Theorem cosatan 24648

Description: The cosine of an arctangent. (Contributed by Mario Carneiro, 3-Apr-2015.)

Assertion

Ref	Expression
cosatan	|- ( A e. dom arctan -> ( cos ` (arctan ` A ) ) = ( 1 / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )

Proof of Theorem cosatan

Step	Hyp	Ref	Expression
1		atancl 24608	. . 3 |- ( A e. dom arctan -> (arctan ` A ) e. CC )
2		cosval 14853	. . 3 |- ( (arctan ` A ) e. CC -> ( cos ` (arctan ` A ) ) = ( ( ( exp ` ( _i x. (arctan ` A ) ) ) + ( exp ` ( -u _i x. (arctan ` A ) ) ) ) / 2 ) )
3	1, 2	syl 17	. 2 |- ( A e. dom arctan -> ( cos ` (arctan ` A ) ) = ( ( ( exp ` ( _i x. (arctan ` A ) ) ) + ( exp ` ( -u _i x. (arctan ` A ) ) ) ) / 2 ) )
4		efiatan2 24644	. . . . 5 |- ( A e. dom arctan -> ( exp ` ( _i x. (arctan ` A ) ) ) = ( ( 1 + ( _i x. A ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )
5		ax-icn 9995	. . . . . . . . 9 |- _i e. CC
6		mulneg12 10468	. . . . . . . . 9 |- ( ( _i e. CC /\ (arctan ` A ) e. CC ) -> ( -u _i x. (arctan ` A ) ) = ( _i x. -u (arctan ` A ) ) )
7	5, 1, 6	sylancr 695	. . . . . . . 8 |- ( A e. dom arctan -> ( -u _i x. (arctan ` A ) ) = ( _i x. -u (arctan ` A ) ) )
8		atanneg 24634	. . . . . . . . 9 |- ( A e. dom arctan -> (arctan ` -u A ) = -u (arctan ` A ) )
9	8	oveq2d 6666	. . . . . . . 8 |- ( A e. dom arctan -> ( _i x. (arctan ` -u A ) ) = ( _i x. -u (arctan ` A ) ) )
10	7, 9	eqtr4d 2659	. . . . . . 7 |- ( A e. dom arctan -> ( -u _i x. (arctan ` A ) ) = ( _i x. (arctan ` -u A ) ) )
11	10	fveq2d 6195	. . . . . 6 |- ( A e. dom arctan -> ( exp ` ( -u _i x. (arctan ` A ) ) ) = ( exp ` ( _i x. (arctan ` -u A ) ) ) )
12		atandmneg 24633	. . . . . . 7 |- ( A e. dom arctan -> -u A e. dom arctan )
13		efiatan2 24644	. . . . . . 7 |- ( -u A e. dom arctan -> ( exp ` ( _i x. (arctan ` -u A ) ) ) = ( ( 1 + ( _i x. -u A ) ) / ( sqr ` ( 1 + ( -u A ^ 2 ) ) ) ) )
14	12, 13	syl 17	. . . . . 6 |- ( A e. dom arctan -> ( exp ` ( _i x. (arctan ` -u A ) ) ) = ( ( 1 + ( _i x. -u A ) ) / ( sqr ` ( 1 + ( -u A ^ 2 ) ) ) ) )
15		atandm4 24606	. . . . . . . . . . 11 |- ( A e. dom arctan <-> ( A e. CC /\ ( 1 + ( A ^ 2 ) ) =/= 0 ) )
16	15	simplbi 476	. . . . . . . . . 10 |- ( A e. dom arctan -> A e. CC )
17		mulneg2 10467	. . . . . . . . . 10 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. -u A ) = -u ( _i x. A ) )
18	5, 16, 17	sylancr 695	. . . . . . . . 9 |- ( A e. dom arctan -> ( _i x. -u A ) = -u ( _i x. A ) )
19	18	oveq2d 6666	. . . . . . . 8 |- ( A e. dom arctan -> ( 1 + ( _i x. -u A ) ) = ( 1 + -u ( _i x. A ) ) )
20		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
21		mulcl 10020	. . . . . . . . . 10 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
22	5, 16, 21	sylancr 695	. . . . . . . . 9 |- ( A e. dom arctan -> ( _i x. A ) e. CC )
23		negsub 10329	. . . . . . . . 9 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + -u ( _i x. A ) ) = ( 1 - ( _i x. A ) ) )
24	20, 22, 23	sylancr 695	. . . . . . . 8 |- ( A e. dom arctan -> ( 1 + -u ( _i x. A ) ) = ( 1 - ( _i x. A ) ) )
25	19, 24	eqtrd 2656	. . . . . . 7 |- ( A e. dom arctan -> ( 1 + ( _i x. -u A ) ) = ( 1 - ( _i x. A ) ) )
26		sqneg 12923	. . . . . . . . . 10 |- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
27	16, 26	syl 17	. . . . . . . . 9 |- ( A e. dom arctan -> ( -u A ^ 2 ) = ( A ^ 2 ) )
28	27	oveq2d 6666	. . . . . . . 8 |- ( A e. dom arctan -> ( 1 + ( -u A ^ 2 ) ) = ( 1 + ( A ^ 2 ) ) )
29	28	fveq2d 6195	. . . . . . 7 |- ( A e. dom arctan -> ( sqr ` ( 1 + ( -u A ^ 2 ) ) ) = ( sqr ` ( 1 + ( A ^ 2 ) ) ) )
30	25, 29	oveq12d 6668	. . . . . 6 |- ( A e. dom arctan -> ( ( 1 + ( _i x. -u A ) ) / ( sqr ` ( 1 + ( -u A ^ 2 ) ) ) ) = ( ( 1 - ( _i x. A ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )
31	11, 14, 30	3eqtrd 2660	. . . . 5 |- ( A e. dom arctan -> ( exp ` ( -u _i x. (arctan ` A ) ) ) = ( ( 1 - ( _i x. A ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )
32	4, 31	oveq12d 6668	. . . 4 |- ( A e. dom arctan -> ( ( exp ` ( _i x. (arctan ` A ) ) ) + ( exp ` ( -u _i x. (arctan ` A ) ) ) ) = ( ( ( 1 + ( _i x. A ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) + ( ( 1 - ( _i x. A ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) ) )
33		addcl 10018	. . . . . 6 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 + ( _i x. A ) ) e. CC )
34	20, 22, 33	sylancr 695	. . . . 5 |- ( A e. dom arctan -> ( 1 + ( _i x. A ) ) e. CC )
35		subcl 10280	. . . . . 6 |- ( ( 1 e. CC /\ ( _i x. A ) e. CC ) -> ( 1 - ( _i x. A ) ) e. CC )
36	20, 22, 35	sylancr 695	. . . . 5 |- ( A e. dom arctan -> ( 1 - ( _i x. A ) ) e. CC )
37	16	sqcld 13006	. . . . . . 7 |- ( A e. dom arctan -> ( A ^ 2 ) e. CC )
38		addcl 10018	. . . . . . 7 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 + ( A ^ 2 ) ) e. CC )
39	20, 37, 38	sylancr 695	. . . . . 6 |- ( A e. dom arctan -> ( 1 + ( A ^ 2 ) ) e. CC )
40	39	sqrtcld 14176	. . . . 5 |- ( A e. dom arctan -> ( sqr ` ( 1 + ( A ^ 2 ) ) ) e. CC )
41	39	sqsqrtd 14178	. . . . . . 7 |- ( A e. dom arctan -> ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) = ( 1 + ( A ^ 2 ) ) )
42	15	simprbi 480	. . . . . . 7 |- ( A e. dom arctan -> ( 1 + ( A ^ 2 ) ) =/= 0 )
43	41, 42	eqnetrd 2861	. . . . . 6 |- ( A e. dom arctan -> ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) =/= 0 )
44		sqne0 12930	. . . . . . 7 |- ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) e. CC -> ( ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) =/= 0 <-> ( sqr ` ( 1 + ( A ^ 2 ) ) ) =/= 0 ) )
45	40, 44	syl 17	. . . . . 6 |- ( A e. dom arctan -> ( ( ( sqr ` ( 1 + ( A ^ 2 ) ) ) ^ 2 ) =/= 0 <-> ( sqr ` ( 1 + ( A ^ 2 ) ) ) =/= 0 ) )
46	43, 45	mpbid 222	. . . . 5 |- ( A e. dom arctan -> ( sqr ` ( 1 + ( A ^ 2 ) ) ) =/= 0 )
47	34, 36, 40, 46	divdird 10839	. . . 4 |- ( A e. dom arctan -> ( ( ( 1 + ( _i x. A ) ) + ( 1 - ( _i x. A ) ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) = ( ( ( 1 + ( _i x. A ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) + ( ( 1 - ( _i x. A ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) ) )
48	20	a1i 11	. . . . . . 7 |- ( A e. dom arctan -> 1 e. CC )
49	48, 22, 48	ppncand 10432	. . . . . 6 |- ( A e. dom arctan -> ( ( 1 + ( _i x. A ) ) + ( 1 - ( _i x. A ) ) ) = ( 1 + 1 ) )
50		df-2 11079	. . . . . 6 |- 2 = ( 1 + 1 )
51	49, 50	syl6eqr 2674	. . . . 5 |- ( A e. dom arctan -> ( ( 1 + ( _i x. A ) ) + ( 1 - ( _i x. A ) ) ) = 2 )
52	51	oveq1d 6665	. . . 4 |- ( A e. dom arctan -> ( ( ( 1 + ( _i x. A ) ) + ( 1 - ( _i x. A ) ) ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) = ( 2 / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )
53	32, 47, 52	3eqtr2d 2662	. . 3 |- ( A e. dom arctan -> ( ( exp ` ( _i x. (arctan ` A ) ) ) + ( exp ` ( -u _i x. (arctan ` A ) ) ) ) = ( 2 / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )
54	53	oveq1d 6665	. 2 |- ( A e. dom arctan -> ( ( ( exp ` ( _i x. (arctan ` A ) ) ) + ( exp ` ( -u _i x. (arctan ` A ) ) ) ) / 2 ) = ( ( 2 / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) / 2 ) )
55		2cnd 11093	. . . 4 |- ( A e. dom arctan -> 2 e. CC )
56		2ne0 11113	. . . . 5 |- 2 =/= 0
57	56	a1i 11	. . . 4 |- ( A e. dom arctan -> 2 =/= 0 )
58	55, 40, 55, 46, 57	divdiv32d 10826	. . 3 |- ( A e. dom arctan -> ( ( 2 / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) / 2 ) = ( ( 2 / 2 ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )
59		2div2e1 11150	. . . 4 |- ( 2 / 2 ) = 1
60	59	oveq1i 6660	. . 3 |- ( ( 2 / 2 ) / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) = ( 1 / ( sqr ` ( 1 + ( A ^ 2 ) ) ) )
61	58, 60	syl6eq 2672	. 2 |- ( A e. dom arctan -> ( ( 2 / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) / 2 ) = ( 1 / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )
62	3, 54, 61	3eqtrd 2660	1 |- ( A e. dom arctan -> ( cos ` (arctan ` A ) ) = ( 1 / ( sqr ` ( 1 + ( A ^ 2 ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = 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   ^cexp 12860   sqrcsqrt 13973   expce 14792   cosccos 14795  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-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-cxp 24304  df-atan 24594

This theorem is referenced by:  cosatanne0  24649