Theorem tanval3 14864

Description: Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.)

Assertion

Ref	Expression
tanval3	|- ( ( 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 tanval3

Step	Hyp	Ref	Expression
1		ax-icn 9995	. . . . . 6 |- _i e. CC
2		simpl 473	. . . . . 6 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> A e. CC )
3		mulcl 10020	. . . . . 6 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
4	1, 2, 3	sylancr 695	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> ( _i x. A ) e. CC )
5		efcl 14813	. . . . 5 |- ( ( _i x. A ) e. CC -> ( exp ` ( _i x. A ) ) e. CC )
6	4, 5	syl 17	. . . 4 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> ( exp ` ( _i x. A ) ) e. CC )
7		negicn 10282	. . . . . 6 |- -u _i e. CC
8		mulcl 10020	. . . . . 6 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
9	7, 2, 8	sylancr 695	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> ( -u _i x. A ) e. CC )
10		efcl 14813	. . . . 5 |- ( ( -u _i x. A ) e. CC -> ( exp ` ( -u _i x. A ) ) e. CC )
11	9, 10	syl 17	. . . 4 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> ( exp ` ( -u _i x. A ) ) e. CC )
12	6, 11	subcld 10392	. . 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 10059	. . . 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 10020	. . . 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 695	. . 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 11409	. . . . . . . . . . 11 |- 2 e. ZZ
17		efexp 14831	. . . . . . . . . . 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 694	. . . . . . . . . 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 13005	. . . . . . . . . 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 2656	. . . . . . . . 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 10466	. . . . . . . . . . . . 13 |- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. A ) = -u ( _i x. A ) )
22	1, 2, 21	sylancr 695	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> ( -u _i x. A ) = -u ( _i x. A ) )
23	22	fveq2d 6195	. . . . . . . . . . 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 6666	. . . . . . . . . 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 14826	. . . . . . . . . . 11 |- ( ( _i x. A ) e. CC -> ( ( exp ` ( _i x. A ) ) x. ( exp ` -u ( _i x. A ) ) ) = 1 )
26	4, 25	syl 17	. . . . . . . . . 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 2657	. . . . . . . . 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 6668	. . . . . . . 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 10064	. . . . . . . 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 2659	. . . . . . 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 6666	. . . . . 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 11	. . . . . . 7 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> _i e. CC )
33	32, 6, 13	mul12d 10245	. . . . . 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 2656	. . . . 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 11091	. . . . . . . . 9 |- 2 e. CC
36		mulcl 10020	. . . . . . . . 9 |- ( ( 2 e. CC /\ ( _i x. A ) e. CC ) -> ( 2 x. ( _i x. A ) ) e. CC )
37	35, 4, 36	sylancr 695	. . . . . . . 8 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> ( 2 x. ( _i x. A ) ) e. CC )
38		efcl 14813	. . . . . . . 8 |- ( ( 2 x. ( _i x. A ) ) e. CC -> ( exp ` ( 2 x. ( _i x. A ) ) ) e. CC )
39	37, 38	syl 17	. . . . . . 7 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> ( exp ` ( 2 x. ( _i x. A ) ) ) e. CC )
40		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
41		addcl 10018	. . . . . . 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 694	. . . . . 6 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) e. CC )
43		ine0 10465	. . . . . . 7 |- _i =/= 0
44	43	a1i 11	. . . . . 6 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> _i =/= 0 )
45		simpr 477	. . . . . 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	mulne0d 10679	. . . . 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	eqnetrrd 2862	. . . 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	mulne0bbd 10683	. . 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		efne0 14827	. . . 4 |- ( ( _i x. A ) e. CC -> ( exp ` ( _i x. A ) ) =/= 0 )
50	4, 49	syl 17	. . 3 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> ( exp ` ( _i x. A ) ) =/= 0 )
51	12, 15, 6, 48, 50	divcan5d 10827	. 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 6668	. . . 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 10486	. . . 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 2659	. . 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 6668	. 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 14853	. . . . 5 |- ( A e. CC -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) )
57	56	adantr 481	. . . 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 11093	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> 2 e. CC )
59	32, 13, 48	mulne0bbd 10683	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) =/= 0 )
60		2ne0 11113	. . . . . 6 |- 2 =/= 0
61	60	a1i 11	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> 2 =/= 0 )
62	13, 58, 59, 61	divne0d 10817	. . . 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	eqnetrd 2861	. . 3 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) =/= 0 ) -> ( cos ` A ) =/= 0 )
64		tanval2 14863	. . 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 487	. 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 2667	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 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   ZZcz 11377   ^cexp 12860   expce 14792   cosccos 14795   tanctan 14796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-tan 14802

This theorem is referenced by:  tanarg  24365  tanatan  24646