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Theorem tanval2 13850
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 13845 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
2 2cn 10613 . . . . . . 7  |-  2  e.  CC
3 ax-icn 9554 . . . . . . 7  |-  _i  e.  CC
42, 3mulcomi 9605 . . . . . 6  |-  ( 2  x.  _i )  =  ( _i  x.  2 )
54oveq2i 6292 . . . . 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 13839 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
76adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sin `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
8 simpl 457 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  A  e.  CC )
9 mulcl 9579 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
103, 8, 9sylancr 663 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( _i  x.  A
)  e.  CC )
11 efcl 13800 . . . . . . . 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 )
13 negicn 9826 . . . . . . . . 9  |-  -u _i  e.  CC
14 mulcl 9579 . . . . . . . . 9  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
1513, 8, 14sylancr 663 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( -u _i  x.  A
)  e.  CC )
16 efcl 13800 . . . . . . . 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 9936 . . . . . 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 9999 . . . . . . 7  |-  _i  =/=  0
2221a1i 11 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  _i  =/=  0 )
23 2ne0 10635 . . . . . . 7  |-  2  =/=  0
2423a1i 11 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
2  =/=  0 )
2518, 19, 20, 22, 24divdiv1d 10358 . . . . 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 2510 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sin `  A
)  =  ( ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  / 
2 ) )
27 cosval 13840 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
2827adantr 465 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( cos `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
2926, 28oveq12d 6299 . . 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 2484 . 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 10327 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  e.  CC )
3212, 17addcld 9618 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
33 simpr 461 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( cos `  A
)  =/=  0 )
3428, 33eqnetrrd 2737 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =/=  0 )
3532, 20, 24diveq0ad 10337 . . . . 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 2701 . . . 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 210 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  =/=  0 )
3831, 32, 20, 37, 24divcan7d 10355 . 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 10358 . 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 2488 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   _ici 9497    + caddc 9498    x. cmul 9500    - cmin 9810   -ucneg 9811    / cdiv 10213   2c2 10592   expce 13779   sincsin 13781   cosccos 13782   tanctan 13783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-ico 11546  df-fz 11684  df-fzo 11807  df-fl 11911  df-seq 12090  df-exp 12149  df-fac 12336  df-hash 12388  df-shft 12882  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-limsup 13276  df-clim 13293  df-rlim 13294  df-sum 13491  df-ef 13785  df-sin 13787  df-cos 13788  df-tan 13789
This theorem is referenced by:  tanval3  13851
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