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

Step	Hyp	Ref	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
4	2, 3	mulcomi 7980	. . . . . 6 |- ( 2 x. _i ) = ( _i x. 2 )
5	4	oveq2i 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 ) ) )
7	6	adantr 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 )
10	3, 8, 9	sylancr 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 )
12	10, 11	syl 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 )
15	13, 8, 14	sylancr 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 )
17	15, 16	syl 14	. . . . . . 7 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( exp ` ( -u _i x. A ) ) e. CC )
18	12, 17	subcld 8285	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC )
19	3	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> _i e. CC )
20	2	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> 2 e. CC )
21		iap0 9159	. . . . . . 7 |- _i # 0
22	21	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> _i # 0 )
23		2ap0 9029	. . . . . . 7 |- 2 # 0
24	23	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> 2 # 0 )
25	18, 19, 20, 22, 24	divdivap1d 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 ) ) )
26	5, 7, 25	3eqtr4a 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 ) )
28	27	adantr 276	. . . 4 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) )
29	26, 28	oveq12d 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 ) ) )
30	1, 29	eqtrd 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 ) ) )
31	18, 19, 22	divclapd 8764	. . 3 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) e. CC )
32	12, 17	addcld 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 )
34	28, 33	eqbrtrrd 4041	. . . 4 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) # 0 )
35	32, 20, 24	divap0bd 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 ) )
36	34, 35	mpbird 167	. . 3 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) # 0 )
37	31, 32, 20, 36, 24	divcanap7d 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 ) ) ) ) )
38	18, 19, 32, 22, 36	divdivap1d 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 ) ) ) ) ) )
39	30, 37, 38	3eqtrd 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 ) ) ) ) ) )

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