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Theorem sinhval 12434
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 9397 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
21oveq1i 5868 . . . . . . . 8  |-  ( ( _i  x.  _i )  x.  A )  =  ( -u 1  x.  A )
3 ax-icn 8796 . . . . . . . . 9  |-  _i  e.  CC
4 mulass 8825 . . . . . . . . 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 9221 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
72, 5, 63eqtr3a 2339 . . . . . . 7  |-  ( A  e.  CC  ->  (
_i  x.  ( _i  x.  A ) )  = 
-u A )
87fveq2d 5529 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( _i  x.  A
) ) )  =  ( exp `  -u A
) )
93, 3mulneg1i 9225 . . . . . . . . . 10  |-  ( -u _i  x.  _i )  = 
-u ( _i  x.  _i )
101negeqi 9045 . . . . . . . . . . 11  |-  -u (
_i  x.  _i )  =  -u -u 1
11 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
1211negnegi 9116 . . . . . . . . . . 11  |-  -u -u 1  =  1
1310, 12eqtri 2303 . . . . . . . . . 10  |-  -u (
_i  x.  _i )  =  1
149, 13eqtri 2303 . . . . . . . . 9  |-  ( -u _i  x.  _i )  =  1
1514oveq1i 5868 . . . . . . . 8  |-  ( (
-u _i  x.  _i )  x.  A )  =  ( 1  x.  A )
163negcli 9114 . . . . . . . . 9  |-  -u _i  e.  CC
17 mulass 8825 . . . . . . . . 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 8836 . . . . . . . 8  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2015, 18, 193eqtr3a 2339 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  ( _i  x.  A ) )  =  A )
2120fveq2d 5529 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  ( _i  x.  A
) ) )  =  ( exp `  A
) )
228, 21oveq12d 5876 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  ( _i  x.  A ) ) )  -  ( exp `  ( -u _i  x.  ( _i  x.  A ) ) ) )  =  ( ( exp `  -u A
)  -  ( exp `  A ) ) )
2322oveq1d 5873 . . . 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 8821 . . . . . 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 12402 . . . . 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 11202 . . . . . . . 8  |-  ( 1  /  _i )  = 
-u _i
2928negeqi 9045 . . . . . . 7  |-  -u (
1  /  _i )  =  -u -u _i
303negnegi 9116 . . . . . . 7  |-  -u -u _i  =  _i
3129, 30eqtri 2303 . . . . . 6  |-  -u (
1  /  _i )  =  _i
3231oveq1i 5868 . . . . 5  |-  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( _i  x.  ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) )
33 ine0 9215 . . . . . . . 8  |-  _i  =/=  0
343, 33reccli 9490 . . . . . . 7  |-  ( 1  /  _i )  e.  CC
35 efcl 12364 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
36 negcl 9052 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u A  e.  CC )
37 efcl 12364 . . . . . . . . . 10  |-  ( -u A  e.  CC  ->  ( exp `  -u A
)  e.  CC )
3836, 37syl 15 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  -u A )  e.  CC )
3935, 38subcld 9157 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
4039halfcld 9956 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  e.  CC )
41 mulneg12 9218 . . . . . . 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 9816 . . . . . . . . . . 11  |-  2  e.  CC
4443a1i 10 . . . . . . . . . 10  |-  ( A  e.  CC  ->  2  e.  CC )
45 2ne0 9829 . . . . . . . . . . 11  |-  2  =/=  0
4645a1i 10 . . . . . . . . . 10  |-  ( A  e.  CC  ->  2  =/=  0 )
4739, 44, 46divnegd 9549 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( -u ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
4835, 38negsubdi2d 9173 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u (
( exp `  A
)  -  ( exp `  -u A ) )  =  ( ( exp `  -u A )  -  ( exp `  A ) ) )
4948oveq1d 5873 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( -u ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 ) )
5047, 49eqtrd 2315 . . . . . . . 8  |-  ( A  e.  CC  ->  -u (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 ) )
5150oveq2d 5874 . . . . . . 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 9157 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( exp `  -u A
)  -  ( exp `  A ) )  e.  CC )
5352halfcld 9956 . . . . . . . 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 9540 . . . . . . 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 9567 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 )  /  _i )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  ( 2  x.  _i ) ) )
5851, 56, 573eqtr2d 2321 . . . . . 6  |-  ( A  e.  CC  ->  (
( 1  /  _i )  x.  -u ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
5942, 58eqtrd 2315 . . . . 5  |-  ( A  e.  CC  ->  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
6032, 59syl5eqr 2329 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
6123, 27, 603eqtr4d 2325 . . 3  |-  ( A  e.  CC  ->  ( sin `  ( _i  x.  A ) )  =  ( _i  x.  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) ) )
6261oveq1d 5873 . 2  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( _i  x.  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )  /  _i ) )
6340, 54, 55divcan3d 9541 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) )
6462, 63eqtrd 2315 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 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738   _ici 8739    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   expce 12343   sincsin 12345
This theorem is referenced by:  resinhcl  12436  tanhlt1  12440  sinhpcosh  28210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351
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