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Theorem taylfval 23312
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 23318 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  Dn 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  Dn
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 23308 . . . . 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  Dn f ) `
 k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( s  Dn 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  Dn f ) `  k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( s  Dn
f ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) ) ) ) )
4 eqidd 2423 . . . . 5  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
( NN0  u.  { +oo } )  =  ( NN0 
u.  { +oo } ) )
5 oveq12 6314 . . . . . . . . 9  |-  ( ( s  =  S  /\  f  =  F )  ->  ( s  Dn
f )  =  ( S  Dn F ) )
65ad2antlr 731 . . . . . . . 8  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
s  Dn f )  =  ( S  Dn F ) )
76fveq1d 5883 . . . . . . 7  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( s  Dn
f ) `  k
)  =  ( ( S  Dn F ) `  k ) )
87dmeqd 5056 . . . . . 6  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  dom  ( ( s  Dn f ) `  k )  =  dom  ( ( S  Dn F ) `  k ) )
98iineq2dv 4322 . . . . 5  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  ->  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( s  Dn f ) `  k )  =  |^|_ k  e.  ( (
0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `  k ) )
107fveq1d 5883 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( ( s  Dn f ) `  k ) `  a
)  =  ( ( ( S  Dn
F ) `  k
) `  a )
)
1110oveq1d 6320 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( ( ( s  Dn f ) `
 k ) `  a )  /  ( ! `  k )
)  =  ( ( ( ( S  Dn F ) `  k ) `  a
)  /  ( ! `
 k ) ) )
1211oveq1d 6320 . . . . . . . . 9  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( ( ( ( s  Dn f ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) )  =  ( ( ( ( ( S  Dn F ) `
 k ) `  a )  /  ( ! `  k )
)  x.  ( ( x  -  a ) ^ k ) ) )
1312mpteq2dva 4510 . . . . . . . 8  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( s  Dn
f ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) )  =  ( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) )
1413oveq2d 6321 . . . . . . 7  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
(fld tsums  ( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( s  Dn
f ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) )  =  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) )
1514xpeq2d 4877 . . . . . 6  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( s  Dn
f ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) )  =  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  a )  /  ( ! `  k )
)  x.  ( ( x  -  a ) ^ k ) ) ) ) ) )
1615iuneq2d 4326 . . . . 5  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  ->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( s  Dn f ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) )  =  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) ) )
174, 9, 16mpt2eq123dv 6367 . . . 4  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
( n  e.  ( NN0  u.  { +oo } ) ,  a  e. 
|^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( s  Dn f ) `  k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( s  Dn
f ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) ) )  =  ( n  e.  ( NN0 
u.  { +oo } ) ,  a  e.  |^|_ k  e.  ( (
0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `  k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) ) ) )
18 simpr 462 . . . . 5  |-  ( (
ph  /\  s  =  S )  ->  s  =  S )
1918oveq2d 6321 . . . 4  |-  ( (
ph  /\  s  =  S )  ->  ( CC  ^pm  s )  =  ( CC  ^pm  S
) )
20 taylfval.s . . . 4  |-  ( ph  ->  S  e.  { RR ,  CC } )
21 cnex 9627 . . . . . 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 7500 . . . . 5  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
2622, 20, 23, 24, 25syl22anc 1265 . . . 4  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
27 nn0ex 10882 . . . . . . 7  |-  NN0  e.  _V
28 snex 4662 . . . . . . 7  |-  { +oo }  e.  _V
2927, 28unex 6603 . . . . . 6  |-  ( NN0 
u.  { +oo } )  e.  _V
30 0xr 9694 . . . . . . . . . . 11  |-  0  e.  RR*
3130a1i 11 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { +oo } )  -> 
0  e.  RR* )
32 nn0ssre 10880 . . . . . . . . . . . . 13  |-  NN0  C_  RR
33 ressxr 9691 . . . . . . . . . . . . 13  |-  RR  C_  RR*
3432, 33sstri 3473 . . . . . . . . . . . 12  |-  NN0  C_  RR*
35 pnfxr 11419 . . . . . . . . . . . . 13  |- +oo  e.  RR*
36 snssi 4144 . . . . . . . . . . . . 13  |-  ( +oo  e.  RR*  ->  { +oo }  C_ 
RR* )
3735, 36ax-mp 5 . . . . . . . . . . . 12  |-  { +oo } 
C_  RR*
3834, 37unssi 3641 . . . . . . . . . . 11  |-  ( NN0 
u.  { +oo } ) 
C_  RR*
3938sseli 3460 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { +oo } )  ->  n  e.  RR* )
40 elun 3606 . . . . . . . . . . 11  |-  ( n  e.  ( NN0  u.  { +oo } )  <->  ( n  e.  NN0  \/  n  e. 
{ +oo } ) )
41 nn0ge0 10902 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  0  <_  n )
42 0lepnf 11440 . . . . . . . . . . . . 13  |-  0  <_ +oo
43 elsni 4023 . . . . . . . . . . . . 13  |-  ( n  e.  { +oo }  ->  n  = +oo )
4442, 43syl5breqr 4460 . . . . . . . . . . . 12  |-  ( n  e.  { +oo }  ->  0  <_  n )
4541, 44jaoi 380 . . . . . . . . . . 11  |-  ( ( n  e.  NN0  \/  n  e.  { +oo }
)  ->  0  <_  n )
4640, 45sylbi 198 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { +oo } )  -> 
0  <_  n )
47 lbicc2 11755 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  n  e.  RR*  /\  0  <_  n )  ->  0  e.  ( 0 [,] n
) )
4831, 39, 46, 47syl3anc 1264 . . . . . . . . 9  |-  ( n  e.  ( NN0  u.  { +oo } )  -> 
0  e.  ( 0 [,] n ) )
49 0z 10955 . . . . . . . . 9  |-  0  e.  ZZ
50 inelcm 3849 . . . . . . . . 9  |-  ( ( 0  e.  ( 0 [,] n )  /\  0  e.  ZZ )  ->  ( ( 0 [,] n )  i^i  ZZ )  =/=  (/) )
5148, 49, 50sylancl 666 . . . . . . . 8  |-  ( n  e.  ( NN0  u.  { +oo } )  -> 
( ( 0 [,] n )  i^i  ZZ )  =/=  (/) )
52 fvex 5891 . . . . . . . . . 10  |-  ( ( S  Dn F ) `  k )  e.  _V
5352dmex 6740 . . . . . . . . 9  |-  dom  (
( S  Dn
F ) `  k
)  e.  _V
5453rgenw 2783 . . . . . . . 8  |-  A. k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  e. 
_V
55 iinexg 4584 . . . . . . . 8  |-  ( ( ( ( 0 [,] n )  i^i  ZZ )  =/=  (/)  /\  A. k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  e. 
_V )  ->  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  e. 
_V )
5651, 54, 55sylancl 666 . . . . . . 7  |-  ( n  e.  ( NN0  u.  { +oo } )  ->  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `  k )  e.  _V )
5756rgen 2781 . . . . . 6  |-  A. n  e.  ( NN0  u.  { +oo } ) |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  e. 
_V
58 eqid 2422 . . . . . . 7  |-  ( n  e.  ( NN0  u.  { +oo } ) ,  a  e.  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) ) )  =  ( n  e.  ( NN0  u.  { +oo } ) ,  a  e.  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) ) )
5958mpt2exxg 6881 . . . . . 6  |-  ( ( ( NN0  u.  { +oo } )  e.  _V  /\ 
A. n  e.  ( NN0  u.  { +oo } ) |^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( S  Dn F ) `  k )  e.  _V )  ->  ( n  e.  ( NN0  u.  { +oo } ) ,  a  e.  |^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( S  Dn F ) `  k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) ) )  e.  _V )
6029, 57, 59mp2an 676 . . . . 5  |-  ( n  e.  ( NN0  u.  { +oo } ) ,  a  e.  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) ) )  e.  _V
6160a1i 11 . . . 4  |-  ( ph  ->  ( n  e.  ( NN0  u.  { +oo } ) ,  a  e. 
|^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( S  Dn F ) `  k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) ) )  e.  _V )
623, 17, 19, 20, 26, 61ovmpt2dx 6437 . . 3  |-  ( ph  ->  ( S Tayl  F )  =  ( n  e.  ( NN0  u.  { +oo } ) ,  a  e.  |^|_ k  e.  ( ( 0 [,] n
)  i^i  ZZ ) dom  ( ( S  Dn F ) `  k )  |->  U_ x  e.  CC  ( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) ) ) )
63 simprl 762 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  ->  n  =  N )
6463oveq2d 6321 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( 0 [,] n
)  =  ( 0 [,] N ) )
6564ineq1d 3663 . . . . . . 7  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( 0 [,] n )  i^i  ZZ )  =  ( (
0 [,] N )  i^i  ZZ ) )
66 simprr 764 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
a  =  B )
6766fveq2d 5885 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( ( S  Dn F ) `
 k ) `  a )  =  ( ( ( S  Dn F ) `  k ) `  B
) )
6867oveq1d 6320 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( ( ( S  Dn F ) `  k ) `
 a )  / 
( ! `  k
) )  =  ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
) )
6966oveq2d 6321 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( x  -  a
)  =  ( x  -  B ) )
7069oveq1d 6320 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( x  -  a ) ^ k
)  =  ( ( x  -  B ) ^ k ) )
7168, 70oveq12d 6323 . . . . . . 7  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( ( ( ( S  Dn
F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) )  =  ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) )
7265, 71mpteq12dv 4502 . . . . . 6  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) )  =  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) )
7372oveq2d 6321 . . . . 5  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
(fld tsums  ( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) )  =  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) )
7473xpeq2d 4877 . . . 4  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] n
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  a )  /  ( ! `  k ) )  x.  ( ( x  -  a ) ^ k
) ) ) ) )  =  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) ) ) ) )
7574iuneq2d 4326 . . 3  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  ->  U_ x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] n )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `
 a )  / 
( ! `  k
) )  x.  (
( x  -  a
) ^ k ) ) ) ) )  =  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
) ) ) ) ) )
76 simpr 462 . . . . . 6  |-  ( (
ph  /\  n  =  N )  ->  n  =  N )
7776oveq2d 6321 . . . . 5  |-  ( (
ph  /\  n  =  N )  ->  (
0 [,] n )  =  ( 0 [,] N ) )
7877ineq1d 3663 . . . 4  |-  ( (
ph  /\  n  =  N )  ->  (
( 0 [,] n
)  i^i  ZZ )  =  ( ( 0 [,] N )  i^i 
ZZ ) )
79 iineq1 4314 . . . 4  |-  ( ( ( 0 [,] n
)  i^i  ZZ )  =  ( ( 0 [,] N )  i^i 
ZZ )  ->  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  = 
|^|_ k  e.  ( ( 0 [,] N
)  i^i  ZZ ) dom  ( ( S  Dn F ) `  k ) )
8078, 79syl 17 . . 3  |-  ( (
ph  /\  n  =  N )  ->  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  = 
|^|_ k  e.  ( ( 0 [,] N
)  i^i  ZZ ) dom  ( ( S  Dn F ) `  k ) )
81 taylfval.n . . . . 5  |-  ( ph  ->  ( N  e.  NN0  \/  N  = +oo )
)
82 pnfex 11420 . . . . . . 7  |- +oo  e.  _V
8382elsnc2 4029 . . . . . 6  |-  ( N  e.  { +oo }  <->  N  = +oo )
8483orbi2i 521 . . . . 5  |-  ( ( N  e.  NN0  \/  N  e.  { +oo }
)  <->  ( N  e. 
NN0  \/  N  = +oo ) )
8581, 84sylibr 215 . . . 4  |-  ( ph  ->  ( N  e.  NN0  \/  N  e.  { +oo } ) )
86 elun 3606 . . . 4  |-  ( N  e.  ( NN0  u.  { +oo } )  <->  ( N  e.  NN0  \/  N  e. 
{ +oo } ) )
8785, 86sylibr 215 . . 3  |-  ( ph  ->  N  e.  ( NN0 
u.  { +oo } ) )
88 taylfval.b . . . . 5  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  Dn F ) `
 k ) )
8988ralrimiva 2836 . . . 4  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) )
90 oveq2 6313 . . . . . . . . . 10  |-  ( n  =  N  ->  (
0 [,] n )  =  ( 0 [,] N ) )
9190ineq1d 3663 . . . . . . . . 9  |-  ( n  =  N  ->  (
( 0 [,] n
)  i^i  ZZ )  =  ( ( 0 [,] N )  i^i 
ZZ ) )
9291neeq1d 2697 . . . . . . . 8  |-  ( n  =  N  ->  (
( ( 0 [,] n )  i^i  ZZ )  =/=  (/)  <->  ( ( 0 [,] N )  i^i 
ZZ )  =/=  (/) ) )
9392, 51vtoclga 3145 . . . . . . 7  |-  ( N  e.  ( NN0  u.  { +oo } )  -> 
( ( 0 [,] N )  i^i  ZZ )  =/=  (/) )
9487, 93syl 17 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  =/=  (/) )
95 r19.2z 3888 . . . . . 6  |-  ( ( ( ( 0 [,] N )  i^i  ZZ )  =/=  (/)  /\  A. k  e.  ( ( 0 [,] N )  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) )  ->  E. k  e.  (
( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) )
9694, 89, 95syl2anc 665 . . . . 5  |-  ( ph  ->  E. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) )
97 elex 3089 . . . . . 6  |-  ( B  e.  dom  ( ( S  Dn F ) `  k )  ->  B  e.  _V )
9897rexlimivw 2911 . . . . 5  |-  ( E. k  e.  ( ( 0 [,] N )  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k )  ->  B  e.  _V )
99 eliin 4305 . . . . 5  |-  ( B  e.  _V  ->  ( B  e.  |^|_ k  e.  ( ( 0 [,] N )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  <->  A. k  e.  ( ( 0 [,] N )  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) ) )
10096, 98, 993syl 18 . . . 4  |-  ( ph  ->  ( B  e.  |^|_ k  e.  ( (
0 [,] N )  i^i  ZZ ) dom  ( ( S  Dn F ) `  k )  <->  A. k  e.  ( ( 0 [,] N )  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) ) )
10189, 100mpbird 235 . . 3  |-  ( ph  ->  B  e.  |^|_ k  e.  ( ( 0 [,] N )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k ) )
102 snssi 4144 . . . . . . . 8  |-  ( x  e.  CC  ->  { x }  C_  CC )
103102adantl 467 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  { x }  C_  CC )
10420, 23, 24, 81, 88taylfvallem 23311 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) ) )  C_  CC )
105 xpss12 4959 . . . . . . 7  |-  ( ( { x }  C_  CC  /\  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) 
C_  CC )  -> 
( { x }  X.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) ) )  C_  ( CC  X.  CC ) )
106103, 104, 105syl2anc 665 . . . . . 6  |-  ( (
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 ) ) ) ) )  C_  ( CC  X.  CC ) )
107106ralrimiva 2836 . . . . 5  |-  ( ph  ->  A. x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) ) ) 
C_  ( CC  X.  CC ) )
108 iunss 4340 . . . . 5  |-  ( 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 ) ) ) ) ) 
C_  ( CC  X.  CC )  <->  A. x  e.  CC  ( { x }  X.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) ) ) 
C_  ( CC  X.  CC ) )
109107, 108sylibr 215 . . . 4  |-  ( 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 ) ) ) ) ) 
C_  ( CC  X.  CC ) )
11021, 21xpex 6609 . . . . 5  |-  ( CC 
X.  CC )  e. 
_V
111110ssex 4568 . . . 4  |-  ( 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 ) ) ) ) ) 
C_  ( CC  X.  CC )  ->  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
) ) ) ) )  e.  _V )
112109, 111syl 17 . . 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 ) ) ) ) )  e.  _V )
11362, 75, 80, 87, 101, 112ovmpt2dx 6437 . 2  |-  ( ph  ->  ( N ( S Tayl 
F ) B )  =  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
) ) ) ) ) )
1141, 113syl5eq 2475 1  |-  ( 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
) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   _Vcvv 3080    u. cun 3434    i^i cin 3435    C_ wss 3436   (/)c0 3761   {csn 3998   {cpr 4000   U_ciun 4299   |^|_ciin 4300   class class class wbr 4423    |-> cmpt 4482    X. cxp 4851   dom cdm 4853   -->wf 5597   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307    ^pm cpm 7484   CCcc 9544   RRcr 9545   0cc0 9546    x. cmul 9551   +oocpnf 9679   RR*cxr 9681    <_ cle 9683    - cmin 9867    / cdiv 10276   NN0cn0 10876   ZZcz 10944   [,]cicc 11645   ^cexp 12278   !cfa 12465  ℂfldccnfld 18969   tsums ctsu 21138    Dncdvn 22817   Tayl ctayl 23306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-inf2 8155  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624  ax-addf 9625  ax-mulf 9626
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-fsupp 7893  df-fi 7934  df-sup 7965  df-inf 7966  df-oi 8034  df-card 8381  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-icc 11649  df-fz 11792  df-fzo 11923  df-seq 12220  df-exp 12279  df-fac 12466  df-hash 12522  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-plusg 15202  df-mulr 15203  df-starv 15204  df-tset 15208  df-ple 15209  df-ds 15211  df-unif 15212  df-rest 15320  df-topn 15321  df-0g 15339  df-gsum 15340  df-topgen 15341  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-grp 16672  df-minusg 16673  df-cntz 16970  df-cmn 17431  df-abl 17432  df-mgp 17723  df-ur 17735  df-ring 17781  df-cring 17782  df-psmet 18961  df-xmet 18962  df-met 18963  df-bl 18964  df-mopn 18965  df-fbas 18966  df-fg 18967  df-cnfld 18970  df-top 19919  df-bases 19920  df-topon 19921  df-topsp 19922  df-cld 20032  df-ntr 20033  df-cls 20034  df-nei 20112  df-lp 20150  df-perf 20151  df-cnp 20242  df-haus 20329  df-fil 20859  df-fm 20951  df-flim 20952  df-flf 20953  df-tsms 21139  df-xms 21333  df-ms 21334  df-limc 22819  df-dv 22820  df-dvn 22821  df-tayl 23308
This theorem is referenced by:  eltayl  23313  taylf  23314  taylpfval  23318
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