MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efival Structured version   Unicode version

Theorem efival 13547
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 9445 . . . . . 6  |-  _i  e.  CC
2 mulcl 9470 . . . . . 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 13479 . . . . 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 9715 . . . . . 6  |-  -u _i  e.  CC
7 mulcl 9470 . . . . . 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 13479 . . . . 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 9509 . . 3  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
125, 10subcld 9823 . . 3  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
13 2cn 10496 . . . . 5  |-  2  e.  CC
14 2ne0 10518 . . . . 5  |-  2  =/=  0
1513, 14pm3.2i 455 . . . 4  |-  ( 2  e.  CC  /\  2  =/=  0 )
16 divdir 10121 . . . 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 1304 . . 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 9826 . . . . . 6  |-  ( A  e.  CC  ->  (
( exp `  ( -u _i  x.  A ) )  +  ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  =  ( exp `  (
_i  x.  A )
) )
2019oveq2d 6209 . . . . 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 9512 . . . . 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 10671 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( exp `  ( _i  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( _i  x.  A
) ) ) )
2320, 21, 223eqtr4d 2502 . . . 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 6208 . . 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 10122 . . . . 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 1307 . . . 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 2493 . 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 13518 . . 3  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
30 2mulicn 10652 . . . . . . 7  |-  ( 2  x.  _i )  e.  CC
31 2muline0 10653 . . . . . . 7  |-  ( 2  x.  _i )  =/=  0
3230, 31pm3.2i 455 . . . . . 6  |-  ( ( 2  x.  _i )  e.  CC  /\  (
2  x.  _i )  =/=  0 )
33 div12 10120 . . . . . 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 1306 . . . . 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 13517 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
3736oveq2d 6209 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  ( sin `  A ) )  =  ( _i  x.  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) ) )
38 divrec 10114 . . . . . . 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 1307 . . . . . 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 9493 . . . . . . . 8  |-  ( 1  x.  _i )  =  _i
4241oveq1i 6203 . . . . . . 7  |-  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )  =  ( _i  /  (
2  x.  _i ) )
43 ine0 9884 . . . . . . . . . . 11  |-  _i  =/=  0
441, 43dividi 10168 . . . . . . . . . 10  |-  ( _i 
/  _i )  =  1
4544oveq2i 6204 . . . . . . . . 9  |-  ( ( 1  /  2 )  x.  ( _i  /  _i ) )  =  ( ( 1  /  2
)  x.  1 )
46 ax-1cn 9444 . . . . . . . . . 10  |-  1  e.  CC
4746, 13, 1, 1, 14, 43divmuldivi 10195 . . . . . . . . 9  |-  ( ( 1  /  2 )  x.  ( _i  /  _i ) )  =  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )
4845, 47eqtr3i 2482 . . . . . . . 8  |-  ( ( 1  /  2 )  x.  1 )  =  ( ( 1  x.  _i )  /  (
2  x.  _i ) )
49 halfcn 10645 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
5049mulid1i 9492 . . . . . . . 8  |-  ( ( 1  /  2 )  x.  1 )  =  ( 1  /  2
)
5148, 50eqtr3i 2482 . . . . . . 7  |-  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )  =  ( 1  /  2
)
5242, 51eqtr3i 2482 . . . . . 6  |-  ( _i 
/  ( 2  x.  _i ) )  =  ( 1  /  2
)
5352oveq2i 6204 . . . . 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 2510 . . . 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 2502 . . 3  |-  ( A  e.  CC  ->  (
_i  x.  ( sin `  A ) )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
5629, 55oveq12d 6211 . 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 2502 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 1370    e. wcel 1758    =/= wne 2644   ` cfv 5519  (class class class)co 6193   CCcc 9384   0cc0 9386   1c1 9387   _ici 9388    + caddc 9389    x. cmul 9391    - cmin 9699   -ucneg 9700    / cdiv 10097   2c2 10475   expce 13458   sincsin 13460   cosccos 13461
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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-ico 11410  df-fz 11548  df-fzo 11659  df-fl 11752  df-seq 11917  df-exp 11976  df-fac 12162  df-hash 12214  df-shft 12667  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-limsup 13060  df-clim 13077  df-rlim 13078  df-sum 13275  df-ef 13464  df-sin 13466  df-cos 13467
This theorem is referenced by:  efmival  13548  efeul  13557  efieq  13558  sinadd  13559  cosadd  13560  absefi  13591  demoivre  13595  efhalfpi  22059  efipi  22061  ef2pi  22065  efimpi  22079  efif1olem4  22127  1cubrlem  22362  asinsin  22413  atantan  22444
  Copyright terms: Public domain W3C validator