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Theorem atantayl3 22293
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 10592 . . . . . . . . . . . 12  |-  2  e.  NN0
3 simpr 458 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
4 nn0mulcl 10612 . . . . . . . . . . . 12  |-  ( ( 2  e.  NN0  /\  n  e.  NN0 )  -> 
( 2  x.  n
)  e.  NN0 )
52, 3, 4sylancr 658 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( 2  x.  n )  e.  NN0 )
65nn0cnd 10634 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( 2  x.  n )  e.  CC )
7 ax-1cn 9336 . . . . . . . . . 10  |-  1  e.  CC
8 pncan 9612 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2  x.  n )  +  1 )  -  1 )  =  ( 2  x.  n ) )
96, 7, 8sylancl 657 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( ( 2  x.  n )  +  1 )  - 
1 )  =  ( 2  x.  n ) )
109oveq1d 6105 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 )  =  ( ( 2  x.  n
)  /  2 ) )
11 nn0cn 10585 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  n  e.  CC )
1211adantl 463 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  e.  CC )
13 2cnd 10390 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  2  e.  CC )
14 2ne0 10410 . . . . . . . . . 10  |-  2  =/=  0
1514a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  2  =/=  0
)
1612, 13, 15divcan3d 10108 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  / 
2 )  =  n )
1710, 16eqtr2d 2474 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  =  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )
1817oveq2d 6106 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  =  ( -u 1 ^ ( ( ( ( 2  x.  n )  +  1 )  - 
1 )  /  2
) ) )
1918oveq1d 6105 . . . . 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 4375 . . . 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 2485 . . 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 11810 . 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 2441 . . . 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 22292 . . 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 10421 . . . . . . 7  |-  -u 1  e.  CC
26 expcl 11879 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  e.  CC )
2725, 3, 26sylancr 658 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  e.  CC )
28 simpll 748 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  A  e.  CC )
29 peano2nn0 10616 . . . . . . . . 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 12004 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( A ^
( ( 2  x.  n )  +  1 ) )  e.  CC )
32 nn0p1nn 10615 . . . . . . . . 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 10334 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  e.  CC )
3533nnne0d 10362 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  =/=  0
)
3631, 34, 35divcld 10103 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) )  e.  CC )
3727, 36mulcld 9402 . . . . 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 2516 . . . 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 6097 . . . . . . 7  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
k  -  1 )  =  ( ( ( 2  x.  n )  +  1 )  - 
1 ) )
4039oveq1d 6105 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( k  -  1 )  /  2 )  =  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )
4140oveq2d 6106 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  ( -u 1 ^ ( ( k  -  1 )  /  2 ) )  =  ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) ) )
42 oveq2 6098 . . . . . 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 6108 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( A ^ k
)  /  k )  =  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) ) )
4541, 44oveq12d 6108 . . . 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 13898 . . 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 4309 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 1364    e. wcel 1761    =/= wne 2604   ifcif 3788   class class class wbr 4289    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    < clt 9414    - cmin 9591   -ucneg 9592    / cdiv 9989   NNcn 10318   2c2 10367   NN0cn0 10575    seqcseq 11802   ^cexp 11861   abscabs 12719    ~~> cli 12958    || cdivides 13531  arctancatan 22218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-tan 13353  df-pi 13354  df-dvds 13532  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-cmp 18949  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-ulm 21801  df-log 21967  df-atan 22221
This theorem is referenced by:  log2cnv  22298
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