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Theorem taylfval 21793
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 21799 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 21789 . . . . 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 2438 . . . . 5  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
( NN0  u.  { +oo } )  =  ( NN0 
u.  { +oo } ) )
5 oveq12 6095 . . . . . . . . 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 5686 . . . . . . 7  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( s  Dn
f ) `  k
)  =  ( ( S  Dn F ) `  k ) )
87dmeqd 5034 . . . . . 6  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  dom  ( ( s  Dn f ) `  k )  =  dom  ( ( S  Dn F ) `  k ) )
98iineq2dv 4186 . . . . 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 5686 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( ( s  Dn f ) `  k ) `  a
)  =  ( ( ( S  Dn
F ) `  k
) `  a )
)
1110oveq1d 6101 . . . . . . . . . 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 6101 . . . . . . . . 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 4371 . . . . . . . 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 6102 . . . . . . 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 4856 . . . . . 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 4190 . . . . 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 6143 . . . 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 6102 . . . 4  |-  ( (
ph  /\  s  =  S )  ->  ( CC  ^pm  s )  =  ( CC  ^pm  S
) )
20 taylfval.s . . . 4  |-  ( ph  ->  S  e.  { RR ,  CC } )
21 cnex 9355 . . . . . 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 7222 . . . . 5  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
2622, 20, 23, 24, 25syl22anc 1219 . . . 4  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
27 nn0ex 10577 . . . . . . 7  |-  NN0  e.  _V
28 snex 4526 . . . . . . 7  |-  { +oo }  e.  _V
2927, 28unex 6373 . . . . . 6  |-  ( NN0 
u.  { +oo } )  e.  _V
30 0xr 9422 . . . . . . . . . . 11  |-  0  e.  RR*
3130a1i 11 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { +oo } )  -> 
0  e.  RR* )
32 nn0ssre 10575 . . . . . . . . . . . . 13  |-  NN0  C_  RR
33 ressxr 9419 . . . . . . . . . . . . 13  |-  RR  C_  RR*
3432, 33sstri 3358 . . . . . . . . . . . 12  |-  NN0  C_  RR*
35 pnfxr 11084 . . . . . . . . . . . . 13  |- +oo  e.  RR*
36 snssi 4010 . . . . . . . . . . . . 13  |-  ( +oo  e.  RR*  ->  { +oo }  C_ 
RR* )
3735, 36ax-mp 5 . . . . . . . . . . . 12  |-  { +oo } 
C_  RR*
3834, 37unssi 3524 . . . . . . . . . . 11  |-  ( NN0 
u.  { +oo } ) 
C_  RR*
3938sseli 3345 . . . . . . . . . 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 10597 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  0  <_  n )
42 0lepnf 11103 . . . . . . . . . . . . 13  |-  0  <_ +oo
43 elsni 3895 . . . . . . . . . . . . 13  |-  ( n  e.  { +oo }  ->  n  = +oo )
4442, 43syl5breqr 4321 . . . . . . . . . . . 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 11393 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  n  e.  RR*  /\  0  <_  n )  ->  0  e.  ( 0 [,] n
) )
4831, 39, 46, 47syl3anc 1218 . . . . . . . . 9  |-  ( n  e.  ( NN0  u.  { +oo } )  -> 
0  e.  ( 0 [,] n ) )
49 0z 10649 . . . . . . . . 9  |-  0  e.  ZZ
50 inelcm 3726 . . . . . . . . 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 5694 . . . . . . . . . 10  |-  ( ( S  Dn F ) `  k )  e.  _V
5352dmex 6506 . . . . . . . . 9  |-  dom  (
( S  Dn
F ) `  k
)  e.  _V
5453rgenw 2777 . . . . . . . 8  |-  A. k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  e. 
_V
55 iinexg 4445 . . . . . . . 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 2775 . . . . . 6  |-  A. n  e.  ( NN0  u.  { +oo } ) |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  Dn F ) `
 k )  e. 
_V
58 eqid 2437 . . . . . . 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 6642 . . . . . 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 6212 . . 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 6102 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( 0 [,] n
)  =  ( 0 [,] N ) )
6564ineq1d 3544 . . . . . . 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 5688 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( ( S  Dn F ) `
 k ) `  a )  =  ( ( ( S  Dn F ) `  k ) `  B
) )
6867oveq1d 6101 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( ( ( S  Dn F ) `  k ) `
 a )  / 
( ! `  k
) )  =  ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
) )
6966oveq2d 6102 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( x  -  a
)  =  ( x  -  B ) )
7069oveq1d 6101 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( x  -  a ) ^ k
)  =  ( ( x  -  B ) ^ k ) )
7168, 70oveq12d 6104 . . . . . . 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 4363 . . . . . 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 6102 . . . . 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 4856 . . . 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 4190 . . 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 6102 . . . . 5  |-  ( (
ph  /\  n  =  N )  ->  (
0 [,] n )  =  ( 0 [,] N ) )
7877ineq1d 3544 . . . 4  |-  ( (
ph  /\  n  =  N )  ->  (
( 0 [,] n
)  i^i  ZZ )  =  ( ( 0 [,] N )  i^i 
ZZ ) )
79 iineq1 4178 . . . 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 11085 . . . . . . 7  |- +oo  e.  _V
8382elsnc2 3901 . . . . . 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 3490 . . . 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 2793 . . . 4  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) )
90 oveq2 6094 . . . . . . . . . 10  |-  ( n  =  N  ->  (
0 [,] n )  =  ( 0 [,] N ) )
9190ineq1d 3544 . . . . . . . . 9  |-  ( n  =  N  ->  (
( 0 [,] n
)  i^i  ZZ )  =  ( ( 0 [,] N )  i^i 
ZZ ) )
9291neeq1d 2615 . . . . . . . 8  |-  ( n  =  N  ->  (
( ( 0 [,] n )  i^i  ZZ )  =/=  (/)  <->  ( ( 0 [,] N )  i^i 
ZZ )  =/=  (/) ) )
9392, 51vtoclga 3029 . . . . . . 7  |-  ( N  e.  ( NN0  u.  { +oo } )  -> 
( ( 0 [,] N )  i^i  ZZ )  =/=  (/) )
9487, 93syl 16 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  =/=  (/) )
95 r19.2z 3762 . . . . . 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 2975 . . . . . 6  |-  ( B  e.  dom  ( ( S  Dn F ) `  k )  ->  B  e.  _V )
9897rexlimivw 2831 . . . . 5  |-  ( E. k  e.  ( ( 0 [,] N )  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k )  ->  B  e.  _V )
99 eliin 4169 . . . . 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 4010 . . . . . . . 8  |-  ( x  e.  CC  ->  { x }  C_  CC )
103102adantl 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  { x }  C_  CC )
10420, 23, 24, 81, 88taylfvallem 21792 . . . . . . 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 4937 . . . . . . 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 2793 . . . . 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 4204 . . . . 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 6503 . . . . 5  |-  ( CC 
X.  CC )  e. 
_V
111110ssex 4429 . . . 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 6212 . 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 2481 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 1369    e. wcel 1756    =/= wne 2600   A.wral 2709   E.wrex 2710   _Vcvv 2966    u. cun 3319    i^i cin 3320    C_ wss 3321   (/)c0 3630   {csn 3870   {cpr 3872   U_ciun 4164   |^|_ciin 4165   class class class wbr 4285    e. cmpt 4343    X. cxp 4830   dom cdm 4832   -->wf 5407   ` cfv 5411  (class class class)co 6086    e. cmpt2 6088    ^pm cpm 7207   CCcc 9272   RRcr 9273   0cc0 9274    x. cmul 9279   +oocpnf 9407   RR*cxr 9409    <_ cle 9411    - cmin 9587    / cdiv 9985   NN0cn0 10571   ZZcz 10638   [,]cicc 11295   ^cexp 11857   !cfa 12043  ℂfldccnfld 17787   tsums ctsu 19665    Dncdvn 21308   Tayl ctayl 21787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-iin 4167  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-se 4672  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-isom 5420  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-icc 11299  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-fac 12044  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-mulr 14244  df-starv 14245  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-mnd 15407  df-grp 15534  df-minusg 15535  df-cntz 15824  df-cmn 16268  df-abl 16269  df-mgp 16578  df-ur 16590  df-rng 16633  df-cring 16634  df-psmet 17778  df-xmet 17779  df-met 17780  df-bl 17781  df-mopn 17782  df-fbas 17783  df-fg 17784  df-cnfld 17788  df-top 18472  df-bases 18474  df-topon 18475  df-topsp 18476  df-cld 18592  df-ntr 18593  df-cls 18594  df-nei 18671  df-lp 18709  df-perf 18710  df-cnp 18801  df-haus 18888  df-fil 19388  df-fm 19480  df-flim 19481  df-flf 19482  df-tsms 19666  df-xms 19864  df-ms 19865  df-limc 21310  df-dv 21311  df-dvn 21312  df-tayl 21789
This theorem is referenced by:  eltayl  21794  taylf  21795  taylpfval  21799
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