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Theorem sinhval 13453
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 9980 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
21oveq1i 6116 . . . . . . . 8  |-  ( ( _i  x.  _i )  x.  A )  =  ( -u 1  x.  A )
3 ax-icn 9356 . . . . . . . . 9  |-  _i  e.  CC
4 mulass 9385 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  _i  e.  CC  /\  A  e.  CC )  ->  (
( _i  x.  _i )  x.  A )  =  ( _i  x.  ( _i  x.  A
) ) )
53, 3, 4mp3an12 1304 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( _i  x.  _i )  x.  A )  =  ( _i  x.  ( _i  x.  A
) ) )
6 mulm1 9801 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
72, 5, 63eqtr3a 2499 . . . . . . 7  |-  ( A  e.  CC  ->  (
_i  x.  ( _i  x.  A ) )  = 
-u A )
87fveq2d 5710 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( _i  x.  A
) ) )  =  ( exp `  -u A
) )
93, 3mulneg1i 9805 . . . . . . . . . 10  |-  ( -u _i  x.  _i )  = 
-u ( _i  x.  _i )
101negeqi 9618 . . . . . . . . . . 11  |-  -u (
_i  x.  _i )  =  -u -u 1
11 negneg1e1 10444 . . . . . . . . . . 11  |-  -u -u 1  =  1
1210, 11eqtri 2463 . . . . . . . . . 10  |-  -u (
_i  x.  _i )  =  1
139, 12eqtri 2463 . . . . . . . . 9  |-  ( -u _i  x.  _i )  =  1
1413oveq1i 6116 . . . . . . . 8  |-  ( (
-u _i  x.  _i )  x.  A )  =  ( 1  x.  A )
15 negicn 9626 . . . . . . . . 9  |-  -u _i  e.  CC
16 mulass 9385 . . . . . . . . 9  |-  ( (
-u _i  e.  CC  /\  _i  e.  CC  /\  A  e.  CC )  ->  ( ( -u _i  x.  _i )  x.  A
)  =  ( -u _i  x.  ( _i  x.  A ) ) )
1715, 3, 16mp3an12 1304 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( -u _i  x.  _i )  x.  A )  =  ( -u _i  x.  ( _i  x.  A
) ) )
18 mulid2 9399 . . . . . . . 8  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
1914, 17, 183eqtr3a 2499 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  ( _i  x.  A ) )  =  A )
2019fveq2d 5710 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  ( _i  x.  A
) ) )  =  ( exp `  A
) )
218, 20oveq12d 6124 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  ( _i  x.  A ) ) )  -  ( exp `  ( -u _i  x.  ( _i  x.  A ) ) ) )  =  ( ( exp `  -u A
)  -  ( exp `  A ) ) )
2221oveq1d 6121 . . . 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 ) ) )
23 mulcl 9381 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
243, 23mpan 670 . . . . 5  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
25 sinval 13421 . . . . 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 ) ) )
2624, 25syl 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 ) ) )
27 irec 11980 . . . . . . . 8  |-  ( 1  /  _i )  = 
-u _i
2827negeqi 9618 . . . . . . 7  |-  -u (
1  /  _i )  =  -u -u _i
293negnegi 9693 . . . . . . 7  |-  -u -u _i  =  _i
3028, 29eqtri 2463 . . . . . 6  |-  -u (
1  /  _i )  =  _i
3130oveq1i 6116 . . . . 5  |-  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( _i  x.  ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) )
32 ine0 9795 . . . . . . . 8  |-  _i  =/=  0
333, 32reccli 10076 . . . . . . 7  |-  ( 1  /  _i )  e.  CC
34 efcl 13383 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
35 negcl 9625 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u A  e.  CC )
36 efcl 13383 . . . . . . . . . 10  |-  ( -u A  e.  CC  ->  ( exp `  -u A
)  e.  CC )
3735, 36syl 16 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  -u A )  e.  CC )
3834, 37subcld 9734 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
3938halfcld 10584 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  e.  CC )
40 mulneg12 9798 . . . . . . 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 ) ) )
4133, 39, 40sylancr 663 . . . . . 6  |-  ( A  e.  CC  ->  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( 1  /  _i )  x.  -u ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) ) )
42 2cnd 10409 . . . . . . . . . 10  |-  ( A  e.  CC  ->  2  e.  CC )
43 2ne0 10429 . . . . . . . . . . 11  |-  2  =/=  0
4443a1i 11 . . . . . . . . . 10  |-  ( A  e.  CC  ->  2  =/=  0 )
4538, 42, 44divnegd 10135 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( -u ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
4634, 37negsubdi2d 9750 . . . . . . . . . 10  |-  ( A  e.  CC  ->  -u (
( exp `  A
)  -  ( exp `  -u A ) )  =  ( ( exp `  -u A )  -  ( exp `  A ) ) )
4746oveq1d 6121 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( -u ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 ) )
4845, 47eqtrd 2475 . . . . . . . 8  |-  ( A  e.  CC  ->  -u (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 ) )
4948oveq2d 6122 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 1  /  _i )  x.  -u ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( 1  /  _i )  x.  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 ) ) )
5037, 34subcld 9734 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( exp `  -u A
)  -  ( exp `  A ) )  e.  CC )
5150halfcld 10584 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( exp `  -u A
)  -  ( exp `  A ) )  / 
2 )  e.  CC )
523a1i 11 . . . . . . . 8  |-  ( A  e.  CC  ->  _i  e.  CC )
5332a1i 11 . . . . . . . 8  |-  ( A  e.  CC  ->  _i  =/=  0 )
5451, 52, 53divrec2d 10126 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 )  /  _i )  =  ( ( 1  /  _i )  x.  (
( ( exp `  -u A
)  -  ( exp `  A ) )  / 
2 ) ) )
5550, 42, 52, 44, 53divdiv1d 10153 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( ( exp `  -u A )  -  ( exp `  A ) )  /  2 )  /  _i )  =  ( ( ( exp `  -u A )  -  ( exp `  A ) )  /  ( 2  x.  _i ) ) )
5649, 54, 553eqtr2d 2481 . . . . . 6  |-  ( A  e.  CC  ->  (
( 1  /  _i )  x.  -u ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
5741, 56eqtrd 2475 . . . . 5  |-  ( A  e.  CC  ->  ( -u ( 1  /  _i )  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
5831, 57syl5eqr 2489 . . . 4  |-  ( A  e.  CC  ->  (
_i  x.  ( (
( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  =  ( ( ( exp `  -u A
)  -  ( exp `  A ) )  / 
( 2  x.  _i ) ) )
5922, 26, 583eqtr4d 2485 . . 3  |-  ( A  e.  CC  ->  ( sin `  ( _i  x.  A ) )  =  ( _i  x.  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) ) )
6059oveq1d 6121 . 2  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( _i  x.  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )  /  _i ) )
6139, 52, 53divcan3d 10127 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 ) )
6260, 61eqtrd 2475 1  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    =/= wne 2620   ` cfv 5433  (class class class)co 6106   CCcc 9295   0cc0 9297   1c1 9298   _ici 9299    x. cmul 9302    - cmin 9610   -ucneg 9611    / cdiv 10008   2c2 10386   expce 13362   sincsin 13364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375  ax-addf 9376  ax-mulf 9377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-pm 7232  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-sup 7706  df-oi 7739  df-card 8124  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-n0 10595  df-z 10662  df-uz 10877  df-rp 11007  df-ico 11321  df-fz 11453  df-fzo 11564  df-fl 11657  df-seq 11822  df-exp 11881  df-fac 12067  df-hash 12119  df-shft 12571  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-limsup 12964  df-clim 12981  df-rlim 12982  df-sum 13179  df-ef 13368  df-sin 13370
This theorem is referenced by:  resinhcl  13455  tanhlt1  13459  sinhpcosh  31094
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