Theorem resin4p 12702

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 12699	. 2 |- ( A e. RR -> ( sin ` A ) = ( Im ` ( exp ` ( _i x. A ) ) ) )
2		recn 9044	. . . . 5 |- ( A e. RR -> A e. CC )
3		efi4p.1	. . . . . 6 |- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )
4	3	efi4p 12701	. . . . 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 16	. . . 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 5699	. . 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 9054	. . . . . . 7 |- 1 e. RR
8		resqcl 11412	. . . . . . . 8 |- ( A e. RR -> ( A ^ 2 ) e. RR )
9	8	rehalfcld 10178	. . . . . . 7 |- ( A e. RR -> ( ( A ^ 2 ) / 2 ) e. RR )
10		resubcl 9329	. . . . . . 7 |- ( ( 1 e. RR /\ ( ( A ^ 2 ) / 2 ) e. RR ) -> ( 1 - ( ( A ^ 2 ) / 2 ) ) e. RR )
11	7, 9, 10	sylancr 645	. . . . . 6 |- ( A e. RR -> ( 1 - ( ( A ^ 2 ) / 2 ) ) e. RR )
12	11	recnd 9078	. . . . 5 |- ( A e. RR -> ( 1 - ( ( A ^ 2 ) / 2 ) ) e. CC )
13		ax-icn 9013	. . . . . 6 |- _i e. CC
14		3nn0 10203	. . . . . . . . . 10 |- 3 e. NN0
15		reexpcl 11361	. . . . . . . . . 10 |- ( ( A e. RR /\ 3 e. NN0 ) -> ( A ^ 3 ) e. RR )
16	14, 15	mpan2 653	. . . . . . . . 9 |- ( A e. RR -> ( A ^ 3 ) e. RR )
17		6re 10040	. . . . . . . . . 10 |- 6 e. RR
18		6pos 10052	. . . . . . . . . . 11 |- 0 < 6
19	17, 18	gt0ne0ii 9527	. . . . . . . . . 10 |- 6 =/= 0
20		redivcl 9697	. . . . . . . . . 10 |- ( ( ( A ^ 3 ) e. RR /\ 6 e. RR /\ 6 =/= 0 ) -> ( ( A ^ 3 ) / 6 ) e. RR )
21	17, 19, 20	mp3an23 1271	. . . . . . . . 9 |- ( ( A ^ 3 ) e. RR -> ( ( A ^ 3 ) / 6 ) e. RR )
22	16, 21	syl 16	. . . . . . . 8 |- ( A e. RR -> ( ( A ^ 3 ) / 6 ) e. RR )
23		resubcl 9329	. . . . . . . 8 |- ( ( A e. RR /\ ( ( A ^ 3 ) / 6 ) e. RR ) -> ( A - ( ( A ^ 3 ) / 6 ) ) e. RR )
24	22, 23	mpdan 650	. . . . . . 7 |- ( A e. RR -> ( A - ( ( A ^ 3 ) / 6 ) ) e. RR )
25	24	recnd 9078	. . . . . 6 |- ( A e. RR -> ( A - ( ( A ^ 3 ) / 6 ) ) e. CC )
26		mulcl 9038	. . . . . 6 |- ( ( _i e. CC /\ ( A - ( ( A ^ 3 ) / 6 ) ) e. CC ) -> ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) e. CC )
27	13, 25, 26	sylancr 645	. . . . 5 |- ( A e. RR -> ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) e. CC )
28	12, 27	addcld 9071	. . . 4 |- ( A e. RR -> ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) e. CC )
29		mulcl 9038	. . . . . 6 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
30	13, 2, 29	sylancr 645	. . . . 5 |- ( A e. RR -> ( _i x. A ) e. CC )
31		4nn0 10204	. . . . 5 |- 4 e. NN0
32	3	eftlcl 12671	. . . . 5 |- ( ( ( _i x. A ) e. CC /\ 4 e. NN0 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
33	30, 31, 32	sylancl 644	. . . 4 |- ( A e. RR -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
34	28, 33	imaddd 11983	. . 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 12000	. . . 4 |- ( A e. RR -> ( Im ` ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) ) = ( A - ( ( A ^ 3 ) / 6 ) ) )
36	35	oveq1d 6063	. . 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 2448	. 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 2444	1 |- ( A e. RR -> ( sin ` A ) = ( ( A - ( ( A ^ 3 ) / 6 ) ) + ( Im ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )

Syntax hints:   -> wi 4   = wceq 1649   e. wcel 1721   =/= wne 2575   e. cmpt 4234   ` cfv 5421  (class class class)co 6048   CCcc 8952   RRcr 8953   0cc0 8954   1c1 8955   _ici 8956   + caddc 8957   x. cmul 8959   - cmin 9255   / cdiv 9641   2c2 10013   3c3 10014   4c4 10015   6c6 10017   NN0cn0 10185   ZZ>=cuz 10452   ^cexp 11345   !cfa 11529   Imcim 11866   sum_csu 12442   expce 12627   sincsin 12629

This theorem is referenced by:  sin01bnd  12749

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-ico 10886  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-fac 11530  df-hash 11582  df-shft 11845  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-limsup 12228  df-clim 12245  df-rlim 12246  df-sum 12443  df-ef 12633  df-sin 12635