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Theorem tayl0 22519
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 22071 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
42, 3syl 16 . . . . 5  |-  ( ph  ->  S  C_  CC )
51, 4sstrd 3514 . . . 4  |-  ( ph  ->  A  C_  CC )
6 0xr 9640 . . . . . . . . 9  |-  0  e.  RR*
76a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR* )
8 taylfval.n . . . . . . . . 9  |-  ( ph  ->  ( N  e.  NN0  \/  N  = +oo )
)
9 nn0re 10804 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  RR )
109rexrd 9643 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e. 
RR* )
11 id 22 . . . . . . . . . . 11  |-  ( N  = +oo  ->  N  = +oo )
12 pnfxr 11321 . . . . . . . . . . 11  |- +oo  e.  RR*
1311, 12syl6eqel 2563 . . . . . . . . . 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 11341 . . . . . . . . 9  |-  ( ( N  e.  NN0  \/  N  = +oo )  ->  0  <_  N )
178, 16syl 16 . . . . . . . 8  |-  ( ph  ->  0  <_  N )
18 lbicc2 11636 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  N  e.  RR*  /\  0  <_  N )  ->  0  e.  ( 0 [,] N
) )
197, 15, 17, 18syl3anc 1228 . . . . . . 7  |-  ( ph  ->  0  e.  ( 0 [,] N ) )
20 0zd 10876 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
2119, 20elind 3688 . . . . . 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 2878 . . . . . 6  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  Dn F ) `  k ) )
24 fveq2 5866 . . . . . . . . 9  |-  ( k  =  0  ->  (
( S  Dn
F ) `  k
)  =  ( ( S  Dn F ) `  0 ) )
2524dmeqd 5205 . . . . . . . 8  |-  ( k  =  0  ->  dom  ( ( S  Dn F ) `  k )  =  dom  ( ( S  Dn F ) ` 
0 ) )
2625eleq2d 2537 . . . . . . 7  |-  ( k  =  0  ->  ( B  e.  dom  ( ( S  Dn F ) `  k )  <-> 
B  e.  dom  (
( S  Dn
F ) `  0
) ) )
2726rspcv 3210 . . . . . 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 9573 . . . . . . . . . 10  |-  CC  e.  _V
3029a1i 11 . . . . . . . . 9  |-  ( ph  ->  CC  e.  _V )
31 taylfval.f . . . . . . . . 9  |-  ( ph  ->  F : A --> CC )
32 elpm2r 7436 . . . . . . . . 9  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
3330, 2, 31, 1, 32syl22anc 1229 . . . . . . . 8  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
34 dvn0 22090 . . . . . . . 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 5205 . . . . . 6  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 0 )  =  dom  F )
37 fdm 5735 . . . . . . 7  |-  ( F : A --> CC  ->  dom 
F  =  A )
3831, 37syl 16 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
3936, 38eqtrd 2508 . . . . 5  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 0 )  =  A )
4028, 39eleqtrd 2557 . . . 4  |-  ( ph  ->  B  e.  A )
415, 40sseldd 3505 . . 3  |-  ( ph  ->  B  e.  CC )
42 cnfldbas 18223 . . . . . . 7  |-  CC  =  ( Base ` fld )
43 cnfld0 18241 . . . . . . 7  |-  0  =  ( 0g ` fld )
44 cnrng 18239 . . . . . . . 8  |-fld  e.  Ring
45 rngmnd 17009 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
4644, 45mp1i 12 . . . . . . 7  |-  ( ph  ->fld  e. 
Mnd )
47 ovex 6309 . . . . . . . . 9  |-  ( 0 [,] N )  e. 
_V
4847inex1 4588 . . . . . . . 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 3719 . . . . . . . . . . . . . 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 3502 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
55 inss1 3718 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
5655, 53sseldi 3502 . . . . . . . . . . . . . . 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 11573 . . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  <_  k )
62 elnn0z 10877 . . . . . . . . . . . . 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 22093 . . . . . . . . . . . 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 1228 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( S  Dn
F ) `  k
) : dom  (
( S  Dn
F ) `  k
) --> CC )
6665, 22ffvelrnd 6022 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( S  Dn F ) `  k ) `  B
)  e.  CC )
67 faccl 12331 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
6863, 67syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  NN )
6968nncnd 10552 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
7068nnne0d 10580 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
7166, 69, 70divcld 10320 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
72 0cnd 9589 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  e.  CC )
7372, 63expcld 12278 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
0 ^ k )  e.  CC )
7471, 73mulcld 9616 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  e.  CC )
75 eqid 2467 . . . . . . . 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 6045 . . . . . . 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 3626 . . . . . . . . . . . . 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 4153 . . . . . . . . . . . . 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 10809 . . . . . . . . . . . 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 12294 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( 0 ^ k )  =  0 )
8483oveq2d 6300 . . . . . . . . 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 9778 . . . . . . . . . 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 2508 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) )  =  0 )
88 zex 10873 . . . . . . . . . 10  |-  ZZ  e.  _V
8988inex2 4589 . . . . . . . . 9  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
9089a1i 11 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  e.  _V )
9187, 90suppss2 6934 . . . . . . 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 16791 . . . . . 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 5868 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( ( S  Dn F ) `  k ) `  B
)  =  ( ( ( S  Dn
F ) `  0
) `  B )
)
94 fveq2 5866 . . . . . . . . . . 11  |-  ( k  =  0  ->  ( ! `  k )  =  ( ! ` 
0 ) )
95 fac0 12324 . . . . . . . . . . 11  |-  ( ! `
 0 )  =  1
9694, 95syl6eq 2524 . . . . . . . . . 10  |-  ( k  =  0  ->  ( ! `  k )  =  1 )
9793, 96oveq12d 6302 . . . . . . . . 9  |-  ( k  =  0  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  =  ( ( ( ( S  Dn F ) ` 
0 ) `  B
)  /  1 ) )
98 oveq2 6292 . . . . . . . . . 10  |-  ( k  =  0  ->  (
0 ^ k )  =  ( 0 ^ 0 ) )
99 0exp0e1 12139 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
10098, 99syl6eq 2524 . . . . . . . . 9  |-  ( k  =  0  ->  (
0 ^ k )  =  1 )
10197, 100oveq12d 6302 . . . . . . . 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 5950 . . . . . . 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 5868 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( S  Dn F ) `
 0 ) `  B )  =  ( F `  B ) )
106105oveq1d 6299 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( S  Dn F ) `  0 ) `
 B )  / 
1 )  =  ( ( F `  B
)  /  1 ) )
10731, 40ffvelrnd 6022 . . . . . . . . . 10  |-  ( ph  ->  ( F `  B
)  e.  CC )
108107div1d 10312 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  /  1
)  =  ( F `
 B ) )
109106, 108eqtrd 2508 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( S  Dn F ) `  0 ) `
 B )  / 
1 )  =  ( F `  B ) )
110109oveq1d 6299 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( ( F `  B
)  x.  1 ) )
111107mulid1d 9613 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  1 )  =  ( F `
 B ) )
112110, 111eqtrd 2508 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( F `  B ) )
11392, 104, 1123eqtrd 2512 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  =  ( F `  B ) )
114 rngcmn 17030 . . . . . . 7  |-  (fld  e.  Ring  ->fld  e. CMnd )
11544, 114mp1i 12 . . . . . 6  |-  ( ph  ->fld  e. CMnd
)
116 cnfldtps 21048 . . . . . . 7  |-fld  e.  TopSp
117116a1i 11 . . . . . 6  |-  ( ph  ->fld  e. 
TopSp )
118 mptexg 6130 . . . . . . . 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 5624 . . . . . . . 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 9590 . . . . . . . 8  |-  0  e.  _V
123122a1i 11 . . . . . . 7  |-  ( ph  ->  0  e.  _V )
124 snfi 7596 . . . . . . . 8  |-  { 0 }  e.  Fin
125124a1i 11 . . . . . . 7  |-  ( ph  ->  { 0 }  e.  Fin )
126 suppssfifsupp 7844 . . . . . . 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 1236 . . . . . 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 20401 . . . . 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 2556 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( 0 ^ k ) ) ) ) )
13041subidd 9918 . . . . . . . 8  |-  ( ph  ->  ( B  -  B
)  =  0 )
131130oveq1d 6299 . . . . . . 7  |-  ( ph  ->  ( ( B  -  B ) ^ k
)  =  ( 0 ^ k ) )
132131oveq2d 6300 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) )  =  ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) ) )
133132mpteq2dv 4534 . . . . 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 6300 . . . 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 2558 . . 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 22517 . . 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 920 . 2  |-  ( ph  ->  B T ( F `
 B ) )
1392, 31, 1, 8, 22, 136taylf 22518 . . 3  |-  ( ph  ->  T : dom  T --> CC )
140 ffun 5733 . . 3  |-  ( T : dom  T --> CC  ->  Fun 
T )
141 funbrfv2b 5912 . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    \ cdif 3473    i^i cin 3475    C_ wss 3476   {csn 4027   {cpr 4029   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   Fun wfun 5582   -->wf 5584   ` cfv 5588  (class class class)co 6284   supp csupp 6901    ^pm cpm 7421   Fincfn 7516   finSupp cfsupp 7829   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    x. cmul 9497   +oocpnf 9625   RR*cxr 9627    <_ cle 9629    - cmin 9805    / cdiv 10206   NNcn 10536   NN0cn0 10795   ZZcz 10864   [,]cicc 11532   ^cexp 12134   !cfa 12321    gsumg cgsu 14696   Mndcmnd 15726  CMndccmn 16604   Ringcrg 17000  ℂfldccnfld 18219   TopSpctps 19192   tsums ctsu 20387    Dncdvn 22031   Tayl ctayl 22510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-icc 11536  df-fz 11673  df-fzo 11793  df-seq 12076  df-exp 12135  df-fac 12322  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-grp 15867  df-minusg 15868  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cnp 19523  df-haus 19610  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-tsms 20388  df-xms 20586  df-ms 20587  df-limc 22033  df-dv 22034  df-dvn 22035  df-tayl 22512
This theorem is referenced by:  dvntaylp0  22529
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