Theorem tanval2ap 12424

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 12419	. . 3 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
2		2cn 9325	. . . . . . 7 |- 2 e. CC
3		ax-icn 8238	. . . . . . 7 |- _i e. CC
4	2, 3	mulcomi 8296	. . . . . 6 |- ( 2 x. _i ) = ( _i x. 2 )
5	4	oveq2i 6069	. . . . 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 12413	. . . . . 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 8270	. . . . . . . . 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 12375	. . . . . . . 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 8490	. . . . . . . . 9 |- -u _i e. CC
14		mulcl 8270	. . . . . . . . 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 12375	. . . . . . . 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 8600	. . . . . 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 9478	. . . . . . 7 |- _i # 0
22	21	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> _i # 0 )
23		2ap0 9347	. . . . . . 7 |- 2 # 0
24	23	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> 2 # 0 )
25	18, 19, 20, 22, 24	divdivap1d 9113	. . . . 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 2293	. . . 4 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( sin ` A ) = ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) / 2 ) )
27		cosval 12414	. . . . 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 6076	. . 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 2267	. 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 9081	. . 3 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) e. CC )
32	12, 17	addcld 8309	. . 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 4138	. . . 4 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) # 0 )
35	32, 20, 24	divap0bd 9093	. . . 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 9110	. 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 9113	. 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 2271	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 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   _ici 8145   + caddc 8146   x. cmul 8148   - cmin 8460   -ucneg 8461   # cap 8872   / cdiv 8963   2c2 9305   expce 12353   sincsin 12355   cosccos 12356   tanctan 12357

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-ico 10246  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-fac 11113  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064  df-ef 12359  df-sin 12361  df-cos 12362  df-tan 12363

This theorem is referenced by:  tanval3ap  12425