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Theorem tayl0 21849
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 21401 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
42, 3syl 16 . . . . 5  |-  ( ph  ->  S  C_  CC )
51, 4sstrd 3387 . . . 4  |-  ( ph  ->  A  C_  CC )
6 0xr 9451 . . . . . . . . 9  |-  0  e.  RR*
76a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR* )
8 taylfval.n . . . . . . . . 9  |-  ( ph  ->  ( N  e.  NN0  \/  N  = +oo )
)
9 nn0re 10609 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  RR )
109rexrd 9454 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e. 
RR* )
11 id 22 . . . . . . . . . . 11  |-  ( N  = +oo  ->  N  = +oo )
12 pnfxr 11113 . . . . . . . . . . 11  |- +oo  e.  RR*
1311, 12syl6eqel 2531 . . . . . . . . . 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 11133 . . . . . . . . 9  |-  ( ( N  e.  NN0  \/  N  = +oo )  ->  0  <_  N )
178, 16syl 16 . . . . . . . 8  |-  ( ph  ->  0  <_  N )
18 lbicc2 11422 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  N  e.  RR*  /\  0  <_  N )  ->  0  e.  ( 0 [,] N
) )
197, 15, 17, 18syl3anc 1218 . . . . . . 7  |-  ( ph  ->  0  e.  ( 0 [,] N ) )
20 0zd 10679 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
2119, 20elind 3561 . . . . . 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 2820 . . . . . 6  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) )
24 fveq2 5712 . . . . . . . . 9  |-  ( k  =  0  ->  (
( S  Dn
F ) `  k
)  =  ( ( S  Dn F ) `  0 ) )
2524dmeqd 5063 . . . . . . . 8  |-  ( k  =  0  ->  dom  ( ( S  Dn F ) `  k )  =  dom  ( ( S  Dn F ) ` 
0 ) )
2625eleq2d 2510 . . . . . . 7  |-  ( k  =  0  ->  ( B  e.  dom  ( ( S  Dn F ) `  k )  <-> 
B  e.  dom  (
( S  Dn
F ) `  0
) ) )
2726rspcv 3090 . . . . . 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 9384 . . . . . . . . . 10  |-  CC  e.  _V
3029a1i 11 . . . . . . . . 9  |-  ( ph  ->  CC  e.  _V )
31 taylfval.f . . . . . . . . 9  |-  ( ph  ->  F : A --> CC )
32 elpm2r 7251 . . . . . . . . 9  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
3330, 2, 31, 1, 32syl22anc 1219 . . . . . . . 8  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
34 dvn0 21420 . . . . . . . 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 5063 . . . . . 6  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 0 )  =  dom  F )
37 fdm 5584 . . . . . . 7  |-  ( F : A --> CC  ->  dom 
F  =  A )
3831, 37syl 16 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
3936, 38eqtrd 2475 . . . . 5  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 0 )  =  A )
4028, 39eleqtrd 2519 . . . 4  |-  ( ph  ->  B  e.  A )
415, 40sseldd 3378 . . 3  |-  ( ph  ->  B  e.  CC )
42 cnfldbas 17844 . . . . . . 7  |-  CC  =  ( Base ` fld )
43 cnfld0 17862 . . . . . . 7  |-  0  =  ( 0g ` fld )
44 cnrng 17860 . . . . . . . 8  |-fld  e.  Ring
45 rngmnd 16676 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
4644, 45mp1i 12 . . . . . . 7  |-  ( ph  ->fld  e. 
Mnd )
47 ovex 6137 . . . . . . . . 9  |-  ( 0 [,] N )  e. 
_V
4847inex1 4454 . . . . . . . 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 3592 . . . . . . . . . . . . . 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 3375 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
55 inss1 3591 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
5655, 53sseldi 3375 . . . . . . . . . . . . . . 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 11365 . . . . . . . . . . . . . . . 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 1001 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  <_  k )
62 elnn0z 10680 . . . . . . . . . . . . 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 21423 . . . . . . . . . . . 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 1218 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( S  Dn
F ) `  k
) : dom  (
( S  Dn
F ) `  k
) --> CC )
6665, 22ffvelrnd 5865 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( S  Dn F ) `  k ) `  B
)  e.  CC )
67 faccl 12082 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
6863, 67syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  NN )
6968nncnd 10359 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
7068nnne0d 10387 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
7166, 69, 70divcld 10128 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
72 0cnd 9400 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  e.  CC )
7372, 63expcld 12029 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
0 ^ k )  e.  CC )
7471, 73mulcld 9427 . . . . . . . 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 5888 . . . . . . 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 3499 . . . . . . . . . . . . 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 4022 . . . . . . . . . . . . 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 10614 . . . . . . . . . . . 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 12045 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( 0 ^ k )  =  0 )
8483oveq2d 6128 . . . . . . . . 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 9589 . . . . . . . . . 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 2475 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) )  =  0 )
88 zex 10676 . . . . . . . . . 10  |-  ZZ  e.  _V
8988inex2 4455 . . . . . . . . 9  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
9089a1i 11 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  e.  _V )
9187, 90suppss2 6744 . . . . . . 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 16476 . . . . . 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 5714 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( ( S  Dn F ) `  k ) `  B
)  =  ( ( ( S  Dn
F ) `  0
) `  B )
)
94 fveq2 5712 . . . . . . . . . . 11  |-  ( k  =  0  ->  ( ! `  k )  =  ( ! ` 
0 ) )
95 fac0 12075 . . . . . . . . . . 11  |-  ( ! `
 0 )  =  1
9694, 95syl6eq 2491 . . . . . . . . . 10  |-  ( k  =  0  ->  ( ! `  k )  =  1 )
9793, 96oveq12d 6130 . . . . . . . . 9  |-  ( k  =  0  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  =  ( ( ( ( S  Dn F ) ` 
0 ) `  B
)  /  1 ) )
98 oveq2 6120 . . . . . . . . . 10  |-  ( k  =  0  ->  (
0 ^ k )  =  ( 0 ^ 0 ) )
99 0exp0e1 11891 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
10098, 99syl6eq 2491 . . . . . . . . 9  |-  ( k  =  0  ->  (
0 ^ k )  =  1 )
10197, 100oveq12d 6130 . . . . . . . 8  |-  ( k  =  0  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  =  ( ( ( ( ( S  Dn F ) `
 0 ) `  B )  /  1
)  x.  1 ) )
102 ovex 6137 . . . . . . . 8  |-  ( ( ( ( ( S  Dn F ) `
 0 ) `  B )  /  1
)  x.  1 )  e.  _V
103101, 75, 102fvmpt 5795 . . . . . . 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 5714 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( S  Dn F ) `
 0 ) `  B )  =  ( F `  B ) )
106105oveq1d 6127 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( S  Dn F ) `  0 ) `
 B )  / 
1 )  =  ( ( F `  B
)  /  1 ) )
10731, 40ffvelrnd 5865 . . . . . . . . . 10  |-  ( ph  ->  ( F `  B
)  e.  CC )
108107div1d 10120 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  /  1
)  =  ( F `
 B ) )
109106, 108eqtrd 2475 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( S  Dn F ) `  0 ) `
 B )  / 
1 )  =  ( F `  B ) )
110109oveq1d 6127 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( ( F `  B
)  x.  1 ) )
111107mulid1d 9424 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  1 )  =  ( F `
 B ) )
112110, 111eqtrd 2475 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( F `  B ) )
11392, 104, 1123eqtrd 2479 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  =  ( F `  B ) )
114 rngcmn 16697 . . . . . . 7  |-  (fld  e.  Ring  ->fld  e. CMnd )
11544, 114mp1i 12 . . . . . 6  |-  ( ph  ->fld  e. CMnd
)
116 cnfldtps 20379 . . . . . . 7  |-fld  e.  TopSp
117116a1i 11 . . . . . 6  |-  ( ph  ->fld  e. 
TopSp )
118 mptexg 5968 . . . . . . . 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 5475 . . . . . . . 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 9401 . . . . . . . 8  |-  0  e.  _V
123122a1i 11 . . . . . . 7  |-  ( ph  ->  0  e.  _V )
124 snfi 7411 . . . . . . . 8  |-  { 0 }  e.  Fin
125124a1i 11 . . . . . . 7  |-  ( ph  ->  { 0 }  e.  Fin )
126 suppssfifsupp 7656 . . . . . . 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 1226 . . . . . 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 19732 . . . . 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 2518 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) ) ) ) )
13041subidd 9728 . . . . . . . 8  |-  ( ph  ->  ( B  -  B
)  =  0 )
131130oveq1d 6127 . . . . . . 7  |-  ( ph  ->  ( ( B  -  B ) ^ k
)  =  ( 0 ^ k ) )
132131oveq2d 6128 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) )  =  ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) ) )
133132mpteq2dv 4400 . . . . 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 6128 . . . 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 2520 . . 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 21847 . . 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 913 . 2  |-  ( ph  ->  B T ( F `
 B ) )
1392, 31, 1, 8, 22, 136taylf 21848 . . 3  |-  ( ph  ->  T : dom  T --> CC )
140 ffun 5582 . . 3  |-  ( T : dom  T --> CC  ->  Fun 
T )
141 funbrfv2b 5757 . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   _Vcvv 2993    \ cdif 3346    i^i cin 3348    C_ wss 3349   {csn 3898   {cpr 3900   class class class wbr 4313    e. cmpt 4371   dom cdm 4861   Fun wfun 5433   -->wf 5435   ` cfv 5439  (class class class)co 6112   supp csupp 6711    ^pm cpm 7236   Fincfn 7331   finSupp cfsupp 7641   CCcc 9301   RRcr 9302   0cc0 9303   1c1 9304    x. cmul 9308   +oocpnf 9436   RR*cxr 9438    <_ cle 9440    - cmin 9616    / cdiv 10014   NNcn 10343   NN0cn0 10600   ZZcz 10667   [,]cicc 11324   ^cexp 11886   !cfa 12072    gsumg cgsu 14400   Mndcmnd 15430  CMndccmn 16298   Ringcrg 16667  ℂfldccnfld 17840   TopSpctps 18523   tsums ctsu 19718    Dncdvn 21361   Tayl ctayl 21840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-fi 7682  df-sup 7712  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-icc 11328  df-fz 11459  df-fzo 11570  df-seq 11828  df-exp 11887  df-fac 12073  df-hash 12125  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-rest 14382  df-topn 14383  df-0g 14401  df-gsum 14402  df-topgen 14403  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-submnd 15486  df-grp 15566  df-minusg 15567  df-mulg 15569  df-cntz 15856  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-cring 16670  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-fbas 17836  df-fg 17837  df-cnfld 17841  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cld 18645  df-ntr 18646  df-cls 18647  df-nei 18724  df-lp 18762  df-perf 18763  df-cnp 18854  df-haus 18941  df-fil 19441  df-fm 19533  df-flim 19534  df-flf 19535  df-tsms 19719  df-xms 19917  df-ms 19918  df-limc 21363  df-dv 21364  df-dvn 21365  df-tayl 21842
This theorem is referenced by:  dvntaylp0  21859
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