e.  CC  ->  (
1  +  ( ( ( _i  x.  A
) ^ 2 )  /  2 ) )  =  ( 1  -  ( ( A ^
2 )  /  2
) ) )
47 mulexp 11895 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC  /\  3  e.  NN0 )  ->  (
( _i  x.  A
) ^ 3 )  =  ( ( _i
^ 3 )  x.  ( A ^ 3 ) ) )
481, 13, 47mp3an13 1305 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 3 )  =  ( ( _i
^ 3 )  x.  ( A ^ 3 ) ) )
49 i3 11959 . . . . . . . . . . 11  |-  ( _i
^ 3 )  = 
-u _i
5049oveq1i 6096 . . . . . . . . . 10  |-  ( ( _i ^ 3 )  x.  ( A ^
3 ) )  =  ( -u _i  x.  ( A ^ 3 ) )
5148, 50syl6eq 2486 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 3 )  =  ( -u _i  x.  ( A ^ 3 ) ) )
5251oveq1d 6101 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  =  ( ( -u _i  x.  ( A ^
3 ) )  / 
6 ) )
53 expcl 11875 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  CC )
5413, 53mpan2 671 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A ^ 3 )  e.  CC )
55 negicn 9603 . . . . . . . . . 10  |-  -u _i  e.  CC
5616, 19pm3.2i 455 . . . . . . . . . 10  |-  ( 6  e.  CC  /\  6  =/=  0 )
57 divass 10004 . . . . . . . . . 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
) ) )
5855, 56, 57mp3an13 1305 . . . . . . . . 9  |-  ( ( A ^ 3 )  e.  CC  ->  (
( -u _i  x.  ( A ^ 3 ) )  /  6 )  =  ( -u _i  x.  ( ( A ^
3 )  /  6
) ) )
5954, 58syl 16 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( -u _i  x.  ( A ^ 3 ) )  /  6 )  =  ( -u _i  x.  ( ( A ^
3 )  /  6
) ) )
60 divcl 9992 . . . . . . . . . . 11  |-  ( ( ( A ^ 3 )  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
( A ^ 3 )  /  6 )  e.  CC )
6116, 19, 60mp3an23 1306 . . . . . . . . . 10  |-  ( ( A ^ 3 )  e.  CC  ->  (
( A ^ 3 )  /  6 )  e.  CC )
6254, 61syl 16 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 3 )  /  6 )  e.  CC )
63 mulneg12 9775 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  ( ( A ^
3 )  /  6
)  e.  CC )  ->  ( -u _i  x.  ( ( A ^
3 )  /  6
) )  =  ( _i  x.  -u (
( A ^ 3 )  /  6 ) ) )
641, 62, 63sylancr 663 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u _i  x.  ( ( A ^ 3 )  /  6 ) )  =  ( _i  x.  -u ( ( A ^
3 )  /  6
) ) )
6552, 59, 643eqtrd 2474 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  =  ( _i  x.  -u ( ( A ^
3 )  /  6
) ) )
6665oveq2d 6102 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( ( ( _i  x.  A
) ^ 3 )  /  6 ) )  =  ( ( _i  x.  A )  +  ( _i  x.  -u (
( A ^ 3 )  /  6 ) ) ) )
6762negcld 9698 . . . . . . 7  |-  ( A  e.  CC  ->  -u (
( A ^ 3 )  /  6 )  e.  CC )
68 adddi 9363 . . . . . . . 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
) ) ) )
691, 68mp3an1 1301 . . . . . . 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 ) ) ) )
7067, 69mpdan 668 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( A  +  -u ( ( A ^ 3 )  / 
6 ) ) )  =  ( ( _i  x.  A )  +  ( _i  x.  -u (
( A ^ 3 )  /  6 ) ) ) )
71 negsub 9649 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( A ^
3 )  /  6
)  e.  CC )  ->  ( A  +  -u ( ( A ^
3 )  /  6
) )  =  ( A  -  ( ( A ^ 3 )  /  6 ) ) )
7262, 71mpdan 668 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  +  -u ( ( A ^ 3 )  /  6 ) )  =  ( A  -  ( ( A ^
3 )  /  6
) ) )
7372oveq2d 6102 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( A  +  -u ( ( A ^ 3 )  / 
6 ) ) )  =  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) )
7466, 70, 733eqtr2d 2476 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( ( ( _i  x.  A
) ^ 3 )  /  6 ) )  =  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) )
7546, 74oveq12d 6104 . . . 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 ) ) ) ) )
7623, 25, 753eqtrd 2474 . . 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 ) ) ) ) )
7776oveq1d 6101 . 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
) ) )
786, 77eqtrd 2470 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274   1c1 9275   _ici 9276    + caddc 9277    x. cmul 9279    - cmin 9587   -ucneg 9588    / cdiv 9985   2c2 10363   3c3 10364   4c4 10365   6c6 10367   NN0cn0 10571   ZZ>=cuz 10853   ^cexp 11857   !cfa 12043   sum_csu 13155   expce 13339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-ico 11298  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-fac 12044  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345
This theorem is referenced by:  resin4p  13414  recos4p  13415
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