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Theorem tayl0 19741
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 19254 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
42, 3syl 15 . . . . 5  |-  ( ph  ->  S  C_  CC )
51, 4sstrd 3189 . . . 4  |-  ( ph  ->  A  C_  CC )
6 0xr 8878 . . . . . . . . 9  |-  0  e.  RR*
76a1i 10 . . . . . . . 8  |-  ( ph  ->  0  e.  RR* )
8 taylfval.n . . . . . . . . 9  |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo )
)
9 nn0re 9974 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  RR )
109rexrd 8881 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e. 
RR* )
11 id 19 . . . . . . . . . . 11  |-  ( N  =  +oo  ->  N  =  +oo )
12 pnfxr 10455 . . . . . . . . . . 11  |-  +oo  e.  RR*
1311, 12syl6eqel 2371 . . . . . . . . . 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 9991 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  0  <_  N )
17 pnfge 10469 . . . . . . . . . . . 12  |-  ( 0  e.  RR*  ->  0  <_  +oo )
186, 17ax-mp 8 . . . . . . . . . . 11  |-  0  <_  +oo
1918, 11syl5breqr 4059 . . . . . . . . . 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 10752 . . . . . . . 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 10035 . . . . . . . 8  |-  0  e.  ZZ
2524a1i 10 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
26 elin 3358 . . . . . . 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 2626 . . . . . 6  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k ) )
30 fveq2 5525 . . . . . . . . 9  |-  ( k  =  0  ->  (
( S  D n F ) `  k
)  =  ( ( S  D n F ) `  0 ) )
3130dmeqd 4881 . . . . . . . 8  |-  ( k  =  0  ->  dom  ( ( S  D n F ) `  k
)  =  dom  (
( S  D n F ) `  0
) )
3231eleq2d 2350 . . . . . . 7  |-  ( k  =  0  ->  ( B  e.  dom  ( ( S  D n F ) `  k )  <-> 
B  e.  dom  (
( S  D n F ) `  0
) ) )
3332rspcv 2880 . . . . . 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 8818 . . . . . . . . . 10  |-  CC  e.  _V
3635a1i 10 . . . . . . . . 9  |-  ( ph  ->  CC  e.  _V )
37 taylfval.f . . . . . . . . 9  |-  ( ph  ->  F : A --> CC )
38 elpm2r 6788 . . . . . . . . 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 19273 . . . . . . . 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 4881 . . . . . 6  |-  ( ph  ->  dom  ( ( S  D n F ) `
 0 )  =  dom  F )
43 fdm 5393 . . . . . . 7  |-  ( F : A --> CC  ->  dom 
F  =  A )
4437, 43syl 15 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
4542, 44eqtrd 2315 . . . . 5  |-  ( ph  ->  dom  ( ( S  D n F ) `
 0 )  =  A )
4634, 45eleqtrd 2359 . . . 4  |-  ( ph  ->  B  e.  A )
475, 46sseldd 3181 . . 3  |-  ( ph  ->  B  e.  CC )
48 cnfldbas 16383 . . . . . . 7  |-  CC  =  ( Base ` fld )
49 cnfld0 16398 . . . . . . 7  |-  0  =  ( 0g ` fld )
50 cnrng 16396 . . . . . . . 8  |-fld  e.  Ring
51 rngmnd 15350 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
5250, 51mp1i 11 . . . . . . 7  |-  ( ph  ->fld  e. 
Mnd )
53 ovex 5883 . . . . . . . . 9  |-  ( 0 [,] N )  e. 
_V
5453inex1 4155 . . . . . . . 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 3390 . . . . . . . . . . . . . 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 3178 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
61 inss1 3389 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
6261, 59sseldi 3178 . . . . . . . . . . . . . . 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 10700 . . . . . . . . . . . . . . . 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 10036 . . . . . . . . . . . . 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 19276 . . . . . . . . . . . 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 5666 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( S  D n F ) `  k
) `  B )  e.  CC )
73 faccl 11298 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
7469, 73syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  NN )
7574nncnd 9762 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
7674nnne0d 9790 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
7772, 75, 76divcld 9536 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
78 0cn 8831 . . . . . . . . . . 11  |-  0  e.  CC
7978a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  e.  CC )
8079, 69expcld 11245 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
0 ^ k )  e.  CC )
8177, 80mulcld 8855 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  e.  CC )
82 eqid 2283 . . . . . . . 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 5684 . . . . . . 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 3298 . . . . . . . . . . . . 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 3750 . . . . . . . . . . . . 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 9979 . . . . . . . . . . . 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 11261 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( 0 ^ k )  =  0 )
9190oveq2d 5874 . . . . . . . . 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 9011 . . . . . . . . . 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 2315 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) )  =  0 )
9594suppss2 6073 . . . . . . 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 15222 . . . . . 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 5527 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( ( S  D n F ) `  k
) `  B )  =  ( ( ( S  D n F ) `  0 ) `
 B ) )
98 fveq2 5525 . . . . . . . . . . 11  |-  ( k  =  0  ->  ( ! `  k )  =  ( ! ` 
0 ) )
99 fac0 11291 . . . . . . . . . . 11  |-  ( ! `
 0 )  =  1
10098, 99syl6eq 2331 . . . . . . . . . 10  |-  ( k  =  0  ->  ( ! `  k )  =  1 )
10197, 100oveq12d 5876 . . . . . . . . 9  |-  ( k  =  0  ->  (
( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  =  ( ( ( ( S  D n F ) `  0
) `  B )  /  1 ) )
102 oveq2 5866 . . . . . . . . . 10  |-  ( k  =  0  ->  (
0 ^ k )  =  ( 0 ^ 0 ) )
103 exp0 11108 . . . . . . . . . . 11  |-  ( 0  e.  CC  ->  (
0 ^ 0 )  =  1 )
10478, 103ax-mp 8 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
105102, 104syl6eq 2331 . . . . . . . . 9  |-  ( k  =  0  ->  (
0 ^ k )  =  1 )
106101, 105oveq12d 5876 . . . . . . . 8  |-  ( k  =  0  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  =  ( ( ( ( ( S  D n F ) `
 0 ) `  B )  /  1
)  x.  1 ) )
107 ovex 5883 . . . . . . . 8  |-  ( ( ( ( ( S  D n F ) `
 0 ) `  B )  /  1
)  x.  1 )  e.  _V
108106, 82, 107fvmpt 5602 . . . . . . 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 5527 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( S  D n F ) `
 0 ) `  B )  =  ( F `  B ) )
111110oveq1d 5873 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( S  D n F ) `  0 ) `
 B )  / 
1 )  =  ( ( F `  B
)  /  1 ) )
11237, 46ffvelrnd 5666 . . . . . . . . . 10  |-  ( ph  ->  ( F `  B
)  e.  CC )
113112div1d 9528 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  /  1
)  =  ( F `
 B ) )
114111, 113eqtrd 2315 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( S  D n F ) `  0 ) `
 B )  / 
1 )  =  ( F `  B ) )
115114oveq1d 5873 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( S  D n F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( ( F `  B
)  x.  1 ) )
116112mulid1d 8852 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  1 )  =  ( F `
 B ) )
117115, 116eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  D n F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( F `  B ) )
11896, 109, 1173eqtrd 2319 . . . . 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 15371 . . . . . . 7  |-  (fld  e.  Ring  ->fld  e. CMnd )
12050, 119mp1i 11 . . . . . 6  |-  ( ph  ->fld  e. CMnd
)
121 cnfldtps 18287 . . . . . . 7  |-fld  e.  TopSp
122121a1i 10 . . . . . 6  |-  ( ph  ->fld  e. 
TopSp )
123 snfi 6941 . . . . . . 7  |-  { 0 }  e.  Fin
124 ssfi 7083 . . . . . . 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 17822 . . . . 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 2358 . . . 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 9145 . . . . . . . 8  |-  ( ph  ->  ( B  -  B
)  =  0 )
129128oveq1d 5873 . . . . . . 7  |-  ( ph  ->  ( ( B  -  B ) ^ k
)  =  ( 0 ^ k ) )
130129oveq2d 5874 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) )  =  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) ) )
131130mpteq2dv 4107 . . . . 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 5874 . . . 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 2360 . . 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 19739 . . 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 19740 . . 3  |-  ( ph  ->  T : dom  T --> CC )
138 ffun 5391 . . 3  |-  ( T : dom  T --> CC  ->  Fun 
T )
139 funbrfv2b 5567 . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   {cpr 3641   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   "cima 4692   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    +oocpnf 8864   RR*cxr 8866    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   [,]cicc 10659   ^cexp 11104   !cfa 11288    gsumg cgsu 13401   Mndcmnd 14361  CMndccmn 15089   Ringcrg 15337  ℂfldccnfld 16377   TopSpctps 16634   tsums ctsu 17808    D ncdvn 19214   Tayl ctayl 19732
This theorem is referenced by:  dvntaylp0  19751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-fac 11289  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cnp 16958  df-haus 17043  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-tsms 17809  df-xms 17885  df-ms 17886  df-limc 19216  df-dv 19217  df-dvn 19218  df-tayl 19734
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