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Theorem recos4p 11727
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
StepHypRef Expression
1 recosval 11724 . 2  |-  ( A  e.  RR  ->  ( cos `  A )  =  ( Re `  ( exp `  ( _i  x.  A ) ) ) )
2 recn 7944 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
3 efi4p.1 . . . . . 6  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
43efi4p 11725 . . . . 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 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
) ) )
65fveq2d 5520 . . 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 7956 . . . . . . 7  |-  1  e.  RR
8 resqcl 10588 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A ^ 2 )  e.  RR )
98rehalfcld 9165 . . . . . . 7  |-  ( A  e.  RR  ->  (
( A ^ 2 )  /  2 )  e.  RR )
10 resubcl 8221 . . . . . . 7  |-  ( ( 1  e.  RR  /\  ( ( A ^
2 )  /  2
)  e.  RR )  ->  ( 1  -  ( ( A ^
2 )  /  2
) )  e.  RR )
117, 9, 10sylancr 414 . . . . . 6  |-  ( A  e.  RR  ->  (
1  -  ( ( A ^ 2 )  /  2 ) )  e.  RR )
1211recnd 7986 . . . . 5  |-  ( A  e.  RR  ->  (
1  -  ( ( A ^ 2 )  /  2 ) )  e.  CC )
13 ax-icn 7906 . . . . . 6  |-  _i  e.  CC
14 3nn0 9194 . . . . . . . . . 10  |-  3  e.  NN0
15 reexpcl 10537 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  RR )
1614, 15mpan2 425 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A ^ 3 )  e.  RR )
17 6re 9000 . . . . . . . . . 10  |-  6  e.  RR
18 6pos 9020 . . . . . . . . . . 11  |-  0  <  6
1917, 18gt0ap0ii 8585 . . . . . . . . . 10  |-  6 #  0
20 redivclap 8688 . . . . . . . . . 10  |-  ( ( ( A ^ 3 )  e.  RR  /\  6  e.  RR  /\  6 #  0 )  ->  (
( A ^ 3 )  /  6 )  e.  RR )
2117, 19, 20mp3an23 1329 . . . . . . . . 9  |-  ( ( A ^ 3 )  e.  RR  ->  (
( A ^ 3 )  /  6 )  e.  RR )
2216, 21syl 14 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( A ^ 3 )  /  6 )  e.  RR )
23 resubcl 8221 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( A ^
3 )  /  6
)  e.  RR )  ->  ( A  -  ( ( A ^
3 )  /  6
) )  e.  RR )
2422, 23mpdan 421 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  -  ( ( A ^ 3 )  / 
6 ) )  e.  RR )
2524recnd 7986 . . . . . 6  |-  ( A  e.  RR  ->  ( A  -  ( ( A ^ 3 )  / 
6 ) )  e.  CC )
26 mulcl 7938 . . . . . 6  |-  ( ( _i  e.  CC  /\  ( A  -  (
( A ^ 3 )  /  6 ) )  e.  CC )  ->  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) )  e.  CC )
2713, 25, 26sylancr 414 . . . . 5  |-  ( A  e.  RR  ->  (
_i  x.  ( A  -  ( ( A ^ 3 )  / 
6 ) ) )  e.  CC )
2812, 27addcld 7977 . . . 4  |-  ( A  e.  RR  ->  (
( 1  -  (
( A ^ 2 )  /  2 ) )  +  ( _i  x.  ( A  -  ( ( A ^
3 )  /  6
) ) ) )  e.  CC )
29 mulcl 7938 . . . . . 6  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
3013, 2, 29sylancr 414 . . . . 5  |-  ( A  e.  RR  ->  (
_i  x.  A )  e.  CC )
31 4nn0 9195 . . . . 5  |-  4  e.  NN0
323eftlcl 11696 . . . . 5  |-  ( ( ( _i  x.  A
)  e.  CC  /\  4  e.  NN0 )  ->  sum_ k  e.  ( ZZ>= ` 
4 ) ( F `
 k )  e.  CC )
3330, 31, 32sylancl 413 . . . 4  |-  ( A  e.  RR  ->  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
)  e.  CC )
3428, 33readdd 10968 . . 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 ) ) ) )
3511, 24crred 10985 . . . 4  |-  ( A  e.  RR  ->  (
Re `  ( (
1  -  ( ( A ^ 2 )  /  2 ) )  +  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) ) )  =  ( 1  -  ( ( A ^
2 )  /  2
) ) )
3635oveq1d 5890 . . 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 ) ) ) )
376, 34, 363eqtrd 2214 . 2  |-  ( A  e.  RR  ->  (
Re `  ( exp `  ( _i  x.  A
) ) )  =  ( ( 1  -  ( ( A ^
2 )  /  2
) )  +  ( Re `  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
) ) ) )
381, 37eqtrd 2210 1  |-  ( A  e.  RR  ->  ( cos `  A )  =  ( ( 1  -  ( ( A ^
2 )  /  2
) )  +  ( Re `  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   class class class wbr 4004    |-> cmpt 4065   ` cfv 5217  (class class class)co 5875   CCcc 7809   RRcr 7810   0cc0 7811   1c1 7812   _ici 7813    + caddc 7814    x. cmul 7816    - cmin 8128   # cap 8538    / cdiv 8629   2c2 8970   3c3 8971   4c4 8972   6c6 8974   NN0cn0 9176   ZZ>=cuz 9528   ^cexp 10519   !cfa 10705   Recre 10849   sum_csu 11361   expce 11650   cosccos 11653
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-frec 6392  df-1o 6417  df-oadd 6421  df-er 6535  df-en 6741  df-dom 6742  df-fin 6743  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-5 8981  df-6 8982  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-ico 9894  df-fz 10009  df-fzo 10143  df-seqfrec 10446  df-exp 10520  df-fac 10706  df-ihash 10756  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-clim 11287  df-sumdc 11362  df-ef 11656  df-cos 11659
This theorem is referenced by:  cos01bnd  11766
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