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Theorem taylpfval 22627
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  Dn
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  Dn 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 392 . . . 4  |-  ( ph  ->  ( N  e.  NN0  \/  N  = +oo )
)
6 taylpfval.b . . . . 5  |-  ( ph  ->  B  e.  dom  (
( S  Dn
F ) `  N
) )
71, 2, 3, 4, 6taylplem1 22625 . . . 4  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  Dn F ) `
 k ) )
8 taylpfval.t . . . 4  |-  T  =  ( N ( S Tayl 
F ) B )
91, 2, 3, 5, 7, 8taylfval 22621 . . 3  |-  ( ph  ->  T  =  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) ) )
10 cnfldbas 18294 . . . . . . 7  |-  CC  =  ( Base ` fld )
11 cnfld0 18312 . . . . . . 7  |-  0  =  ( 0g ` fld )
12 cnring 18310 . . . . . . . 8  |-fld  e.  Ring
13 ringcmn 17101 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e. CMnd )
1412, 13mp1i 12 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->fld  e. CMnd )
15 cnfldtps 21153 . . . . . . . 8  |-fld  e.  TopSp
1615a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->fld  e.  TopSp )
17 ovex 6320 . . . . . . . . 9  |-  ( 0 [,] N )  e. 
_V
1817inex1 4594 . . . . . . . 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 22619 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) )  e.  CC )
21 eqid 2467 . . . . . . . 8  |-  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) )  =  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) )
2220, 21fmptd 6056 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) : ( ( 0 [,] N )  i^i  ZZ ) --> CC )
23 0z 10887 . . . . . . . . . . 11  |-  0  e.  ZZ
244nn0zd 10976 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ZZ )
25 fzval2 11687 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0 ... N
)  =  ( ( 0 [,] N )  i^i  ZZ ) )
2623, 24, 25sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 0 ... N
)  =  ( ( 0 [,] N )  i^i  ZZ ) )
2726adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0 ... N )  =  ( ( 0 [,] N )  i^i  ZZ ) )
28 fzfid 12063 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0 ... N )  e. 
Fin )
2927, 28eqeltrrd 2556 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( 0 [,] N )  i^i  ZZ )  e. 
Fin )
30 ovex 6320 . . . . . . . . 9  |-  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) )  e.  _V
3130a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  CC )  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) )  e.  _V )
32 c0ex 9602 . . . . . . . . 9  |-  0  e.  _V
3332a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  0  e. 
_V )
3421, 29, 31, 33fsuppmptdm 7852 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) finSupp  0 )
35 eqid 2467 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3635cnfldhaus 21160 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  Haus
3736a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  ( TopOpen ` fld )  e.  Haus )
3810, 11, 14, 16, 19, 22, 34, 35, 37haustsmsid 20507 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) ) )  =  { (fld  gsumg  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) ) } )
3929, 20gsumfsum 18354 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  (fld  gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) )  =  sum_ k  e.  ( ( 0 [,] N
)  i^i  ZZ )
( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) )
4027sumeq1d 13503 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) )  = 
sum_ k  e.  ( ( 0 [,] N
)  i^i  ZZ )
( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) )
4139, 40eqtr4d 2511 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  (fld  gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) )  =  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) )
4241sneqd 4045 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  { (fld  gsumg  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) ) }  =  { sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) } )
4338, 42eqtrd 2508 . . . . 5  |-  ( (
ph  /\  x  e.  CC )  ->  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) ) )  =  { sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) } )
4443xpeq2d 5029 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) ) )  =  ( { x }  X.  { sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) } ) )
4544iuneq2dv 4353 . . 3  |-  ( ph  ->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) ) )  =  U_ x  e.  CC  ( { x }  X.  { sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) } ) )
469, 45eqtrd 2508 . 2  |-  ( ph  ->  T  =  U_ x  e.  CC  ( { x }  X.  { sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) } ) )
47 dfmpt3 5709 . 2  |-  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) )  =  U_ x  e.  CC  ( { x }  X.  { sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) } )
4846, 47syl6eqr 2526 1  |-  ( ph  ->  T  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480    C_ wss 3481   {csn 4033   {cpr 4035   U_ciun 4331    |-> cmpt 4511    X. cxp 5003   dom cdm 5005   -->wf 5590   ` cfv 5594  (class class class)co 6295   Fincfn 7528   CCcc 9502   RRcr 9503   0cc0 9504    x. cmul 9509   +oocpnf 9637    - cmin 9817    / cdiv 10218   NN0cn0 10807   ZZcz 10876   [,]cicc 11544   ...cfz 11684   ^cexp 12146   !cfa 12333   sum_csu 13488   TopOpenctopn 14694    gsumg cgsu 14713  CMndccmn 16671   Ringcrg 17070  ℂfldccnfld 18290   TopSpctps 19266   Hauscha 19677   tsums ctsu 20492    Dncdvn 22136   Tayl ctayl 22615
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-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  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-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-icc 11548  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-fac 12334  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-cntz 16227  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-cring 17073  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-cnp 19597  df-haus 19684  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-tsms 20493  df-xms 20691  df-ms 20692  df-limc 22138  df-dv 22139  df-dvn 22140  df-tayl 22617
This theorem is referenced by:  taylpf  22628  taylpval  22629  taylply2  22630  dvtaylp  22632
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