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Theorem tanval2ap 11738
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 11733 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( tan `  A )  =  ( ( sin `  A
)  /  ( cos `  A ) ) )
2 2cn 9007 . . . . . . 7  |-  2  e.  CC
3 ax-icn 7923 . . . . . . 7  |-  _i  e.  CC
42, 3mulcomi 7980 . . . . . 6  |-  ( 2  x.  _i )  =  ( _i  x.  2 )
54oveq2i 5901 . . . . 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 11727 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
76adantr 276 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
8 simpl 109 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  A  e.  CC )
9 mulcl 7955 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
103, 8, 9sylancr 414 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  (
_i  x.  A )  e.  CC )
11 efcl 11689 . . . . . . . 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 8175 . . . . . . . . 9  |-  -u _i  e.  CC
14 mulcl 7955 . . . . . . . . 9  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
1513, 8, 14sylancr 414 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( -u _i  x.  A )  e.  CC )
16 efcl 11689 . . . . . . . 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 8285 . . . . . 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 9159 . . . . . . 7  |-  _i #  0
2221a1i 9 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  _i #  0 )
23 2ap0 9029 . . . . . . 7  |-  2 #  0
2423a1i 9 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  2 #  0 )
2518, 19, 20, 22, 24divdivap1d 8796 . . . . 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 2247 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( sin `  A )  =  ( ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  /  2 ) )
27 cosval 11728 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
2827adantr 276 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
2926, 28oveq12d 5908 . . 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 2221 . 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 8764 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  e.  CC )
3212, 17addcld 7994 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
33 simpr 110 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( cos `  A ) #  0 )
3428, 33eqbrtrrd 4041 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  (
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) #  0 )
3532, 20, 24divap0bd 8776 . . . 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 167 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  (
( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) ) #  0 )
3731, 32, 20, 36, 24divcanap7d 8793 . 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 8796 . 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 2225 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 104    = wceq 1363    e. wcel 2159   class class class wbr 4017   ` cfv 5230  (class class class)co 5890   CCcc 7826   0cc0 7828   _ici 7830    + caddc 7831    x. cmul 7833    - cmin 8145   -ucneg 8146   # cap 8555    / cdiv 8646   2c2 8987   expce 11667   sincsin 11669   cosccos 11670   tanctan 11671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947  ax-caucvg 7948
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-isom 5239  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-irdg 6388  df-frec 6409  df-1o 6434  df-oadd 6438  df-er 6552  df-en 6758  df-dom 6759  df-fin 6760  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-n0 9194  df-z 9271  df-uz 9546  df-q 9637  df-rp 9671  df-ico 9911  df-fz 10026  df-fzo 10160  df-seqfrec 10463  df-exp 10537  df-fac 10723  df-ihash 10773  df-cj 10868  df-re 10869  df-im 10870  df-rsqrt 11024  df-abs 11025  df-clim 11304  df-sumdc 11379  df-ef 11673  df-sin 11675  df-cos 11676  df-tan 11677
This theorem is referenced by:  tanval3ap  11739
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