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Theorem atantayl3 22339
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 10601 . . . . . . . . . . . 12  |-  2  e.  NN0
3 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
4 nn0mulcl 10621 . . . . . . . . . . . 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 10643 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( 2  x.  n )  e.  CC )
7 ax-1cn 9345 . . . . . . . . . 10  |-  1  e.  CC
8 pncan 9621 . . . . . . . . . 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 6111 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 )  =  ( ( 2  x.  n
)  /  2 ) )
11 nn0cn 10594 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  n  e.  CC )
1211adantl 466 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  e.  CC )
13 2cnd 10399 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  2  e.  CC )
14 2ne0 10419 . . . . . . . . . 10  |-  2  =/=  0
1514a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  2  =/=  0
)
1612, 13, 15divcan3d 10117 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  / 
2 )  =  n )
1710, 16eqtr2d 2476 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  n  =  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )
1817oveq2d 6112 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  =  ( -u 1 ^ ( ( ( ( 2  x.  n )  +  1 )  - 
1 )  /  2
) ) )
1918oveq1d 6111 . . . . 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 4383 . . . 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 2487 . . 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 11819 . 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 2443 . . . 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 22338 . . 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 10430 . . . . . . 7  |-  -u 1  e.  CC
26 expcl 11888 . . . . . . 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 10625 . . . . . . . . 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 12013 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( A ^
( ( 2  x.  n )  +  1 ) )  e.  CC )
32 nn0p1nn 10624 . . . . . . . . 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 10343 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  e.  CC )
3533nnne0d 10371 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( 2  x.  n )  +  1 )  =/=  0
)
3631, 34, 35divcld 10112 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( abs `  A
)  <  1 )  /\  n  e.  NN0 )  ->  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) )  e.  CC )
3727, 36mulcld 9411 . . . . 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 2518 . . . 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 6103 . . . . . . 7  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
k  -  1 )  =  ( ( ( 2  x.  n )  +  1 )  - 
1 ) )
4039oveq1d 6111 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( k  -  1 )  /  2 )  =  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )
4140oveq2d 6112 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  ( -u 1 ^ ( ( k  -  1 )  /  2 ) )  =  ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) ) )
42 oveq2 6104 . . . . . 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 6114 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( A ^ k
)  /  k )  =  ( ( A ^ ( ( 2  x.  n )  +  1 ) )  / 
( ( 2  x.  n )  +  1 ) ) )
4541, 44oveq12d 6114 . . . 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 13907 . . 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 4317 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 1369    e. wcel 1756    =/= wne 2611   ifcif 3796   class class class wbr 4297    e. cmpt 4355   ` cfv 5423  (class class class)co 6096   CCcc 9285   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292    < clt 9423    - cmin 9600   -ucneg 9601    / cdiv 9998   NNcn 10327   2c2 10376   NN0cn0 10584    seqcseq 11811   ^cexp 11870   abscabs 12728    ~~> cli 12967    || cdivides 13540  arctancatan 22264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-ioc 11310  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-fac 12057  df-bc 12084  df-hash 12109  df-shft 12561  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-limsup 12954  df-clim 12971  df-rlim 12972  df-sum 13169  df-ef 13358  df-sin 13360  df-cos 13361  df-tan 13362  df-pi 13363  df-dvds 13541  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cn 18836  df-cnp 18837  df-haus 18924  df-cmp 18995  df-tx 19140  df-hmeo 19333  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-xms 19900  df-ms 19901  df-tms 19902  df-cncf 20459  df-limc 21346  df-dv 21347  df-ulm 21847  df-log 22013  df-atan 22267
This theorem is referenced by:  log2cnv  22344
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