Theorem tanval2ap 11724

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 11719	. . 3 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
2		2cn 8993	. . . . . . 7 |- 2 e. CC
3		ax-icn 7909	. . . . . . 7 |- _i e. CC
4	2, 3	mulcomi 7966	. . . . . 6 |- ( 2 x. _i ) = ( _i x. 2 )
5	4	oveq2i 5889	. . . . 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 11713	. . . . . 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 7941	. . . . . . . . 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 11675	. . . . . . . 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 8161	. . . . . . . . 9 |- -u _i e. CC
14		mulcl 7941	. . . . . . . . 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 11675	. . . . . . . 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 8271	. . . . . 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 9145	. . . . . . 7 |- _i # 0
22	21	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> _i # 0 )
23		2ap0 9015	. . . . . . 7 |- 2 # 0
24	23	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> 2 # 0 )
25	18, 19, 20, 22, 24	divdivap1d 8782	. . . . 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 2236	. . . 4 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( sin ` A ) = ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) / 2 ) )
27		cosval 11714	. . . . 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 5896	. . 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 2210	. 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 8750	. . 3 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / _i ) e. CC )
32	12, 17	addcld 7980	. . 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 4029	. . . 4 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) # 0 )
35	32, 20, 24	divap0bd 8762	. . . 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 8779	. 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 8782	. 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 2214	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 1353   e. wcel 2148   class class class wbr 4005   ` cfv 5218  (class class class)co 5878   CCcc 7812   0cc0 7814   _ici 7816   + caddc 7817   x. cmul 7819   - cmin 8131   -ucneg 8132   # cap 8541   / cdiv 8632   2c2 8973   expce 11653   sincsin 11655   cosccos 11656   tanctan 11657

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-frec 6395  df-1o 6420  df-oadd 6424  df-er 6538  df-en 6744  df-dom 6745  df-fin 6746  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-q 9623  df-rp 9657  df-ico 9897  df-fz 10012  df-fzo 10146  df-seqfrec 10449  df-exp 10523  df-fac 10709  df-ihash 10759  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-clim 11290  df-sumdc 11365  df-ef 11659  df-sin 11661  df-cos 11662  df-tan 11663

This theorem is referenced by:  tanval3ap  11725