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Theorem efi4p 13542
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 9455 . . . 4  |-  _i  e.  CC
2 mulcl 9480 . . . 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 13518 . . 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 9454 . . . . . 6  |-  1  e.  CC
8 addcl 9478 . . . . . 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 12048 . . . . . . 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 10683 . . . . 5  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  e.  CC )
13 3nn0 10711 . . . . . . 7  |-  3  e.  NN0
14 expcl 12003 . . . . . . 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 10517 . . . . . . 7  |-  6  e.  CC
17 6re 10516 . . . . . . . 8  |-  6  e.  RR
18 6pos 10534 . . . . . . . 8  |-  0  <  6
1917, 18gt0ne0ii 9990 . . . . . . 7  |-  6  =/=  0
20 divcl 10114 . . . . . . 7  |-  ( ( ( ( _i  x.  A ) ^ 3 )  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  e.  CC )
2116, 19, 20mp3an23 1307 . . . . . 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 9522 . . . 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 9707 . . . 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 10710 . . . . . . . . . . 11  |-  2  e.  NN0
27 mulexp 12023 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC  /\  2  e.  NN0 )  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
281, 26, 27mp3an13 1306 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  ( ( _i
^ 2 )  x.  ( A ^ 2 ) ) )
29 i2 12086 . . . . . . . . . . . 12  |-  ( _i
^ 2 )  = 
-u 1
3029oveq1i 6213 . . . . . . . . . . 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 12048 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
3332mulm1d 9910 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( -u 1  x.  ( A ^ 2 ) )  =  -u ( A ^
2 ) )
3428, 31, 333eqtrd 2499 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 2 )  =  -u ( A ^
2 ) )
3534oveq1d 6218 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
36 2cn 10506 . . . . . . . . . 10  |-  2  e.  CC
37 2ne0 10528 . . . . . . . . . 10  |-  2  =/=  0
38 divneg 10140 . . . . . . . . . 10  |-  ( ( ( A ^ 2 )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
( A ^ 2 )  /  2 )  =  ( -u ( A ^ 2 )  / 
2 ) )
3936, 37, 38mp3an23 1307 . . . . . . . . 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 2498 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 2 )  /  2 )  =  -u ( ( A ^ 2 )  / 
2 ) )
4241oveq2d 6219 . . . . . 6  |-  ( A  e.  CC  ->  (
1  +  ( ( ( _i  x.  A
) ^ 2 )  /  2 ) )  =  ( 1  + 
-u ( ( A ^ 2 )  / 
2 ) ) )
4332halfcld 10683 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 2 )  /  2 )  e.  CC )
44 negsub 9771 . . . . . . 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 2495 . . . . 5  |-  ( A  e.  CC  ->  (
1  +  ( ( ( _i  x.  A
) ^ 2 )  /  2 ) )  =  ( 1  -  ( ( A ^
2 )  /  2
) ) )
47 mulexp 12023 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC  /\  3  e.  NN0 )  ->  (
( _i  x.  A
) ^ 3 )  =  ( ( _i
^ 3 )  x.  ( A ^ 3 ) ) )
481, 13, 47mp3an13 1306 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 3 )  =  ( ( _i
^ 3 )  x.  ( A ^ 3 ) ) )
49 i3 12087 . . . . . . . . . . 11  |-  ( _i
^ 3 )  = 
-u _i
5049oveq1i 6213 . . . . . . . . . 10  |-  ( ( _i ^ 3 )  x.  ( A ^
3 ) )  =  ( -u _i  x.  ( A ^ 3 ) )
5148, 50syl6eq 2511 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( _i  x.  A
) ^ 3 )  =  ( -u _i  x.  ( A ^ 3 ) ) )
5251oveq1d 6218 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  =  ( ( -u _i  x.  ( A ^
3 ) )  / 
6 ) )
53 expcl 12003 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  CC )
5413, 53mpan2 671 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A ^ 3 )  e.  CC )
55 negicn 9725 . . . . . . . . . 10  |-  -u _i  e.  CC
5616, 19pm3.2i 455 . . . . . . . . . 10  |-  ( 6  e.  CC  /\  6  =/=  0 )
57 divass 10126 . . . . . . . . . 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 1306 . . . . . . . . 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 10114 . . . . . . . . . . 11  |-  ( ( ( A ^ 3 )  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
( A ^ 3 )  /  6 )  e.  CC )
6116, 19, 60mp3an23 1307 . . . . . . . . . 10  |-  ( ( A ^ 3 )  e.  CC  ->  (
( A ^ 3 )  /  6 )  e.  CC )
6254, 61syl 16 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 3 )  /  6 )  e.  CC )
63 mulneg12 9897 . . . . . . . . 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 2499 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( _i  x.  A ) ^ 3 )  /  6 )  =  ( _i  x.  -u ( ( A ^
3 )  /  6
) ) )
6665oveq2d 6219 . . . . . 6  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( ( ( _i  x.  A
) ^ 3 )  /  6 ) )  =  ( ( _i  x.  A )  +  ( _i  x.  -u (
( A ^ 3 )  /  6 ) ) ) )
6762negcld 9820 . . . . . . 7  |-  ( A  e.  CC  ->  -u (
( A ^ 3 )  /  6 )  e.  CC )
68 adddi 9485 . . . . . . . 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 1302 . . . . . . 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 9771 . . . . . . . 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 6219 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( A  +  -u ( ( A ^ 3 )  / 
6 ) ) )  =  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) )
7466, 70, 733eqtr2d 2501 . . . . 5  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( ( ( _i  x.  A
) ^ 3 )  /  6 ) )  =  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) )
7546, 74oveq12d 6221 . . . 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 2499 . . 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 6218 . 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 2495 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 1370    e. wcel 1758    =/= wne 2648    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203   CCcc 9394   0cc0 9396   1c1 9397   _ici 9398    + caddc 9399    x. cmul 9401    - cmin 9709   -ucneg 9710    / cdiv 10107   2c2 10485   3c3 10486   4c4 10487   6c6 10489   NN0cn0 10693   ZZ>=cuz 10975   ^cexp 11985   !cfa 12171   sum_csu 13284   expce 13468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474  ax-addf 9475  ax-mulf 9476
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-ico 11420  df-fz 11558  df-fzo 11669  df-fl 11762  df-seq 11927  df-exp 11986  df-fac 12172  df-hash 12224  df-shft 12677  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-limsup 13070  df-clim 13087  df-rlim 13088  df-sum 13285  df-ef 13474
This theorem is referenced by:  resin4p  13543  recos4p  13544
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