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Theorem recos4p 11698
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 11695 . 2  |-  ( A  e.  RR  ->  ( cos `  A )  =  ( Re `  ( exp `  ( _i  x.  A ) ) ) )
2 recn 7922 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
3 efi4p.1 . . . . . 6  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
43efi4p 11696 . . . . 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 5514 . . 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 7934 . . . . . . 7  |-  1  e.  RR
8 resqcl 10560 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A ^ 2 )  e.  RR )
98rehalfcld 9141 . . . . . . 7  |-  ( A  e.  RR  ->  (
( A ^ 2 )  /  2 )  e.  RR )
10 resubcl 8198 . . . . . . 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 7963 . . . . 5  |-  ( A  e.  RR  ->  (
1  -  ( ( A ^ 2 )  /  2 ) )  e.  CC )
13 ax-icn 7884 . . . . . 6  |-  _i  e.  CC
14 3nn0 9170 . . . . . . . . . 10  |-  3  e.  NN0
15 reexpcl 10510 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  RR )
1614, 15mpan2 425 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A ^ 3 )  e.  RR )
17 6re 8976 . . . . . . . . . 10  |-  6  e.  RR
18 6pos 8996 . . . . . . . . . . 11  |-  0  <  6
1917, 18gt0ap0ii 8562 . . . . . . . . . 10  |-  6 #  0
20 redivclap 8664 . . . . . . . . . 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 8198 . . . . . . . 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 7963 . . . . . 6  |-  ( A  e.  RR  ->  ( A  -  ( ( A ^ 3 )  / 
6 ) )  e.  CC )
26 mulcl 7916 . . . . . 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 7954 . . . 4  |-  ( A  e.  RR  ->  (
( 1  -  (
( A ^ 2 )  /  2 ) )  +  ( _i  x.  ( A  -  ( ( A ^
3 )  /  6
) ) ) )  e.  CC )
29 mulcl 7916 . . . . . 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 9171 . . . . 5  |-  4  e.  NN0
323eftlcl 11667 . . . . 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 10939 . . 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 10956 . . . 4  |-  ( A  e.  RR  ->  (
Re `  ( (
1  -  ( ( A ^ 2 )  /  2 ) )  +  ( _i  x.  ( A  -  (
( A ^ 3 )  /  6 ) ) ) ) )  =  ( 1  -  ( ( A ^
2 )  /  2
) ) )
3635oveq1d 5883 . . 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 4000    |-> cmpt 4061   ` cfv 5211  (class class class)co 5868   CCcc 7787   RRcr 7788   0cc0 7789   1c1 7790   _ici 7791    + caddc 7792    x. cmul 7794    - cmin 8105   # cap 8515    / cdiv 8605   2c2 8946   3c3 8947   4c4 8948   6c6 8950   NN0cn0 9152   ZZ>=cuz 9504   ^cexp 10492   !cfa 10676   Recre 10820   sum_csu 11332   expce 11621   cosccos 11624
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907  ax-arch 7908  ax-caucvg 7909
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-isom 5220  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-frec 6385  df-1o 6410  df-oadd 6414  df-er 6528  df-en 6734  df-dom 6735  df-fin 6736  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-2 8954  df-3 8955  df-4 8956  df-5 8957  df-6 8958  df-n0 9153  df-z 9230  df-uz 9505  df-q 9596  df-rp 9628  df-ico 9868  df-fz 9983  df-fzo 10116  df-seqfrec 10419  df-exp 10493  df-fac 10677  df-ihash 10727  df-cj 10822  df-re 10823  df-im 10824  df-rsqrt 10978  df-abs 10979  df-clim 11258  df-sumdc 11333  df-ef 11627  df-cos 11630
This theorem is referenced by:  cos01bnd  11737
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