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Theorem tanval2 12736
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 12731 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
2 2cn 10072 . . . . . . 7  |-  2  e.  CC
3 ax-icn 9051 . . . . . . 7  |-  _i  e.  CC
42, 3mulcomi 9098 . . . . . 6  |-  ( 2  x.  _i )  =  ( _i  x.  2 )
54oveq2i 6094 . . . . 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 12725 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
76adantr 453 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sin `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
8 simpl 445 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  A  e.  CC )
9 mulcl 9076 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
103, 8, 9sylancr 646 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( _i  x.  A
)  e.  CC )
11 efcl 12687 . . . . . . . 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 9370 . . . . . . . . 9  |-  -u _i  e.  CC
14 mulcl 9076 . . . . . . . . 9  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
1513, 8, 14sylancr 646 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( -u _i  x.  A
)  e.  CC )
16 efcl 12687 . . . . . . . 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 9413 . . . . . 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 9471 . . . . . . 7  |-  _i  =/=  0
2221a1i 11 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  _i  =/=  0 )
23 2ne0 10085 . . . . . . 7  |-  2  =/=  0
2423a1i 11 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
2  =/=  0 )
2518, 19, 20, 22, 24divdiv1d 9823 . . . . 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 2496 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sin `  A
)  =  ( ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  / 
2 ) )
27 cosval 12726 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
2827adantr 453 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( cos `  A
)  =  ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
2926, 28oveq12d 6101 . . 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 2470 . 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 9792 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  _i )  e.  CC )
3212, 17addcld 9109 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
33 simpr 449 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( cos `  A
)  =/=  0 )
3428, 33eqnetrrd 2623 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  =/=  0 )
3532, 20, 24diveq0ad 9802 . . . . 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 2638 . . . 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 203 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  =/=  0 )
3831, 32, 20, 37, 24divcan7d 9820 . 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 9823 . 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 2474 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 360    = wceq 1653    e. wcel 1726    =/= wne 2601   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992   _ici 8994    + caddc 8995    x. cmul 8997    - cmin 9293   -ucneg 9294    / cdiv 9679   2c2 10051   expce 12666   sincsin 12668   cosccos 12669   tanctan 12670
This theorem is referenced by:  tanval3  12737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-ico 10924  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-fac 11569  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482  df-ef 12672  df-sin 12674  df-cos 12675  df-tan 12676
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