Theorem tanval3ap 11722

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
tanval3ap	|- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( tan ` A ) = ( ( ( exp ` ( 2 x. ( _i x. A ) ) ) - 1 ) / ( _i x. ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) ) ) )

Proof of Theorem tanval3ap

Step	Hyp	Ref	Expression
1		ax-icn 7906	. . . . . 6 |- _i e. CC
2		simpl 109	. . . . . 6 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> A e. CC )
3		mulcl 7938	. . . . . 6 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
4	1, 2, 3	sylancr 414	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( _i x. A ) e. CC )
5		efcl 11672	. . . . 5 |- ( ( _i x. A ) e. CC -> ( exp ` ( _i x. A ) ) e. CC )
6	4, 5	syl 14	. . . 4 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( exp ` ( _i x. A ) ) e. CC )
7		negicn 8158	. . . . . 6 |- -u _i e. CC
8		mulcl 7938	. . . . . 6 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
9	7, 2, 8	sylancr 414	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( -u _i x. A ) e. CC )
10		efcl 11672	. . . . 5 |- ( ( -u _i x. A ) e. CC -> ( exp ` ( -u _i x. A ) ) e. CC )
11	9, 10	syl 14	. . . 4 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( exp ` ( -u _i x. A ) ) e. CC )
12	6, 11	subcld 8268	. . 3 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC )
13	6, 11	addcld 7977	. . . 4 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) e. CC )
14		mulcl 7938	. . . 4 |- ( ( _i e. CC /\ ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) e. CC ) -> ( _i x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) e. CC )
15	1, 13, 14	sylancr 414	. . 3 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( _i x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) e. CC )
16		2z 9281	. . . . . . . . . . 11 |- 2 e. ZZ
17		efexp 11690	. . . . . . . . . . 11 |- ( ( ( _i x. A ) e. CC /\ 2 e. ZZ ) -> ( exp ` ( 2 x. ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) ^ 2 ) )
18	4, 16, 17	sylancl 413	. . . . . . . . . 10 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( exp ` ( 2 x. ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) ^ 2 ) )
19	6	sqvald 10651	. . . . . . . . . 10 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( _i x. A ) ) ^ 2 ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) )
20	18, 19	eqtrd 2210	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( exp ` ( 2 x. ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) )
21		mulneg1 8352	. . . . . . . . . . . . 13 |- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. A ) = -u ( _i x. A ) )
22	1, 2, 21	sylancr 414	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( -u _i x. A ) = -u ( _i x. A ) )
23	22	fveq2d 5520	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( exp ` ( -u _i x. A ) ) = ( exp ` -u ( _i x. A ) ) )
24	23	oveq2d 5891	. . . . . . . . . 10 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` -u ( _i x. A ) ) ) )
25		efcan 11684	. . . . . . . . . . 11 |- ( ( _i x. A ) e. CC -> ( ( exp ` ( _i x. A ) ) x. ( exp ` -u ( _i x. A ) ) ) = 1 )
26	4, 25	syl 14	. . . . . . . . . 10 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( _i x. A ) ) x. ( exp ` -u ( _i x. A ) ) ) = 1 )
27	24, 26	eqtr2d 2211	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> 1 = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) )
28	20, 27	oveq12d 5893	. . . . . . . 8 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) = ( ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) + ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) ) )
29	6, 6, 11	adddid 7982	. . . . . . . 8 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( _i x. A ) ) x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) = ( ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) + ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) ) )
30	28, 29	eqtr4d 2213	. . . . . . 7 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) = ( ( exp ` ( _i x. A ) ) x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) )
31	30	oveq2d 5891	. . . . . 6 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( _i x. ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) ) = ( _i x. ( ( exp ` ( _i x. A ) ) x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) ) )
32	1	a1i 9	. . . . . . 7 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> _i e. CC )
33	32, 6, 13	mul12d 8109	. . . . . 6 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( _i x. ( ( exp ` ( _i x. A ) ) x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( _i x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) ) )
34	31, 33	eqtrd 2210	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( _i x. ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) ) = ( ( exp ` ( _i x. A ) ) x. ( _i x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) ) )
35		2cn 8990	. . . . . . . . 9 |- 2 e. CC
36		mulcl 7938	. . . . . . . . 9 |- ( ( 2 e. CC /\ ( _i x. A ) e. CC ) -> ( 2 x. ( _i x. A ) ) e. CC )
37	35, 4, 36	sylancr 414	. . . . . . . 8 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( 2 x. ( _i x. A ) ) e. CC )
38		efcl 11672	. . . . . . . 8 |- ( ( 2 x. ( _i x. A ) ) e. CC -> ( exp ` ( 2 x. ( _i x. A ) ) ) e. CC )
39	37, 38	syl 14	. . . . . . 7 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( exp ` ( 2 x. ( _i x. A ) ) ) e. CC )
40		ax-1cn 7904	. . . . . . 7 |- 1 e. CC
41		addcl 7936	. . . . . . 7 |- ( ( ( exp ` ( 2 x. ( _i x. A ) ) ) e. CC /\ 1 e. CC ) -> ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) e. CC )
42	39, 40, 41	sylancl 413	. . . . . 6 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) e. CC )
43		iap0 9142	. . . . . . 7 |- _i # 0
44	43	a1i 9	. . . . . 6 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> _i # 0 )
45		simpr 110	. . . . . 6 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 )
46	32, 42, 44, 45	mulap0d 8615	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( _i x. ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) ) # 0 )
47	34, 46	eqbrtrrd 4028	. . . 4 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( _i x. A ) ) x. ( _i x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) ) # 0 )
48	6, 15, 47	mulap0bbd 8617	. . 3 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( _i x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) # 0 )
49		efap0 11685	. . . 4 |- ( ( _i x. A ) e. CC -> ( exp ` ( _i x. A ) ) # 0 )
50	4, 49	syl 14	. . 3 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( exp ` ( _i x. A ) ) # 0 )
51	12, 15, 6, 48, 50	divcanap5d 8774	. 2 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( ( exp ` ( _i x. A ) ) x. ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) / ( ( exp ` ( _i x. A ) ) x. ( _i x. ( ( 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 ) ) ) ) ) )
52	20, 27	oveq12d 5893	. . . 4 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( 2 x. ( _i x. A ) ) ) - 1 ) = ( ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) - ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) ) )
53	6, 6, 11	subdid 8371	. . . 4 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( _i x. A ) ) x. ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) = ( ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) - ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) ) )
54	52, 53	eqtr4d 2213	. . 3 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( 2 x. ( _i x. A ) ) ) - 1 ) = ( ( exp ` ( _i x. A ) ) x. ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) )
55	54, 34	oveq12d 5893	. 2 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( ( exp ` ( 2 x. ( _i x. A ) ) ) - 1 ) / ( _i x. ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) ) ) = ( ( ( exp ` ( _i x. A ) ) x. ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) / ( ( exp ` ( _i x. A ) ) x. ( _i x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) ) ) )
56		cosval 11711	. . . . 5 |- ( A e. CC -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) )
57	56	adantr 276	. . . 4 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) )
58		2cnd 8992	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> 2 e. CC )
59	32, 13, 48	mulap0bbd 8617	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) # 0 )
60		2ap0 9012	. . . . . 6 |- 2 # 0
61	60	a1i 9	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> 2 # 0 )
62	13, 58, 59, 61	divap0d 8763	. . . 4 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) # 0 )
63	57, 62	eqbrtrd 4026	. . 3 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( cos ` A ) # 0 )
64		tanval2ap 11721	. . 3 |- ( ( 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 ) ) ) ) ) )
65	63, 64	syldan 282	. 2 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( tan ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( _i x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) ) )
66	51, 55, 65	3eqtr4rd 2221	1 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( tan ` A ) = ( ( ( exp ` ( 2 x. ( _i x. A ) ) ) - 1 ) / ( _i x. ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   class class class wbr 4004   ` cfv 5217  (class class class)co 5875   CCcc 7809   0cc0 7811   1c1 7812   _ici 7813   + caddc 7814   x. cmul 7816   - cmin 8128   -ucneg 8129   # cap 8538   / cdiv 8629   2c2 8970   ZZcz 9253   ^cexp 10519   expce 11650   cosccos 11653   tanctan 11654

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 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931

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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-disj 3982  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-frec 6392  df-1o 6417  df-oadd 6421  df-er 6535  df-en 6741  df-dom 6742  df-fin 6743  df-sup 6983  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-ico 9894  df-fz 10009  df-fzo 10143  df-seqfrec 10446  df-exp 10520  df-fac 10706  df-bc 10728  df-ihash 10756  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-clim 11287  df-sumdc 11362  df-ef 11656  df-sin 11658  df-cos 11659  df-tan 11660

This theorem is referenced by: (None)