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Theorem taylfval 22481
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 22487 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 22477 . . . . 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 2461 . . . . 5  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
( NN0  u.  { +oo } )  =  ( NN0 
u.  { +oo } ) )
5 oveq12 6284 . . . . . . . . 9  |-  ( ( s  =  S  /\  f  =  F )  ->  ( s  Dn
f )  =  ( S  Dn F ) )
65ad2antlr 726 . . . . . . . 8  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
s  Dn f )  =  ( S  Dn F ) )
76fveq1d 5859 . . . . . . 7  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( s  Dn
f ) `  k
)  =  ( ( S  Dn F ) `  k ) )
87dmeqd 5196 . . . . . 6  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  dom  ( ( s  Dn f ) `  k )  =  dom  ( ( S  Dn F ) `  k ) )
98iineq2dv 4341 . . . . 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 5859 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( ( s  Dn f ) `  k ) `  a
)  =  ( ( ( S  Dn
F ) `  k
) `  a )
)
1110oveq1d 6290 . . . . . . . . . 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 6290 . . . . . . . . 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 4526 . . . . . . . 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 6291 . . . . . . 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 5016 . . . . . 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 4345 . . . . 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 6334 . . . 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 461 . . . . 5  |-  ( (
ph  /\  s  =  S )  ->  s  =  S )
1918oveq2d 6291 . . . 4  |-  ( (
ph  /\  s  =  S )  ->  ( CC  ^pm  s )  =  ( CC  ^pm  S
) )
20 taylfval.s . . . 4  |-  ( ph  ->  S  e.  { RR ,  CC } )
21 cnex 9562 . . . . . 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 7426 . . . . 5  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
2622, 20, 23, 24, 25syl22anc 1224 . . . 4  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
27 nn0ex 10790 . . . . . . 7  |-  NN0  e.  _V
28 snex 4681 . . . . . . 7  |-  { +oo }  e.  _V
2927, 28unex 6573 . . . . . 6  |-  ( NN0 
u.  { +oo } )  e.  _V
30 0xr 9629 . . . . . . . . . . 11  |-  0  e.  RR*
3130a1i 11 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { +oo } )  -> 
0  e.  RR* )
32 nn0ssre 10788 . . . . . . . . . . . . 13  |-  NN0  C_  RR
33 ressxr 9626 . . . . . . . . . . . . 13  |-  RR  C_  RR*
3432, 33sstri 3506 . . . . . . . . . . . 12  |-  NN0  C_  RR*
35 pnfxr 11310 . . . . . . . . . . . . 13  |- +oo  e.  RR*
36 snssi 4164 . . . . . . . . . . . . 13  |-  ( +oo  e.  RR*  ->  { +oo }  C_ 
RR* )
3735, 36ax-mp 5 . . . . . . . . . . . 12  |-  { +oo } 
C_  RR*
3834, 37unssi 3672 . . . . . . . . . . 11  |-  ( NN0 
u.  { +oo } ) 
C_  RR*
3938sseli 3493 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { +oo } )  ->  n  e.  RR* )
40 elun 3638 . . . . . . . . . . 11  |-  ( n  e.  ( NN0  u.  { +oo } )  <->  ( n  e.  NN0  \/  n  e. 
{ +oo } ) )
41 nn0ge0 10810 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  0  <_  n )
42 0lepnf 11329 . . . . . . . . . . . . 13  |-  0  <_ +oo
43 elsni 4045 . . . . . . . . . . . . 13  |-  ( n  e.  { +oo }  ->  n  = +oo )
4442, 43syl5breqr 4476 . . . . . . . . . . . 12  |-  ( n  e.  { +oo }  ->  0  <_  n )
4541, 44jaoi 379 . . . . . . . . . . 11  |-  ( ( n  e.  NN0  \/  n  e.  { +oo }
)  ->  0  <_  n )
4640, 45sylbi 195 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { +oo } )  -> 
0  <_  n )
47 lbicc2 11625 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  n  e.  RR*  /\  0  <_  n )  ->  0  e.  ( 0 [,] n
) )
4831, 39, 46, 47syl3anc 1223 . . . . . . . . 9  |-  ( n  e.  ( NN0  u.  { +oo } )  -> 
0  e.  ( 0 [,] n ) )
49 0z 10864 . . . . . . . . 9  |-  0  e.  ZZ
50 inelcm 3874 . . . . . . . . 9  |-  ( ( 0  e.  ( 0 [,] n )  /\  0  e.  ZZ )  ->  ( ( 0 [,] n )  i^i  ZZ )  =/=  (/) )
5148, 49, 50sylancl 662 . . . . . . . 8  |-  ( n  e.  ( NN0  u.  { +oo } )  -> 
( ( 0 [,] n )  i^i  ZZ )  =/=  (/) )
52 fvex 5867 . . . . . . . . . 10  |-  ( ( S  Dn F ) `  k )  e.  _V
5352dmex 6707 . . . . . . . . 9  |-  dom  (
( S  Dn
F ) `  k
)  e.  _V
5453rgenw 2818 . . . . . . . 8  |-  A. k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  e. 
_V
55 iinexg 4600 . . . . . . . 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 662 . . . . . . 7  |-  ( n  e.  ( NN0  u.  { +oo } )  ->  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `  k )  e.  _V )
5756rgen 2817 . . . . . 6  |-  A. n  e.  ( NN0  u.  { +oo } ) |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  e. 
_V
58 eqid 2460 . . . . . . 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 6847 . . . . . 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 672 . . . . 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 6404 . . 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 755 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  ->  n  =  N )
6463oveq2d 6291 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( 0 [,] n
)  =  ( 0 [,] N ) )
6564ineq1d 3692 . . . . . . 7  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( 0 [,] n )  i^i  ZZ )  =  ( (
0 [,] N )  i^i  ZZ ) )
66 simprr 756 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
a  =  B )
6766fveq2d 5861 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( ( S  Dn F ) `
 k ) `  a )  =  ( ( ( S  Dn F ) `  k ) `  B
) )
6867oveq1d 6290 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( ( ( S  Dn F ) `  k ) `
 a )  / 
( ! `  k
) )  =  ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
) )
6966oveq2d 6291 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( x  -  a
)  =  ( x  -  B ) )
7069oveq1d 6290 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( x  -  a ) ^ k
)  =  ( ( x  -  B ) ^ k ) )
7168, 70oveq12d 6293 . . . . . . 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 4518 . . . . . 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 6291 . . . . 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 5016 . . . 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 4345 . . 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 461 . . . . . 6  |-  ( (
ph  /\  n  =  N )  ->  n  =  N )
7776oveq2d 6291 . . . . 5  |-  ( (
ph  /\  n  =  N )  ->  (
0 [,] n )  =  ( 0 [,] N ) )
7877ineq1d 3692 . . . 4  |-  ( (
ph  /\  n  =  N )  ->  (
( 0 [,] n
)  i^i  ZZ )  =  ( ( 0 [,] N )  i^i 
ZZ ) )
79 iineq1 4333 . . . 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 16 . . 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 11311 . . . . . . 7  |- +oo  e.  _V
8382elsnc2 4051 . . . . . 6  |-  ( N  e.  { +oo }  <->  N  = +oo )
8483orbi2i 519 . . . . 5  |-  ( ( N  e.  NN0  \/  N  e.  { +oo }
)  <->  ( N  e. 
NN0  \/  N  = +oo ) )
8581, 84sylibr 212 . . . 4  |-  ( ph  ->  ( N  e.  NN0  \/  N  e.  { +oo } ) )
86 elun 3638 . . . 4  |-  ( N  e.  ( NN0  u.  { +oo } )  <->  ( N  e.  NN0  \/  N  e. 
{ +oo } ) )
8785, 86sylibr 212 . . 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 2871 . . . 4  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) )
90 oveq2 6283 . . . . . . . . . 10  |-  ( n  =  N  ->  (
0 [,] n )  =  ( 0 [,] N ) )
9190ineq1d 3692 . . . . . . . . 9  |-  ( n  =  N  ->  (
( 0 [,] n
)  i^i  ZZ )  =  ( ( 0 [,] N )  i^i 
ZZ ) )
9291neeq1d 2737 . . . . . . . 8  |-  ( n  =  N  ->  (
( ( 0 [,] n )  i^i  ZZ )  =/=  (/)  <->  ( ( 0 [,] N )  i^i 
ZZ )  =/=  (/) ) )
9392, 51vtoclga 3170 . . . . . . 7  |-  ( N  e.  ( NN0  u.  { +oo } )  -> 
( ( 0 [,] N )  i^i  ZZ )  =/=  (/) )
9487, 93syl 16 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  =/=  (/) )
95 r19.2z 3910 . . . . . 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 661 . . . . 5  |-  ( ph  ->  E. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) )
97 elex 3115 . . . . . 6  |-  ( B  e.  dom  ( ( S  Dn F ) `  k )  ->  B  e.  _V )
9897rexlimivw 2945 . . . . 5  |-  ( E. k  e.  ( ( 0 [,] N )  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k )  ->  B  e.  _V )
99 eliin 4324 . . . . 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 20 . . . 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 232 . . 3  |-  ( ph  ->  B  e.  |^|_ k  e.  ( ( 0 [,] N )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k ) )
102 snssi 4164 . . . . . . . 8  |-  ( x  e.  CC  ->  { x }  C_  CC )
103102adantl 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  { x }  C_  CC )
10420, 23, 24, 81, 88taylfvallem 22480 . . . . . . 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 5099 . . . . . . 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 661 . . . . . 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 2871 . . . . 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 4359 . . . . 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 212 . . . 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 6704 . . . . 5  |-  ( CC 
X.  CC )  e. 
_V
111110ssex 4584 . . . 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 16 . . 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 6404 . 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 2513 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 184    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808   _Vcvv 3106    u. cun 3467    i^i cin 3468    C_ wss 3469   (/)c0 3778   {csn 4020   {cpr 4022   U_ciun 4318   |^|_ciin 4319   class class class wbr 4440    |-> cmpt 4498    X. cxp 4990   dom cdm 4992   -->wf 5575   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277    ^pm cpm 7411   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486   +oocpnf 9614   RR*cxr 9616    <_ cle 9618    - cmin 9794    / cdiv 10195   NN0cn0 10784   ZZcz 10853   [,]cicc 11521   ^cexp 12122   !cfa 12308  ℂfldccnfld 18184   tsums ctsu 20352    Dncdvn 21996   Tayl ctayl 22475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-fac 12309  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-mulr 14558  df-starv 14559  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-mnd 15721  df-grp 15851  df-minusg 15852  df-cntz 16143  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cnp 19488  df-haus 19575  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-tsms 20353  df-xms 20551  df-ms 20552  df-limc 21998  df-dv 21999  df-dvn 22000  df-tayl 22477
This theorem is referenced by:  eltayl  22482  taylf  22483  taylpfval  22487
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