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Theorem sinhval 12525
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 9484 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
21oveq1i 5952 . . . . . . . 8  |-  ( ( _i  x.  _i )  x.  A )  =  ( -u 1  x.  A )
3 ax-icn 8883 . . . . . . . . 9  |-  _i  e.  CC
4 mulass 8912 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  _i  e.  CC  /\  A  e.  CC )  ->  (
( _i  x.  _i )  x.  A )  =  ( _i  x.  ( _i  x.  A
) ) )
53, 3, 4mp3an12 1267 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( _i  x.  _i )  x.  A )  =  ( _i  x.  ( _i  x.  A
) ) )
6 mulm1 9308 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
72, 5, 63eqtr3a 2414 . . . . . . 7  |-  ( A  e.  CC  ->  (
_i  x.  ( _i  x.  A ) )  = 
-u A )
87fveq2d 5609 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( _i  x.  A
) ) )  =  ( exp `  -u A
) )
93, 3mulneg1i 9312 . . . . . . . . . 10  |-  ( -u _i  x.  _i )  = 
-u ( _i  x.  _i )
101negeqi 9132 . . . . . . . . . . 11  |-  -u (
_i  x.  _i )  =  -u -u 1
11 ax-1cn 8882 . . . . . . . . . . . 12  |-  1  e.  CC
1211negnegi 9203 . . . . . . . . . . 11  |-  -u -u 1  =  1
1310, 12eqtri 2378 . . . . . . . . . 10  |-  -u (
_i  x.  _i )  =  1
149, 13eqtri 2378 . . . . . . . . 9  |-  ( -u _i  x.  _i )  =  1
1514oveq1i 5952 . . . . . . . 8  |-  ( (
-u _i  x.  _i )  x.  A )  =  ( 1  x.  A )
163negcli 9201 . . . . . . . . 9  |-  -u _i  e.  CC
17 mulass 8912 . . . . . . . . 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 1267 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( -u _i  x.  _i )  x.  A )  =  ( -u _i  x.  ( _i  x.  A
) ) )
19 mulid2 8923 . . . . . . . 8  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2015, 18, 193eqtr3a 2414 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  ( _i  x.  A ) )  =  A )
2120fveq2d 5609 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  ( _i  x.  A
) ) )  =  ( exp `  A
) )
228, 21oveq12d 5960 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  ( _i  x.  A ) ) )  -  ( exp `  ( -u _i  x.  ( _i  x.  A ) ) ) )  =  ( ( exp `  -u A
)  -  ( exp `  A ) ) )
2322oveq1d 5957 . . . 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 8908 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
253, 24mpan 651 . . . . 5  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
26 sinval 12493 . . . . 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 15 . . . 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 11292 . . . . . . . 8  |-  ( 1  /  _i )  = 
-u _i
2928negeqi 9132 . . . . . . 7  |-  -u (
1  /  _i )  =  -u -u _i
303negnegi 9203 . . . . . . 7  |-  -u -u _i  =  _i
3129, 30eqtri 2378 . . . . . 6  |-  -u (
1  /  _i )  =  _i
3231oveq1i 5952 . . . . 5  |-  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( _i  x.  ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) )
33 ine0 9302 . . . . . . . 8  |-  _i  =/=  0
343, 33reccli 9577 . . . . . . 7  |-  ( 1  /  _i )  e.  CC
35 efcl 12455 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
36 negcl 9139 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u A  e.  CC )
37 efcl 12455 . . . . . . . . . 10  |-  ( -u A  e.  CC  ->  ( exp `  -u A
)  e.  CC )
3836, 37syl 15 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  -u A )  e.  CC )
3935, 38subcld 9244 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
4039halfcld 10045 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  e.  CC )
41 mulneg12 9305 . . . . . . 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 644 . . . . . 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 9903 . . . . . . . . . . 11  |-  2  e.  CC
4443a1i 10 . . . . . . . . . 10  |-  ( A  e.  CC  ->  2  e.  CC )
45 2ne0 9916 . . . . . . . . . . 11  |-  2  =/=  0
4645a1i 10 . . . . . . . . . 10  |-  ( A  e.  CC  ->  2  =/=  0 )
4739, 44, 46divnegd 9636 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( -u ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
4835, 38negsubdi2d 9260 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u (
( exp `  A
)  -  ( exp `  -u A ) )  =  ( ( exp `  -u A )  -  ( exp `  A ) ) )
4948oveq1d 5957 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( -u ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 ) )
5047, 49eqtrd 2390 . . . . . . . 8  |-  ( A  e.  CC  ->  -u (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 ) )
5150oveq2d 5958 . . . . . . 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 9244 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( exp `  -u A
)  -  ( exp `  A ) )  e.  CC )
5352halfcld 10045 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( exp `  -u A
)  -  ( exp `  A ) )  / 
2 )  e.  CC )
543a1i 10 . . . . . . . 8  |-  ( A  e.  CC  ->  _i  e.  CC )
5533a1i 10 . . . . . . . 8  |-  ( A  e.  CC  ->  _i  =/=  0 )
5653, 54, 55divrec2d 9627 . . . . . . 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 9654 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 )  /  _i )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  ( 2  x.  _i ) ) )
5851, 56, 573eqtr2d 2396 . . . . . 6  |-  ( A  e.  CC  ->  (
( 1  /  _i )  x.  -u ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
5942, 58eqtrd 2390 . . . . 5  |-  ( A  e.  CC  ->  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
6032, 59syl5eqr 2404 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
6123, 27, 603eqtr4d 2400 . . 3  |-  ( A  e.  CC  ->  ( sin `  ( _i  x.  A ) )  =  ( _i  x.  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) ) )
6261oveq1d 5957 . 2  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( _i  x.  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )  /  _i ) )
6340, 54, 55divcan3d 9628 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) )
6462, 63eqtrd 2390 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 1642    e. wcel 1710    =/= wne 2521   ` cfv 5334  (class class class)co 5942   CCcc 8822   0cc0 8824   1c1 8825   _ici 8826    x. cmul 8829    - cmin 9124   -ucneg 9125    / cdiv 9510   2c2 9882   expce 12434   sincsin 12436
This theorem is referenced by:  resinhcl  12527  tanhlt1  12531  sinhpcosh  27893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902  ax-addf 8903  ax-mulf 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-oi 7312  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-ico 10751  df-fz 10872  df-fzo 10960  df-fl 11014  df-seq 11136  df-exp 11195  df-fac 11379  df-hash 11428  df-shft 11652  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-limsup 12035  df-clim 12052  df-rlim 12053  df-sum 12250  df-ef 12440  df-sin 12442
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