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Theorem tayl0 19757
Description: The Taylor series is always defined at the basepoint, with value equal to the value 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
tayl0  |-  ( ph  ->  ( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) )
Distinct variable groups:    B, k    k, F    ph, k    k, N    S, k
Allowed substitution hints:    A( k)    T( k)

Proof of Theorem tayl0
StepHypRef Expression
1 taylfval.a . . . . 5  |-  ( ph  ->  A  C_  S )
2 taylfval.s . . . . . 6  |-  ( ph  ->  S  e.  { RR ,  CC } )
3 recnprss 19270 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
42, 3syl 15 . . . . 5  |-  ( ph  ->  S  C_  CC )
51, 4sstrd 3202 . . . 4  |-  ( ph  ->  A  C_  CC )
6 0xr 8894 . . . . . . . . 9  |-  0  e.  RR*
76a1i 10 . . . . . . . 8  |-  ( ph  ->  0  e.  RR* )
8 taylfval.n . . . . . . . . 9  |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo )
)
9 nn0re 9990 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  RR )
109rexrd 8897 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e. 
RR* )
11 id 19 . . . . . . . . . . 11  |-  ( N  =  +oo  ->  N  =  +oo )
12 pnfxr 10471 . . . . . . . . . . 11  |-  +oo  e.  RR*
1311, 12syl6eqel 2384 . . . . . . . . . 10  |-  ( N  =  +oo  ->  N  e.  RR* )
1410, 13jaoi 368 . . . . . . . . 9  |-  ( ( N  e.  NN0  \/  N  =  +oo )  ->  N  e.  RR* )
158, 14syl 15 . . . . . . . 8  |-  ( ph  ->  N  e.  RR* )
16 nn0ge0 10007 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  0  <_  N )
17 pnfge 10485 . . . . . . . . . . . 12  |-  ( 0  e.  RR*  ->  0  <_  +oo )
186, 17ax-mp 8 . . . . . . . . . . 11  |-  0  <_  +oo
1918, 11syl5breqr 4075 . . . . . . . . . 10  |-  ( N  =  +oo  ->  0  <_  N )
2016, 19jaoi 368 . . . . . . . . 9  |-  ( ( N  e.  NN0  \/  N  =  +oo )  -> 
0  <_  N )
218, 20syl 15 . . . . . . . 8  |-  ( ph  ->  0  <_  N )
22 lbicc2 10768 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  N  e.  RR*  /\  0  <_  N )  ->  0  e.  ( 0 [,] N
) )
237, 15, 21, 22syl3anc 1182 . . . . . . 7  |-  ( ph  ->  0  e.  ( 0 [,] N ) )
24 0z 10051 . . . . . . . 8  |-  0  e.  ZZ
2524a1i 10 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
26 elin 3371 . . . . . . 7  |-  ( 0  e.  ( ( 0 [,] N )  i^i 
ZZ )  <->  ( 0  e.  ( 0 [,] N )  /\  0  e.  ZZ ) )
2723, 25, 26sylanbrc 645 . . . . . 6  |-  ( ph  ->  0  e.  ( ( 0 [,] N )  i^i  ZZ ) )
28 taylfval.b . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `
 k ) )
2928ralrimiva 2639 . . . . . 6  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k ) )
30 fveq2 5541 . . . . . . . . 9  |-  ( k  =  0  ->  (
( S  D n F ) `  k
)  =  ( ( S  D n F ) `  0 ) )
3130dmeqd 4897 . . . . . . . 8  |-  ( k  =  0  ->  dom  ( ( S  D n F ) `  k
)  =  dom  (
( S  D n F ) `  0
) )
3231eleq2d 2363 . . . . . . 7  |-  ( k  =  0  ->  ( B  e.  dom  ( ( S  D n F ) `  k )  <-> 
B  e.  dom  (
( S  D n F ) `  0
) ) )
3332rspcv 2893 . . . . . 6  |-  ( 0  e.  ( ( 0 [,] N )  i^i 
ZZ )  ->  ( A. k  e.  (
( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k )  ->  B  e.  dom  ( ( S  D n F ) `  0
) ) )
3427, 29, 33sylc 56 . . . . 5  |-  ( ph  ->  B  e.  dom  (
( S  D n F ) `  0
) )
35 cnex 8834 . . . . . . . . . 10  |-  CC  e.  _V
3635a1i 10 . . . . . . . . 9  |-  ( ph  ->  CC  e.  _V )
37 taylfval.f . . . . . . . . 9  |-  ( ph  ->  F : A --> CC )
38 elpm2r 6804 . . . . . . . . 9  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
3936, 2, 37, 1, 38syl22anc 1183 . . . . . . . 8  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
40 dvn0 19289 . . . . . . . 8  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  (
( S  D n F ) `  0
)  =  F )
414, 39, 40syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( S  D n F ) `  0
)  =  F )
4241dmeqd 4897 . . . . . 6  |-  ( ph  ->  dom  ( ( S  D n F ) `
 0 )  =  dom  F )
43 fdm 5409 . . . . . . 7  |-  ( F : A --> CC  ->  dom 
F  =  A )
4437, 43syl 15 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
4542, 44eqtrd 2328 . . . . 5  |-  ( ph  ->  dom  ( ( S  D n F ) `
 0 )  =  A )
4634, 45eleqtrd 2372 . . . 4  |-  ( ph  ->  B  e.  A )
475, 46sseldd 3194 . . 3  |-  ( ph  ->  B  e.  CC )
48 cnfldbas 16399 . . . . . . 7  |-  CC  =  ( Base ` fld )
49 cnfld0 16414 . . . . . . 7  |-  0  =  ( 0g ` fld )
50 cnrng 16412 . . . . . . . 8  |-fld  e.  Ring
51 rngmnd 15366 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
5250, 51mp1i 11 . . . . . . 7  |-  ( ph  ->fld  e. 
Mnd )
53 ovex 5899 . . . . . . . . 9  |-  ( 0 [,] N )  e. 
_V
5453inex1 4171 . . . . . . . 8  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
5554a1i 10 . . . . . . 7  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  e.  _V )
562adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  S  e.  { RR ,  CC } )
5739adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  F  e.  ( CC  ^pm  S
) )
58 inss2 3403 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ZZ
59 simpr 447 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )
6058, 59sseldi 3191 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
61 inss1 3402 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
6261, 59sseldi 3191 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( 0 [,] N
) )
6315adantr 451 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  N  e.  RR* )
64 elicc1 10716 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR*  /\  N  e.  RR* )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
656, 63, 64sylancr 644 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
6662, 65mpbid 201 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  RR*  /\  0  <_  k  /\  k  <_  N ) )
6766simp2d 968 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  <_  k )
68 elnn0z 10052 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  <->  ( k  e.  ZZ  /\  0  <_ 
k ) )
6960, 67, 68sylanbrc 645 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  NN0 )
70 dvnf 19292 . . . . . . . . . . . 12  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  k  e.  NN0 )  ->  ( ( S  D n F ) `
 k ) : dom  ( ( S  D n F ) `
 k ) --> CC )
7156, 57, 69, 70syl3anc 1182 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( S  D n F ) `  k
) : dom  (
( S  D n F ) `  k
) --> CC )
7271, 28ffvelrnd 5682 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( S  D n F ) `  k
) `  B )  e.  CC )
73 faccl 11314 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
7469, 73syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  NN )
7574nncnd 9778 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
7674nnne0d 9806 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
7772, 75, 76divcld 9552 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
78 0cn 8847 . . . . . . . . . . 11  |-  0  e.  CC
7978a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  e.  CC )
8079, 69expcld 11261 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
0 ^ k )  e.  CC )
8177, 80mulcld 8871 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  e.  CC )
82 eqid 2296 . . . . . . . 8  |-  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) )  =  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) )
8381, 82fmptd 5700 . . . . . . 7  |-  ( ph  ->  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) : ( ( 0 [,] N )  i^i  ZZ )
--> CC )
84 eldifi 3311 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( ( 0 [,] N )  i^i  ZZ )  \  { 0 } )  ->  k  e.  ( ( 0 [,] N
)  i^i  ZZ )
)
8584, 69sylan2 460 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  e.  NN0 )
86 eldifsni 3763 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( ( 0 [,] N )  i^i  ZZ )  \  { 0 } )  ->  k  =/=  0
)
8786adantl 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  =/=  0
)
88 elnnne0 9995 . . . . . . . . . . . 12  |-  ( k  e.  NN  <->  ( k  e.  NN0  /\  k  =/=  0 ) )
8985, 87, 88sylanbrc 645 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  e.  NN )
90890expd 11277 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( 0 ^ k )  =  0 )
9190oveq2d 5890 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) )  =  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  0 ) )
9277mul01d 9027 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  0 )  =  0 )
9384, 92sylan2 460 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  0 )  =  0 )
9491, 93eqtrd 2328 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) )  =  0 )
9594suppss2 6089 . . . . . . 7  |-  ( ph  ->  ( `' ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) ) " ( _V 
\  { 0 } ) )  C_  { 0 } )
9648, 49, 52, 55, 27, 83, 95gsumpt 15238 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  =  ( ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) ) `  0 ) )
9730fveq1d 5543 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( ( S  D n F ) `  k
) `  B )  =  ( ( ( S  D n F ) `  0 ) `
 B ) )
98 fveq2 5541 . . . . . . . . . . 11  |-  ( k  =  0  ->  ( ! `  k )  =  ( ! ` 
0 ) )
99 fac0 11307 . . . . . . . . . . 11  |-  ( ! `
 0 )  =  1
10098, 99syl6eq 2344 . . . . . . . . . 10  |-  ( k  =  0  ->  ( ! `  k )  =  1 )
10197, 100oveq12d 5892 . . . . . . . . 9  |-  ( k  =  0  ->  (
( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  =  ( ( ( ( S  D n F ) `  0
) `  B )  /  1 ) )
102 oveq2 5882 . . . . . . . . . 10  |-  ( k  =  0  ->  (
0 ^ k )  =  ( 0 ^ 0 ) )
103 exp0 11124 . . . . . . . . . . 11  |-  ( 0  e.  CC  ->  (
0 ^ 0 )  =  1 )
10478, 103ax-mp 8 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
105102, 104syl6eq 2344 . . . . . . . . 9  |-  ( k  =  0  ->  (
0 ^ k )  =  1 )
106101, 105oveq12d 5892 . . . . . . . 8  |-  ( k  =  0  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  =  ( ( ( ( ( S  D n F ) `
 0 ) `  B )  /  1
)  x.  1 ) )
107 ovex 5899 . . . . . . . 8  |-  ( ( ( ( ( S  D n F ) `
 0 ) `  B )  /  1
)  x.  1 )  e.  _V
108106, 82, 107fvmpt 5618 . . . . . . 7  |-  ( 0  e.  ( ( 0 [,] N )  i^i 
ZZ )  ->  (
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ` 
0 )  =  ( ( ( ( ( S  D n F ) `  0 ) `
 B )  / 
1 )  x.  1 ) )
10927, 108syl 15 . . . . . 6  |-  ( ph  ->  ( ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ` 
0 )  =  ( ( ( ( ( S  D n F ) `  0 ) `
 B )  / 
1 )  x.  1 ) )
11041fveq1d 5543 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( S  D n F ) `
 0 ) `  B )  =  ( F `  B ) )
111110oveq1d 5889 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( S  D n F ) `  0 ) `
 B )  / 
1 )  =  ( ( F `  B
)  /  1 ) )
11237, 46ffvelrnd 5682 . . . . . . . . . 10  |-  ( ph  ->  ( F `  B
)  e.  CC )
113112div1d 9544 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  /  1
)  =  ( F `
 B ) )
114111, 113eqtrd 2328 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( S  D n F ) `  0 ) `
 B )  / 
1 )  =  ( F `  B ) )
115114oveq1d 5889 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( S  D n F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( ( F `  B
)  x.  1 ) )
116112mulid1d 8868 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  1 )  =  ( F `
 B ) )
117115, 116eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  D n F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( F `  B ) )
11896, 109, 1173eqtrd 2332 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  =  ( F `  B ) )
119 rngcmn 15387 . . . . . . 7  |-  (fld  e.  Ring  ->fld  e. CMnd )
12050, 119mp1i 11 . . . . . 6  |-  ( ph  ->fld  e. CMnd
)
121 cnfldtps 18303 . . . . . . 7  |-fld  e.  TopSp
122121a1i 10 . . . . . 6  |-  ( ph  ->fld  e. 
TopSp )
123 snfi 6957 . . . . . . 7  |-  { 0 }  e.  Fin
124 ssfi 7099 . . . . . . 7  |-  ( ( { 0 }  e.  Fin  /\  ( `' ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) ) ) " ( _V  \  { 0 } ) )  C_  { 0 } )  ->  ( `' ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) "
( _V  \  {
0 } ) )  e.  Fin )
125123, 95, 124sylancr 644 . . . . . 6  |-  ( ph  ->  ( `' ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) ) " ( _V 
\  { 0 } ) )  e.  Fin )
12648, 49, 120, 122, 55, 83, 125tsmsid 17838 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ) )
127118, 126eqeltrrd 2371 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ) )
12847subidd 9161 . . . . . . . 8  |-  ( ph  ->  ( B  -  B
)  =  0 )
129128oveq1d 5889 . . . . . . 7  |-  ( ph  ->  ( ( B  -  B ) ^ k
)  =  ( 0 ^ k ) )
130129oveq2d 5890 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) )  =  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) ) )
131130mpteq2dv 4123 . . . . 5  |-  ( ph  ->  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) ) )  =  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )
132131oveq2d 5890 . . . 4  |-  ( ph  ->  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) ) ) )  =  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ) )
133127, 132eleqtrrd 2373 . . 3  |-  ( ph  ->  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) ) ) ) )
134 taylfval.t . . . 4  |-  T  =  ( N ( S Tayl 
F ) B )
1352, 37, 1, 8, 28, 134eltayl 19755 . . 3  |-  ( ph  ->  ( B T ( F `  B )  <-> 
( B  e.  CC  /\  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) ) ) ) ) ) )
13647, 133, 135mpbir2and 888 . 2  |-  ( ph  ->  B T ( F `
 B ) )
1372, 37, 1, 8, 28, 134taylf 19756 . . 3  |-  ( ph  ->  T : dom  T --> CC )
138 ffun 5407 . . 3  |-  ( T : dom  T --> CC  ->  Fun 
T )
139 funbrfv2b 5583 . . 3  |-  ( Fun 
T  ->  ( B T ( F `  B )  <->  ( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) ) )
140137, 138, 1393syl 18 . 2  |-  ( ph  ->  ( B T ( F `  B )  <-> 
( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) ) )
141136, 140mpbid 201 1  |-  ( ph  ->  ( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   {csn 3653   {cpr 3654   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   "cima 4708   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    +oocpnf 8880   RR*cxr 8882    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   [,]cicc 10675   ^cexp 11120   !cfa 11304    gsumg cgsu 13417   Mndcmnd 14377  CMndccmn 15105   Ringcrg 15353  ℂfldccnfld 16393   TopSpctps 16650   tsums ctsu 17824    D ncdvn 19230   Tayl ctayl 19748
This theorem is referenced by:  dvntaylp0  19767
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-ress 13171  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-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-mulg 14508  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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