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Theorem tanval2ap 11058
Description: Express the tangent function directly in terms of  exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
Assertion
Ref Expression
tanval2ap  |-  ( ( 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 tanval2ap
StepHypRef Expression
1 tanvalap 11053 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( tan `  A )  =  ( ( sin `  A
)  /  ( cos `  A ) ) )
2 2cn 8547 . . . . . . 7  |-  2  e.  CC
3 ax-icn 7494 . . . . . . 7  |-  _i  e.  CC
42, 3mulcomi 7548 . . . . . 6  |-  ( 2  x.  _i )  =  ( _i  x.  2 )
54oveq2i 5677 . . . . 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 11047 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
76adantr 271 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
8 simpl 108 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  A  e.  CC )
9 mulcl 7523 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
103, 8, 9sylancr 406 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  (
_i  x.  A )  e.  CC )
11 efcl 11008 . . . . . . . 8  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
1210, 11syl 14 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
13 negicn 7737 . . . . . . . . 9  |-  -u _i  e.  CC
14 mulcl 7523 . . . . . . . . 9  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
1513, 8, 14sylancr 406 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( -u _i  x.  A )  e.  CC )
16 efcl 11008 . . . . . . . 8  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
1715, 16syl 14 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
1812, 17subcld 7847 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
193a1i 9 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  _i  e.  CC )
202a1i 9 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  2  e.  CC )
21 iap0 8693 . . . . . . 7  |-  _i #  0
2221a1i 9 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  _i #  0 )
23 2ap0 8569 . . . . . . 7  |-  2 #  0
2423a1i 9 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  2 #  0 )
2518, 19, 20, 22, 24divdivap1d 8343 . . . . 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 2147 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( sin `  A )  =  ( ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  /  2 ) )
27 cosval 11048 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
2827adantr 271 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
2926, 28oveq12d 5684 . . 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 2121 . 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, 22divclapd 8311 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  e.  CC )
3212, 17addcld 7561 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
33 simpr 109 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( cos `  A ) #  0 )
3428, 33eqbrtrrd 3873 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  (
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) #  0 )
3532, 20, 24divap0bd 8323 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  (
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) #  0  <->  ( ( ( exp `  ( _i  x.  A ) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2
) #  0 ) )
3634, 35mpbird 166 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) #  0 )
3731, 32, 20, 36, 24divcanap7d 8340 . 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 ) ) ) ) )
3818, 19, 32, 22, 36divdivap1d 8343 . 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 ) ) ) ) ) )
3930, 37, 383eqtrd 2125 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 103    = wceq 1290    e. wcel 1439   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   CCcc 7402   0cc0 7404   _ici 7406    + caddc 7407    x. cmul 7409    - cmin 7707   -ucneg 7708   # cap 8112    / cdiv 8193   2c2 8527   expce 10986   sincsin 10988   cosccos 10989   tanctan 10990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517  ax-arch 7518  ax-caucvg 7519
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113  df-div 8194  df-inn 8477  df-2 8535  df-3 8536  df-4 8537  df-n0 8728  df-z 8805  df-uz 9074  df-q 9159  df-rp 9189  df-ico 9366  df-fz 9479  df-fzo 9608  df-iseq 9907  df-seq3 9908  df-exp 10009  df-fac 10188  df-ihash 10238  df-cj 10330  df-re 10331  df-im 10332  df-rsqrt 10485  df-abs 10486  df-clim 10721  df-isum 10797  df-ef 10992  df-sin 10994  df-cos 10995  df-tan 10996
This theorem is referenced by:  tanval3ap  11059
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