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Theorem sinhval 12755
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 9651 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
21oveq1i 6091 . . . . . . . 8  |-  ( ( _i  x.  _i )  x.  A )  =  ( -u 1  x.  A )
3 ax-icn 9049 . . . . . . . . 9  |-  _i  e.  CC
4 mulass 9078 . . . . . . . . 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 9475 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
72, 5, 63eqtr3a 2492 . . . . . . 7  |-  ( A  e.  CC  ->  (
_i  x.  ( _i  x.  A ) )  = 
-u A )
87fveq2d 5732 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( _i  x.  A
) ) )  =  ( exp `  -u A
) )
93, 3mulneg1i 9479 . . . . . . . . . 10  |-  ( -u _i  x.  _i )  = 
-u ( _i  x.  _i )
101negeqi 9299 . . . . . . . . . . 11  |-  -u (
_i  x.  _i )  =  -u -u 1
11 ax-1cn 9048 . . . . . . . . . . . 12  |-  1  e.  CC
1211negnegi 9370 . . . . . . . . . . 11  |-  -u -u 1  =  1
1310, 12eqtri 2456 . . . . . . . . . 10  |-  -u (
_i  x.  _i )  =  1
149, 13eqtri 2456 . . . . . . . . 9  |-  ( -u _i  x.  _i )  =  1
1514oveq1i 6091 . . . . . . . 8  |-  ( (
-u _i  x.  _i )  x.  A )  =  ( 1  x.  A )
163negcli 9368 . . . . . . . . 9  |-  -u _i  e.  CC
17 mulass 9078 . . . . . . . . 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 9089 . . . . . . . 8  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2015, 18, 193eqtr3a 2492 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  ( _i  x.  A ) )  =  A )
2120fveq2d 5732 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  ( _i  x.  A
) ) )  =  ( exp `  A
) )
228, 21oveq12d 6099 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  ( _i  x.  A ) ) )  -  ( exp `  ( -u _i  x.  ( _i  x.  A ) ) ) )  =  ( ( exp `  -u A
)  -  ( exp `  A ) ) )
2322oveq1d 6096 . . . 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 9074 . . . . . 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 12723 . . . . 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 11480 . . . . . . . 8  |-  ( 1  /  _i )  = 
-u _i
2928negeqi 9299 . . . . . . 7  |-  -u (
1  /  _i )  =  -u -u _i
303negnegi 9370 . . . . . . 7  |-  -u -u _i  =  _i
3129, 30eqtri 2456 . . . . . 6  |-  -u (
1  /  _i )  =  _i
3231oveq1i 6091 . . . . 5  |-  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( _i  x.  ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) )
33 ine0 9469 . . . . . . . 8  |-  _i  =/=  0
343, 33reccli 9744 . . . . . . 7  |-  ( 1  /  _i )  e.  CC
35 efcl 12685 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
36 negcl 9306 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u A  e.  CC )
37 efcl 12685 . . . . . . . . . 10  |-  ( -u A  e.  CC  ->  ( exp `  -u A
)  e.  CC )
3836, 37syl 16 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  -u A )  e.  CC )
3935, 38subcld 9411 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
4039halfcld 10212 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  e.  CC )
41 mulneg12 9472 . . . . . . 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 10070 . . . . . . . . . . 11  |-  2  e.  CC
4443a1i 11 . . . . . . . . . 10  |-  ( A  e.  CC  ->  2  e.  CC )
45 2ne0 10083 . . . . . . . . . . 11  |-  2  =/=  0
4645a1i 11 . . . . . . . . . 10  |-  ( A  e.  CC  ->  2  =/=  0 )
4739, 44, 46divnegd 9803 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( -u ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
4835, 38negsubdi2d 9427 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u (
( exp `  A
)  -  ( exp `  -u A ) )  =  ( ( exp `  -u A )  -  ( exp `  A ) ) )
4948oveq1d 6096 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( -u ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 ) )
5047, 49eqtrd 2468 . . . . . . . 8  |-  ( A  e.  CC  ->  -u (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 ) )
5150oveq2d 6097 . . . . . . 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 9411 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( exp `  -u A
)  -  ( exp `  A ) )  e.  CC )
5352halfcld 10212 . . . . . . . 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 9794 . . . . . . 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 9821 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 )  /  _i )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  ( 2  x.  _i ) ) )
5851, 56, 573eqtr2d 2474 . . . . . 6  |-  ( A  e.  CC  ->  (
( 1  /  _i )  x.  -u ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
5942, 58eqtrd 2468 . . . . 5  |-  ( A  e.  CC  ->  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
6032, 59syl5eqr 2482 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
6123, 27, 603eqtr4d 2478 . . 3  |-  ( A  e.  CC  ->  ( sin `  ( _i  x.  A ) )  =  ( _i  x.  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) ) )
6261oveq1d 6096 . 2  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( _i  x.  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )  /  _i ) )
6340, 54, 55divcan3d 9795 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) )
6462, 63eqtrd 2468 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 1652    e. wcel 1725    =/= wne 2599   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991   _ici 8992    x. cmul 8995    - cmin 9291   -ucneg 9292    / cdiv 9677   2c2 10049   expce 12664   sincsin 12666
This theorem is referenced by:  resinhcl  12757  tanhlt1  12761  sinhpcosh  28483
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ico 10922  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-fac 11567  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672
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