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Theorem tayl0 22629
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 22181 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
42, 3syl 16 . . . . 5  |-  ( ph  ->  S  C_  CC )
51, 4sstrd 3499 . . . 4  |-  ( ph  ->  A  C_  CC )
6 0xr 9643 . . . . . . . . 9  |-  0  e.  RR*
76a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR* )
8 taylfval.n . . . . . . . . 9  |-  ( ph  ->  ( N  e.  NN0  \/  N  = +oo )
)
9 nn0re 10810 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  RR )
109rexrd 9646 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e. 
RR* )
11 id 22 . . . . . . . . . . 11  |-  ( N  = +oo  ->  N  = +oo )
12 pnfxr 11330 . . . . . . . . . . 11  |- +oo  e.  RR*
1311, 12syl6eqel 2539 . . . . . . . . . 10  |-  ( N  = +oo  ->  N  e.  RR* )
1410, 13jaoi 379 . . . . . . . . 9  |-  ( ( N  e.  NN0  \/  N  = +oo )  ->  N  e.  RR* )
158, 14syl 16 . . . . . . . 8  |-  ( ph  ->  N  e.  RR* )
16 nn0pnfge0 11350 . . . . . . . . 9  |-  ( ( N  e.  NN0  \/  N  = +oo )  ->  0  <_  N )
178, 16syl 16 . . . . . . . 8  |-  ( ph  ->  0  <_  N )
18 lbicc2 11645 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  N  e.  RR*  /\  0  <_  N )  ->  0  e.  ( 0 [,] N
) )
197, 15, 17, 18syl3anc 1229 . . . . . . 7  |-  ( ph  ->  0  e.  ( 0 [,] N ) )
20 0zd 10882 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
2119, 20elind 3673 . . . . . 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 2857 . . . . . 6  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) )
24 fveq2 5856 . . . . . . . . 9  |-  ( k  =  0  ->  (
( S  Dn
F ) `  k
)  =  ( ( S  Dn F ) `  0 ) )
2524dmeqd 5195 . . . . . . . 8  |-  ( k  =  0  ->  dom  ( ( S  Dn F ) `  k )  =  dom  ( ( S  Dn F ) ` 
0 ) )
2625eleq2d 2513 . . . . . . 7  |-  ( k  =  0  ->  ( B  e.  dom  ( ( S  Dn F ) `  k )  <-> 
B  e.  dom  (
( S  Dn
F ) `  0
) ) )
2726rspcv 3192 . . . . . 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 60 . . . . 5  |-  ( ph  ->  B  e.  dom  (
( S  Dn
F ) `  0
) )
29 cnex 9576 . . . . . . . . . 10  |-  CC  e.  _V
3029a1i 11 . . . . . . . . 9  |-  ( ph  ->  CC  e.  _V )
31 taylfval.f . . . . . . . . 9  |-  ( ph  ->  F : A --> CC )
32 elpm2r 7438 . . . . . . . . 9  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
3330, 2, 31, 1, 32syl22anc 1230 . . . . . . . 8  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
34 dvn0 22200 . . . . . . . 8  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  (
( S  Dn
F ) `  0
)  =  F )
354, 33, 34syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( S  Dn F ) ` 
0 )  =  F )
3635dmeqd 5195 . . . . . 6  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 0 )  =  dom  F )
37 fdm 5725 . . . . . . 7  |-  ( F : A --> CC  ->  dom 
F  =  A )
3831, 37syl 16 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
3936, 38eqtrd 2484 . . . . 5  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 0 )  =  A )
4028, 39eleqtrd 2533 . . . 4  |-  ( ph  ->  B  e.  A )
415, 40sseldd 3490 . . 3  |-  ( ph  ->  B  e.  CC )
42 cnfldbas 18298 . . . . . . 7  |-  CC  =  ( Base ` fld )
43 cnfld0 18316 . . . . . . 7  |-  0  =  ( 0g ` fld )
44 cnring 18314 . . . . . . . 8  |-fld  e.  Ring
45 ringmnd 17081 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
4644, 45mp1i 12 . . . . . . 7  |-  ( ph  ->fld  e. 
Mnd )
47 ovex 6309 . . . . . . . . 9  |-  ( 0 [,] N )  e. 
_V
4847inex1 4578 . . . . . . . 8  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
4948a1i 11 . . . . . . 7  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  e.  _V )
502adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  S  e.  { RR ,  CC } )
5133adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  F  e.  ( CC  ^pm  S
) )
52 inss2 3704 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ZZ
53 simpr 461 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )
5452, 53sseldi 3487 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
55 inss1 3703 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
5655, 53sseldi 3487 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( 0 [,] N
) )
5715adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  N  e.  RR* )
58 elicc1 11582 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR*  /\  N  e.  RR* )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
596, 57, 58sylancr 663 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
6056, 59mpbid 210 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  RR*  /\  0  <_  k  /\  k  <_  N ) )
6160simp2d 1010 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  <_  k )
62 elnn0z 10883 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  <->  ( k  e.  ZZ  /\  0  <_ 
k ) )
6354, 61, 62sylanbrc 664 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  NN0 )
64 dvnf 22203 . . . . . . . . . . . 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 1229 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( S  Dn
F ) `  k
) : dom  (
( S  Dn
F ) `  k
) --> CC )
6665, 22ffvelrnd 6017 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( S  Dn F ) `  k ) `  B
)  e.  CC )
67 faccl 12342 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
6863, 67syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  NN )
6968nncnd 10558 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
7068nnne0d 10586 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
7166, 69, 70divcld 10326 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
72 0cnd 9592 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  e.  CC )
7372, 63expcld 12289 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
0 ^ k )  e.  CC )
7471, 73mulcld 9619 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  e.  CC )
75 eqid 2443 . . . . . . . 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 6040 . . . . . . 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 3611 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( ( 0 [,] N )  i^i  ZZ )  \  { 0 } )  ->  k  e.  ( ( 0 [,] N
)  i^i  ZZ )
)
7877, 63sylan2 474 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  e.  NN0 )
79 eldifsni 4141 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( ( 0 [,] N )  i^i  ZZ )  \  { 0 } )  ->  k  =/=  0
)
8079adantl 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  =/=  0
)
81 elnnne0 10815 . . . . . . . . . . . 12  |-  ( k  e.  NN  <->  ( k  e.  NN0  /\  k  =/=  0 ) )
8278, 80, 81sylanbrc 664 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  e.  NN )
83820expd 12305 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( 0 ^ k )  =  0 )
8483oveq2d 6297 . . . . . . . . 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 9782 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  0 )  =  0 )
8677, 85sylan2 474 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  0 )  =  0 )
8784, 86eqtrd 2484 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) )  =  0 )
88 zex 10879 . . . . . . . . . 10  |-  ZZ  e.  _V
8988inex2 4579 . . . . . . . . 9  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
9089a1i 11 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  e.  _V )
9187, 90suppss2 6936 . . . . . . 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 16862 . . . . . 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 5858 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( ( S  Dn F ) `  k ) `  B
)  =  ( ( ( S  Dn
F ) `  0
) `  B )
)
94 fveq2 5856 . . . . . . . . . . 11  |-  ( k  =  0  ->  ( ! `  k )  =  ( ! ` 
0 ) )
95 fac0 12335 . . . . . . . . . . 11  |-  ( ! `
 0 )  =  1
9694, 95syl6eq 2500 . . . . . . . . . 10  |-  ( k  =  0  ->  ( ! `  k )  =  1 )
9793, 96oveq12d 6299 . . . . . . . . 9  |-  ( k  =  0  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  =  ( ( ( ( S  Dn F ) ` 
0 ) `  B
)  /  1 ) )
98 oveq2 6289 . . . . . . . . . 10  |-  ( k  =  0  ->  (
0 ^ k )  =  ( 0 ^ 0 ) )
99 0exp0e1 12150 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
10098, 99syl6eq 2500 . . . . . . . . 9  |-  ( k  =  0  ->  (
0 ^ k )  =  1 )
10197, 100oveq12d 6299 . . . . . . . 8  |-  ( k  =  0  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  =  ( ( ( ( ( S  Dn F ) `
 0 ) `  B )  /  1
)  x.  1 ) )
102 ovex 6309 . . . . . . . 8  |-  ( ( ( ( ( S  Dn F ) `
 0 ) `  B )  /  1
)  x.  1 )  e.  _V
103101, 75, 102fvmpt 5941 . . . . . . 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 16 . . . . . 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 5858 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( S  Dn F ) `
 0 ) `  B )  =  ( F `  B ) )
106105oveq1d 6296 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( S  Dn F ) `  0 ) `
 B )  / 
1 )  =  ( ( F `  B
)  /  1 ) )
10731, 40ffvelrnd 6017 . . . . . . . . . 10  |-  ( ph  ->  ( F `  B
)  e.  CC )
108107div1d 10318 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  /  1
)  =  ( F `
 B ) )
109106, 108eqtrd 2484 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( S  Dn F ) `  0 ) `
 B )  / 
1 )  =  ( F `  B ) )
110109oveq1d 6296 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( ( F `  B
)  x.  1 ) )
111107mulid1d 9616 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  1 )  =  ( F `
 B ) )
112110, 111eqtrd 2484 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( F `  B ) )
11392, 104, 1123eqtrd 2488 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  =  ( F `  B ) )
114 ringcmn 17103 . . . . . . 7  |-  (fld  e.  Ring  ->fld  e. CMnd )
11544, 114mp1i 12 . . . . . 6  |-  ( ph  ->fld  e. CMnd
)
116 cnfldtps 21158 . . . . . . 7  |-fld  e.  TopSp
117116a1i 11 . . . . . 6  |-  ( ph  ->fld  e. 
TopSp )
118 mptexg 6127 . . . . . . . 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 12 . . . . . . 7  |-  ( ph  ->  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) )  e. 
_V )
120 funmpt 5614 . . . . . . . 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 9593 . . . . . . . 8  |-  0  e.  _V
123122a1i 11 . . . . . . 7  |-  ( ph  ->  0  e.  _V )
124 snfi 7598 . . . . . . . 8  |-  { 0 }  e.  Fin
125124a1i 11 . . . . . . 7  |-  ( ph  ->  { 0 }  e.  Fin )
126 suppssfifsupp 7846 . . . . . . 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 1237 . . . . . 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 20511 . . . . 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 2532 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) ) ) ) )
13041subidd 9924 . . . . . . . 8  |-  ( ph  ->  ( B  -  B
)  =  0 )
131130oveq1d 6296 . . . . . . 7  |-  ( ph  ->  ( ( B  -  B ) ^ k
)  =  ( 0 ^ k ) )
132131oveq2d 6297 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) )  =  ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) ) )
133132mpteq2dv 4524 . . . . 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 6297 . . . 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 2534 . . 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 22627 . . 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 922 . 2  |-  ( ph  ->  B T ( F `
 B ) )
1392, 31, 1, 8, 22, 136taylf 22628 . . 3  |-  ( ph  ->  T : dom  T --> CC )
140 ffun 5723 . . 3  |-  ( T : dom  T --> CC  ->  Fun 
T )
141 funbrfv2b 5902 . . 3  |-  ( Fun 
T  ->  ( B T ( F `  B )  <->  ( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) ) )
142139, 140, 1413syl 20 . 2  |-  ( ph  ->  ( B T ( F `  B )  <-> 
( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) ) )
143138, 142mpbid 210 1  |-  ( ph  ->  ( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   _Vcvv 3095    \ cdif 3458    i^i cin 3460    C_ wss 3461   {csn 4014   {cpr 4016   class class class wbr 4437    |-> cmpt 4495   dom cdm 4989   Fun wfun 5572   -->wf 5574   ` cfv 5578  (class class class)co 6281   supp csupp 6903    ^pm cpm 7423   Fincfn 7518   finSupp cfsupp 7831   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    x. cmul 9500   +oocpnf 9628   RR*cxr 9630    <_ cle 9632    - cmin 9810    / cdiv 10212   NNcn 10542   NN0cn0 10801   ZZcz 10870   [,]cicc 11541   ^cexp 12145   !cfa 12332    gsumg cgsu 14715   Mndcmnd 15793  CMndccmn 16672   Ringcrg 17072  ℂfldccnfld 18294   TopSpctps 19270   tsums ctsu 20497    Dncdvn 22141   Tayl ctayl 22620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-icc 11545  df-fz 11682  df-fzo 11804  df-seq 12087  df-exp 12146  df-fac 12333  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-grp 15931  df-minusg 15932  df-mulg 15934  df-cntz 16229  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-ring 17074  df-cring 17075  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-lp 19510  df-perf 19511  df-cnp 19602  df-haus 19689  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-tsms 20498  df-xms 20696  df-ms 20697  df-limc 22143  df-dv 22144  df-dvn 22145  df-tayl 22622
This theorem is referenced by:  dvntaylp0  22639
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