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Theorem tayl0 23317
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  Dn 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 22859 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
42, 3syl 17 . . . . 5  |-  ( ph  ->  S  C_  CC )
51, 4sstrd 3442 . . . 4  |-  ( ph  ->  A  C_  CC )
6 0xr 9687 . . . . . . . . 9  |-  0  e.  RR*
76a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR* )
8 taylfval.n . . . . . . . . 9  |-  ( ph  ->  ( N  e.  NN0  \/  N  = +oo )
)
9 nn0re 10878 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  RR )
109rexrd 9690 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e. 
RR* )
11 id 22 . . . . . . . . . . 11  |-  ( N  = +oo  ->  N  = +oo )
12 pnfxr 11412 . . . . . . . . . . 11  |- +oo  e.  RR*
1311, 12syl6eqel 2537 . . . . . . . . . 10  |-  ( N  = +oo  ->  N  e.  RR* )
1410, 13jaoi 381 . . . . . . . . 9  |-  ( ( N  e.  NN0  \/  N  = +oo )  ->  N  e.  RR* )
158, 14syl 17 . . . . . . . 8  |-  ( ph  ->  N  e.  RR* )
16 nn0pnfge0 11434 . . . . . . . . 9  |-  ( ( N  e.  NN0  \/  N  = +oo )  ->  0  <_  N )
178, 16syl 17 . . . . . . . 8  |-  ( ph  ->  0  <_  N )
18 lbicc2 11748 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  N  e.  RR*  /\  0  <_  N )  ->  0  e.  ( 0 [,] N
) )
197, 15, 17, 18syl3anc 1268 . . . . . . 7  |-  ( ph  ->  0  e.  ( 0 [,] N ) )
20 0zd 10949 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
2119, 20elind 3618 . . . . . 6  |-  ( ph  ->  0  e.  ( ( 0 [,] N )  i^i  ZZ ) )
22 taylfval.b . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  Dn F ) `
 k ) )
2322ralrimiva 2802 . . . . . 6  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) )
24 fveq2 5865 . . . . . . . . 9  |-  ( k  =  0  ->  (
( S  Dn
F ) `  k
)  =  ( ( S  Dn F ) `  0 ) )
2524dmeqd 5037 . . . . . . . 8  |-  ( k  =  0  ->  dom  ( ( S  Dn F ) `  k )  =  dom  ( ( S  Dn F ) ` 
0 ) )
2625eleq2d 2514 . . . . . . 7  |-  ( k  =  0  ->  ( B  e.  dom  ( ( S  Dn F ) `  k )  <-> 
B  e.  dom  (
( S  Dn
F ) `  0
) ) )
2726rspcv 3146 . . . . . 6  |-  ( 0  e.  ( ( 0 [,] N )  i^i 
ZZ )  ->  ( A. k  e.  (
( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k )  ->  B  e.  dom  ( ( S  Dn F ) ` 
0 ) ) )
2821, 23, 27sylc 62 . . . . 5  |-  ( ph  ->  B  e.  dom  (
( S  Dn
F ) `  0
) )
29 cnex 9620 . . . . . . . . . 10  |-  CC  e.  _V
3029a1i 11 . . . . . . . . 9  |-  ( ph  ->  CC  e.  _V )
31 taylfval.f . . . . . . . . 9  |-  ( ph  ->  F : A --> CC )
32 elpm2r 7489 . . . . . . . . 9  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
3330, 2, 31, 1, 32syl22anc 1269 . . . . . . . 8  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
34 dvn0 22878 . . . . . . . 8  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  (
( S  Dn
F ) `  0
)  =  F )
354, 33, 34syl2anc 667 . . . . . . 7  |-  ( ph  ->  ( ( S  Dn F ) ` 
0 )  =  F )
3635dmeqd 5037 . . . . . 6  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 0 )  =  dom  F )
37 fdm 5733 . . . . . . 7  |-  ( F : A --> CC  ->  dom 
F  =  A )
3831, 37syl 17 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
3936, 38eqtrd 2485 . . . . 5  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 0 )  =  A )
4028, 39eleqtrd 2531 . . . 4  |-  ( ph  ->  B  e.  A )
415, 40sseldd 3433 . . 3  |-  ( ph  ->  B  e.  CC )
42 cnfldbas 18974 . . . . . . 7  |-  CC  =  ( Base ` fld )
43 cnfld0 18992 . . . . . . 7  |-  0  =  ( 0g ` fld )
44 cnring 18990 . . . . . . . 8  |-fld  e.  Ring
45 ringmnd 17789 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
4644, 45mp1i 13 . . . . . . 7  |-  ( ph  ->fld  e. 
Mnd )
47 ovex 6318 . . . . . . . . 9  |-  ( 0 [,] N )  e. 
_V
4847inex1 4544 . . . . . . . 8  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
4948a1i 11 . . . . . . 7  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  e.  _V )
502adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  S  e.  { RR ,  CC } )
5133adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  F  e.  ( CC  ^pm  S
) )
52 inss2 3653 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ZZ
53 simpr 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )
5452, 53sseldi 3430 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
55 inss1 3652 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
5655, 53sseldi 3430 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( 0 [,] N
) )
5715adantr 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  N  e.  RR* )
58 elicc1 11680 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR*  /\  N  e.  RR* )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
596, 57, 58sylancr 669 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
6056, 59mpbid 214 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  RR*  /\  0  <_  k  /\  k  <_  N ) )
6160simp2d 1021 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  <_  k )
62 elnn0z 10950 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  <->  ( k  e.  ZZ  /\  0  <_ 
k ) )
6354, 61, 62sylanbrc 670 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  NN0 )
64 dvnf 22881 . . . . . . . . . . . 12  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  k  e.  NN0 )  ->  ( ( S  Dn F ) `
 k ) : dom  ( ( S  Dn F ) `
 k ) --> CC )
6550, 51, 63, 64syl3anc 1268 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( S  Dn
F ) `  k
) : dom  (
( S  Dn
F ) `  k
) --> CC )
6665, 22ffvelrnd 6023 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( S  Dn F ) `  k ) `  B
)  e.  CC )
67 faccl 12469 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
6863, 67syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  NN )
6968nncnd 10625 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
7068nnne0d 10654 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
7166, 69, 70divcld 10383 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
72 0cnd 9636 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  e.  CC )
7372, 63expcld 12416 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
0 ^ k )  e.  CC )
7471, 73mulcld 9663 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  e.  CC )
75 eqid 2451 . . . . . . . 8  |-  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) )  =  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) )
7674, 75fmptd 6046 . . . . . . 7  |-  ( ph  ->  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) : ( ( 0 [,] N )  i^i  ZZ )
--> CC )
77 eldifi 3555 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( ( 0 [,] N )  i^i  ZZ )  \  { 0 } )  ->  k  e.  ( ( 0 [,] N
)  i^i  ZZ )
)
7877, 63sylan2 477 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  e.  NN0 )
79 eldifsni 4098 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( ( 0 [,] N )  i^i  ZZ )  \  { 0 } )  ->  k  =/=  0
)
8079adantl 468 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  =/=  0
)
81 elnnne0 10883 . . . . . . . . . . . 12  |-  ( k  e.  NN  <->  ( k  e.  NN0  /\  k  =/=  0 ) )
8278, 80, 81sylanbrc 670 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  e.  NN )
83820expd 12432 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( 0 ^ k )  =  0 )
8483oveq2d 6306 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) )  =  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  0 ) )
8571mul01d 9832 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  0 )  =  0 )
8677, 85sylan2 477 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  0 )  =  0 )
8784, 86eqtrd 2485 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) )  =  0 )
88 zex 10946 . . . . . . . . . 10  |-  ZZ  e.  _V
8988inex2 4545 . . . . . . . . 9  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
9089a1i 11 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  e.  _V )
9187, 90suppss2 6949 . . . . . . 7  |-  ( ph  ->  ( ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) ) ) supp  0 )  C_  { 0 } )
9242, 43, 46, 49, 21, 76, 91gsumpt 17594 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  =  ( ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) ) `  0 ) )
9324fveq1d 5867 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( ( S  Dn F ) `  k ) `  B
)  =  ( ( ( S  Dn
F ) `  0
) `  B )
)
94 fveq2 5865 . . . . . . . . . . 11  |-  ( k  =  0  ->  ( ! `  k )  =  ( ! ` 
0 ) )
95 fac0 12462 . . . . . . . . . . 11  |-  ( ! `
 0 )  =  1
9694, 95syl6eq 2501 . . . . . . . . . 10  |-  ( k  =  0  ->  ( ! `  k )  =  1 )
9793, 96oveq12d 6308 . . . . . . . . 9  |-  ( k  =  0  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  =  ( ( ( ( S  Dn F ) ` 
0 ) `  B
)  /  1 ) )
98 oveq2 6298 . . . . . . . . . 10  |-  ( k  =  0  ->  (
0 ^ k )  =  ( 0 ^ 0 ) )
99 0exp0e1 12277 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
10098, 99syl6eq 2501 . . . . . . . . 9  |-  ( k  =  0  ->  (
0 ^ k )  =  1 )
10197, 100oveq12d 6308 . . . . . . . 8  |-  ( k  =  0  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  =  ( ( ( ( ( S  Dn F ) `
 0 ) `  B )  /  1
)  x.  1 ) )
102 ovex 6318 . . . . . . . 8  |-  ( ( ( ( ( S  Dn F ) `
 0 ) `  B )  /  1
)  x.  1 )  e.  _V
103101, 75, 102fvmpt 5948 . . . . . . 7  |-  ( 0  e.  ( ( 0 [,] N )  i^i 
ZZ )  ->  (
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ` 
0 )  =  ( ( ( ( ( S  Dn F ) `  0 ) `
 B )  / 
1 )  x.  1 ) )
10421, 103syl 17 . . . . . 6  |-  ( ph  ->  ( ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) ) ) `
 0 )  =  ( ( ( ( ( S  Dn
F ) `  0
) `  B )  /  1 )  x.  1 ) )
10535fveq1d 5867 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( S  Dn F ) `
 0 ) `  B )  =  ( F `  B ) )
106105oveq1d 6305 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( S  Dn F ) `  0 ) `
 B )  / 
1 )  =  ( ( F `  B
)  /  1 ) )
10731, 40ffvelrnd 6023 . . . . . . . . . 10  |-  ( ph  ->  ( F `  B
)  e.  CC )
108107div1d 10375 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  /  1
)  =  ( F `
 B ) )
109106, 108eqtrd 2485 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( S  Dn F ) `  0 ) `
 B )  / 
1 )  =  ( F `  B ) )
110109oveq1d 6305 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( ( F `  B
)  x.  1 ) )
111107mulid1d 9660 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  1 )  =  ( F `
 B ) )
112110, 111eqtrd 2485 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( F `  B ) )
11392, 104, 1123eqtrd 2489 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  =  ( F `  B ) )
114 ringcmn 17811 . . . . . . 7  |-  (fld  e.  Ring  ->fld  e. CMnd )
11544, 114mp1i 13 . . . . . 6  |-  ( ph  ->fld  e. CMnd
)
116 cnfldtps 21798 . . . . . . 7  |-fld  e.  TopSp
117116a1i 11 . . . . . 6  |-  ( ph  ->fld  e. 
TopSp )
118 mptexg 6135 . . . . . . . 8  |-  ( ( ( 0 [,] N
)  i^i  ZZ )  e.  _V  ->  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) ) )  e.  _V )
11989, 118mp1i 13 . . . . . . 7  |-  ( ph  ->  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) )  e. 
_V )
120 funmpt 5618 . . . . . . . 8  |-  Fun  (
k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) ) )
121120a1i 11 . . . . . . 7  |-  ( ph  ->  Fun  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) ) ) )
122 c0ex 9637 . . . . . . . 8  |-  0  e.  _V
123122a1i 11 . . . . . . 7  |-  ( ph  ->  0  e.  _V )
124 snfi 7650 . . . . . . . 8  |-  { 0 }  e.  Fin
125124a1i 11 . . . . . . 7  |-  ( ph  ->  { 0 }  e.  Fin )
126 suppssfifsupp 7898 . . . . . . 7  |-  ( ( ( ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) ) )  e.  _V  /\  Fun  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) )  /\  0  e.  _V )  /\  ( { 0 }  e.  Fin  /\  (
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) supp  0
)  C_  { 0 } ) )  -> 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) finSupp  0
)
127119, 121, 123, 125, 91, 126syl32anc 1276 . . . . . 6  |-  ( ph  ->  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) finSupp  0
)
12842, 43, 115, 117, 49, 76, 127tsmsid 21154 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ) )
129113, 128eqeltrrd 2530 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) ) ) ) )
13041subidd 9974 . . . . . . . 8  |-  ( ph  ->  ( B  -  B
)  =  0 )
131130oveq1d 6305 . . . . . . 7  |-  ( ph  ->  ( ( B  -  B ) ^ k
)  =  ( 0 ^ k ) )
132131oveq2d 6306 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) )  =  ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) ) )
133132mpteq2dv 4490 . . . . 5  |-  ( ph  ->  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) ) )  =  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )
134133oveq2d 6306 . . . 4  |-  ( ph  ->  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) ) ) )  =  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ) )
135129, 134eleqtrrd 2532 . . 3  |-  ( ph  ->  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( B  -  B ) ^
k ) ) ) ) )
136 taylfval.t . . . 4  |-  T  =  ( N ( S Tayl 
F ) B )
1372, 31, 1, 8, 22, 136eltayl 23315 . . 3  |-  ( ph  ->  ( B T ( F `  B )  <-> 
( B  e.  CC  /\  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( B  -  B ) ^
k ) ) ) ) ) ) )
13841, 135, 137mpbir2and 933 . 2  |-  ( ph  ->  B T ( F `
 B ) )
1392, 31, 1, 8, 22, 136taylf 23316 . . 3  |-  ( ph  ->  T : dom  T --> CC )
140 ffun 5731 . . 3  |-  ( T : dom  T --> CC  ->  Fun 
T )
141 funbrfv2b 5909 . . 3  |-  ( Fun 
T  ->  ( B T ( F `  B )  <->  ( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) ) )
142139, 140, 1413syl 18 . 2  |-  ( ph  ->  ( B T ( F `  B )  <-> 
( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) ) )
143138, 142mpbid 214 1  |-  ( ph  ->  ( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   _Vcvv 3045    \ cdif 3401    i^i cin 3403    C_ wss 3404   {csn 3968   {cpr 3970   class class class wbr 4402    |-> cmpt 4461   dom cdm 4834   Fun wfun 5576   -->wf 5578   ` cfv 5582  (class class class)co 6290   supp csupp 6914    ^pm cpm 7473   Fincfn 7569   finSupp cfsupp 7883   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544   +oocpnf 9672   RR*cxr 9674    <_ cle 9676    - cmin 9860    / cdiv 10269   NNcn 10609   NN0cn0 10869   ZZcz 10937   [,]cicc 11638   ^cexp 12272   !cfa 12459    gsumg cgsu 15339   Mndcmnd 16535  CMndccmn 17430   Ringcrg 17780  ℂfldccnfld 18970   TopSpctps 19919   tsums ctsu 21140    Dncdvn 22819   Tayl ctayl 23308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-fac 12460  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-grp 16673  df-minusg 16674  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-cring 17783  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cnp 20244  df-haus 20331  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-tsms 21141  df-xms 21335  df-ms 21336  df-limc 22821  df-dv 22822  df-dvn 22823  df-tayl 23310
This theorem is referenced by:  dvntaylp0  23327
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