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Theorem recos4p 14085
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 14082 . 2  |-  ( A  e.  RR  ->  ( cos `  A )  =  ( Re `  ( exp `  ( _i  x.  A ) ) ) )
2 recn 9614 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
3 efi4p.1 . . . . . 6  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
43efi4p 14083 . . . . 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 17 . . . 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 5855 . . 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 9627 . . . . . . 7  |-  1  e.  RR
8 resqcl 12282 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A ^ 2 )  e.  RR )
98rehalfcld 10828 . . . . . . 7  |-  ( A  e.  RR  ->  (
( A ^ 2 )  /  2 )  e.  RR )
10 resubcl 9921 . . . . . . 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 9654 . . . . 5  |-  ( A  e.  RR  ->  (
1  -  ( ( A ^ 2 )  /  2 ) )  e.  CC )
13 ax-icn 9583 . . . . . 6  |-  _i  e.  CC
14 3nn0 10856 . . . . . . . . . 10  |-  3  e.  NN0
15 reexpcl 12229 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  RR )
1614, 15mpan2 671 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A ^ 3 )  e.  RR )
17 6re 10659 . . . . . . . . . 10  |-  6  e.  RR
18 6pos 10677 . . . . . . . . . . 11  |-  0  <  6
1917, 18gt0ne0ii 10131 . . . . . . . . . 10  |-  6  =/=  0
20 redivcl 10306 . . . . . . . . . 10  |-  ( ( ( A ^ 3 )  e.  RR  /\  6  e.  RR  /\  6  =/=  0 )  ->  (
( A ^ 3 )  /  6 )  e.  RR )
2117, 19, 20mp3an23 1320 . . . . . . . . 9  |-  ( ( A ^ 3 )  e.  RR  ->  (
( A ^ 3 )  /  6 )  e.  RR )
2216, 21syl 17 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( A ^ 3 )  /  6 )  e.  RR )
23 resubcl 9921 . . . . . . . 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 9654 . . . . . 6  |-  ( A  e.  RR  ->  ( A  -  ( ( A ^ 3 )  / 
6 ) )  e.  CC )
26 mulcl 9608 . . . . . 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 9647 . . . 4  |-  ( A  e.  RR  ->  (
( 1  -  (
( A ^ 2 )  /  2 ) )  +  ( _i  x.  ( A  -  ( ( A ^
3 )  /  6
) ) ) )  e.  CC )
29 mulcl 9608 . . . . . 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 10857 . . . . 5  |-  4  e.  NN0
323eftlcl 14053 . . . . 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, 33readdd 13198 . . 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 13215 . . . 4  |-  ( A  e.  RR  ->  (
Re `  ( (
1  -  ( ( A ^ 2 )  /  2 ) )  +  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) ) )  =  ( 1  -  ( ( A ^
2 )  /  2
) ) )
3635oveq1d 6295 . . 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 2449 . 2  |-  ( A  e.  RR  ->  (
Re `  ( exp `  ( _i  x.  A
) ) )  =  ( ( 1  -  ( ( A ^
2 )  /  2
) )  +  ( Re `  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
) ) ) )
381, 37eqtrd 2445 1  |-  ( A  e.  RR  ->  ( cos `  A )  =  ( ( 1  -  ( ( A ^
2 )  /  2
) )  +  ( Re `  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    e. wcel 1844    =/= wne 2600    |-> cmpt 4455   ` cfv 5571  (class class class)co 6280   CCcc 9522   RRcr 9523   0cc0 9524   1c1 9525   _ici 9526    + caddc 9527    x. cmul 9529    - cmin 9843    / cdiv 10249   2c2 10628   3c3 10629   4c4 10630   6c6 10632   NN0cn0 10838   ZZ>=cuz 11129   ^cexp 12212   !cfa 12399   Recre 13081   sum_csu 13659   expce 14008   cosccos 14011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602  ax-addf 9603  ax-mulf 9604
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-oi 7971  df-card 8354  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-n0 10839  df-z 10908  df-uz 11130  df-rp 11268  df-ico 11590  df-fz 11729  df-fzo 11857  df-fl 11968  df-seq 12154  df-exp 12213  df-fac 12400  df-hash 12455  df-shft 13051  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-limsup 13445  df-clim 13462  df-rlim 13463  df-sum 13660  df-ef 14014  df-cos 14017
This theorem is referenced by:  cos01bnd  14132
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