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Theorem atantayl3 23136
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 10824 . . . . . . . . . . . 12  |-  2  e.  NN0
3 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
4 nn0mulcl 10844 . . . . . . . . . . . 12  |-  ( ( 2  e.  NN0  /\  n  e.  NN0 )  -> 
( 2  x.  n
)  e.  NN0 )
52, 3, 4sylancr 663 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( 2  x.  n )  e.  NN0 )
65nn0cnd 10866 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( 2  x.  n )  e.  CC )
7 ax-1cn 9562 . . . . . . . . . 10  |-  1  e.  CC
8 pncan 9838 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2  x.  n )  +  1 )  -  1 )  =  ( 2  x.  n ) )
96, 7, 8sylancl 662 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( ( 2  x.  n )  +  1 )  - 
1 )  =  ( 2  x.  n ) )
109oveq1d 6310 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 )  =  ( ( 2  x.  n
)  /  2 ) )
11 nn0cn 10817 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  n  e.  CC )
1211adantl 466 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  e.  CC )
13 2cnd 10620 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  2  e.  CC )
14 2ne0 10640 . . . . . . . . . 10  |-  2  =/=  0
1514a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  2  =/=  0
)
1612, 13, 15divcan3d 10337 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  / 
2 )  =  n )
1710, 16eqtr2d 2509 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  =  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )
1817oveq2d 6311 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  =  ( -u 1 ^ ( ( ( ( 2  x.  n )  +  1 )  - 
1 )  /  2
) ) )
1918oveq1d 6310 . . . . 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 ) ) ) )
2019mpteq2dva 4539 . . . 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 ) ) ) ) )
211, 20syl5eq 2520 . . 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 ) ) ) ) )
2221seqeq3d 12095 . 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 ) ) ) ) ) )
23 eqid 2467 . . . 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 ) ) ) )
2423atantayl2 23135 . . 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 )
)
25 neg1cn 10651 . . . . . . 7  |-  -u 1  e.  CC
26 expcl 12164 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  e.  CC )
2725, 3, 26sylancr 663 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  e.  CC )
28 simpll 753 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  A  e.  CC )
29 peano2nn0 10848 . . . . . . . . 9  |-  ( ( 2  x.  n )  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e. 
NN0 )
305, 29syl 16 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  e.  NN0 )
3128, 30expcld 12290 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( A ^
( ( 2  x.  n )  +  1 ) )  e.  CC )
32 nn0p1nn 10847 . . . . . . . . 9  |-  ( ( 2  x.  n )  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
335, 32syl 16 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
3433nncnd 10564 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  e.  CC )
3533nnne0d 10592 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  =/=  0
)
3631, 34, 35divcld 10332 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) )  e.  CC )
3727, 36mulcld 9628 . . . . 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 )
3819, 37eqeltrrd 2556 . . . 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 )
39 oveq1 6302 . . . . . . 7  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
k  -  1 )  =  ( ( ( 2  x.  n )  +  1 )  - 
1 ) )
4039oveq1d 6310 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( k  -  1 )  /  2 )  =  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )
4140oveq2d 6311 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  ( -u 1 ^ ( ( k  -  1 )  /  2 ) )  =  ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) ) )
42 oveq2 6303 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  ( A ^ k )  =  ( A ^ (
( 2  x.  n
)  +  1 ) ) )
43 id 22 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  k  =  ( ( 2  x.  n )  +  1 ) )
4442, 43oveq12d 6313 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( A ^ k
)  /  k )  =  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) ) )
4541, 44oveq12d 6313 . . . 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 ) ) ) )
4638, 45iserodd 14235 . . 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 )
) )
4724, 46mpbird 232 . 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 )
)
4822, 47eqbrtrd 4473 1  |-  ( ( A  e.  CC  /\  ( abs `  A )  <  1 )  ->  seq 0 (  +  ,  F )  ~~>  (arctan `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ifcif 3945   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    < clt 9640    - cmin 9817   -ucneg 9818    / cdiv 10218   NNcn 10548   2c2 10597   NN0cn0 10807    seqcseq 12087   ^cexp 12146   abscabs 13047    ~~> cli 13287    || cdivides 13864  arctancatan 23061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12880  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-limsup 13274  df-clim 13291  df-rlim 13292  df-sum 13489  df-ef 13682  df-sin 13684  df-cos 13685  df-tan 13686  df-pi 13687  df-dvds 13865  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-cmp 19755  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-limc 22138  df-dv 22139  df-ulm 22639  df-log 22810  df-atan 23064
This theorem is referenced by:  log2cnv  23141
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