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

Theorem tanval2 12689
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 12684 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
2 2cn 10026 . . . . . . 7  |-  2  e.  CC
3 ax-icn 9005 . . . . . . 7  |-  _i  e.  CC
42, 3mulcomi 9052 . . . . . 6  |-  ( 2  x.  _i )  =  ( _i  x.  2 )
54oveq2i 6051 . . . . 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 12678 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
76adantr 452 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sin `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
8 simpl 444 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  A  e.  CC )
9 mulcl 9030 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
103, 8, 9sylancr 645 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( _i  x.  A
)  e.  CC )
11 efcl 12640 . . . . . . . 8  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
1210, 11syl 16 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( exp `  (
_i  x.  A )
)  e.  CC )
133negcli 9324 . . . . . . . . 9  |-  -u _i  e.  CC
14 mulcl 9030 . . . . . . . . 9  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
1513, 8, 14sylancr 645 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( -u _i  x.  A
)  e.  CC )
16 efcl 12640 . . . . . . . 8  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
1715, 16syl 16 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( exp `  ( -u _i  x.  A ) )  e.  CC )
1812, 17subcld 9367 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
193a1i 11 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  _i  e.  CC )
202a1i 11 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
2  e.  CC )
21 ine0 9425 . . . . . . 7  |-  _i  =/=  0
2221a1i 11 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  _i  =/=  0 )
23 2ne0 10039 . . . . . . 7  |-  2  =/=  0
2423a1i 11 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
2  =/=  0 )
2518, 19, 20, 22, 24divdiv1d 9777 . . . . 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 2462 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sin `  A
)  =  ( ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  / 
2 ) )
27 cosval 12679 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
2827adantr 452 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( cos `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
2926, 28oveq12d 6058 . . 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 2436 . 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 9746 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  e.  CC )
3212, 17addcld 9063 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
33 simpr 448 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( cos `  A
)  =/=  0 )
3428, 33eqnetrrd 2587 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =/=  0 )
3532, 20, 24diveq0ad 9756 . . . . 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 ) )
3635necon3bid 2602 . . . 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 ) )
3734, 36mpbid 202 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  =/=  0 )
3831, 32, 20, 37, 24divcan7d 9774 . 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 ) ) ) ) )
3918, 19, 32, 22, 37divdiv1d 9777 . 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 ) ) ) ) ) )
4030, 38, 393eqtrd 2440 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    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   _ici 8948    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248    / cdiv 9633   2c2 10005   expce 12619   sincsin 12621   cosccos 12622   tanctan 12623
This theorem is referenced by:  tanval3  12690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-fac 11522  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-tan 12629
  Copyright terms: Public domain W3C validator