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Theorem resin4p 13544
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
StepHypRef Expression
1 resinval 13541 . 2  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( Im `  ( exp `  ( _i  x.  A ) ) ) )
2 recn 9487 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
3 efi4p.1 . . . . . 6  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
43efi4p 13543 . . . . 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
) ) )
52, 4syl 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
) ) )
65fveq2d 5806 . . 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 9500 . . . . . . 7  |-  1  e.  RR
8 resqcl 12054 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A ^ 2 )  e.  RR )
98rehalfcld 10686 . . . . . . 7  |-  ( A  e.  RR  ->  (
( A ^ 2 )  /  2 )  e.  RR )
10 resubcl 9788 . . . . . . 7  |-  ( ( 1  e.  RR  /\  ( ( A ^
2 )  /  2
)  e.  RR )  ->  ( 1  -  ( ( A ^
2 )  /  2
) )  e.  RR )
117, 9, 10sylancr 663 . . . . . 6  |-  ( A  e.  RR  ->  (
1  -  ( ( A ^ 2 )  /  2 ) )  e.  RR )
1211recnd 9527 . . . . 5  |-  ( A  e.  RR  ->  (
1  -  ( ( A ^ 2 )  /  2 ) )  e.  CC )
13 ax-icn 9456 . . . . . 6  |-  _i  e.  CC
14 3nn0 10712 . . . . . . . . . 10  |-  3  e.  NN0
15 reexpcl 12003 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  RR )
1614, 15mpan2 671 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A ^ 3 )  e.  RR )
17 6re 10517 . . . . . . . . . 10  |-  6  e.  RR
18 6pos 10535 . . . . . . . . . . 11  |-  0  <  6
1917, 18gt0ne0ii 9991 . . . . . . . . . 10  |-  6  =/=  0
20 redivcl 10165 . . . . . . . . . 10  |-  ( ( ( A ^ 3 )  e.  RR  /\  6  e.  RR  /\  6  =/=  0 )  ->  (
( A ^ 3 )  /  6 )  e.  RR )
2117, 19, 20mp3an23 1307 . . . . . . . . 9  |-  ( ( A ^ 3 )  e.  RR  ->  (
( A ^ 3 )  /  6 )  e.  RR )
2216, 21syl 16 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( A ^ 3 )  /  6 )  e.  RR )
23 resubcl 9788 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( A ^
3 )  /  6
)  e.  RR )  ->  ( A  -  ( ( A ^
3 )  /  6
) )  e.  RR )
2422, 23mpdan 668 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  -  ( ( A ^ 3 )  / 
6 ) )  e.  RR )
2524recnd 9527 . . . . . 6  |-  ( A  e.  RR  ->  ( A  -  ( ( A ^ 3 )  / 
6 ) )  e.  CC )
26 mulcl 9481 . . . . . 6  |-  ( ( _i  e.  CC  /\  ( A  -  (
( A ^ 3 )  /  6 ) )  e.  CC )  ->  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) )  e.  CC )
2713, 25, 26sylancr 663 . . . . 5  |-  ( A  e.  RR  ->  (
_i  x.  ( A  -  ( ( A ^ 3 )  / 
6 ) ) )  e.  CC )
2812, 27addcld 9520 . . . 4  |-  ( A  e.  RR  ->  (
( 1  -  (
( A ^ 2 )  /  2 ) )  +  ( _i  x.  ( A  -  ( ( A ^
3 )  /  6
) ) ) )  e.  CC )
29 mulcl 9481 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
3013, 2, 29sylancr 663 . . . . 5  |-  ( A  e.  RR  ->  (
_i  x.  A )  e.  CC )
31 4nn0 10713 . . . . 5  |-  4  e.  NN0
323eftlcl 13513 . . . . 5  |-  ( ( ( _i  x.  A
)  e.  CC  /\  4  e.  NN0 )  ->  sum_ k  e.  ( ZZ>= ` 
4 ) ( F `
 k )  e.  CC )
3330, 31, 32sylancl 662 . . . 4  |-  ( A  e.  RR  ->  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
)  e.  CC )
3428, 33imaddd 12826 . . 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 ) ) ) )
3511, 24crimd 12843 . . . 4  |-  ( A  e.  RR  ->  (
Im `  ( (
1  -  ( ( A ^ 2 )  /  2 ) )  +  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) ) )  =  ( A  -  ( ( A ^
3 )  /  6
) ) )
3635oveq1d 6218 . . 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 ) ) ) )
376, 34, 363eqtrd 2499 . 2  |-  ( A  e.  RR  ->  (
Im `  ( exp `  ( _i  x.  A
) ) )  =  ( ( A  -  ( ( A ^
3 )  /  6
) )  +  ( Im `  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
) ) ) )
381, 37eqtrd 2495 1  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( ( A  -  ( ( A ^
3 )  /  6
) )  +  ( Im `  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    =/= wne 2648    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203   CCcc 9395   RRcr 9396   0cc0 9397   1c1 9398   _ici 9399    + caddc 9400    x. cmul 9402    - cmin 9710    / cdiv 10108   2c2 10486   3c3 10487   4c4 10488   6c6 10490   NN0cn0 10694   ZZ>=cuz 10976   ^cexp 11986   !cfa 12172   Imcim 12709   sum_csu 13285   expce 13469   sincsin 13471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475  ax-addf 9476  ax-mulf 9477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-ico 11421  df-fz 11559  df-fzo 11670  df-fl 11763  df-seq 11928  df-exp 11987  df-fac 12173  df-hash 12225  df-shft 12678  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-limsup 13071  df-clim 13088  df-rlim 13089  df-sum 13286  df-ef 13475  df-sin 13477
This theorem is referenced by:  sin01bnd  13591
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