MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tanval2 Unicode version

Theorem tanval2 12429
Description: Express the tangent function directly in terms of  exp. (Contributed by Mario Carneiro, 25-Feb-2015.)
Assertion
Ref Expression
tanval2  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( _i  x.  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) ) ) )

Proof of Theorem tanval2
StepHypRef Expression
1 tanval 12424 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
2 2cn 9832 . . . . . . 7  |-  2  e.  CC
3 ax-icn 8812 . . . . . . 7  |-  _i  e.  CC
42, 3mulcomi 8859 . . . . . 6  |-  ( 2  x.  _i )  =  ( _i  x.  2 )
54oveq2i 5885 . . . . 5  |-  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( _i  x.  2 ) )
6 sinval 12418 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
76adantr 451 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sin `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
8 simpl 443 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  A  e.  CC )
9 mulcl 8837 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
103, 8, 9sylancr 644 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( _i  x.  A
)  e.  CC )
11 efcl 12380 . . . . . . . 8  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
1210, 11syl 15 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( exp `  (
_i  x.  A )
)  e.  CC )
133negcli 9130 . . . . . . . . 9  |-  -u _i  e.  CC
14 mulcl 8837 . . . . . . . . 9  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
1513, 8, 14sylancr 644 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( -u _i  x.  A
)  e.  CC )
16 efcl 12380 . . . . . . . 8  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
1715, 16syl 15 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( exp `  ( -u _i  x.  A ) )  e.  CC )
1812, 17subcld 9173 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
193a1i 10 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  _i  e.  CC )
202a1i 10 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
2  e.  CC )
21 ine0 9231 . . . . . . 7  |-  _i  =/=  0
2221a1i 10 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  _i  =/=  0 )
23 2ne0 9845 . . . . . . 7  |-  2  =/=  0
2423a1i 10 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
2  =/=  0 )
2518, 19, 20, 22, 24divdiv1d 9583 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  /  2 )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( _i  x.  2 ) ) )
265, 7, 253eqtr4a 2354 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sin `  A
)  =  ( ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  / 
2 ) )
27 cosval 12419 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
2827adantr 451 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( cos `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
2926, 28oveq12d 5892 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( sin `  A
)  /  ( cos `  A ) )  =  ( ( ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  / 
2 )  /  (
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) ) )
301, 29eqtrd 2328 . 2  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  / 
2 )  /  (
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) ) )
3118, 19, 22divcld 9552 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  e.  CC )
3212, 17addcld 8870 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
33 simpr 447 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( cos `  A
)  =/=  0 )
3428, 33eqnetrrd 2479 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =/=  0 )
35 diveq0 9450 . . . . . 6  |-  ( ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =  0  <->  ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) )  =  0 ) )
3632, 20, 24, 35syl3anc 1182 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2
)  =  0  <->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  =  0 ) )
3736necon3bid 2494 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2
)  =/=  0  <->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  =/=  0 ) )
3834, 37mpbid 201 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  =/=  0 )
3931, 32, 20, 38, 24divcan7d 9580 . 2  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  / 
2 )  /  (
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )  =  ( ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  / 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) ) )
4018, 19, 32, 22, 38divdiv1d 9583 . 2  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  /  ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) ) )  =  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( _i  x.  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) ) ) )
4130, 39, 403eqtrd 2332 1  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( _i  x.  ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   _ici 8755    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   expce 12359   sincsin 12361   cosccos 12362   tanctan 12363
This theorem is referenced by:  tanval3  12430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-tan 12369
  Copyright terms: Public domain W3C validator