Theorem recos4p 12267

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 12264	. 2 |- ( A e. RR -> ( cos ` A ) = ( Re ` ( exp ` ( _i x. A ) ) ) )
2		recn 8153	. . . . 5 |- ( A e. RR -> A e. CC )
3		efi4p.1	. . . . . 6 |- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )
4	3	efi4p 12265	. . . . 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 5637	. . 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 8166	. . . . . . 7 |- 1 e. RR
8		resqcl 10857	. . . . . . . 8 |- ( A e. RR -> ( A ^ 2 ) e. RR )
9	8	rehalfcld 9379	. . . . . . 7 |- ( A e. RR -> ( ( A ^ 2 ) / 2 ) e. RR )
10		resubcl 8431	. . . . . . 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 8196	. . . . 5 |- ( A e. RR -> ( 1 - ( ( A ^ 2 ) / 2 ) ) e. CC )
13		ax-icn 8115	. . . . . 6 |- _i e. CC
14		3nn0 9408	. . . . . . . . . 10 |- 3 e. NN0
15		reexpcl 10806	. . . . . . . . . 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 9212	. . . . . . . . . 10 |- 6 e. RR
18		6pos 9232	. . . . . . . . . . 11 |- 0 < 6
19	17, 18	gt0ap0ii 8796	. . . . . . . . . 10 |- 6 # 0
20		redivclap 8899	. . . . . . . . . 10 |- ( ( ( A ^ 3 ) e. RR /\ 6 e. RR /\ 6 # 0 ) -> ( ( A ^ 3 ) / 6 ) e. RR )
21	17, 19, 20	mp3an23 1363	. . . . . . . . 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 8431	. . . . . . . 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 8196	. . . . . 6 |- ( A e. RR -> ( A - ( ( A ^ 3 ) / 6 ) ) e. CC )
26		mulcl 8147	. . . . . 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 8187	. . . 4 |- ( A e. RR -> ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) e. CC )
29		mulcl 8147	. . . . . 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 9409	. . . . 5 |- 4 e. NN0
32	3	eftlcl 12236	. . . . 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 11507	. . 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 11524	. . . 4 |- ( A e. RR -> ( Re ` ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) ) = ( 1 - ( ( A ^ 2 ) / 2 ) ) )
36	35	oveq1d 6026	. . 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 2266	. 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 2262	1 |- ( A e. RR -> ( cos ` A ) = ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( Re ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   class class class wbr 4084   |-> cmpt 4146   ` cfv 5322  (class class class)co 6011   CCcc 8018   RRcr 8019   0cc0 8020   1c1 8021   _ici 8022   + caddc 8023   x. cmul 8025   - cmin 8338   # cap 8749   / cdiv 8840   2c2 9182   3c3 9183   4c4 9184   6c6 9186   NN0cn0 9390   ZZ>=cuz 9743   ^cexp 10788   !cfa 10975   Recre 11388   sum_csu 11901   expce 12190   cosccos 12193

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-mulrcl 8119  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-precex 8130  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136  ax-pre-mulgt0 8137  ax-pre-mulext 8138  ax-arch 8139  ax-caucvg 8140

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-po 4389  df-iso 4390  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-isom 5331  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-irdg 6529  df-frec 6550  df-1o 6575  df-oadd 6579  df-er 6695  df-en 6903  df-dom 6904  df-fin 6905  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-reap 8743  df-ap 8750  df-div 8841  df-inn 9132  df-2 9190  df-3 9191  df-4 9192  df-5 9193  df-6 9194  df-n0 9391  df-z 9468  df-uz 9744  df-q 9842  df-rp 9877  df-ico 10117  df-fz 10232  df-fzo 10366  df-seqfrec 10698  df-exp 10789  df-fac 10976  df-ihash 11026  df-cj 11390  df-re 11391  df-im 11392  df-rsqrt 11546  df-abs 11547  df-clim 11827  df-sumdc 11902  df-ef 12196  df-cos 12199

This theorem is referenced by:  cos01bnd  12306