Theorem recos4p 11462

Description: Separate out the first four terms of the infinite series expansion of the cosine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Hypothesis

Ref	Expression
efi4p.1	|- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )

Assertion

Ref	Expression
recos4p	|- ( A e. RR -> ( cos ` A ) = ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( Re ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )

Distinct variable groups:   A, k, n   k, F

Allowed substitution hint:   F( n)

Proof of Theorem recos4p

Step	Hyp	Ref	Expression
1		recosval 11459	. 2 |- ( A e. RR -> ( cos ` A ) = ( Re ` ( exp ` ( _i x. A ) ) ) )
2		recn 7777	. . . . 5 |- ( A e. RR -> A e. CC )
3		efi4p.1	. . . . . 6 |- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )
4	3	efi4p 11460	. . . . 5 |- ( A e. CC -> ( exp ` ( _i x. A ) ) = ( ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) + sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) )
5	2, 4	syl 14	. . . 4 |- ( A e. RR -> ( exp ` ( _i x. A ) ) = ( ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) + sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) )
6	5	fveq2d 5433	. . 3 |- ( A e. RR -> ( Re ` ( exp ` ( _i x. A ) ) ) = ( Re ` ( ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) + sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )
7		1re 7789	. . . . . . 7 |- 1 e. RR
8		resqcl 10391	. . . . . . . 8 |- ( A e. RR -> ( A ^ 2 ) e. RR )
9	8	rehalfcld 8990	. . . . . . 7 |- ( A e. RR -> ( ( A ^ 2 ) / 2 ) e. RR )
10		resubcl 8050	. . . . . . 7 |- ( ( 1 e. RR /\ ( ( A ^ 2 ) / 2 ) e. RR ) -> ( 1 - ( ( A ^ 2 ) / 2 ) ) e. RR )
11	7, 9, 10	sylancr 411	. . . . . 6 |- ( A e. RR -> ( 1 - ( ( A ^ 2 ) / 2 ) ) e. RR )
12	11	recnd 7818	. . . . 5 |- ( A e. RR -> ( 1 - ( ( A ^ 2 ) / 2 ) ) e. CC )
13		ax-icn 7739	. . . . . 6 |- _i e. CC
14		3nn0 9019	. . . . . . . . . 10 |- 3 e. NN0
15		reexpcl 10341	. . . . . . . . . 10 |- ( ( A e. RR /\ 3 e. NN0 ) -> ( A ^ 3 ) e. RR )
16	14, 15	mpan2 422	. . . . . . . . 9 |- ( A e. RR -> ( A ^ 3 ) e. RR )
17		6re 8825	. . . . . . . . . 10 |- 6 e. RR
18		6pos 8845	. . . . . . . . . . 11 |- 0 < 6
19	17, 18	gt0ap0ii 8414	. . . . . . . . . 10 |- 6 # 0
20		redivclap 8515	. . . . . . . . . 10 |- ( ( ( A ^ 3 ) e. RR /\ 6 e. RR /\ 6 # 0 ) -> ( ( A ^ 3 ) / 6 ) e. RR )
21	17, 19, 20	mp3an23 1308	. . . . . . . . 9 |- ( ( A ^ 3 ) e. RR -> ( ( A ^ 3 ) / 6 ) e. RR )
22	16, 21	syl 14	. . . . . . . 8 |- ( A e. RR -> ( ( A ^ 3 ) / 6 ) e. RR )
23		resubcl 8050	. . . . . . . 8 |- ( ( A e. RR /\ ( ( A ^ 3 ) / 6 ) e. RR ) -> ( A - ( ( A ^ 3 ) / 6 ) ) e. RR )
24	22, 23	mpdan 418	. . . . . . 7 |- ( A e. RR -> ( A - ( ( A ^ 3 ) / 6 ) ) e. RR )
25	24	recnd 7818	. . . . . 6 |- ( A e. RR -> ( A - ( ( A ^ 3 ) / 6 ) ) e. CC )
26		mulcl 7771	. . . . . 6 |- ( ( _i e. CC /\ ( A - ( ( A ^ 3 ) / 6 ) ) e. CC ) -> ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) e. CC )
27	13, 25, 26	sylancr 411	. . . . 5 |- ( A e. RR -> ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) e. CC )
28	12, 27	addcld 7809	. . . 4 |- ( A e. RR -> ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) e. CC )
29		mulcl 7771	. . . . . 6 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
30	13, 2, 29	sylancr 411	. . . . 5 |- ( A e. RR -> ( _i x. A ) e. CC )
31		4nn0 9020	. . . . 5 |- 4 e. NN0
32	3	eftlcl 11431	. . . . 5 |- ( ( ( _i x. A ) e. CC /\ 4 e. NN0 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
33	30, 31, 32	sylancl 410	. . . 4 |- ( A e. RR -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
34	28, 33	readdd 10763	. . 3 |- ( A e. RR -> ( Re ` ( ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) + sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) = ( ( Re ` ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) ) + ( Re ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )
35	11, 24	crred 10780	. . . 4 |- ( A e. RR -> ( Re ` ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
36	35	oveq1d 5797	. . 3 |- ( A e. RR -> ( ( Re ` ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) ) + ( Re ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) = ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( Re ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )
37	6, 34, 36	3eqtrd 2177	. 2 |- ( A e. RR -> ( Re ` ( exp ` ( _i x. A ) ) ) = ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( Re ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )
38	1, 37	eqtrd 2173	1 |- ( A e. RR -> ( cos ` A ) = ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( Re ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   class class class wbr 3937   |-> cmpt 3997   ` cfv 5131  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645   _ici 7646   + caddc 7647   x. cmul 7649   - cmin 7957   # cap 8367   / cdiv 8456   2c2 8795   3c3 8796   4c4 8797   6c6 8799   NN0cn0 9001   ZZ>=cuz 9350   ^cexp 10323   !cfa 10503   Recre 10644   sum_csu 11154   expce 11385   cosccos 11388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-5 8806  df-6 8807  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-ico 9707  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-fac 10504  df-ihash 10554  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155  df-ef 11391  df-cos 11394

This theorem is referenced by:  cos01bnd  11501