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Theorem sinhval 12714
Description: Value of the hyperbolic sine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)
Assertion
Ref Expression
sinhval  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )

Proof of Theorem sinhval
StepHypRef Expression
1 ixi 9611 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
21oveq1i 6054 . . . . . . . 8  |-  ( ( _i  x.  _i )  x.  A )  =  ( -u 1  x.  A )
3 ax-icn 9009 . . . . . . . . 9  |-  _i  e.  CC
4 mulass 9038 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  _i  e.  CC  /\  A  e.  CC )  ->  (
( _i  x.  _i )  x.  A )  =  ( _i  x.  ( _i  x.  A
) ) )
53, 3, 4mp3an12 1269 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( _i  x.  _i )  x.  A )  =  ( _i  x.  ( _i  x.  A
) ) )
6 mulm1 9435 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
72, 5, 63eqtr3a 2464 . . . . . . 7  |-  ( A  e.  CC  ->  (
_i  x.  ( _i  x.  A ) )  = 
-u A )
87fveq2d 5695 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( _i  x.  A
) ) )  =  ( exp `  -u A
) )
93, 3mulneg1i 9439 . . . . . . . . . 10  |-  ( -u _i  x.  _i )  = 
-u ( _i  x.  _i )
101negeqi 9259 . . . . . . . . . . 11  |-  -u (
_i  x.  _i )  =  -u -u 1
11 ax-1cn 9008 . . . . . . . . . . . 12  |-  1  e.  CC
1211negnegi 9330 . . . . . . . . . . 11  |-  -u -u 1  =  1
1310, 12eqtri 2428 . . . . . . . . . 10  |-  -u (
_i  x.  _i )  =  1
149, 13eqtri 2428 . . . . . . . . 9  |-  ( -u _i  x.  _i )  =  1
1514oveq1i 6054 . . . . . . . 8  |-  ( (
-u _i  x.  _i )  x.  A )  =  ( 1  x.  A )
163negcli 9328 . . . . . . . . 9  |-  -u _i  e.  CC
17 mulass 9038 . . . . . . . . 9  |-  ( (
-u _i  e.  CC  /\  _i  e.  CC  /\  A  e.  CC )  ->  ( ( -u _i  x.  _i )  x.  A
)  =  ( -u _i  x.  ( _i  x.  A ) ) )
1816, 3, 17mp3an12 1269 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( -u _i  x.  _i )  x.  A )  =  ( -u _i  x.  ( _i  x.  A
) ) )
19 mulid2 9049 . . . . . . . 8  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2015, 18, 193eqtr3a 2464 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  ( _i  x.  A ) )  =  A )
2120fveq2d 5695 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  ( _i  x.  A
) ) )  =  ( exp `  A
) )
228, 21oveq12d 6062 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  ( _i  x.  A ) ) )  -  ( exp `  ( -u _i  x.  ( _i  x.  A ) ) ) )  =  ( ( exp `  -u A
)  -  ( exp `  A ) ) )
2322oveq1d 6059 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  ( _i  x.  A ) ) )  -  ( exp `  ( -u _i  x.  ( _i  x.  A ) ) ) )  /  (
2  x.  _i ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
24 mulcl 9034 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
253, 24mpan 652 . . . . 5  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
26 sinval 12682 . . . . 5  |-  ( ( _i  x.  A )  e.  CC  ->  ( sin `  ( _i  x.  A ) )  =  ( ( ( exp `  ( _i  x.  (
_i  x.  A )
) )  -  ( exp `  ( -u _i  x.  ( _i  x.  A
) ) ) )  /  ( 2  x.  _i ) ) )
2725, 26syl 16 . . . 4  |-  ( A  e.  CC  ->  ( sin `  ( _i  x.  A ) )  =  ( ( ( exp `  ( _i  x.  (
_i  x.  A )
) )  -  ( exp `  ( -u _i  x.  ( _i  x.  A
) ) ) )  /  ( 2  x.  _i ) ) )
28 irec 11439 . . . . . . . 8  |-  ( 1  /  _i )  = 
-u _i
2928negeqi 9259 . . . . . . 7  |-  -u (
1  /  _i )  =  -u -u _i
303negnegi 9330 . . . . . . 7  |-  -u -u _i  =  _i
3129, 30eqtri 2428 . . . . . 6  |-  -u (
1  /  _i )  =  _i
3231oveq1i 6054 . . . . 5  |-  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( _i  x.  ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) )
33 ine0 9429 . . . . . . . 8  |-  _i  =/=  0
343, 33reccli 9704 . . . . . . 7  |-  ( 1  /  _i )  e.  CC
35 efcl 12644 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
36 negcl 9266 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u A  e.  CC )
37 efcl 12644 . . . . . . . . . 10  |-  ( -u A  e.  CC  ->  ( exp `  -u A
)  e.  CC )
3836, 37syl 16 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  -u A )  e.  CC )
3935, 38subcld 9371 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
4039halfcld 10172 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  e.  CC )
41 mulneg12 9432 . . . . . . 7  |-  ( ( ( 1  /  _i )  e.  CC  /\  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  e.  CC )  ->  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( 1  /  _i )  x.  -u ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) ) )
4234, 40, 41sylancr 645 . . . . . 6  |-  ( A  e.  CC  ->  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( 1  /  _i )  x.  -u ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) ) )
43 2cn 10030 . . . . . . . . . . 11  |-  2  e.  CC
4443a1i 11 . . . . . . . . . 10  |-  ( A  e.  CC  ->  2  e.  CC )
45 2ne0 10043 . . . . . . . . . . 11  |-  2  =/=  0
4645a1i 11 . . . . . . . . . 10  |-  ( A  e.  CC  ->  2  =/=  0 )
4739, 44, 46divnegd 9763 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( -u ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
4835, 38negsubdi2d 9387 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u (
( exp `  A
)  -  ( exp `  -u A ) )  =  ( ( exp `  -u A )  -  ( exp `  A ) ) )
4948oveq1d 6059 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( -u ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 ) )
5047, 49eqtrd 2440 . . . . . . . 8  |-  ( A  e.  CC  ->  -u (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 ) )
5150oveq2d 6060 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 1  /  _i )  x.  -u ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( 1  /  _i )  x.  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 ) ) )
5238, 35subcld 9371 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( exp `  -u A
)  -  ( exp `  A ) )  e.  CC )
5352halfcld 10172 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( exp `  -u A
)  -  ( exp `  A ) )  / 
2 )  e.  CC )
543a1i 11 . . . . . . . 8  |-  ( A  e.  CC  ->  _i  e.  CC )
5533a1i 11 . . . . . . . 8  |-  ( A  e.  CC  ->  _i  =/=  0 )
5653, 54, 55divrec2d 9754 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 )  /  _i )  =  ( ( 1  /  _i )  x.  (
( ( exp `  -u A
)  -  ( exp `  A ) )  / 
2 ) ) )
5752, 44, 54, 46, 55divdiv1d 9781 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 )  /  _i )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  ( 2  x.  _i ) ) )
5851, 56, 573eqtr2d 2446 . . . . . 6  |-  ( A  e.  CC  ->  (
( 1  /  _i )  x.  -u ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
5942, 58eqtrd 2440 . . . . 5  |-  ( A  e.  CC  ->  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
6032, 59syl5eqr 2454 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
6123, 27, 603eqtr4d 2450 . . 3  |-  ( A  e.  CC  ->  ( sin `  ( _i  x.  A ) )  =  ( _i  x.  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) ) )
6261oveq1d 6059 . 2  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( _i  x.  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )  /  _i ) )
6340, 54, 55divcan3d 9755 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) )
6462, 63eqtrd 2440 1  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2571   ` cfv 5417  (class class class)co 6044   CCcc 8948   0cc0 8950   1c1 8951   _ici 8952    x. cmul 8955    - cmin 9251   -ucneg 9252    / cdiv 9637   2c2 10009   expce 12623   sincsin 12625
This theorem is referenced by:  resinhcl  12716  tanhlt1  12720  sinhpcosh  28201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028  ax-addf 9029  ax-mulf 9030
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-ico 10882  df-fz 11004  df-fzo 11095  df-fl 11161  df-seq 11283  df-exp 11342  df-fac 11526  df-hash 11578  df-shft 11841  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-limsup 12224  df-clim 12241  df-rlim 12242  df-sum 12439  df-ef 12629  df-sin 12631
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