MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  taylpfval Structured version   Unicode version

Theorem taylpfval 22926
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 390 . . . 4  |-  ( ph  ->  ( N  e.  NN0  \/  N  = +oo )
)
6 taylpfval.b . . . . 5  |-  ( ph  ->  B  e.  dom  (
( S  Dn
F ) `  N
) )
71, 2, 3, 4, 6taylplem1 22924 . . . 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 22920 . . 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 18619 . . . . . . 7  |-  CC  =  ( Base ` fld )
11 cnfld0 18637 . . . . . . 7  |-  0  =  ( 0g ` fld )
12 cnring 18635 . . . . . . . 8  |-fld  e.  Ring
13 ringcmn 17424 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e. CMnd )
1412, 13mp1i 12 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->fld  e. CMnd )
15 cnfldtps 21451 . . . . . . . 8  |-fld  e.  TopSp
1615a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->fld  e.  TopSp )
17 ovex 6298 . . . . . . . . 9  |-  ( 0 [,] N )  e. 
_V
1817inex1 4578 . . . . . . . 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 22918 . . . . . . . 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 2454 . . . . . . . 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 6031 . . . . . . 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 10871 . . . . . . . . . . 11  |-  0  e.  ZZ
244nn0zd 10963 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ZZ )
25 fzval2 11678 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0 ... N
)  =  ( ( 0 [,] N )  i^i  ZZ ) )
2623, 24, 25sylancr 661 . . . . . . . . . 10  |-  ( ph  ->  ( 0 ... N
)  =  ( ( 0 [,] N )  i^i  ZZ ) )
2726adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0 ... N )  =  ( ( 0 [,] N )  i^i  ZZ ) )
28 fzfid 12065 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0 ... N )  e. 
Fin )
2927, 28eqeltrrd 2543 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( 0 [,] N )  i^i  ZZ )  e. 
Fin )
30 ovex 6298 . . . . . . . . 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 9579 . . . . . . . . 9  |-  0  e.  _V
3332a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  0  e. 
_V )
3421, 29, 31, 33fsuppmptdm 7832 . . . . . . 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 2454 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3635cnfldhaus 21458 . . . . . . . 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 20805 . . . . . 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 18679 . . . . . . . 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 13605 . . . . . . . 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 2498 . . . . . . 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 4028 . . . . . 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 2495 . . . . 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 5012 . . . 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 4337 . . 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 2495 . 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 5685 . 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 2513 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    i^i cin 3460    C_ wss 3461   {csn 4016   {cpr 4018   U_ciun 4315    |-> cmpt 4497    X. cxp 4986   dom cdm 4988   -->wf 5566   ` cfv 5570  (class class class)co 6270   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486   +oocpnf 9614    - cmin 9796    / cdiv 10202   NN0cn0 10791   ZZcz 10860   [,]cicc 11535   ...cfz 11675   ^cexp 12148   !cfa 12335   sum_csu 13590   TopOpenctopn 14911    gsumg cgsu 14930  CMndccmn 16997   Ringcrg 17393  ℂfldccnfld 18615   TopSpctps 19564   Hauscha 19976   tsums ctsu 20790    Dncdvn 22434   Tayl ctayl 22914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-fac 12336  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-plusg 14797  df-mulr 14798  df-starv 14799  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cnp 19896  df-haus 19983  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-tsms 20791  df-xms 20989  df-ms 20990  df-limc 22436  df-dv 22437  df-dvn 22438  df-tayl 22916
This theorem is referenced by:  taylpf  22927  taylpval  22928  taylply2  22929  dvtaylp  22931
  Copyright terms: Public domain W3C validator