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

Theorem taylfval 20277
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.

This "extended" version of taylpfval 20283 additionally handles the case  N  =  +oo, in which case this is not a polynomial but an infinite series, the Taylor series of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)

Hypotheses
Ref Expression
taylfval.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
taylfval.f  |-  ( ph  ->  F : A --> CC )
taylfval.a  |-  ( ph  ->  A  C_  S )
taylfval.n  |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo )
)
taylfval.b  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `
 k ) )
taylfval.t  |-  T  =  ( N ( S Tayl 
F ) B )
Assertion
Ref Expression
taylfval  |-  ( 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
) ) ) ) ) )
Distinct variable groups:    x, k, B    k, F, x    ph, k, x    k, N, x    S, k, x    x, T
Allowed substitution hints:    A( x, k)    T( k)

Proof of Theorem taylfval
Dummy variables  a  n  f  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 taylfval.t . 2  |-  T  =  ( N ( S Tayl 
F ) B )
2 df-tayl 20273 . . . . 5  |- Tayl  =  ( s  e.  { RR ,  CC } ,  f  e.  ( CC  ^pm  s )  |->  ( n  e.  ( NN0  u.  { 
+oo } ) ,  a  e.  |^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( s  D n f ) `  k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( s  D n
f ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) ) ) )
32a1i 11 . . . 4  |-  ( ph  -> Tayl  =  ( s  e. 
{ RR ,  CC } ,  f  e.  ( CC  ^pm  s ) 
|->  ( n  e.  ( NN0  u.  {  +oo } ) ,  a  e. 
|^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( s  D n f ) `  k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( s  D n
f ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) ) ) ) )
4 eqidd 2439 . . . . 5  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
( NN0  u.  {  +oo } )  =  ( NN0 
u.  {  +oo } ) )
5 oveq12 6092 . . . . . . . . 9  |-  ( ( s  =  S  /\  f  =  F )  ->  ( s  D n
f )  =  ( S  D n F ) )
65ad2antlr 709 . . . . . . . 8  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
s  D n f )  =  ( S  D n F ) )
76fveq1d 5732 . . . . . . 7  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( s  D n
f ) `  k
)  =  ( ( S  D n F ) `  k ) )
87dmeqd 5074 . . . . . 6  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  dom  ( ( s  D n f ) `  k )  =  dom  ( ( S  D n F ) `  k
) )
98iineq2dv 4117 . . . . 5  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  ->  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( s  D n f ) `  k )  =  |^|_ k  e.  ( (
0 [,] n )  i^i  ZZ ) dom  ( ( S  D n F ) `  k
) )
107fveq1d 5732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( ( s  D n f ) `  k ) `  a
)  =  ( ( ( S  D n F ) `  k
) `  a )
)
1110oveq1d 6098 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( ( ( s  D n f ) `
 k ) `  a )  /  ( ! `  k )
)  =  ( ( ( ( S  D n F ) `  k
) `  a )  /  ( ! `  k ) ) )
1211oveq1d 6098 . . . . . . . . 9  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( ( ( ( s  D n f ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) )  =  ( ( ( ( ( S  D n F ) `
 k ) `  a )  /  ( ! `  k )
)  x.  ( ( x  -  a ) ^ k ) ) )
1312mpteq2dva 4297 . . . . . . . 8  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( s  D n
f ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) )  =  ( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) )
1413oveq2d 6099 . . . . . . 7  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
(fld tsums  ( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( s  D n
f ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) )  =  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) )
1514xpeq2d 4904 . . . . . 6  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( s  D n
f ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) )  =  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  a )  /  ( ! `  k )
)  x.  ( ( x  -  a ) ^ k ) ) ) ) ) )
1615iuneq2d 4120 . . . . 5  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  ->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( s  D n f ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) )  =  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) ) )
174, 9, 16mpt2eq123dv 6138 . . . 4  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
( n  e.  ( NN0  u.  {  +oo } ) ,  a  e. 
|^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( s  D n f ) `  k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( s  D n
f ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) ) )  =  ( n  e.  ( NN0 
u.  {  +oo } ) ,  a  e.  |^|_ k  e.  ( (
0 [,] n )  i^i  ZZ ) dom  ( ( S  D n F ) `  k
)  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) ) ) )
18 simpr 449 . . . . 5  |-  ( (
ph  /\  s  =  S )  ->  s  =  S )
1918oveq2d 6099 . . . 4  |-  ( (
ph  /\  s  =  S )  ->  ( CC  ^pm  s )  =  ( CC  ^pm  S
) )
20 taylfval.s . . . 4  |-  ( ph  ->  S  e.  { RR ,  CC } )
21 cnex 9073 . . . . . 6  |-  CC  e.  _V
2221a1i 11 . . . . 5  |-  ( ph  ->  CC  e.  _V )
23 taylfval.f . . . . 5  |-  ( ph  ->  F : A --> CC )
24 taylfval.a . . . . 5  |-  ( ph  ->  A  C_  S )
25 elpm2r 7036 . . . . 5  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
2622, 20, 23, 24, 25syl22anc 1186 . . . 4  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
27 nn0ex 10229 . . . . . . 7  |-  NN0  e.  _V
28 snex 4407 . . . . . . 7  |-  {  +oo }  e.  _V
2927, 28unex 4709 . . . . . 6  |-  ( NN0 
u.  {  +oo } )  e.  _V
30 0xr 9133 . . . . . . . . . . 11  |-  0  e.  RR*
3130a1i 11 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { 
+oo } )  ->  0  e.  RR* )
32 nn0ssre 10227 . . . . . . . . . . . . 13  |-  NN0  C_  RR
33 ressxr 9131 . . . . . . . . . . . . 13  |-  RR  C_  RR*
3432, 33sstri 3359 . . . . . . . . . . . 12  |-  NN0  C_  RR*
35 pnfxr 10715 . . . . . . . . . . . . 13  |-  +oo  e.  RR*
36 snssi 3944 . . . . . . . . . . . . 13  |-  (  +oo  e.  RR*  ->  {  +oo }  C_ 
RR* )
3735, 36ax-mp 8 . . . . . . . . . . . 12  |-  {  +oo } 
C_  RR*
3834, 37unssi 3524 . . . . . . . . . . 11  |-  ( NN0 
u.  {  +oo } ) 
C_  RR*
3938sseli 3346 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { 
+oo } )  ->  n  e.  RR* )
40 elun 3490 . . . . . . . . . . 11  |-  ( n  e.  ( NN0  u.  { 
+oo } )  <->  ( n  e.  NN0  \/  n  e. 
{  +oo } ) )
41 nn0ge0 10249 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  0  <_  n )
42 pnfge 10729 . . . . . . . . . . . . . 14  |-  ( 0  e.  RR*  ->  0  <_  +oo )
4330, 42ax-mp 8 . . . . . . . . . . . . 13  |-  0  <_  +oo
44 elsni 3840 . . . . . . . . . . . . 13  |-  ( n  e.  {  +oo }  ->  n  =  +oo )
4543, 44syl5breqr 4250 . . . . . . . . . . . 12  |-  ( n  e.  {  +oo }  ->  0  <_  n )
4641, 45jaoi 370 . . . . . . . . . . 11  |-  ( ( n  e.  NN0  \/  n  e.  {  +oo }
)  ->  0  <_  n )
4740, 46sylbi 189 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { 
+oo } )  ->  0  <_  n )
48 lbicc2 11015 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  n  e.  RR*  /\  0  <_  n )  ->  0  e.  ( 0 [,] n
) )
4931, 39, 47, 48syl3anc 1185 . . . . . . . . 9  |-  ( n  e.  ( NN0  u.  { 
+oo } )  ->  0  e.  ( 0 [,] n
) )
50 0z 10295 . . . . . . . . 9  |-  0  e.  ZZ
51 inelcm 3684 . . . . . . . . 9  |-  ( ( 0  e.  ( 0 [,] n )  /\  0  e.  ZZ )  ->  ( ( 0 [,] n )  i^i  ZZ )  =/=  (/) )
5249, 50, 51sylancl 645 . . . . . . . 8  |-  ( n  e.  ( NN0  u.  { 
+oo } )  ->  (
( 0 [,] n
)  i^i  ZZ )  =/=  (/) )
53 fvex 5744 . . . . . . . . . 10  |-  ( ( S  D n F ) `  k )  e.  _V
5453dmex 5134 . . . . . . . . 9  |-  dom  (
( S  D n F ) `  k
)  e.  _V
5554rgenw 2775 . . . . . . . 8  |-  A. k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k )  e. 
_V
56 iinexg 4362 . . . . . . . 8  |-  ( ( ( ( 0 [,] n )  i^i  ZZ )  =/=  (/)  /\  A. k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k )  e. 
_V )  ->  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k )  e. 
_V )
5752, 55, 56sylancl 645 . . . . . . 7  |-  ( n  e.  ( NN0  u.  { 
+oo } )  ->  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k )  e. 
_V )
5857rgen 2773 . . . . . 6  |-  A. n  e.  ( NN0  u.  {  +oo } ) |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k )  e. 
_V
59 eqid 2438 . . . . . . 7  |-  ( n  e.  ( NN0  u.  { 
+oo } ) ,  a  e.  |^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( S  D n F ) `  k
)  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) ) )  =  ( n  e.  ( NN0  u.  { 
+oo } ) ,  a  e.  |^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( S  D n F ) `  k
)  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) ) )
6059mpt2exxg 6424 . . . . . 6  |-  ( ( ( NN0  u.  {  +oo } )  e.  _V  /\ 
A. n  e.  ( NN0  u.  {  +oo } ) |^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( S  D n F ) `  k
)  e.  _V )  ->  ( n  e.  ( NN0  u.  {  +oo } ) ,  a  e. 
|^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( S  D n F ) `  k
)  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) ) )  e.  _V )
6129, 58, 60mp2an 655 . . . . 5  |-  ( n  e.  ( NN0  u.  { 
+oo } ) ,  a  e.  |^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( S  D n F ) `  k
)  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) ) )  e.  _V
6261a1i 11 . . . 4  |-  ( ph  ->  ( n  e.  ( NN0  u.  {  +oo } ) ,  a  e. 
|^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( S  D n F ) `  k
)  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) ) )  e.  _V )
633, 17, 19, 20, 26, 62ovmpt2dx 6202 . . 3  |-  ( ph  ->  ( S Tayl  F )  =  ( n  e.  ( NN0  u.  {  +oo } ) ,  a  e.  |^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( S  D n F ) `  k
)  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) ) ) )
64 simprl 734 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  ->  n  =  N )
6564oveq2d 6099 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( 0 [,] n
)  =  ( 0 [,] N ) )
6665ineq1d 3543 . . . . . . 7  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( 0 [,] n )  i^i  ZZ )  =  ( (
0 [,] N )  i^i  ZZ ) )
67 simprr 735 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
a  =  B )
6867fveq2d 5734 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( ( S  D n F ) `
 k ) `  a )  =  ( ( ( S  D n F ) `  k
) `  B )
)
6968oveq1d 6098 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( ( ( S  D n F ) `  k ) `
 a )  / 
( ! `  k
) )  =  ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
) )
7067oveq2d 6099 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( x  -  a
)  =  ( x  -  B ) )
7170oveq1d 6098 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( x  -  a ) ^ k
)  =  ( ( x  -  B ) ^ k ) )
7269, 71oveq12d 6101 . . . . . . 7  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( ( ( ( S  D n F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) )  =  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) )
7366, 72mpteq12dv 4289 . . . . . 6  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) )  =  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) )
7473oveq2d 6099 . . . . 5  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
(fld tsums  ( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) )  =  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) )
7574xpeq2d 4904 . . . 4  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) )  =  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) ) ) )
7675iuneq2d 4120 . . 3  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  ->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) )  =  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
) ) ) ) ) )
77 simpr 449 . . . . . 6  |-  ( (
ph  /\  n  =  N )  ->  n  =  N )
7877oveq2d 6099 . . . . 5  |-  ( (
ph  /\  n  =  N )  ->  (
0 [,] n )  =  ( 0 [,] N ) )
7978ineq1d 3543 . . . 4  |-  ( (
ph  /\  n  =  N )  ->  (
( 0 [,] n
)  i^i  ZZ )  =  ( ( 0 [,] N )  i^i 
ZZ ) )
80 iineq1 4109 . . . 4  |-  ( ( ( 0 [,] n
)  i^i  ZZ )  =  ( ( 0 [,] N )  i^i 
ZZ )  ->  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k )  = 
|^|_ k  e.  ( ( 0 [,] N
)  i^i  ZZ ) dom  ( ( S  D n F ) `  k
) )
8179, 80syl 16 . . 3  |-  ( (
ph  /\  n  =  N )  ->  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k )  = 
|^|_ k  e.  ( ( 0 [,] N
)  i^i  ZZ ) dom  ( ( S  D n F ) `  k
) )
82 taylfval.n . . . . 5  |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo )
)
8335elexi 2967 . . . . . . 7  |-  +oo  e.  _V
8483elsnc2 3845 . . . . . 6  |-  ( N  e.  {  +oo }  <->  N  =  +oo )
8584orbi2i 507 . . . . 5  |-  ( ( N  e.  NN0  \/  N  e.  {  +oo }
)  <->  ( N  e. 
NN0  \/  N  =  +oo ) )
8682, 85sylibr 205 . . . 4  |-  ( ph  ->  ( N  e.  NN0  \/  N  e.  {  +oo } ) )
87 elun 3490 . . . 4  |-  ( N  e.  ( NN0  u.  { 
+oo } )  <->  ( N  e.  NN0  \/  N  e. 
{  +oo } ) )
8886, 87sylibr 205 . . 3  |-  ( ph  ->  N  e.  ( NN0 
u.  {  +oo } ) )
89 taylfval.b . . . . 5  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `
 k ) )
9089ralrimiva 2791 . . . 4  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k ) )
91 oveq2 6091 . . . . . . . . . 10  |-  ( n  =  N  ->  (
0 [,] n )  =  ( 0 [,] N ) )
9291ineq1d 3543 . . . . . . . . 9  |-  ( n  =  N  ->  (
( 0 [,] n
)  i^i  ZZ )  =  ( ( 0 [,] N )  i^i 
ZZ ) )
9392neeq1d 2616 . . . . . . . 8  |-  ( n  =  N  ->  (
( ( 0 [,] n )  i^i  ZZ )  =/=  (/)  <->  ( ( 0 [,] N )  i^i 
ZZ )  =/=  (/) ) )
9493, 52vtoclga 3019 . . . . . . 7  |-  ( N  e.  ( NN0  u.  { 
+oo } )  ->  (
( 0 [,] N
)  i^i  ZZ )  =/=  (/) )
9588, 94syl 16 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  =/=  (/) )
96 r19.2z 3719 . . . . . 6  |-  ( ( ( ( 0 [,] N )  i^i  ZZ )  =/=  (/)  /\  A. k  e.  ( ( 0 [,] N )  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k
) )  ->  E. k  e.  ( ( 0 [,] N )  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k
) )
9795, 90, 96syl2anc 644 . . . . 5  |-  ( ph  ->  E. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k ) )
98 elex 2966 . . . . . 6  |-  ( B  e.  dom  ( ( S  D n F ) `  k )  ->  B  e.  _V )
9998rexlimivw 2828 . . . . 5  |-  ( E. k  e.  ( ( 0 [,] N )  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k )  ->  B  e.  _V )
100 eliin 4100 . . . . 5  |-  ( B  e.  _V  ->  ( B  e.  |^|_ k  e.  ( ( 0 [,] N )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k )  <->  A. k  e.  ( ( 0 [,] N )  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k
) ) )
10197, 99, 1003syl 19 . . . 4  |-  ( ph  ->  ( B  e.  |^|_ k  e.  ( (
0 [,] N )  i^i  ZZ ) dom  ( ( S  D n F ) `  k
)  <->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k ) ) )
10290, 101mpbird 225 . . 3  |-  ( ph  ->  B  e.  |^|_ k  e.  ( ( 0 [,] N )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k ) )
103 snssi 3944 . . . . . . . 8  |-  ( x  e.  CC  ->  { x }  C_  CC )
104103adantl 454 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  { x }  C_  CC )
10520, 23, 24, 82, 89taylfvallem 20276 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) 
C_  CC )
106 xpss12 4983 . . . . . . 7  |-  ( ( { x }  C_  CC  /\  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) 
C_  CC )  -> 
( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) )  C_  ( CC  X.  CC ) )
107104, 105, 106syl2anc 644 . . . . . 6  |-  ( (
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 ) ) ) ) )  C_  ( CC  X.  CC ) )
108107ralrimiva 2791 . . . . 5  |-  ( ph  ->  A. 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 ) ) ) ) ) 
C_  ( CC  X.  CC ) )
109 iunss 4134 . . . . 5  |-  ( 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 ) ) ) ) ) 
C_  ( CC  X.  CC )  <->  A. 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 ) ) ) ) ) 
C_  ( CC  X.  CC ) )
110108, 109sylibr 205 . . . 4  |-  ( 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 ) ) ) ) ) 
C_  ( CC  X.  CC ) )
11121, 21xpex 4992 . . . . 5  |-  ( CC 
X.  CC )  e. 
_V
112111ssex 4349 . . . 4  |-  ( 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 ) ) ) ) ) 
C_  ( CC  X.  CC )  ->  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
) ) ) ) )  e.  _V )
113110, 112syl 16 . . 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 ) ) ) ) )  e.  _V )
11463, 76, 81, 88, 102, 113ovmpt2dx 6202 . 2  |-  ( ph  ->  ( N ( S Tayl 
F ) B )  =  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
) ) ) ) ) )
1151, 114syl5eq 2482 1  |-  ( 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
) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816   {cpr 3817   U_ciun 4095   |^|_ciin 4096   class class class wbr 4214    e. cmpt 4268    X. cxp 4878   dom cdm 4880   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085    ^pm cpm 7021   CCcc 8990   RRcr 8991   0cc0 8992    x. cmul 8997    +oocpnf 9119   RR*cxr 9121    <_ cle 9123    - cmin 9293    / cdiv 9679   NN0cn0 10223   ZZcz 10284   [,]cicc 10921   ^cexp 11384   !cfa 11568  ℂfldccnfld 16705   tsums ctsu 18157    D ncdvn 19753   Tayl ctayl 20271
This theorem is referenced by:  eltayl  20278  taylf  20279  taylpfval  20283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-icc 10925  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-fac 11569  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-mulr 13545  df-starv 13546  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-rest 13652  df-topn 13653  df-topgen 13669  df-0g 13729  df-gsum 13730  df-mnd 14692  df-grp 14814  df-minusg 14815  df-cntz 15118  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-cring 15666  df-ur 15667  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cnp 17294  df-haus 17381  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-tsms 18158  df-xms 18352  df-ms 18353  df-limc 19755  df-dv 19756  df-dvn 19757  df-tayl 20273
  Copyright terms: Public domain W3C validator