Theorem tanval2ap 11878

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 11873	. . 3 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
2		2cn 9061	. . . . . . 7 |- 2 e. CC
3		ax-icn 7974	. . . . . . 7 |- _i e. CC
4	2, 3	mulcomi 8032	. . . . . 6 |- ( 2 x. _i ) = ( _i x. 2 )
5	4	oveq2i 5933	. . . . 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 11867	. . . . . 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 8006	. . . . . . . . 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 11829	. . . . . . . 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 8227	. . . . . . . . 9 |- -u _i e. CC
14		mulcl 8006	. . . . . . . . 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 11829	. . . . . . . 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 8337	. . . . . 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 9214	. . . . . . 7 |- _i # 0
22	21	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> _i # 0 )
23		2ap0 9083	. . . . . . 7 |- 2 # 0
24	23	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> 2 # 0 )
25	18, 19, 20, 22, 24	divdivap1d 8849	. . . . 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 2255	. . . 4 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( sin ` A ) = ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) / 2 ) )
27		cosval 11868	. . . . 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 5940	. . 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 2229	. 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 8817	. . 3 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) e. CC )
32	12, 17	addcld 8046	. . 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 4057	. . . 4 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) # 0 )
35	32, 20, 24	divap0bd 8829	. . . 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 8846	. 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 8849	. 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 2233	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 1364   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   CCcc 7877   0cc0 7879   _ici 7881   + caddc 7882   x. cmul 7884   - cmin 8197   -ucneg 8198   # cap 8608   / cdiv 8699   2c2 9041   expce 11807   sincsin 11809   cosccos 11810   tanctan 11811

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-ico 9969  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-fac 10818  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519  df-ef 11813  df-sin 11815  df-cos 11816  df-tan 11817

This theorem is referenced by:  tanval3ap  11879