Theorem tanval3ap 12025

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 8020	. . . . . 6 |- _i e. CC
2		simpl 109	. . . . . 6 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> A e. CC )
3		mulcl 8052	. . . . . 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 11975	. . . . 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 8273	. . . . . 6 |- -u _i e. CC
8		mulcl 8052	. . . . . 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 11975	. . . . 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 8383	. . 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 8092	. . . 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 8052	. . . 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 9400	. . . . . . . . . . 11 |- 2 e. ZZ
17		efexp 11993	. . . . . . . . . . 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 10815	. . . . . . . . . 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 2238	. . . . . . . . 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 8467	. . . . . . . . . . . . 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 5580	. . . . . . . . . . 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 5960	. . . . . . . . . 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 11987	. . . . . . . . . . 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 2239	. . . . . . . . 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 5962	. . . . . . . 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 8097	. . . . . . . 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 2241	. . . . . . 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 5960	. . . . . 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 8224	. . . . . 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 2238	. . . . 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 9107	. . . . . . . . 9 |- 2 e. CC
36		mulcl 8052	. . . . . . . . 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 11975	. . . . . . . 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 8018	. . . . . . 7 |- 1 e. CC
41		addcl 8050	. . . . . . 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 9260	. . . . . . 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 8731	. . . . 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 4068	. . . 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 8733	. . 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 11988	. . . 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 8890	. 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 5962	. . . 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 8486	. . . 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 2241	. . 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 5962	. 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 12014	. . . . 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 9109	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> 2 e. CC )
59	32, 13, 48	mulap0bbd 8733	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) # 0 )
60		2ap0 9129	. . . . . 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 8879	. . . 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 4066	. . 3 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( cos ` A ) # 0 )
64		tanval2ap 12024	. . 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 2249	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 1373   e. wcel 2176   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   CCcc 7923   0cc0 7925   1c1 7926   _ici 7927   + caddc 7928   x. cmul 7930   - cmin 8243   -ucneg 8244   # cap 8654   / cdiv 8745   2c2 9087   ZZcz 9372   ^cexp 10683   expce 11953   cosccos 11956   tanctan 11957

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-disj 4022  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-frec 6477  df-1o 6502  df-oadd 6506  df-er 6620  df-en 6828  df-dom 6829  df-fin 6830  df-sup 7086  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-ico 10016  df-fz 10131  df-fzo 10265  df-seqfrec 10593  df-exp 10684  df-fac 10871  df-bc 10893  df-ihash 10921  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590  df-sumdc 11665  df-ef 11959  df-sin 11961  df-cos 11962  df-tan 11963

This theorem is referenced by: (None)