Theorem resin4p 12297

Description: Separate out the first four terms of the infinite series expansion of the sine 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
resin4p	|- ( A e. RR -> ( sin ` A ) = ( ( A - ( ( A ^ 3 ) / 6 ) ) + ( Im ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )

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

Allowed substitution hint:   F( n)

Proof of Theorem resin4p

Step	Hyp	Ref	Expression
1		resinval 12294	. 2 |- ( A e. RR -> ( sin ` A ) = ( Im ` ( exp ` ( _i x. A ) ) ) )
2		recn 8165	. . . . 5 |- ( A e. RR -> A e. CC )
3		efi4p.1	. . . . . 6 |- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )
4	3	efi4p 12296	. . . . 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 5643	. . 3 |- ( A e. RR -> ( Im ` ( exp ` ( _i x. A ) ) ) = ( Im ` ( ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) + sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )
7		1re 8178	. . . . . . 7 |- 1 e. RR
8		resqcl 10870	. . . . . . . 8 |- ( A e. RR -> ( A ^ 2 ) e. RR )
9	8	rehalfcld 9391	. . . . . . 7 |- ( A e. RR -> ( ( A ^ 2 ) / 2 ) e. RR )
10		resubcl 8443	. . . . . . 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 8208	. . . . 5 |- ( A e. RR -> ( 1 - ( ( A ^ 2 ) / 2 ) ) e. CC )
13		ax-icn 8127	. . . . . 6 |- _i e. CC
14		3nn0 9420	. . . . . . . . . 10 |- 3 e. NN0
15		reexpcl 10819	. . . . . . . . . 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 9224	. . . . . . . . . 10 |- 6 e. RR
18		6pos 9244	. . . . . . . . . . 11 |- 0 < 6
19	17, 18	gt0ap0ii 8808	. . . . . . . . . 10 |- 6 # 0
20		redivclap 8911	. . . . . . . . . 10 |- ( ( ( A ^ 3 ) e. RR /\ 6 e. RR /\ 6 # 0 ) -> ( ( A ^ 3 ) / 6 ) e. RR )
21	17, 19, 20	mp3an23 1365	. . . . . . . . 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 8443	. . . . . . . 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 8208	. . . . . 6 |- ( A e. RR -> ( A - ( ( A ^ 3 ) / 6 ) ) e. CC )
26		mulcl 8159	. . . . . 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 8199	. . . 4 |- ( A e. RR -> ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) e. CC )
29		mulcl 8159	. . . . . 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 9421	. . . . 5 |- 4 e. NN0
32	3	eftlcl 12267	. . . . 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	imaddd 11538	. . 3 |- ( A e. RR -> ( Im ` ( ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) + sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) = ( ( Im ` ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) ) + ( Im ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )
35	11, 24	crimd 11555	. . . 4 |- ( A e. RR -> ( Im ` ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) ) = ( A - ( ( A ^ 3 ) / 6 ) ) )
36	35	oveq1d 6033	. . 3 |- ( A e. RR -> ( ( Im ` ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) ) + ( Im ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) = ( ( A - ( ( A ^ 3 ) / 6 ) ) + ( Im ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )
37	6, 34, 36	3eqtrd 2268	. 2 |- ( A e. RR -> ( Im ` ( exp ` ( _i x. A ) ) ) = ( ( A - ( ( A ^ 3 ) / 6 ) ) + ( Im ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )
38	1, 37	eqtrd 2264	1 |- ( A e. RR -> ( sin ` A ) = ( ( A - ( ( A ^ 3 ) / 6 ) ) + ( Im ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   class class class wbr 4088   |-> cmpt 4150   ` cfv 5326  (class class class)co 6018   CCcc 8030   RRcr 8031   0cc0 8032   1c1 8033   _ici 8034   + caddc 8035   x. cmul 8037   - cmin 8350   # cap 8761   / cdiv 8852   2c2 9194   3c3 9195   4c4 9196   6c6 9198   NN0cn0 9402   ZZ>=cuz 9755   ^cexp 10801   !cfa 10988   Imcim 11419   sum_csu 11931   expce 12221   sincsin 12223

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-ico 10129  df-fz 10244  df-fzo 10378  df-seqfrec 10711  df-exp 10802  df-fac 10989  df-ihash 11039  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-clim 11857  df-sumdc 11932  df-ef 12227  df-sin 12229

This theorem is referenced by:  sin01bnd  12336