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Theorem taylfval 19754
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 19760 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 19750 . . . . 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 10 . . . 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 2297 . . . . 5  |-  ( (
ph  /\  ( s  =  S  /\  f  =  F ) )  -> 
( NN0  u.  {  +oo } )  =  ( NN0 
u.  {  +oo } ) )
5 oveq12 5883 . . . . . . . . 9  |-  ( ( s  =  S  /\  f  =  F )  ->  ( s  D n
f )  =  ( S  D n F ) )
65ad2antlr 707 . . . . . . . 8  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
s  D n f )  =  ( S  D n F ) )
76fveq1d 5543 . . . . . . 7  |-  ( ( ( ph  /\  (
s  =  S  /\  f  =  F )
)  /\  k  e.  ( ( 0 [,] n )  i^i  ZZ ) )  ->  (
( s  D n
f ) `  k
)  =  ( ( S  D n F ) `  k ) )
87dmeqd 4897 . . . . . 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 3943 . . . . 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 5543 . . . . . . . . . . 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 5889 . . . . . . . . . 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 5889 . . . . . . . . 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 4122 . . . . . . . 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 5890 . . . . . . 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 4729 . . . . . 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 3946 . . . . 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 5926 . . . 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 447 . . . . 5  |-  ( (
ph  /\  s  =  S )  ->  s  =  S )
1918oveq2d 5890 . . . 4  |-  ( (
ph  /\  s  =  S )  ->  ( CC  ^pm  s )  =  ( CC  ^pm  S
) )
20 taylfval.s . . . 4  |-  ( ph  ->  S  e.  { RR ,  CC } )
21 cnex 8834 . . . . . 6  |-  CC  e.  _V
2221a1i 10 . . . . 5  |-  ( ph  ->  CC  e.  _V )
23 taylfval.f . . . . 5  |-  ( ph  ->  F : A --> CC )
24 taylfval.a . . . . 5  |-  ( ph  ->  A  C_  S )
25 elpm2r 6804 . . . . 5  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
2622, 20, 23, 24, 25syl22anc 1183 . . . 4  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
27 nn0ex 9987 . . . . . . 7  |-  NN0  e.  _V
28 snex 4232 . . . . . . 7  |-  {  +oo }  e.  _V
2927, 28unex 4534 . . . . . 6  |-  ( NN0 
u.  {  +oo } )  e.  _V
30 0xr 8894 . . . . . . . . . . 11  |-  0  e.  RR*
3130a1i 10 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { 
+oo } )  ->  0  e.  RR* )
32 nn0ssre 9985 . . . . . . . . . . . . 13  |-  NN0  C_  RR
33 ressxr 8892 . . . . . . . . . . . . 13  |-  RR  C_  RR*
3432, 33sstri 3201 . . . . . . . . . . . 12  |-  NN0  C_  RR*
35 pnfxr 10471 . . . . . . . . . . . . 13  |-  +oo  e.  RR*
36 snssi 3775 . . . . . . . . . . . . 13  |-  (  +oo  e.  RR*  ->  {  +oo }  C_ 
RR* )
3735, 36ax-mp 8 . . . . . . . . . . . 12  |-  {  +oo } 
C_  RR*
3834, 37unssi 3363 . . . . . . . . . . 11  |-  ( NN0 
u.  {  +oo } ) 
C_  RR*
3938sseli 3189 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { 
+oo } )  ->  n  e.  RR* )
40 elun 3329 . . . . . . . . . . 11  |-  ( n  e.  ( NN0  u.  { 
+oo } )  <->  ( n  e.  NN0  \/  n  e. 
{  +oo } ) )
41 nn0ge0 10007 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  0  <_  n )
42 pnfge 10485 . . . . . . . . . . . . . 14  |-  ( 0  e.  RR*  ->  0  <_  +oo )
4330, 42ax-mp 8 . . . . . . . . . . . . 13  |-  0  <_  +oo
44 elsni 3677 . . . . . . . . . . . . 13  |-  ( n  e.  {  +oo }  ->  n  =  +oo )
4543, 44syl5breqr 4075 . . . . . . . . . . . 12  |-  ( n  e.  {  +oo }  ->  0  <_  n )
4641, 45jaoi 368 . . . . . . . . . . 11  |-  ( ( n  e.  NN0  \/  n  e.  {  +oo }
)  ->  0  <_  n )
4740, 46sylbi 187 . . . . . . . . . 10  |-  ( n  e.  ( NN0  u.  { 
+oo } )  ->  0  <_  n )
48 lbicc2 10768 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  n  e.  RR*  /\  0  <_  n )  ->  0  e.  ( 0 [,] n
) )
4931, 39, 47, 48syl3anc 1182 . . . . . . . . 9  |-  ( n  e.  ( NN0  u.  { 
+oo } )  ->  0  e.  ( 0 [,] n
) )
50 0z 10051 . . . . . . . . 9  |-  0  e.  ZZ
51 inelcm 3522 . . . . . . . . 9  |-  ( ( 0  e.  ( 0 [,] n )  /\  0  e.  ZZ )  ->  ( ( 0 [,] n )  i^i  ZZ )  =/=  (/) )
5249, 50, 51sylancl 643 . . . . . . . 8  |-  ( n  e.  ( NN0  u.  { 
+oo } )  ->  (
( 0 [,] n
)  i^i  ZZ )  =/=  (/) )
53 fvex 5555 . . . . . . . . . 10  |-  ( ( S  D n F ) `  k )  e.  _V
5453dmex 4957 . . . . . . . . 9  |-  dom  (
( S  D n F ) `  k
)  e.  _V
5554rgenw 2623 . . . . . . . 8  |-  A. k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k )  e. 
_V
56 iinexg 4187 . . . . . . . 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 643 . . . . . . 7  |-  ( n  e.  ( NN0  u.  { 
+oo } )  ->  |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k )  e. 
_V )
5857rgen 2621 . . . . . 6  |-  A. n  e.  ( NN0  u.  {  +oo } ) |^|_ k  e.  ( ( 0 [,] n )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k )  e. 
_V
59 eqid 2296 . . . . . . 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 6211 . . . . . 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 653 . . . . 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 10 . . . 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 5990 . . 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 732 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  ->  n  =  N )
6564oveq2d 5890 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( 0 [,] n
)  =  ( 0 [,] N ) )
6665ineq1d 3382 . . . . . . 7  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( 0 [,] n )  i^i  ZZ )  =  ( (
0 [,] N )  i^i  ZZ ) )
67 simprr 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
a  =  B )
6867fveq2d 5545 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( ( S  D n F ) `
 k ) `  a )  =  ( ( ( S  D n F ) `  k
) `  B )
)
6968oveq1d 5889 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( ( ( S  D n F ) `  k ) `
 a )  / 
( ! `  k
) )  =  ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
) )
7067oveq2d 5890 . . . . . . . . 9  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( x  -  a
)  =  ( x  -  B ) )
7170oveq1d 5889 . . . . . . . 8  |-  ( (
ph  /\  ( n  =  N  /\  a  =  B ) )  -> 
( ( x  -  a ) ^ k
)  =  ( ( x  -  B ) ^ k ) )
7269, 71oveq12d 5892 . . . . . . 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 4114 . . . . . 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 5890 . . . . 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 4729 . . . 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 3946 . . 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 447 . . . . . 6  |-  ( (
ph  /\  n  =  N )  ->  n  =  N )
7877oveq2d 5890 . . . . 5  |-  ( (
ph  /\  n  =  N )  ->  (
0 [,] n )  =  ( 0 [,] N ) )
7978ineq1d 3382 . . . 4  |-  ( (
ph  /\  n  =  N )  ->  (
( 0 [,] n
)  i^i  ZZ )  =  ( ( 0 [,] N )  i^i 
ZZ ) )
80 iineq1 3935 . . . 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 15 . . 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 2810 . . . . . . 7  |-  +oo  e.  _V
8483elsnc2 3682 . . . . . 6  |-  ( N  e.  {  +oo }  <->  N  =  +oo )
8584orbi2i 505 . . . . 5  |-  ( ( N  e.  NN0  \/  N  e.  {  +oo }
)  <->  ( N  e. 
NN0  \/  N  =  +oo ) )
8682, 85sylibr 203 . . . 4  |-  ( ph  ->  ( N  e.  NN0  \/  N  e.  {  +oo } ) )
87 elun 3329 . . . 4  |-  ( N  e.  ( NN0  u.  { 
+oo } )  <->  ( N  e.  NN0  \/  N  e. 
{  +oo } ) )
8886, 87sylibr 203 . . 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 2639 . . . 4  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k ) )
91 oveq2 5882 . . . . . . . . . 10  |-  ( n  =  N  ->  (
0 [,] n )  =  ( 0 [,] N ) )
9291ineq1d 3382 . . . . . . . . 9  |-  ( n  =  N  ->  (
( 0 [,] n
)  i^i  ZZ )  =  ( ( 0 [,] N )  i^i 
ZZ ) )
9392neeq1d 2472 . . . . . . . 8  |-  ( n  =  N  ->  (
( ( 0 [,] n )  i^i  ZZ )  =/=  (/)  <->  ( ( 0 [,] N )  i^i 
ZZ )  =/=  (/) ) )
9493, 52vtoclga 2862 . . . . . . 7  |-  ( N  e.  ( NN0  u.  { 
+oo } )  ->  (
( 0 [,] N
)  i^i  ZZ )  =/=  (/) )
9588, 94syl 15 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  =/=  (/) )
96 r19.2z 3556 . . . . . 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 642 . . . . 5  |-  ( ph  ->  E. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k ) )
98 elex 2809 . . . . . 6  |-  ( B  e.  dom  ( ( S  D n F ) `  k )  ->  B  e.  _V )
9998rexlimivw 2676 . . . . 5  |-  ( E. k  e.  ( ( 0 [,] N )  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k )  ->  B  e.  _V )
100 eliin 3926 . . . . 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 18 . . . 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 223 . . 3  |-  ( ph  ->  B  e.  |^|_ k  e.  ( ( 0 [,] N )  i^i  ZZ ) dom  ( ( S  D n F ) `
 k ) )
103 snssi 3775 . . . . . . . 8  |-  ( x  e.  CC  ->  { x }  C_  CC )
104103adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  CC )  ->  { x }  C_  CC )
10520, 23, 24, 82, 89taylfvallem 19753 . . . . . . 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 4808 . . . . . . 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 642 . . . . . 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 2639 . . . . 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 3959 . . . . 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 203 . . . 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 4817 . . . . 5  |-  ( CC 
X.  CC )  e. 
_V
112111ssex 4174 . . . 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 15 . . 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 5990 . 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 2340 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 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   {cpr 3654   U_ciun 3921   |^|_ciin 3922   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    ^pm cpm 6789   CCcc 8751   RRcr 8752   0cc0 8753    x. cmul 8758    +oocpnf 8880   RR*cxr 8882    <_ cle 8884    - cmin 9053    / cdiv 9439   NN0cn0 9981   ZZcz 10040   [,]cicc 10675   ^cexp 11120   !cfa 11304  ℂfldccnfld 16393   tsums ctsu 17824    D ncdvn 19230   Tayl ctayl 19748
This theorem is referenced by:  eltayl  19755  taylf  19756  taylpfval  19760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-fac 11305  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-0g 13420  df-gsum 13421  df-mnd 14383  df-grp 14505  df-minusg 14506  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cnp 16974  df-haus 17059  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-tsms 17825  df-xms 17901  df-ms 17902  df-limc 19232  df-dv 19233  df-dvn 19234  df-tayl 19750
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