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Theorem atantayl3 20235
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 9982 . . . . . . . . . . . 12  |-  2  e.  NN0
3 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
4 nn0mulcl 10000 . . . . . . . . . . . 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 10020 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( 2  x.  n )  e.  CC )
7 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
8 pncan 9057 . . . . . . . . . 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 5873 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 )  =  ( ( 2  x.  n
)  /  2 ) )
11 nn0cn 9975 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  n  e.  CC )
1211adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  e.  CC )
13 2cn 9816 . . . . . . . . . 10  |-  2  e.  CC
1413a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  2  e.  CC )
15 2ne0 9829 . . . . . . . . . 10  |-  2  =/=  0
1615a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  2  =/=  0
)
1712, 14, 16divcan3d 9541 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  / 
2 )  =  n )
1810, 17eqtr2d 2316 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  =  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )
1918oveq2d 5874 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  =  ( -u 1 ^ ( ( ( ( 2  x.  n )  +  1 )  - 
1 )  /  2
) ) )
2019oveq1d 5873 . . . . 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 4106 . . . 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 2327 . . 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 11054 . 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 2283 . . . 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 20234 . . 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 9813 . . . . . . 7  |-  -u 1  e.  CC
27 expcl 11121 . . . . . . 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 10004 . . . . . . . . 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 11245 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( A ^
( ( 2  x.  n )  +  1 ) )  e.  CC )
33 nn0p1nn 10003 . . . . . . . . 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 9762 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  e.  CC )
3634nnne0d 9790 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  =/=  0
)
3732, 35, 36divcld 9536 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) )  e.  CC )
3828, 37mulcld 8855 . . . . 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 2358 . . . 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 5865 . . . . . . 7  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
k  -  1 )  =  ( ( ( 2  x.  n )  +  1 )  - 
1 ) )
4140oveq1d 5873 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( k  -  1 )  /  2 )  =  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )
4241oveq2d 5874 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  ( -u 1 ^ ( ( k  -  1 )  /  2 ) )  =  ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) ) )
43 oveq2 5866 . . . . . 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 5876 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( A ^ k
)  /  k )  =  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) ) )
4642, 45oveq12d 5876 . . . 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 12888 . . 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 4043 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 1623    e. wcel 1684    =/= wne 2446   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958    || cdivides 12531  arctancatan 20160
This theorem is referenced by:  log2cnv  20240
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-iin 3908  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-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  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-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  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  df-cos 12352  df-tan 12353  df-pi 12354  df-dvds 12532  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-ulm 19756  df-log 19914  df-atan 20163
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