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Theorem efival 13764
Description: The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.)
Assertion
Ref Expression
efival  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( cos `  A
)  +  ( _i  x.  ( sin `  A
) ) ) )

Proof of Theorem efival
StepHypRef Expression
1 ax-icn 9554 . . . . . 6  |-  _i  e.  CC
2 mulcl 9579 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
31, 2mpan 670 . . . . 5  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
4 efcl 13696 . . . . 5  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
53, 4syl 16 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
6 negicn 9826 . . . . . 6  |-  -u _i  e.  CC
7 mulcl 9579 . . . . . 6  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
86, 7mpan 670 . . . . 5  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
9 efcl 13696 . . . . 5  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
108, 9syl 16 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
115, 10addcld 9618 . . 3  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
125, 10subcld 9936 . . 3  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
13 2cn 10612 . . . . 5  |-  2  e.  CC
14 2ne0 10634 . . . . 5  |-  2  =/=  0
1513, 14pm3.2i 455 . . . 4  |-  ( 2  e.  CC  /\  2  =/=  0 )
16 divdir 10236 . . . 4  |-  ( ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  /\  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  /\  (
2  e.  CC  /\  2  =/=  0 ) )  ->  ( ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  /  2 )  =  ( ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  +  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) ) )
1715, 16mp3an3 1314 . . 3  |-  ( ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  /\  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )  -> 
( ( ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) ) )  /  2 )  =  ( ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  +  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) ) )
1811, 12, 17syl2anc 661 . 2  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  / 
2 )  =  ( ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  +  ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  2
) ) )
1910, 5pncan3d 9939 . . . . . 6  |-  ( A  e.  CC  ->  (
( exp `  ( -u _i  x.  A ) )  +  ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  =  ( exp `  (
_i  x.  A )
) )
2019oveq2d 6297 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  +  ( ( exp `  ( -u _i  x.  A ) )  +  ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) ) ) )  =  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( _i  x.  A
) ) ) )
215, 10, 12addassd 9621 . . . . 5  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  =  ( ( exp `  ( _i  x.  A ) )  +  ( ( exp `  ( -u _i  x.  A ) )  +  ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) ) ) ) )
2252timesd 10787 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( exp `  ( _i  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( _i  x.  A
) ) ) )
2320, 21, 223eqtr4d 2494 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  =  ( 2  x.  ( exp `  (
_i  x.  A )
) ) )
2423oveq1d 6296 . . 3  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  / 
2 )  =  ( ( 2  x.  ( exp `  ( _i  x.  A ) ) )  /  2 ) )
25 divcan3 10237 . . . . 5  |-  ( ( ( exp `  (
_i  x.  A )
)  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  ( exp `  ( _i  x.  A ) ) )  /  2 )  =  ( exp `  (
_i  x.  A )
) )
2613, 14, 25mp3an23 1317 . . . 4  |-  ( ( exp `  ( _i  x.  A ) )  e.  CC  ->  (
( 2  x.  ( exp `  ( _i  x.  A ) ) )  /  2 )  =  ( exp `  (
_i  x.  A )
) )
275, 26syl 16 . . 3  |-  ( A  e.  CC  ->  (
( 2  x.  ( exp `  ( _i  x.  A ) ) )  /  2 )  =  ( exp `  (
_i  x.  A )
) )
2824, 27eqtr2d 2485 . 2  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) )  +  ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) ) )  /  2 ) )
29 cosval 13735 . . 3  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
30 2mulicn 10768 . . . . . . 7  |-  ( 2  x.  _i )  e.  CC
31 2muline0 10769 . . . . . . 7  |-  ( 2  x.  _i )  =/=  0
3230, 31pm3.2i 455 . . . . . 6  |-  ( ( 2  x.  _i )  e.  CC  /\  (
2  x.  _i )  =/=  0 )
33 div12 10235 . . . . . 6  |-  ( ( _i  e.  CC  /\  ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  /\  (
( 2  x.  _i )  e.  CC  /\  (
2  x.  _i )  =/=  0 ) )  ->  ( _i  x.  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )  =  ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  (
_i  /  ( 2  x.  _i ) ) ) )
341, 32, 33mp3an13 1316 . . . . 5  |-  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  ->  (
_i  x.  ( (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )  =  ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  (
_i  /  ( 2  x.  _i ) ) ) )
3512, 34syl 16 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  ( (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )  =  ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  (
_i  /  ( 2  x.  _i ) ) ) )
36 sinval 13734 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
3736oveq2d 6297 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  ( sin `  A ) )  =  ( _i  x.  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) ) )
38 divrec 10229 . . . . . . 7  |-  ( ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  ( 1  / 
2 ) ) )
3913, 14, 38mp3an23 1317 . . . . . 6  |-  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  ( 1  / 
2 ) ) )
4012, 39syl 16 . . . . 5  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  ( 1  / 
2 ) ) )
411mulid2i 9602 . . . . . . . 8  |-  ( 1  x.  _i )  =  _i
4241oveq1i 6291 . . . . . . 7  |-  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )  =  ( _i  /  (
2  x.  _i ) )
43 ine0 9998 . . . . . . . . . . 11  |-  _i  =/=  0
441, 43dividi 10283 . . . . . . . . . 10  |-  ( _i 
/  _i )  =  1
4544oveq2i 6292 . . . . . . . . 9  |-  ( ( 1  /  2 )  x.  ( _i  /  _i ) )  =  ( ( 1  /  2
)  x.  1 )
46 ax-1cn 9553 . . . . . . . . . 10  |-  1  e.  CC
4746, 13, 1, 1, 14, 43divmuldivi 10310 . . . . . . . . 9  |-  ( ( 1  /  2 )  x.  ( _i  /  _i ) )  =  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )
4845, 47eqtr3i 2474 . . . . . . . 8  |-  ( ( 1  /  2 )  x.  1 )  =  ( ( 1  x.  _i )  /  (
2  x.  _i ) )
49 halfcn 10761 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
5049mulid1i 9601 . . . . . . . 8  |-  ( ( 1  /  2 )  x.  1 )  =  ( 1  /  2
)
5148, 50eqtr3i 2474 . . . . . . 7  |-  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )  =  ( 1  /  2
)
5242, 51eqtr3i 2474 . . . . . 6  |-  ( _i 
/  ( 2  x.  _i ) )  =  ( 1  /  2
)
5352oveq2i 6292 . . . . 5  |-  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  x.  ( _i  / 
( 2  x.  _i ) ) )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  ( 1  / 
2 ) )
5440, 53syl6eqr 2502 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  x.  ( _i  / 
( 2  x.  _i ) ) ) )
5535, 37, 543eqtr4d 2494 . . 3  |-  ( A  e.  CC  ->  (
_i  x.  ( sin `  A ) )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
5629, 55oveq12d 6299 . 2  |-  ( A  e.  CC  ->  (
( cos `  A
)  +  ( _i  x.  ( sin `  A
) ) )  =  ( ( ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2
)  +  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) ) )
5718, 28, 563eqtr4d 2494 1  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( cos `  A
)  +  ( _i  x.  ( sin `  A
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496   _ici 9497    + caddc 9498    x. cmul 9500    - cmin 9810   -ucneg 9811    / cdiv 10212   2c2 10591   expce 13675   sincsin 13677   cosccos 13678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-ico 11544  df-fz 11682  df-fzo 11804  df-fl 11908  df-seq 12087  df-exp 12146  df-fac 12333  df-hash 12385  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684
This theorem is referenced by:  efmival  13765  efeul  13774  efieq  13775  sinadd  13776  cosadd  13777  absefi  13808  demoivre  13812  efhalfpi  22736  efipi  22738  ef2pi  22742  efimpi  22756  efif1olem4  22804  1cubrlem  23044  asinsin  23095  atantan  23126
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