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Theorem efival 11446
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 7722 . . . . . 6  |-  _i  e.  CC
2 mulcl 7754 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
31, 2mpan 420 . . . . 5  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
4 efcl 11377 . . . . 5  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
53, 4syl 14 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
6 negicn 7970 . . . . . 6  |-  -u _i  e.  CC
7 mulcl 7754 . . . . . 6  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
86, 7mpan 420 . . . . 5  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
9 efcl 11377 . . . . 5  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
108, 9syl 14 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
115, 10addcld 7792 . . 3  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
125, 10subcld 8080 . . 3  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
13 2cn 8798 . . . . 5  |-  2  e.  CC
14 2ap0 8820 . . . . 5  |-  2 #  0
1513, 14pm3.2i 270 . . . 4  |-  ( 2  e.  CC  /\  2 #  0 )
16 divdirap 8464 . . . 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 408 . 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 8083 . . . . . 6  |-  ( A  e.  CC  ->  (
( exp `  ( -u _i  x.  A ) )  +  ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  =  ( exp `  (
_i  x.  A )
) )
2019oveq2d 5790 . . . . 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 7795 . . . . 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 8969 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( exp `  ( _i  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( _i  x.  A
) ) ) )
2320, 21, 223eqtr4d 2182 . . . 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 5789 . . 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 divcanap3 8465 . . . . 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 14 . . 3  |-  ( A  e.  CC  ->  (
( 2  x.  ( exp `  ( _i  x.  A ) ) )  /  2 )  =  ( exp `  (
_i  x.  A )
) )
2824, 27eqtr2d 2173 . 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 11417 . . 3  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
30 2mulicn 8949 . . . . . . 7  |-  ( 2  x.  _i )  e.  CC
31 2muliap0 8951 . . . . . . 7  |-  ( 2  x.  _i ) #  0
3230, 31pm3.2i 270 . . . . . 6  |-  ( ( 2  x.  _i )  e.  CC  /\  (
2  x.  _i ) #  0 )
33 div12ap 8461 . . . . . 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 14 . . . 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 11416 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
3736oveq2d 5790 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  ( sin `  A ) )  =  ( _i  x.  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) ) )
38 divrecap 8455 . . . . . . 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 14 . . . . 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 7776 . . . . . . . 8  |-  ( 1  x.  _i )  =  _i
4241oveq1i 5784 . . . . . . 7  |-  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )  =  ( _i  /  (
2  x.  _i ) )
43 iap0 8950 . . . . . . . . . . 11  |-  _i #  0
441, 43dividapi 8512 . . . . . . . . . 10  |-  ( _i 
/  _i )  =  1
4544oveq2i 5785 . . . . . . . . 9  |-  ( ( 1  /  2 )  x.  ( _i  /  _i ) )  =  ( ( 1  /  2
)  x.  1 )
46 ax-1cn 7720 . . . . . . . . . 10  |-  1  e.  CC
4746, 13, 1, 1, 14, 43divmuldivapi 8539 . . . . . . . . 9  |-  ( ( 1  /  2 )  x.  ( _i  /  _i ) )  =  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )
4845, 47eqtr3i 2162 . . . . . . . 8  |-  ( ( 1  /  2 )  x.  1 )  =  ( ( 1  x.  _i )  /  (
2  x.  _i ) )
49 halfcn 8941 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
5049mulid1i 7775 . . . . . . . 8  |-  ( ( 1  /  2 )  x.  1 )  =  ( 1  /  2
)
5148, 50eqtr3i 2162 . . . . . . 7  |-  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )  =  ( 1  /  2
)
5242, 51eqtr3i 2162 . . . . . 6  |-  ( _i 
/  ( 2  x.  _i ) )  =  ( 1  /  2
)
5352oveq2i 5785 . . . . 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 2190 . . . 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 2182 . . 3  |-  ( A  e.  CC  ->  (
_i  x.  ( sin `  A ) )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
5629, 55oveq12d 5792 . 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 2182 1  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  =  ( ( cos `  A
)  +  ( _i  x.  ( sin `  A
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7625   0cc0 7627   1c1 7628   _ici 7629    + caddc 7630    x. cmul 7632    - cmin 7940   -ucneg 7941   # cap 8350    / cdiv 8439   2c2 8778   expce 11355   sincsin 11357   cosccos 11358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-ico 9684  df-fz 9798  df-fzo 9927  df-seqfrec 10226  df-exp 10300  df-fac 10479  df-ihash 10529  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-clim 11055  df-sumdc 11130  df-ef 11361  df-sin 11363  df-cos 11364
This theorem is referenced by:  efmival  11447  efeul  11448  efieq  11449  sinadd  11450  cosadd  11451  absefi  11482  demoivre  11486  efhalfpi  12893  efipi  12895  ef2pi  12899  efimpi  12913
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