Theorem tanatan 24646

Description: The arctangent function is an inverse to tan. (Contributed by Mario Carneiro, 2-Apr-2015.)

Assertion

Ref	Expression
tanatan	|- ( A e. dom arctan -> ( tan ` (arctan ` A ) ) = A )

Proof of Theorem tanatan

Step	Hyp	Ref	Expression
1		atancl 24608	. . 3 |- ( A e. dom arctan -> (arctan ` A ) e. CC )
2		2efiatan 24645	. . . . . 6 |- ( A e. dom arctan -> ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - 1 ) )
3	2	oveq1d 6665	. . . . 5 |- ( A e. dom arctan -> ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) + 1 ) = ( ( ( ( 2 x. _i ) / ( A + _i ) ) - 1 ) + 1 ) )
4		2mulicn 11255	. . . . . . . 8 |- ( 2 x. _i ) e. CC
5	4	a1i 11	. . . . . . 7 |- ( A e. dom arctan -> ( 2 x. _i ) e. CC )
6		atandm 24603	. . . . . . . . 9 |- ( A e. dom arctan <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) )
7	6	simp1bi 1076	. . . . . . . 8 |- ( A e. dom arctan -> A e. CC )
8		ax-icn 9995	. . . . . . . 8 |- _i e. CC
9		addcl 10018	. . . . . . . 8 |- ( ( A e. CC /\ _i e. CC ) -> ( A + _i ) e. CC )
10	7, 8, 9	sylancl 694	. . . . . . 7 |- ( A e. dom arctan -> ( A + _i ) e. CC )
11		subneg 10330	. . . . . . . . 9 |- ( ( A e. CC /\ _i e. CC ) -> ( A - -u _i ) = ( A + _i ) )
12	7, 8, 11	sylancl 694	. . . . . . . 8 |- ( A e. dom arctan -> ( A - -u _i ) = ( A + _i ) )
13	6	simp2bi 1077	. . . . . . . . 9 |- ( A e. dom arctan -> A =/= -u _i )
14	8	negcli 10349	. . . . . . . . . 10 |- -u _i e. CC
15		subeq0 10307	. . . . . . . . . . 11 |- ( ( A e. CC /\ -u _i e. CC ) -> ( ( A - -u _i ) = 0 <-> A = -u _i ) )
16	15	necon3bid 2838	. . . . . . . . . 10 |- ( ( A e. CC /\ -u _i e. CC ) -> ( ( A - -u _i ) =/= 0 <-> A =/= -u _i ) )
17	7, 14, 16	sylancl 694	. . . . . . . . 9 |- ( A e. dom arctan -> ( ( A - -u _i ) =/= 0 <-> A =/= -u _i ) )
18	13, 17	mpbird 247	. . . . . . . 8 |- ( A e. dom arctan -> ( A - -u _i ) =/= 0 )
19	12, 18	eqnetrrd 2862	. . . . . . 7 |- ( A e. dom arctan -> ( A + _i ) =/= 0 )
20	5, 10, 19	divcld 10801	. . . . . 6 |- ( A e. dom arctan -> ( ( 2 x. _i ) / ( A + _i ) ) e. CC )
21		ax-1cn 9994	. . . . . 6 |- 1 e. CC
22		npcan 10290	. . . . . 6 |- ( ( ( ( 2 x. _i ) / ( A + _i ) ) e. CC /\ 1 e. CC ) -> ( ( ( ( 2 x. _i ) / ( A + _i ) ) - 1 ) + 1 ) = ( ( 2 x. _i ) / ( A + _i ) ) )
23	20, 21, 22	sylancl 694	. . . . 5 |- ( A e. dom arctan -> ( ( ( ( 2 x. _i ) / ( A + _i ) ) - 1 ) + 1 ) = ( ( 2 x. _i ) / ( A + _i ) ) )
24	3, 23	eqtrd 2656	. . . 4 |- ( A e. dom arctan -> ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) + 1 ) = ( ( 2 x. _i ) / ( A + _i ) ) )
25		2muline0 11256	. . . . . 6 |- ( 2 x. _i ) =/= 0
26	25	a1i 11	. . . . 5 |- ( A e. dom arctan -> ( 2 x. _i ) =/= 0 )
27	5, 10, 26, 19	divne0d 10817	. . . 4 |- ( A e. dom arctan -> ( ( 2 x. _i ) / ( A + _i ) ) =/= 0 )
28	24, 27	eqnetrd 2861	. . 3 |- ( A e. dom arctan -> ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) + 1 ) =/= 0 )
29		tanval3 14864	. . 3 |- ( ( (arctan ` A ) e. CC /\ ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) + 1 ) =/= 0 ) -> ( tan ` (arctan ` A ) ) = ( ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) - 1 ) / ( _i x. ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) + 1 ) ) ) )
30	1, 28, 29	syl2anc 693	. 2 |- ( A e. dom arctan -> ( tan ` (arctan ` A ) ) = ( ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) - 1 ) / ( _i x. ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) + 1 ) ) ) )
31	2	oveq1d 6665	. . . . . 6 |- ( A e. dom arctan -> ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) - 1 ) = ( ( ( ( 2 x. _i ) / ( A + _i ) ) - 1 ) - 1 ) )
32	21	a1i 11	. . . . . . . 8 |- ( A e. dom arctan -> 1 e. CC )
33	20, 32, 32	subsub4d 10423	. . . . . . 7 |- ( A e. dom arctan -> ( ( ( ( 2 x. _i ) / ( A + _i ) ) - 1 ) - 1 ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - ( 1 + 1 ) ) )
34		df-2 11079	. . . . . . . 8 |- 2 = ( 1 + 1 )
35	34	oveq2i 6661	. . . . . . 7 |- ( ( ( 2 x. _i ) / ( A + _i ) ) - 2 ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - ( 1 + 1 ) )
36	33, 35	syl6eqr 2674	. . . . . 6 |- ( A e. dom arctan -> ( ( ( ( 2 x. _i ) / ( A + _i ) ) - 1 ) - 1 ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - 2 ) )
37	31, 36	eqtrd 2656	. . . . 5 |- ( A e. dom arctan -> ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) - 1 ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - 2 ) )
38		2cn 11091	. . . . . . . 8 |- 2 e. CC
39		mulcl 10020	. . . . . . . 8 |- ( ( 2 e. CC /\ ( A + _i ) e. CC ) -> ( 2 x. ( A + _i ) ) e. CC )
40	38, 10, 39	sylancr 695	. . . . . . 7 |- ( A e. dom arctan -> ( 2 x. ( A + _i ) ) e. CC )
41	5, 40, 10, 19	divsubdird 10840	. . . . . 6 |- ( A e. dom arctan -> ( ( ( 2 x. _i ) - ( 2 x. ( A + _i ) ) ) / ( A + _i ) ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - ( ( 2 x. ( A + _i ) ) / ( A + _i ) ) ) )
42		mulneg12 10468	. . . . . . . . 9 |- ( ( 2 e. CC /\ A e. CC ) -> ( -u 2 x. A ) = ( 2 x. -u A ) )
43	38, 7, 42	sylancr 695	. . . . . . . 8 |- ( A e. dom arctan -> ( -u 2 x. A ) = ( 2 x. -u A ) )
44		negsub 10329	. . . . . . . . . . . 12 |- ( ( _i e. CC /\ A e. CC ) -> ( _i + -u A ) = ( _i - A ) )
45	8, 7, 44	sylancr 695	. . . . . . . . . . 11 |- ( A e. dom arctan -> ( _i + -u A ) = ( _i - A ) )
46	45	oveq1d 6665	. . . . . . . . . 10 |- ( A e. dom arctan -> ( ( _i + -u A ) - _i ) = ( ( _i - A ) - _i ) )
47	7	negcld 10379	. . . . . . . . . . 11 |- ( A e. dom arctan -> -u A e. CC )
48		pncan2 10288	. . . . . . . . . . 11 |- ( ( _i e. CC /\ -u A e. CC ) -> ( ( _i + -u A ) - _i ) = -u A )
49	8, 47, 48	sylancr 695	. . . . . . . . . 10 |- ( A e. dom arctan -> ( ( _i + -u A ) - _i ) = -u A )
50	8	a1i 11	. . . . . . . . . . 11 |- ( A e. dom arctan -> _i e. CC )
51	50, 7, 50	subsub4d 10423	. . . . . . . . . 10 |- ( A e. dom arctan -> ( ( _i - A ) - _i ) = ( _i - ( A + _i ) ) )
52	46, 49, 51	3eqtr3rd 2665	. . . . . . . . 9 |- ( A e. dom arctan -> ( _i - ( A + _i ) ) = -u A )
53	52	oveq2d 6666	. . . . . . . 8 |- ( A e. dom arctan -> ( 2 x. ( _i - ( A + _i ) ) ) = ( 2 x. -u A ) )
54	38	a1i 11	. . . . . . . . 9 |- ( A e. dom arctan -> 2 e. CC )
55	54, 50, 10	subdid 10486	. . . . . . . 8 |- ( A e. dom arctan -> ( 2 x. ( _i - ( A + _i ) ) ) = ( ( 2 x. _i ) - ( 2 x. ( A + _i ) ) ) )
56	43, 53, 55	3eqtr2rd 2663	. . . . . . 7 |- ( A e. dom arctan -> ( ( 2 x. _i ) - ( 2 x. ( A + _i ) ) ) = ( -u 2 x. A ) )
57	56	oveq1d 6665	. . . . . 6 |- ( A e. dom arctan -> ( ( ( 2 x. _i ) - ( 2 x. ( A + _i ) ) ) / ( A + _i ) ) = ( ( -u 2 x. A ) / ( A + _i ) ) )
58	54, 10, 19	divcan4d 10807	. . . . . . 7 |- ( A e. dom arctan -> ( ( 2 x. ( A + _i ) ) / ( A + _i ) ) = 2 )
59	58	oveq2d 6666	. . . . . 6 |- ( A e. dom arctan -> ( ( ( 2 x. _i ) / ( A + _i ) ) - ( ( 2 x. ( A + _i ) ) / ( A + _i ) ) ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - 2 ) )
60	41, 57, 59	3eqtr3d 2664	. . . . 5 |- ( A e. dom arctan -> ( ( -u 2 x. A ) / ( A + _i ) ) = ( ( ( 2 x. _i ) / ( A + _i ) ) - 2 ) )
61	37, 60	eqtr4d 2659	. . . 4 |- ( A e. dom arctan -> ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) - 1 ) = ( ( -u 2 x. A ) / ( A + _i ) ) )
62	24	oveq2d 6666	. . . . 5 |- ( A e. dom arctan -> ( _i x. ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) + 1 ) ) = ( _i x. ( ( 2 x. _i ) / ( A + _i ) ) ) )
63	8, 38, 8	mul12i 10231	. . . . . . . 8 |- ( _i x. ( 2 x. _i ) ) = ( 2 x. ( _i x. _i ) )
64		ixi 10656	. . . . . . . . 9 |- ( _i x. _i ) = -u 1
65	64	oveq2i 6661	. . . . . . . 8 |- ( 2 x. ( _i x. _i ) ) = ( 2 x. -u 1 )
66	21	negcli 10349	. . . . . . . . 9 |- -u 1 e. CC
67	38	mulm1i 10475	. . . . . . . . 9 |- ( -u 1 x. 2 ) = -u 2
68	66, 38, 67	mulcomli 10047	. . . . . . . 8 |- ( 2 x. -u 1 ) = -u 2
69	63, 65, 68	3eqtri 2648	. . . . . . 7 |- ( _i x. ( 2 x. _i ) ) = -u 2
70	69	oveq1i 6660	. . . . . 6 |- ( ( _i x. ( 2 x. _i ) ) / ( A + _i ) ) = ( -u 2 / ( A + _i ) )
71	50, 5, 10, 19	divassd 10836	. . . . . 6 |- ( A e. dom arctan -> ( ( _i x. ( 2 x. _i ) ) / ( A + _i ) ) = ( _i x. ( ( 2 x. _i ) / ( A + _i ) ) ) )
72	70, 71	syl5eqr 2670	. . . . 5 |- ( A e. dom arctan -> ( -u 2 / ( A + _i ) ) = ( _i x. ( ( 2 x. _i ) / ( A + _i ) ) ) )
73	62, 72	eqtr4d 2659	. . . 4 |- ( A e. dom arctan -> ( _i x. ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) + 1 ) ) = ( -u 2 / ( A + _i ) ) )
74	61, 73	oveq12d 6668	. . 3 |- ( A e. dom arctan -> ( ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) - 1 ) / ( _i x. ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) + 1 ) ) ) = ( ( ( -u 2 x. A ) / ( A + _i ) ) / ( -u 2 / ( A + _i ) ) ) )
75	38	negcli 10349	. . . . . 6 |- -u 2 e. CC
76		mulcl 10020	. . . . . 6 |- ( ( -u 2 e. CC /\ A e. CC ) -> ( -u 2 x. A ) e. CC )
77	75, 7, 76	sylancr 695	. . . . 5 |- ( A e. dom arctan -> ( -u 2 x. A ) e. CC )
78	75	a1i 11	. . . . 5 |- ( A e. dom arctan -> -u 2 e. CC )
79		2ne0 11113	. . . . . . 7 |- 2 =/= 0
80	38, 79	negne0i 10356	. . . . . 6 |- -u 2 =/= 0
81	80	a1i 11	. . . . 5 |- ( A e. dom arctan -> -u 2 =/= 0 )
82	77, 78, 10, 81, 19	divcan7d 10829	. . . 4 |- ( A e. dom arctan -> ( ( ( -u 2 x. A ) / ( A + _i ) ) / ( -u 2 / ( A + _i ) ) ) = ( ( -u 2 x. A ) / -u 2 ) )
83	7, 78, 81	divcan3d 10806	. . . 4 |- ( A e. dom arctan -> ( ( -u 2 x. A ) / -u 2 ) = A )
84	82, 83	eqtrd 2656	. . 3 |- ( A e. dom arctan -> ( ( ( -u 2 x. A ) / ( A + _i ) ) / ( -u 2 / ( A + _i ) ) ) = A )
85	74, 84	eqtrd 2656	. 2 |- ( A e. dom arctan -> ( ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) - 1 ) / ( _i x. ( ( exp ` ( 2 x. ( _i x. (arctan ` A ) ) ) ) + 1 ) ) ) = A )
86	30, 85	eqtrd 2656	1 |- ( A e. dom arctan -> ( tan ` (arctan ` A ) ) = A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   dom cdm 5114   ` 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   expce 14792   tanctan 14796  arctancatan 24591

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-iin 4523  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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  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  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-atan 24594

This theorem is referenced by:  atantanb  24651  atanord  24654