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Theorem resin4p 12418
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 12415 . 2  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( Im `  ( exp `  ( _i  x.  A ) ) ) )
2 recn 8827 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
3 efi4p.1 . . . . . 6  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
43efi4p 12417 . . . . 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 15 . . . 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 5529 . . 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 8837 . . . . . . 7  |-  1  e.  RR
8 resqcl 11171 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A ^ 2 )  e.  RR )
98rehalfcld 9958 . . . . . . 7  |-  ( A  e.  RR  ->  (
( A ^ 2 )  /  2 )  e.  RR )
10 resubcl 9111 . . . . . . 7  |-  ( ( 1  e.  RR  /\  ( ( A ^
2 )  /  2
)  e.  RR )  ->  ( 1  -  ( ( A ^
2 )  /  2
) )  e.  RR )
117, 9, 10sylancr 644 . . . . . 6  |-  ( A  e.  RR  ->  (
1  -  ( ( A ^ 2 )  /  2 ) )  e.  RR )
1211recnd 8861 . . . . 5  |-  ( A  e.  RR  ->  (
1  -  ( ( A ^ 2 )  /  2 ) )  e.  CC )
13 ax-icn 8796 . . . . . 6  |-  _i  e.  CC
14 3nn0 9983 . . . . . . . . . 10  |-  3  e.  NN0
15 reexpcl 11120 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  RR )
1614, 15mpan2 652 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A ^ 3 )  e.  RR )
17 6re 9822 . . . . . . . . . 10  |-  6  e.  RR
18 6pos 9834 . . . . . . . . . . 11  |-  0  <  6
1917, 18gt0ne0ii 9309 . . . . . . . . . 10  |-  6  =/=  0
20 redivcl 9479 . . . . . . . . . 10  |-  ( ( ( A ^ 3 )  e.  RR  /\  6  e.  RR  /\  6  =/=  0 )  ->  (
( A ^ 3 )  /  6 )  e.  RR )
2117, 19, 20mp3an23 1269 . . . . . . . . 9  |-  ( ( A ^ 3 )  e.  RR  ->  (
( A ^ 3 )  /  6 )  e.  RR )
2216, 21syl 15 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( A ^ 3 )  /  6 )  e.  RR )
23 resubcl 9111 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( A ^
3 )  /  6
)  e.  RR )  ->  ( A  -  ( ( A ^
3 )  /  6
) )  e.  RR )
2422, 23mpdan 649 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  -  ( ( A ^ 3 )  / 
6 ) )  e.  RR )
2524recnd 8861 . . . . . 6  |-  ( A  e.  RR  ->  ( A  -  ( ( A ^ 3 )  / 
6 ) )  e.  CC )
26 mulcl 8821 . . . . . 6  |-  ( ( _i  e.  CC  /\  ( A  -  (
( A ^ 3 )  /  6 ) )  e.  CC )  ->  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) )  e.  CC )
2713, 25, 26sylancr 644 . . . . 5  |-  ( A  e.  RR  ->  (
_i  x.  ( A  -  ( ( A ^ 3 )  / 
6 ) ) )  e.  CC )
2812, 27addcld 8854 . . . 4  |-  ( A  e.  RR  ->  (
( 1  -  (
( A ^ 2 )  /  2 ) )  +  ( _i  x.  ( A  -  ( ( A ^
3 )  /  6
) ) ) )  e.  CC )
29 mulcl 8821 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
3013, 2, 29sylancr 644 . . . . 5  |-  ( A  e.  RR  ->  (
_i  x.  A )  e.  CC )
31 4nn0 9984 . . . . 5  |-  4  e.  NN0
323eftlcl 12387 . . . . 5  |-  ( ( ( _i  x.  A
)  e.  CC  /\  4  e.  NN0 )  ->  sum_ k  e.  ( ZZ>= ` 
4 ) ( F `
 k )  e.  CC )
3330, 31, 32sylancl 643 . . . 4  |-  ( A  e.  RR  ->  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
)  e.  CC )
3428, 33imaddd 11700 . . 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 11717 . . . 4  |-  ( A  e.  RR  ->  (
Im `  ( (
1  -  ( ( A ^ 2 )  /  2 ) )  +  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) ) )  =  ( A  -  ( ( A ^
3 )  /  6
) ) )
3635oveq1d 5873 . . 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 2319 . 2  |-  ( A  e.  RR  ->  (
Im `  ( exp `  ( _i  x.  A
) ) )  =  ( ( A  -  ( ( A ^
3 )  /  6
) )  +  ( Im `  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
) ) ) )
381, 37eqtrd 2315 1  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( ( A  -  ( ( A ^
3 )  /  6
) )  +  ( Im `  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037    / cdiv 9423   2c2 9795   3c3 9796   4c4 9797   6c6 9799   NN0cn0 9965   ZZ>=cuz 10230   ^cexp 11104   !cfa 11288   Imcim 11583   sum_csu 12158   expce 12343   sincsin 12345
This theorem is referenced by:  sin01bnd  12465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351
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