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Theorem taylpfval 21966
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 21964 . . . 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 21960 . . 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 17950 . . . . . . 7  |-  CC  =  ( Base ` fld )
11 cnfld0 17968 . . . . . . 7  |-  0  =  ( 0g ` fld )
12 cnrng 17966 . . . . . . . 8  |-fld  e.  Ring
13 rngcmn 16801 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e. CMnd )
1412, 13mp1i 12 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->fld  e. CMnd )
15 cnfldtps 20492 . . . . . . . 8  |-fld  e.  TopSp
1615a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->fld  e.  TopSp )
17 ovex 6228 . . . . . . . . 9  |-  ( 0 [,] N )  e. 
_V
1817inex1 4544 . . . . . . . 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 21958 . . . . . . . 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 5979 . . . . . . 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 10771 . . . . . . . . . . 11  |-  0  e.  ZZ
244nn0zd 10859 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ZZ )
25 fzval2 11560 . . . . . . . . . . 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 11915 . . . . . . . . 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 6228 . . . . . . . . 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 9494 . . . . . . . . 9  |-  0  e.  _V
3332a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  0  e. 
_V )
3421, 29, 31, 33fsuppmptdm 7745 . . . . . . 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 20499 . . . . . . . 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 19846 . . . . . 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 18007 . . . . . . . 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 13299 . . . . . . . 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 4000 . . . . . 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 4975 . . . 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 4303 . . 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 5644 . 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    i^i cin 3438    C_ wss 3439   {csn 3988   {cpr 3990   U_ciun 4282    |-> cmpt 4461    X. cxp 4949   dom cdm 4951   -->wf 5525   ` cfv 5529  (class class class)co 6203   Fincfn 7423   CCcc 9394   RRcr 9395   0cc0 9396    x. cmul 9401   +oocpnf 9529    - cmin 9709    / cdiv 10107   NN0cn0 10693   ZZcz 10760   [,]cicc 11417   ...cfz 11557   ^cexp 11985   !cfa 12171   sum_csu 13284   TopOpenctopn 14482    gsumg cgsu 14501  CMndccmn 16401   Ringcrg 16771  ℂfldccnfld 17946   TopSpctps 18636   Hauscha 19047   tsums ctsu 19831    Dncdvn 21475   Tayl ctayl 21954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474  ax-addf 9475  ax-mulf 9476
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-fi 7775  df-sup 7805  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-q 11068  df-rp 11106  df-xneg 11203  df-xadd 11204  df-xmul 11205  df-icc 11421  df-fz 11558  df-fzo 11669  df-seq 11927  df-exp 11986  df-fac 12172  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-sum 13285  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-plusg 14373  df-mulr 14374  df-starv 14375  df-tset 14379  df-ple 14380  df-ds 14382  df-unif 14383  df-rest 14483  df-topn 14484  df-0g 14502  df-gsum 14503  df-topgen 14504  df-mnd 15537  df-grp 15667  df-minusg 15668  df-cntz 15957  df-cmn 16403  df-abl 16404  df-mgp 16717  df-ur 16729  df-rng 16773  df-cring 16774  df-psmet 17937  df-xmet 17938  df-met 17939  df-bl 17940  df-mopn 17941  df-fbas 17942  df-fg 17943  df-cnfld 17947  df-top 18638  df-bases 18640  df-topon 18641  df-topsp 18642  df-cld 18758  df-ntr 18759  df-cls 18760  df-nei 18837  df-lp 18875  df-perf 18876  df-cnp 18967  df-haus 19054  df-fil 19554  df-fm 19646  df-flim 19647  df-flf 19648  df-tsms 19832  df-xms 20030  df-ms 20031  df-limc 21477  df-dv 21478  df-dvn 21479  df-tayl 21956
This theorem is referenced by:  taylpf  21967  taylpval  21968  taylply2  21969  dvtaylp  21971
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