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Theorem tayl0 23396
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 22938 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
42, 3syl 17 . . . . 5  |-  ( ph  ->  S  C_  CC )
51, 4sstrd 3428 . . . 4  |-  ( ph  ->  A  C_  CC )
6 0xr 9705 . . . . . . . . 9  |-  0  e.  RR*
76a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR* )
8 taylfval.n . . . . . . . . 9  |-  ( ph  ->  ( N  e.  NN0  \/  N  = +oo )
)
9 nn0re 10902 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  RR )
109rexrd 9708 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e. 
RR* )
11 id 22 . . . . . . . . . . 11  |-  ( N  = +oo  ->  N  = +oo )
12 pnfxr 11435 . . . . . . . . . . 11  |- +oo  e.  RR*
1311, 12syl6eqel 2557 . . . . . . . . . 10  |-  ( N  = +oo  ->  N  e.  RR* )
1410, 13jaoi 386 . . . . . . . . 9  |-  ( ( N  e.  NN0  \/  N  = +oo )  ->  N  e.  RR* )
158, 14syl 17 . . . . . . . 8  |-  ( ph  ->  N  e.  RR* )
16 nn0pnfge0 11457 . . . . . . . . 9  |-  ( ( N  e.  NN0  \/  N  = +oo )  ->  0  <_  N )
178, 16syl 17 . . . . . . . 8  |-  ( ph  ->  0  <_  N )
18 lbicc2 11774 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  N  e.  RR*  /\  0  <_  N )  ->  0  e.  ( 0 [,] N
) )
197, 15, 17, 18syl3anc 1292 . . . . . . 7  |-  ( ph  ->  0  e.  ( 0 [,] N ) )
20 0zd 10973 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
2119, 20elind 3609 . . . . . 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 2809 . . . . . 6  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) )
24 fveq2 5879 . . . . . . . . 9  |-  ( k  =  0  ->  (
( S  Dn
F ) `  k
)  =  ( ( S  Dn F ) `  0 ) )
2524dmeqd 5042 . . . . . . . 8  |-  ( k  =  0  ->  dom  ( ( S  Dn F ) `  k )  =  dom  ( ( S  Dn F ) ` 
0 ) )
2625eleq2d 2534 . . . . . . 7  |-  ( k  =  0  ->  ( B  e.  dom  ( ( S  Dn F ) `  k )  <-> 
B  e.  dom  (
( S  Dn
F ) `  0
) ) )
2726rspcv 3132 . . . . . 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 61 . . . . 5  |-  ( ph  ->  B  e.  dom  (
( S  Dn
F ) `  0
) )
29 cnex 9638 . . . . . . . . . 10  |-  CC  e.  _V
3029a1i 11 . . . . . . . . 9  |-  ( ph  ->  CC  e.  _V )
31 taylfval.f . . . . . . . . 9  |-  ( ph  ->  F : A --> CC )
32 elpm2r 7507 . . . . . . . . 9  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
3330, 2, 31, 1, 32syl22anc 1293 . . . . . . . 8  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
34 dvn0 22957 . . . . . . . 8  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  (
( S  Dn
F ) `  0
)  =  F )
354, 33, 34syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( ( S  Dn F ) ` 
0 )  =  F )
3635dmeqd 5042 . . . . . 6  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 0 )  =  dom  F )
37 fdm 5745 . . . . . . 7  |-  ( F : A --> CC  ->  dom 
F  =  A )
3831, 37syl 17 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
3936, 38eqtrd 2505 . . . . 5  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 0 )  =  A )
4028, 39eleqtrd 2551 . . . 4  |-  ( ph  ->  B  e.  A )
415, 40sseldd 3419 . . 3  |-  ( ph  ->  B  e.  CC )
42 cnfldbas 19051 . . . . . . 7  |-  CC  =  ( Base ` fld )
43 cnfld0 19069 . . . . . . 7  |-  0  =  ( 0g ` fld )
44 cnring 19067 . . . . . . . 8  |-fld  e.  Ring
45 ringmnd 17867 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
4644, 45mp1i 13 . . . . . . 7  |-  ( ph  ->fld  e. 
Mnd )
47 ovex 6336 . . . . . . . . 9  |-  ( 0 [,] N )  e. 
_V
4847inex1 4537 . . . . . . . 8  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
4948a1i 11 . . . . . . 7  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  e.  _V )
502adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  S  e.  { RR ,  CC } )
5133adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  F  e.  ( CC  ^pm  S
) )
52 inss2 3644 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ZZ
53 simpr 468 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )
5452, 53sseldi 3416 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
55 inss1 3643 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
5655, 53sseldi 3416 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( 0 [,] N
) )
5715adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  N  e.  RR* )
58 elicc1 11705 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR*  /\  N  e.  RR* )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
596, 57, 58sylancr 676 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
6056, 59mpbid 215 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  RR*  /\  0  <_  k  /\  k  <_  N ) )
6160simp2d 1043 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  <_  k )
62 elnn0z 10974 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  <->  ( k  e.  ZZ  /\  0  <_ 
k ) )
6354, 61, 62sylanbrc 677 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  NN0 )
64 dvnf 22960 . . . . . . . . . . . 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 1292 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( S  Dn
F ) `  k
) : dom  (
( S  Dn
F ) `  k
) --> CC )
6665, 22ffvelrnd 6038 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( S  Dn F ) `  k ) `  B
)  e.  CC )
67 faccl 12507 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
6863, 67syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  NN )
6968nncnd 10647 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
7068nnne0d 10676 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
7166, 69, 70divcld 10405 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
72 0cnd 9654 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  e.  CC )
7372, 63expcld 12454 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
0 ^ k )  e.  CC )
7471, 73mulcld 9681 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  e.  CC )
75 eqid 2471 . . . . . . . 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 6061 . . . . . . 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 3544 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( ( 0 [,] N )  i^i  ZZ )  \  { 0 } )  ->  k  e.  ( ( 0 [,] N
)  i^i  ZZ )
)
7877, 63sylan2 482 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  e.  NN0 )
79 eldifsni 4089 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( ( 0 [,] N )  i^i  ZZ )  \  { 0 } )  ->  k  =/=  0
)
8079adantl 473 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  =/=  0
)
81 elnnne0 10907 . . . . . . . . . . . 12  |-  ( k  e.  NN  <->  ( k  e.  NN0  /\  k  =/=  0 ) )
8278, 80, 81sylanbrc 677 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  e.  NN )
83820expd 12470 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( 0 ^ k )  =  0 )
8483oveq2d 6324 . . . . . . . . 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 9850 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  0 )  =  0 )
8677, 85sylan2 482 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  0 )  =  0 )
8784, 86eqtrd 2505 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) )  =  0 )
88 zex 10970 . . . . . . . . . 10  |-  ZZ  e.  _V
8988inex2 4538 . . . . . . . . 9  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
9089a1i 11 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  e.  _V )
9187, 90suppss2 6968 . . . . . . 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 17672 . . . . . 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 5881 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( ( S  Dn F ) `  k ) `  B
)  =  ( ( ( S  Dn
F ) `  0
) `  B )
)
94 fveq2 5879 . . . . . . . . . . 11  |-  ( k  =  0  ->  ( ! `  k )  =  ( ! ` 
0 ) )
95 fac0 12500 . . . . . . . . . . 11  |-  ( ! `
 0 )  =  1
9694, 95syl6eq 2521 . . . . . . . . . 10  |-  ( k  =  0  ->  ( ! `  k )  =  1 )
9793, 96oveq12d 6326 . . . . . . . . 9  |-  ( k  =  0  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  =  ( ( ( ( S  Dn F ) ` 
0 ) `  B
)  /  1 ) )
98 oveq2 6316 . . . . . . . . . 10  |-  ( k  =  0  ->  (
0 ^ k )  =  ( 0 ^ 0 ) )
99 0exp0e1 12315 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
10098, 99syl6eq 2521 . . . . . . . . 9  |-  ( k  =  0  ->  (
0 ^ k )  =  1 )
10197, 100oveq12d 6326 . . . . . . . 8  |-  ( k  =  0  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  =  ( ( ( ( ( S  Dn F ) `
 0 ) `  B )  /  1
)  x.  1 ) )
102 ovex 6336 . . . . . . . 8  |-  ( ( ( ( ( S  Dn F ) `
 0 ) `  B )  /  1
)  x.  1 )  e.  _V
103101, 75, 102fvmpt 5963 . . . . . . 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 5881 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( S  Dn F ) `
 0 ) `  B )  =  ( F `  B ) )
106105oveq1d 6323 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( S  Dn F ) `  0 ) `
 B )  / 
1 )  =  ( ( F `  B
)  /  1 ) )
10731, 40ffvelrnd 6038 . . . . . . . . . 10  |-  ( ph  ->  ( F `  B
)  e.  CC )
108107div1d 10397 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  /  1
)  =  ( F `
 B ) )
109106, 108eqtrd 2505 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( S  Dn F ) `  0 ) `
 B )  / 
1 )  =  ( F `  B ) )
110109oveq1d 6323 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( ( F `  B
)  x.  1 ) )
111107mulid1d 9678 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  1 )  =  ( F `
 B ) )
112110, 111eqtrd 2505 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( F `  B ) )
11392, 104, 1123eqtrd 2509 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  =  ( F `  B ) )
114 ringcmn 17889 . . . . . . 7  |-  (fld  e.  Ring  ->fld  e. CMnd )
11544, 114mp1i 13 . . . . . 6  |-  ( ph  ->fld  e. CMnd
)
116 cnfldtps 21876 . . . . . . 7  |-fld  e.  TopSp
117116a1i 11 . . . . . 6  |-  ( ph  ->fld  e. 
TopSp )
118 mptexg 6151 . . . . . . . 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 5625 . . . . . . . 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 9655 . . . . . . . 8  |-  0  e.  _V
123122a1i 11 . . . . . . 7  |-  ( ph  ->  0  e.  _V )
124 snfi 7668 . . . . . . . 8  |-  { 0 }  e.  Fin
125124a1i 11 . . . . . . 7  |-  ( ph  ->  { 0 }  e.  Fin )
126 suppssfifsupp 7916 . . . . . . 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 1300 . . . . . 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 21232 . . . . 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 2550 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) ) ) ) )
13041subidd 9993 . . . . . . . 8  |-  ( ph  ->  ( B  -  B
)  =  0 )
131130oveq1d 6323 . . . . . . 7  |-  ( ph  ->  ( ( B  -  B ) ^ k
)  =  ( 0 ^ k ) )
132131oveq2d 6324 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) )  =  ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) ) )
133132mpteq2dv 4483 . . . . 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 6324 . . . 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 2552 . . 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 23394 . . 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 936 . 2  |-  ( ph  ->  B T ( F `
 B ) )
1392, 31, 1, 8, 22, 136taylf 23395 . . 3  |-  ( ph  ->  T : dom  T --> CC )
140 ffun 5742 . . 3  |-  ( T : dom  T --> CC  ->  Fun 
T )
141 funbrfv2b 5923 . . 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 215 1  |-  ( ph  ->  ( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    \ cdif 3387    i^i cin 3389    C_ wss 3390   {csn 3959   {cpr 3961   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   supp csupp 6933    ^pm cpm 7491   Fincfn 7587   finSupp cfsupp 7901   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562   +oocpnf 9690   RR*cxr 9692    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   NN0cn0 10893   ZZcz 10961   [,]cicc 11663   ^cexp 12310   !cfa 12497    gsumg cgsu 15417   Mndcmnd 16613  CMndccmn 17508   Ringcrg 17858  ℂfldccnfld 19047   TopSpctps 19996   tsums ctsu 21218    Dncdvn 22898   Tayl ctayl 23387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-fac 12498  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cnp 20321  df-haus 20408  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-tsms 21219  df-xms 21413  df-ms 21414  df-limc 22900  df-dv 22901  df-dvn 22902  df-tayl 23389
This theorem is referenced by:  dvntaylp0  23406
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