Theorem tanval2ap 11676

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 11671	. . 3 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
2		2cn 8949	. . . . . . 7 |- 2 e. CC
3		ax-icn 7869	. . . . . . 7 |- _i e. CC
4	2, 3	mulcomi 7926	. . . . . 6 |- ( 2 x. _i ) = ( _i x. 2 )
5	4	oveq2i 5864	. . . . 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 11665	. . . . . 6 |- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
7	6	adantr 274	. . . . 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 7901	. . . . . . . . 9 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
10	3, 8, 9	sylancr 412	. . . . . . . 8 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( _i x. A ) e. CC )
11		efcl 11627	. . . . . . . 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 8120	. . . . . . . . 9 |- -u _i e. CC
14		mulcl 7901	. . . . . . . . 9 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
15	13, 8, 14	sylancr 412	. . . . . . . 8 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( -u _i x. A ) e. CC )
16		efcl 11627	. . . . . . . 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 8230	. . . . . 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 9101	. . . . . . 7 |- _i # 0
22	21	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> _i # 0 )
23		2ap0 8971	. . . . . . 7 |- 2 # 0
24	23	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> 2 # 0 )
25	18, 19, 20, 22, 24	divdivap1d 8739	. . . . 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 2229	. . . 4 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( sin ` A ) = ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) / 2 ) )
27		cosval 11666	. . . . 5 |- ( A e. CC -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) )
28	27	adantr 274	. . . 4 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) )
29	26, 28	oveq12d 5871	. . 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 2203	. 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 8707	. . 3 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) e. CC )
32	12, 17	addcld 7939	. . 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 )
34	28, 33	eqbrtrrd 4013	. . . 4 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) # 0 )
35	32, 20, 24	divap0bd 8719	. . . 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 166	. . 3 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) # 0 )
37	31, 32, 20, 36, 24	divcanap7d 8736	. 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 8739	. 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 2207	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 103   = wceq 1348   e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   _ici 7776   + caddc 7777   x. cmul 7779   - cmin 8090   -ucneg 8091   # cap 8500   / cdiv 8589   2c2 8929   expce 11605   sincsin 11607   cosccos 11608   tanctan 11609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-ico 9851  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611  df-sin 11613  df-cos 11614  df-tan 11615

This theorem is referenced by:  tanval3ap  11677