Theorem tanval3ap 12459

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 8264	. . . . . 6 |- _i e. CC
2		simpl 109	. . . . . 6 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> A e. CC )
3		mulcl 8296	. . . . . 6 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
4	1, 2, 3	sylancr 418	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( _i x. A ) e. CC )
5		efcl 12409	. . . . 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 8517	. . . . . 6 |- -u _i e. CC
8		mulcl 8296	. . . . . 6 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
9	7, 2, 8	sylancr 418	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( -u _i x. A ) e. CC )
10		efcl 12409	. . . . 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 8627	. . 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 8335	. . . 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 8296	. . . 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 418	. . 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 9651	. . . . . . . . . . 11 |- 2 e. ZZ
17		efexp 12427	. . . . . . . . . . 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 417	. . . . . . . . . 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 11086	. . . . . . . . . 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 2271	. . . . . . . . 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 8712	. . . . . . . . . . . . 13 |- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. A ) = -u ( _i x. A ) )
22	1, 2, 21	sylancr 418	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( -u _i x. A ) = -u ( _i x. A ) )
23	22	fveq2d 5694	. . . . . . . . . . 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 6091	. . . . . . . . . 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 12421	. . . . . . . . . . 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 2272	. . . . . . . . 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 6093	. . . . . . . 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 8340	. . . . . . . 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 2274	. . . . . . 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 6091	. . . . . 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 8468	. . . . . 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 2271	. . . . 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 9354	. . . . . . . . 9 |- 2 e. CC
36		mulcl 8296	. . . . . . . . 9 |- ( ( 2 e. CC /\ ( _i x. A ) e. CC ) -> ( 2 x. ( _i x. A ) ) e. CC )
37	35, 4, 36	sylancr 418	. . . . . . . 8 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( 2 x. ( _i x. A ) ) e. CC )
38		efcl 12409	. . . . . . . 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 8262	. . . . . . 7 |- 1 e. CC
41		addcl 8294	. . . . . . 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 417	. . . . . 6 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) e. CC )
43		iap0 9507	. . . . . . 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 8976	. . . . 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 4149	. . . 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 8978	. . 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 12422	. . . 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 9137	. 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 6093	. . . 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 8731	. . . 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 2274	. . 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 6093	. 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 12448	. . . . 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 9356	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> 2 e. CC )
59	32, 13, 48	mulap0bbd 8978	. . . . 5 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) # 0 )
60		2ap0 9376	. . . . . 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 9126	. . . 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 4147	. . 3 |- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( cos ` A ) # 0 )
64		tanval2ap 12458	. . 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 2282	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 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   _ici 8171   + caddc 8172   x. cmul 8174   - cmin 8487   -ucneg 8488   # cap 8899   / cdiv 8992   2c2 9334   ZZcz 9623   ^cexp 10953   expce 12387   cosccos 12390   tanctan 12391

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-disj 4102  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-sup 7314  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-ico 10275  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-fac 11142  df-bc 11164  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098  df-ef 12393  df-sin 12395  df-cos 12396  df-tan 12397

This theorem is referenced by: (None)