MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  atantayl3 Unicode version

Theorem atantayl3 20457
Description: The Taylor series for arctan ( A
). (Contributed by Mario Carneiro, 7-Apr-2015.)
Hypothesis
Ref Expression
atantayl3.1  |-  F  =  ( n  e.  NN0  |->  ( ( -u 1 ^ n )  x.  ( ( A ^
( ( 2  x.  n )  +  1 ) )  /  (
( 2  x.  n
)  +  1 ) ) ) )
Assertion
Ref Expression
atantayl3  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  ,  F )  ~~>  (arctan `  A ) )
Distinct variable group:    A, n
Allowed substitution hint:    F( n)

Proof of Theorem atantayl3
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 atantayl3.1 . . . 4  |-  F  =  ( n  e.  NN0  |->  ( ( -u 1 ^ n )  x.  ( ( A ^
( ( 2  x.  n )  +  1 ) )  /  (
( 2  x.  n
)  +  1 ) ) ) )
2 2nn0 10131 . . . . . . . . . . . 12  |-  2  e.  NN0
3 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
4 nn0mulcl 10149 . . . . . . . . . . . 12  |-  ( ( 2  e.  NN0  /\  n  e.  NN0 )  -> 
( 2  x.  n
)  e.  NN0 )
52, 3, 4sylancr 644 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( 2  x.  n )  e.  NN0 )
65nn0cnd 10169 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( 2  x.  n )  e.  CC )
7 ax-1cn 8942 . . . . . . . . . 10  |-  1  e.  CC
8 pncan 9204 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2  x.  n )  +  1 )  -  1 )  =  ( 2  x.  n ) )
96, 7, 8sylancl 643 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( ( 2  x.  n )  +  1 )  - 
1 )  =  ( 2  x.  n ) )
109oveq1d 5996 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 )  =  ( ( 2  x.  n
)  /  2 ) )
11 nn0cn 10124 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  n  e.  CC )
1211adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  e.  CC )
13 2cn 9963 . . . . . . . . . 10  |-  2  e.  CC
1413a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  2  e.  CC )
15 2ne0 9976 . . . . . . . . . 10  |-  2  =/=  0
1615a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  2  =/=  0
)
1712, 14, 16divcan3d 9688 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  / 
2 )  =  n )
1810, 17eqtr2d 2399 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  =  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )
1918oveq2d 5997 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  =  ( -u 1 ^ ( ( ( ( 2  x.  n )  +  1 )  - 
1 )  /  2
) ) )
2019oveq1d 5996 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( -u
1 ^ n )  x.  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) ) )  =  ( ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )  x.  ( ( A ^
( ( 2  x.  n )  +  1 ) )  /  (
( 2  x.  n
)  +  1 ) ) ) )
2120mpteq2dva 4208 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
( n  e.  NN0  |->  ( ( -u 1 ^ n )  x.  ( ( A ^
( ( 2  x.  n )  +  1 ) )  /  (
( 2  x.  n
)  +  1 ) ) ) )  =  ( n  e.  NN0  |->  ( ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )  x.  ( ( A ^
( ( 2  x.  n )  +  1 ) )  /  (
( 2  x.  n
)  +  1 ) ) ) ) )
221, 21syl5eq 2410 . . 3  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  F  =  ( n  e.  NN0  |->  ( ( -u
1 ^ ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  x.  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) ) ) ) )
2322seqeq3d 11218 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  ,  F )  =  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )  x.  ( ( A ^
( ( 2  x.  n )  +  1 ) )  /  (
( 2  x.  n
)  +  1 ) ) ) ) ) )
24 eqid 2366 . . . 4  |-  ( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( (
-u 1 ^ (
( k  -  1 )  /  2 ) )  x.  ( ( A ^ k )  /  k ) ) ) )  =  ( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  x.  (
( A ^ k
)  /  k ) ) ) )
2524atantayl2 20456 . . 3  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  1 (  +  , 
( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  x.  (
( A ^ k
)  /  k ) ) ) ) )  ~~>  (arctan `  A )
)
26 neg1cn 9960 . . . . . . 7  |-  -u 1  e.  CC
27 expcl 11286 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  e.  CC )
2826, 3, 27sylancr 644 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  e.  CC )
29 simpll 730 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  A  e.  CC )
30 peano2nn0 10153 . . . . . . . . 9  |-  ( ( 2  x.  n )  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e. 
NN0 )
315, 30syl 15 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  e.  NN0 )
3229, 31expcld 11410 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( A ^
( ( 2  x.  n )  +  1 ) )  e.  CC )
33 nn0p1nn 10152 . . . . . . . . 9  |-  ( ( 2  x.  n )  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
345, 33syl 15 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
3534nncnd 9909 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  e.  CC )
3634nnne0d 9937 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  =/=  0
)
3732, 35, 36divcld 9683 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) )  e.  CC )
3828, 37mulcld 9002 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( -u
1 ^ n )  x.  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) ) )  e.  CC )
3920, 38eqeltrrd 2441 . . . 4  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( -u
1 ^ ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  x.  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) ) )  e.  CC )
40 oveq1 5988 . . . . . . 7  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
k  -  1 )  =  ( ( ( 2  x.  n )  +  1 )  - 
1 ) )
4140oveq1d 5996 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( k  -  1 )  /  2 )  =  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )
4241oveq2d 5997 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  ( -u 1 ^ ( ( k  -  1 )  /  2 ) )  =  ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) ) )
43 oveq2 5989 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  ( A ^ k )  =  ( A ^ (
( 2  x.  n
)  +  1 ) ) )
44 id 19 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  k  =  ( ( 2  x.  n )  +  1 ) )
4543, 44oveq12d 5999 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( A ^ k
)  /  k )  =  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) ) )
4642, 45oveq12d 5999 . . . 4  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( -u 1 ^ (
( k  -  1 )  /  2 ) )  x.  ( ( A ^ k )  /  k ) )  =  ( ( -u
1 ^ ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  x.  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) ) ) )
4739, 46iserodd 13096 . . 3  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  -> 
(  seq  0 (  +  ,  ( n  e.  NN0  |->  ( (
-u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  x.  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  /  ( ( 2  x.  n )  +  1 ) ) ) ) )  ~~>  (arctan `  A )  <->  seq  1
(  +  ,  ( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  x.  (
( A ^ k
)  /  k ) ) ) ) )  ~~>  (arctan `  A )
) )
4825, 47mpbird 223 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )  x.  ( ( A ^
( ( 2  x.  n )  +  1 ) )  /  (
( 2  x.  n
)  +  1 ) ) ) ) )  ~~>  (arctan `  A )
)
4923, 48eqbrtrd 4145 1  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq  0 (  +  ,  F )  ~~>  (arctan `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   ifcif 3654   class class class wbr 4125    e. cmpt 4179   ` cfv 5358  (class class class)co 5981   CCcc 8882   0cc0 8884   1c1 8885    + caddc 8887    x. cmul 8889    < clt 9014    - cmin 9184   -ucneg 9185    / cdiv 9570   NNcn 9893   2c2 9942   NN0cn0 10114    seq cseq 11210   ^cexp 11269   abscabs 11926    ~~> cli 12165    || cdivides 12739  arctancatan 20382
This theorem is referenced by:  log2cnv  20462
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963  ax-mulf 8964
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-fi 7312  df-sup 7341  df-oi 7372  df-card 7719  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-q 10468  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-ioo 10813  df-ioc 10814  df-ico 10815  df-icc 10816  df-fz 10936  df-fzo 11026  df-fl 11089  df-mod 11138  df-seq 11211  df-exp 11270  df-fac 11454  df-bc 11481  df-hash 11506  df-shft 11769  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-limsup 12152  df-clim 12169  df-rlim 12170  df-sum 12367  df-ef 12557  df-sin 12559  df-cos 12560  df-tan 12561  df-pi 12562  df-dvds 12740  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-starv 13431  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-unif 13439  df-hom 13440  df-cco 13441  df-rest 13537  df-topn 13538  df-topgen 13554  df-pt 13555  df-prds 13558  df-xrs 13613  df-0g 13614  df-gsum 13615  df-qtop 13620  df-imas 13621  df-xps 13623  df-mre 13698  df-mrc 13699  df-acs 13701  df-mnd 14577  df-submnd 14626  df-mulg 14702  df-cntz 15003  df-cmn 15301  df-xmet 16586  df-met 16587  df-bl 16588  df-mopn 16589  df-fbas 16590  df-fg 16591  df-cnfld 16594  df-top 16853  df-bases 16855  df-topon 16856  df-topsp 16857  df-cld 16973  df-ntr 16974  df-cls 16975  df-nei 17052  df-lp 17085  df-perf 17086  df-cn 17174  df-cnp 17175  df-haus 17260  df-cmp 17331  df-tx 17474  df-hmeo 17663  df-fil 17754  df-fm 17846  df-flim 17847  df-flf 17848  df-xms 18098  df-ms 18099  df-tms 18100  df-cncf 18596  df-limc 19431  df-dv 19432  df-ulm 19971  df-log 20132  df-atan 20385
  Copyright terms: Public domain W3C validator