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Theorem efi4p 12433
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 8812 . . . 4  |-  _i  e.  CC
2 mulcl 8837 . . . 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 12409 . . 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 8811 . . . . . 6  |-  1  e.  CC
8 addcl 8835 . . . . . 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 11182 . . . . . . 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 9972 . . . . 5  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  e.  CC )
13 3nn0 9999 . . . . . . 7  |-  3  e.  NN0
14 expcl 11137 . . . . . . 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 9838 . . . . . . . 8  |-  6  e.  RR
1716recni 8865 . . . . . . 7  |-  6  e.  CC
18 6pos 9850 . . . . . . . 8  |-  0  <  6
1916, 18gt0ne0ii 9325 . . . . . . 7  |-  6  =/=  0
20 divcl 9446 . . . . . . 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 8873 . . . 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 9051 . . . 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 9998 . . . . . . . . . . 11  |-  2  e.  NN0
27 mulexp 11157 . . . . . . . . . . 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 11219 . . . . . . . . . . . 12  |-  ( _i
^ 2 )  = 
-u 1
3029oveq1i 5884 . . . . . . . . . . 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 11182 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
3332mulm1d 9247 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( -u 1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
3428, 31, 333eqtrd 2332 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  -u ( A ^
2 ) )
3534oveq1d 5889 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
36 2cn 9832 . . . . . . . . . 10  |-  2  e.  CC
37 2ne0 9845 . . . . . . . . . 10  |-  2  =/=  0
38 divneg 9471 . . . . . . . . . 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 2331 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  =  -u ( ( A ^ 2 )  / 
2 ) )
4241oveq2d 5890 . . . . . 6  |-  ( A  e.  CC  ->  (
1  +  ( ( ( _i  x.  A
) ^ 2 )  /  2 ) )  =  ( 1  + 
-u ( ( A ^ 2 )  / 
2 ) ) )
4332halfcld 9972 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 2 )  /  2 )  e.  CC )
44 negsub 9111 . . . . . . 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 2328 . . . . 5  |-  ( A  e.  CC  ->  (
1  +  ( ( ( _i  x.  A
) ^ 2 )  /  2 ) )  =  ( 1  -  ( ( A ^
2 )  /  2
) ) )
47 mulexp 11157 . . . . . . . . . . 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 11220 . . . . . . . . . . 11  |-  ( _i
^ 3 )  = 
-u _i
5049oveq1i 5884 . . . . . . . . . 10  |-  ( ( _i ^ 3 )  x.  ( A ^
3 ) )  =  ( -u _i  x.  ( A ^ 3 ) )
5148, 50syl6eq 2344 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 3 )  =  ( -u _i  x.  ( A ^ 3 ) ) )
5251oveq1d 5889 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  =  ( ( -u _i  x.  ( A ^
3 ) )  / 
6 ) )
53 expcl 11137 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  CC )
5413, 53mpan2 652 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A ^ 3 )  e.  CC )
551negcli 9130 . . . . . . . . . 10  |-  -u _i  e.  CC
5617, 19pm3.2i 441 . . . . . . . . . 10  |-  ( 6  e.  CC  /\  6  =/=  0 )
57 divass 9458 . . . . . . . . . 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 9446 . . . . . . . . . . 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 9234 . . . . . . . . 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 2332 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  =  ( _i  x.  -u ( ( A ^
3 )  /  6
) ) )
6665oveq2d 5890 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( ( ( _i  x.  A
) ^ 3 )  /  6 ) )  =  ( ( _i  x.  A )  +  ( _i  x.  -u (
( A ^ 3 )  /  6 ) ) ) )
6762negcld 9160 . . . . . . 7  |-  ( A  e.  CC  ->  -u (
( A ^ 3 )  /  6 )  e.  CC )
68 adddi 8842 . . . . . . . 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 9111 . . . . . . . 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 5890 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( A  +  -u ( ( A ^ 3 )  / 
6 ) ) )  =  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) )
7466, 70, 733eqtr2d 2334 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( ( ( _i  x.  A
) ^ 3 )  /  6 ) )  =  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) )
7546, 74oveq12d 5892 . . . 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 2332 . . 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 5889 . 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 2328 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 1632    e. wcel 1696    =/= wne 2459    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   3c3 9812   4c4 9813   6c6 9815   NN0cn0 9981   ZZ>=cuz 10246   ^cexp 11120   !cfa 11304   sum_csu 12174   expce 12359
This theorem is referenced by:  resin4p  12434  recos4p  12435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365
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