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Theorem taylpfval 20281
Description: Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: 
S is the base set with respect to evaluate the derivatives (generally  RR or 
CC),  F is the function we are approximating, at point  B, to order  N. The result is a polynomial function of  x. (Contributed by Mario Carneiro, 31-Dec-2016.)
Hypotheses
Ref Expression
taylpfval.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
taylpfval.f  |-  ( ph  ->  F : A --> CC )
taylpfval.a  |-  ( ph  ->  A  C_  S )
taylpfval.n  |-  ( ph  ->  N  e.  NN0 )
taylpfval.b  |-  ( ph  ->  B  e.  dom  (
( S  D n F ) `  N
) )
taylpfval.t  |-  T  =  ( N ( S Tayl 
F ) B )
Assertion
Ref Expression
taylpfval  |-  ( ph  ->  T  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) )
Distinct variable groups:    x, k, B    k, F, x    k, N, x    ph, k, x    S, k, x    x, T
Allowed substitution hints:    A( x, k)    T( k)

Proof of Theorem taylpfval
StepHypRef Expression
1 taylpfval.s . . . 4  |-  ( ph  ->  S  e.  { RR ,  CC } )
2 taylpfval.f . . . 4  |-  ( ph  ->  F : A --> CC )
3 taylpfval.a . . . 4  |-  ( ph  ->  A  C_  S )
4 taylpfval.n . . . . 5  |-  ( ph  ->  N  e.  NN0 )
54orcd 382 . . . 4  |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo )
)
6 taylpfval.b . . . . 5  |-  ( ph  ->  B  e.  dom  (
( S  D n F ) `  N
) )
71, 2, 3, 4, 6taylplem1 20279 . . . 4  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `
 k ) )
8 taylpfval.t . . . 4  |-  T  =  ( N ( S Tayl 
F ) B )
91, 2, 3, 5, 7, 8taylfval 20275 . . 3  |-  ( ph  ->  T  =  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) )
10 cnfldbas 16707 . . . . . . 7  |-  CC  =  ( Base ` fld )
11 cnfld0 16725 . . . . . . 7  |-  0  =  ( 0g ` fld )
12 cnrng 16723 . . . . . . . 8  |-fld  e.  Ring
13 rngcmn 15694 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e. CMnd )
1412, 13mp1i 12 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->fld  e. CMnd )
15 cnfldtps 18812 . . . . . . . 8  |-fld  e.  TopSp
1615a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->fld  e.  TopSp )
17 ovex 6106 . . . . . . . . 9  |-  ( 0 [,] N )  e. 
_V
1817inex1 4344 . . . . . . . 8  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
1918a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V )
201, 2, 3, 5, 7taylfvallem1 20273 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) )  e.  CC )
21 eqid 2436 . . . . . . . 8  |-  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) )  =  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) )
2220, 21fmptd 5893 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) : ( ( 0 [,] N )  i^i  ZZ ) --> CC )
23 0z 10293 . . . . . . . . . . 11  |-  0  e.  ZZ
244nn0zd 10373 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ZZ )
25 fzval2 11046 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0 ... N
)  =  ( ( 0 [,] N )  i^i  ZZ ) )
2623, 24, 25sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 0 ... N
)  =  ( ( 0 [,] N )  i^i  ZZ ) )
2726adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0 ... N )  =  ( ( 0 [,] N )  i^i  ZZ ) )
28 fzfid 11312 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0 ... N )  e. 
Fin )
2927, 28eqeltrrd 2511 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( 0 [,] N )  i^i  ZZ )  e. 
Fin )
3029, 22fisuppfi 14773 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  ( `' ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) "
( _V  \  {
0 } ) )  e.  Fin )
31 eqid 2436 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3231cnfldhaus 18819 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  Haus
3332a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  ( TopOpen ` fld )  e.  Haus )
3410, 11, 14, 16, 19, 22, 30, 31, 33haustsmsid 18170 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) )  =  { (fld  gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) } )
3529, 20gsumfsum 16766 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  (fld  gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) )  =  sum_ k  e.  ( ( 0 [,] N
)  i^i  ZZ )
( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) )
3627sumeq1d 12495 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) )  =  sum_ k  e.  ( (
0 [,] N )  i^i  ZZ ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) )
3735, 36eqtr4d 2471 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  (fld  gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) )  =  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) )
3837sneqd 3827 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  { (fld  gsumg  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) ) }  =  { sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) } )
3934, 38eqtrd 2468 . . . . 5  |-  ( (
ph  /\  x  e.  CC )  ->  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) )  =  { sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) } )
4039xpeq2d 4902 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) ) )  =  ( { x }  X.  { sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) } ) )
4140iuneq2dv 4114 . . 3  |-  ( ph  ->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) ) )  =  U_ x  e.  CC  ( { x }  X.  { sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) } ) )
429, 41eqtrd 2468 . 2  |-  ( ph  ->  T  =  U_ x  e.  CC  ( { x }  X.  { sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) } ) )
43 dfmpt3 5567 . 2  |-  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) )  = 
U_ x  e.  CC  ( { x }  X.  { sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) } )
4442, 43syl6eqr 2486 1  |-  ( ph  ->  T  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    i^i cin 3319    C_ wss 3320   {csn 3814   {cpr 3815   U_ciun 4093    e. cmpt 4266    X. cxp 4876   dom cdm 4878   -->wf 5450   ` cfv 5454  (class class class)co 6081   Fincfn 7109   CCcc 8988   RRcr 8989   0cc0 8990    x. cmul 8995    +oocpnf 9117    - cmin 9291    / cdiv 9677   NN0cn0 10221   ZZcz 10282   [,]cicc 10919   ...cfz 11043   ^cexp 11382   !cfa 11566   sum_csu 12479   TopOpenctopn 13649    gsumg cgsu 13724  CMndccmn 15412   Ringcrg 15660  ℂfldccnfld 16703   TopSpctps 16961   Hauscha 17372   tsums ctsu 18155    D ncdvn 19751   Tayl ctayl 20269
This theorem is referenced by:  taylpf  20282  taylpval  20283  taylply2  20284  dvtaylp  20286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-icc 10923  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-fac 11567  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-mulr 13543  df-starv 13544  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-rest 13650  df-topn 13651  df-topgen 13667  df-0g 13727  df-gsum 13728  df-mnd 14690  df-grp 14812  df-minusg 14813  df-cntz 15116  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cnp 17292  df-haus 17379  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-tsms 18156  df-xms 18350  df-ms 18351  df-limc 19753  df-dv 19754  df-dvn 19755  df-tayl 20271
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