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Theorem efival 14206
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 9598 . . . . . 6  |-  _i  e.  CC
2 mulcl 9623 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
31, 2mpan 676 . . . . 5  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
4 efcl 14137 . . . . 5  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
53, 4syl 17 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
6 negicn 9876 . . . . . 6  |-  -u _i  e.  CC
7 mulcl 9623 . . . . . 6  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
86, 7mpan 676 . . . . 5  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
9 efcl 14137 . . . . 5  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
108, 9syl 17 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
115, 10addcld 9662 . . 3  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
125, 10subcld 9986 . . 3  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
13 2cn 10680 . . . . 5  |-  2  e.  CC
14 2ne0 10702 . . . . 5  |-  2  =/=  0
1513, 14pm3.2i 457 . . . 4  |-  ( 2  e.  CC  /\  2  =/=  0 )
16 divdir 10293 . . . 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 1353 . . 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 667 . 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 9989 . . . . . 6  |-  ( A  e.  CC  ->  (
( exp `  ( -u _i  x.  A ) )  +  ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) ) )  =  ( exp `  (
_i  x.  A )
) )
2019oveq2d 6306 . . . . 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 9665 . . . . 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 10855 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( exp `  ( _i  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( _i  x.  A
) ) ) )
2320, 21, 223eqtr4d 2495 . . . 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 6305 . . 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 10294 . . . . 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 1356 . . . 4  |-  ( ( exp `  ( _i  x.  A ) )  e.  CC  ->  (
( 2  x.  ( exp `  ( _i  x.  A ) ) )  /  2 )  =  ( exp `  (
_i  x.  A )
) )
275, 26syl 17 . . 3  |-  ( A  e.  CC  ->  (
( 2  x.  ( exp `  ( _i  x.  A ) ) )  /  2 )  =  ( exp `  (
_i  x.  A )
) )
2824, 27eqtr2d 2486 . 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 14177 . . 3  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
30 2mulicn 10836 . . . . . . 7  |-  ( 2  x.  _i )  e.  CC
31 2muline0 10837 . . . . . . 7  |-  ( 2  x.  _i )  =/=  0
3230, 31pm3.2i 457 . . . . . 6  |-  ( ( 2  x.  _i )  e.  CC  /\  (
2  x.  _i )  =/=  0 )
33 div12 10292 . . . . . 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 1355 . . . . 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 17 . . . 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 14176 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
3736oveq2d 6306 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  ( sin `  A ) )  =  ( _i  x.  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) ) )
38 divrec 10286 . . . . . . 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 1356 . . . . . 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 17 . . . . 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 9646 . . . . . . . 8  |-  ( 1  x.  _i )  =  _i
4241oveq1i 6300 . . . . . . 7  |-  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )  =  ( _i  /  (
2  x.  _i ) )
43 ine0 10054 . . . . . . . . . . 11  |-  _i  =/=  0
441, 43dividi 10340 . . . . . . . . . 10  |-  ( _i 
/  _i )  =  1
4544oveq2i 6301 . . . . . . . . 9  |-  ( ( 1  /  2 )  x.  ( _i  /  _i ) )  =  ( ( 1  /  2
)  x.  1 )
46 ax-1cn 9597 . . . . . . . . . 10  |-  1  e.  CC
4746, 13, 1, 1, 14, 43divmuldivi 10367 . . . . . . . . 9  |-  ( ( 1  /  2 )  x.  ( _i  /  _i ) )  =  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )
4845, 47eqtr3i 2475 . . . . . . . 8  |-  ( ( 1  /  2 )  x.  1 )  =  ( ( 1  x.  _i )  /  (
2  x.  _i ) )
49 halfcn 10829 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
5049mulid1i 9645 . . . . . . . 8  |-  ( ( 1  /  2 )  x.  1 )  =  ( 1  /  2
)
5148, 50eqtr3i 2475 . . . . . . 7  |-  ( ( 1  x.  _i )  /  ( 2  x.  _i ) )  =  ( 1  /  2
)
5242, 51eqtr3i 2475 . . . . . 6  |-  ( _i 
/  ( 2  x.  _i ) )  =  ( 1  /  2
)
5352oveq2i 6301 . . . . 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 2503 . . . 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 2495 . . 3  |-  ( A  e.  CC  ->  (
_i  x.  ( sin `  A ) )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
5629, 55oveq12d 6308 . 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 2495 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 371    = wceq 1444    e. wcel 1887    =/= wne 2622   ` cfv 5582  (class class class)co 6290   CCcc 9537   0cc0 9539   1c1 9540   _ici 9541    + caddc 9542    x. cmul 9544    - cmin 9860   -ucneg 9861    / cdiv 10269   2c2 10659   expce 14114   sincsin 14116   cosccos 14117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-fac 12460  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124
This theorem is referenced by:  efmival  14207  efeul  14216  efieq  14217  sinadd  14218  cosadd  14219  absefi  14250  demoivre  14254  efhalfpi  23426  efipi  23428  ef2pi  23432  efimpi  23446  efif1olem4  23494  1cubrlem  23767  asinsin  23818  atantan  23849
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