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Theorem efi4p 12417
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 8796 . . . 4  |-  _i  e.  CC
2 mulcl 8821 . . . 4  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
31, 2mpan 651 . . 3  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
4 efi4p.1 . . . 4  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
54ef4p 12393 . . 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 15 . 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 8795 . . . . . 6  |-  1  e.  CC
8 addcl 8819 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  e.  CC )
97, 3, 8sylancr 644 . . . . 5  |-  ( A  e.  CC  ->  (
1  +  ( _i  x.  A ) )  e.  CC )
10 sqcl 11166 . . . . . . 7  |-  ( ( _i  x.  A )  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  e.  CC )
113, 10syl 15 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  e.  CC )
1211halfcld 9956 . . . . 5  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  e.  CC )
13 3nn0 9983 . . . . . . 7  |-  3  e.  NN0
14 expcl 11121 . . . . . . 7  |-  ( ( ( _i  x.  A
)  e.  CC  /\  3  e.  NN0 )  -> 
( ( _i  x.  A ) ^ 3 )  e.  CC )
153, 13, 14sylancl 643 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 3 )  e.  CC )
16 6re 9822 . . . . . . . 8  |-  6  e.  RR
1716recni 8849 . . . . . . 7  |-  6  e.  CC
18 6pos 9834 . . . . . . . 8  |-  0  <  6
1916, 18gt0ne0ii 9309 . . . . . . 7  |-  6  =/=  0
20 divcl 9430 . . . . . . 7  |-  ( ( ( ( _i  x.  A ) ^ 3 )  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  e.  CC )
2117, 19, 20mp3an23 1269 . . . . . 6  |-  ( ( ( _i  x.  A
) ^ 3 )  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  e.  CC )
2215, 21syl 15 . . . . 5  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  e.  CC )
239, 12, 22addassd 8857 . . . 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 10 . . . . 5  |-  ( A  e.  CC  ->  1  e.  CC )
2524, 3, 12, 22add4d 9035 . . . 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 9982 . . . . . . . . . . 11  |-  2  e.  NN0
27 mulexp 11141 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC  /\  2  e.  NN0 )  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
281, 26, 27mp3an13 1268 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
29 i2 11203 . . . . . . . . . . . 12  |-  ( _i
^ 2 )  = 
-u 1
3029oveq1i 5868 . . . . . . . . . . 11  |-  ( ( _i ^ 2 )  x.  ( A ^
2 ) )  =  ( -u 1  x.  ( A ^ 2 ) )
3130a1i 10 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i ^ 2 )  x.  ( A ^ 2 ) )  =  ( -u 1  x.  ( A ^ 2 ) ) )
32 sqcl 11166 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
3332mulm1d 9231 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( -u 1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
3428, 31, 333eqtrd 2319 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  -u ( A ^
2 ) )
3534oveq1d 5873 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
36 2cn 9816 . . . . . . . . . 10  |-  2  e.  CC
37 2ne0 9829 . . . . . . . . . 10  |-  2  =/=  0
38 divneg 9455 . . . . . . . . . 10  |-  ( ( ( A ^ 2 )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
( A ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
3936, 37, 38mp3an23 1269 . . . . . . . . 9  |-  ( ( A ^ 2 )  e.  CC  ->  -u (
( A ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
4032, 39syl 15 . . . . . . . 8  |-  ( A  e.  CC  ->  -u (
( A ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
4135, 40eqtr4d 2318 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  =  -u ( ( A ^ 2 )  / 
2 ) )
4241oveq2d 5874 . . . . . 6  |-  ( A  e.  CC  ->  (
1  +  ( ( ( _i  x.  A
) ^ 2 )  /  2 ) )  =  ( 1  + 
-u ( ( A ^ 2 )  / 
2 ) ) )
4332halfcld 9956 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 2 )  /  2 )  e.  CC )
44 negsub 9095 . . . . . . 7  |-  ( ( 1  e.  CC  /\  ( ( A ^
2 )  /  2
)  e.  CC )  ->  ( 1  + 
-u ( ( A ^ 2 )  / 
2 ) )  =  ( 1  -  (
( A ^ 2 )  /  2 ) ) )
457, 43, 44sylancr 644 . . . . . 6  |-  ( A  e.  CC  ->  (
1  +  -u (
( A ^ 2 )  /  2 ) )  =  ( 1  -  ( ( A ^ 2 )  / 
2 ) ) )
4642, 45eqtrd 2315 . . . . 5  |-  ( A  e.  CC  ->  (
1  +  ( ( ( _i  x.  A
) ^ 2 )  /  2 ) )  =  ( 1  -  ( ( A ^
2 )  /  2
) ) )
47 mulexp 11141 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC  /\  3  e.  NN0 )  ->  (
( _i  x.  A
) ^ 3 )  =  ( ( _i
^ 3 )  x.  ( A ^ 3 ) ) )
481, 13, 47mp3an13 1268 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 3 )  =  ( ( _i
^ 3 )  x.  ( A ^ 3 ) ) )
49 i3 11204 . . . . . . . . . . 11  |-  ( _i
^ 3 )  = 
-u _i
5049oveq1i 5868 . . . . . . . . . 10  |-  ( ( _i ^ 3 )  x.  ( A ^
3 ) )  =  ( -u _i  x.  ( A ^ 3 ) )
5148, 50syl6eq 2331 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 3 )  =  ( -u _i  x.  ( A ^ 3 ) ) )
5251oveq1d 5873 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  =  ( ( -u _i  x.  ( A ^
3 ) )  / 
6 ) )
53 expcl 11121 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  CC )
5413, 53mpan2 652 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A ^ 3 )  e.  CC )
551negcli 9114 . . . . . . . . . 10  |-  -u _i  e.  CC
5617, 19pm3.2i 441 . . . . . . . . . 10  |-  ( 6  e.  CC  /\  6  =/=  0 )
57 divass 9442 . . . . . . . . . 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 1268 . . . . . . . . 9  |-  ( ( A ^ 3 )  e.  CC  ->  (
( -u _i  x.  ( A ^ 3 ) )  /  6 )  =  ( -u _i  x.  ( ( A ^
3 )  /  6
) ) )
5954, 58syl 15 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( -u _i  x.  ( A ^ 3 ) )  /  6 )  =  ( -u _i  x.  ( ( A ^
3 )  /  6
) ) )
60 divcl 9430 . . . . . . . . . . 11  |-  ( ( ( A ^ 3 )  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
( A ^ 3 )  /  6 )  e.  CC )
6117, 19, 60mp3an23 1269 . . . . . . . . . 10  |-  ( ( A ^ 3 )  e.  CC  ->  (
( A ^ 3 )  /  6 )  e.  CC )
6254, 61syl 15 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 3 )  /  6 )  e.  CC )
63 mulneg12 9218 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  ( ( A ^
3 )  /  6
)  e.  CC )  ->  ( -u _i  x.  ( ( A ^
3 )  /  6
) )  =  ( _i  x.  -u (
( A ^ 3 )  /  6 ) ) )
641, 62, 63sylancr 644 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u _i  x.  ( ( A ^ 3 )  /  6 ) )  =  ( _i  x.  -u ( ( A ^
3 )  /  6
) ) )
6552, 59, 643eqtrd 2319 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  =  ( _i  x.  -u ( ( A ^
3 )  /  6
) ) )
6665oveq2d 5874 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( ( ( _i  x.  A
) ^ 3 )  /  6 ) )  =  ( ( _i  x.  A )  +  ( _i  x.  -u (
( A ^ 3 )  /  6 ) ) ) )
6762negcld 9144 . . . . . . 7  |-  ( A  e.  CC  ->  -u (
( A ^ 3 )  /  6 )  e.  CC )
68 adddi 8826 . . . . . . . 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 1264 . . . . . . 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 649 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( A  +  -u ( ( A ^ 3 )  / 
6 ) ) )  =  ( ( _i  x.  A )  +  ( _i  x.  -u (
( A ^ 3 )  /  6 ) ) ) )
71 negsub 9095 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( A ^
3 )  /  6
)  e.  CC )  ->  ( A  +  -u ( ( A ^
3 )  /  6
) )  =  ( A  -  ( ( A ^ 3 )  /  6 ) ) )
7262, 71mpdan 649 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  +  -u ( ( A ^ 3 )  /  6 ) )  =  ( A  -  ( ( A ^
3 )  /  6
) ) )
7372oveq2d 5874 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( A  +  -u ( ( A ^ 3 )  / 
6 ) ) )  =  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) )
7466, 70, 733eqtr2d 2321 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( ( ( _i  x.  A
) ^ 3 )  /  6 ) )  =  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) )
7546, 74oveq12d 5876 . . . 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 2319 . . 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 5873 . 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 2315 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 358    = wceq 1623    e. wcel 1684    =/= wne 2446    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / 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:  resin4p  12418  recos4p  12419
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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