Theorem recos4p 11718

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 11715	. 2 |- ( A e. RR -> ( cos ` A ) = ( Re ` ( exp ` ( _i x. A ) ) ) )
2		recn 7939	. . . . 5 |- ( A e. RR -> A e. CC )
3		efi4p.1	. . . . . 6 |- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )
4	3	efi4p 11716	. . . . 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 5516	. . 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 7951	. . . . . . 7 |- 1 e. RR
8		resqcl 10580	. . . . . . . 8 |- ( A e. RR -> ( A ^ 2 ) e. RR )
9	8	rehalfcld 9159	. . . . . . 7 |- ( A e. RR -> ( ( A ^ 2 ) / 2 ) e. RR )
10		resubcl 8215	. . . . . . 7 |- ( ( 1 e. RR /\ ( ( A ^ 2 ) / 2 ) e. RR ) -> ( 1 - ( ( A ^ 2 ) / 2 ) ) e. RR )
11	7, 9, 10	sylancr 414	. . . . . 6 |- ( A e. RR -> ( 1 - ( ( A ^ 2 ) / 2 ) ) e. RR )
12	11	recnd 7980	. . . . 5 |- ( A e. RR -> ( 1 - ( ( A ^ 2 ) / 2 ) ) e. CC )
13		ax-icn 7901	. . . . . 6 |- _i e. CC
14		3nn0 9188	. . . . . . . . . 10 |- 3 e. NN0
15		reexpcl 10530	. . . . . . . . . 10 |- ( ( A e. RR /\ 3 e. NN0 ) -> ( A ^ 3 ) e. RR )
16	14, 15	mpan2 425	. . . . . . . . 9 |- ( A e. RR -> ( A ^ 3 ) e. RR )
17		6re 8994	. . . . . . . . . 10 |- 6 e. RR
18		6pos 9014	. . . . . . . . . . 11 |- 0 < 6
19	17, 18	gt0ap0ii 8579	. . . . . . . . . 10 |- 6 # 0
20		redivclap 8682	. . . . . . . . . 10 |- ( ( ( A ^ 3 ) e. RR /\ 6 e. RR /\ 6 # 0 ) -> ( ( A ^ 3 ) / 6 ) e. RR )
21	17, 19, 20	mp3an23 1329	. . . . . . . . 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 8215	. . . . . . . 8 |- ( ( A e. RR /\ ( ( A ^ 3 ) / 6 ) e. RR ) -> ( A - ( ( A ^ 3 ) / 6 ) ) e. RR )
24	22, 23	mpdan 421	. . . . . . 7 |- ( A e. RR -> ( A - ( ( A ^ 3 ) / 6 ) ) e. RR )
25	24	recnd 7980	. . . . . 6 |- ( A e. RR -> ( A - ( ( A ^ 3 ) / 6 ) ) e. CC )
26		mulcl 7933	. . . . . 6 |- ( ( _i e. CC /\ ( A - ( ( A ^ 3 ) / 6 ) ) e. CC ) -> ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) e. CC )
27	13, 25, 26	sylancr 414	. . . . 5 |- ( A e. RR -> ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) e. CC )
28	12, 27	addcld 7971	. . . 4 |- ( A e. RR -> ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) e. CC )
29		mulcl 7933	. . . . . 6 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
30	13, 2, 29	sylancr 414	. . . . 5 |- ( A e. RR -> ( _i x. A ) e. CC )
31		4nn0 9189	. . . . 5 |- 4 e. NN0
32	3	eftlcl 11687	. . . . 5 |- ( ( ( _i x. A ) e. CC /\ 4 e. NN0 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
33	30, 31, 32	sylancl 413	. . . 4 |- ( A e. RR -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
34	28, 33	readdd 10959	. . 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 10976	. . . 4 |- ( A e. RR -> ( Re ` ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
36	35	oveq1d 5885	. . 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 2214	. 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 2210	1 |- ( A e. RR -> ( cos ` A ) = ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( Re ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   class class class wbr 4001   |-> cmpt 4062   ` cfv 5213  (class class class)co 5870   CCcc 7804   RRcr 7805   0cc0 7806   1c1 7807   _ici 7808   + caddc 7809   x. cmul 7811   - cmin 8122   # cap 8532   / cdiv 8623   2c2 8964   3c3 8965   4c4 8966   6c6 8968   NN0cn0 9170   ZZ>=cuz 9522   ^cexp 10512   !cfa 10696   Recre 10840   sum_csu 11352   expce 11641   cosccos 11644

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4116  ax-sep 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-iinf 4585  ax-cnex 7897  ax-resscn 7898  ax-1cn 7899  ax-1re 7900  ax-icn 7901  ax-addcl 7902  ax-addrcl 7903  ax-mulcl 7904  ax-mulrcl 7905  ax-addcom 7906  ax-mulcom 7907  ax-addass 7908  ax-mulass 7909  ax-distr 7910  ax-i2m1 7911  ax-0lt1 7912  ax-1rid 7913  ax-0id 7914  ax-rnegex 7915  ax-precex 7916  ax-cnre 7917  ax-pre-ltirr 7918  ax-pre-ltwlin 7919  ax-pre-lttrn 7920  ax-pre-apti 7921  ax-pre-ltadd 7922  ax-pre-mulgt0 7923  ax-pre-mulext 7924  ax-arch 7925  ax-caucvg 7926

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-iun 3887  df-br 4002  df-opab 4063  df-mpt 4064  df-tr 4100  df-id 4291  df-po 4294  df-iso 4295  df-iord 4364  df-on 4366  df-ilim 4367  df-suc 4369  df-iom 4588  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-isom 5222  df-riota 5826  df-ov 5873  df-oprab 5874  df-mpo 5875  df-1st 6136  df-2nd 6137  df-recs 6301  df-irdg 6366  df-frec 6387  df-1o 6412  df-oadd 6416  df-er 6530  df-en 6736  df-dom 6737  df-fin 6738  df-pnf 7988  df-mnf 7989  df-xr 7990  df-ltxr 7991  df-le 7992  df-sub 8124  df-neg 8125  df-reap 8526  df-ap 8533  df-div 8624  df-inn 8914  df-2 8972  df-3 8973  df-4 8974  df-5 8975  df-6 8976  df-n0 9171  df-z 9248  df-uz 9523  df-q 9614  df-rp 9648  df-ico 9888  df-fz 10003  df-fzo 10136  df-seqfrec 10439  df-exp 10513  df-fac 10697  df-ihash 10747  df-cj 10842  df-re 10843  df-im 10844  df-rsqrt 10998  df-abs 10999  df-clim 11278  df-sumdc 11353  df-ef 11647  df-cos 11650

This theorem is referenced by:  cos01bnd  11757