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Theorem tayl0 20270
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  D n 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 19783 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
42, 3syl 16 . . . . 5  |-  ( ph  ->  S  C_  CC )
51, 4sstrd 3350 . . . 4  |-  ( ph  ->  A  C_  CC )
6 0xr 9123 . . . . . . . . 9  |-  0  e.  RR*
76a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR* )
8 taylfval.n . . . . . . . . 9  |-  ( ph  ->  ( N  e.  NN0  \/  N  =  +oo )
)
9 nn0re 10222 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  RR )
109rexrd 9126 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e. 
RR* )
11 id 20 . . . . . . . . . . 11  |-  ( N  =  +oo  ->  N  =  +oo )
12 pnfxr 10705 . . . . . . . . . . 11  |-  +oo  e.  RR*
1311, 12syl6eqel 2523 . . . . . . . . . 10  |-  ( N  =  +oo  ->  N  e.  RR* )
1410, 13jaoi 369 . . . . . . . . 9  |-  ( ( N  e.  NN0  \/  N  =  +oo )  ->  N  e.  RR* )
158, 14syl 16 . . . . . . . 8  |-  ( ph  ->  N  e.  RR* )
16 nn0pnfge0 10720 . . . . . . . . 9  |-  ( ( N  e.  NN0  \/  N  =  +oo )  -> 
0  <_  N )
178, 16syl 16 . . . . . . . 8  |-  ( ph  ->  0  <_  N )
18 lbicc2 11005 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  N  e.  RR*  /\  0  <_  N )  ->  0  e.  ( 0 [,] N
) )
197, 15, 17, 18syl3anc 1184 . . . . . . 7  |-  ( ph  ->  0  e.  ( 0 [,] N ) )
20 0z 10285 . . . . . . . 8  |-  0  e.  ZZ
2120a1i 11 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
22 elin 3522 . . . . . . 7  |-  ( 0  e.  ( ( 0 [,] N )  i^i 
ZZ )  <->  ( 0  e.  ( 0 [,] N )  /\  0  e.  ZZ ) )
2319, 21, 22sylanbrc 646 . . . . . 6  |-  ( ph  ->  0  e.  ( ( 0 [,] N )  i^i  ZZ ) )
24 taylfval.b . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  B  e.  dom  ( ( S  D n F ) `
 k ) )
2524ralrimiva 2781 . . . . . 6  |-  ( ph  ->  A. k  e.  ( ( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k ) )
26 fveq2 5720 . . . . . . . . 9  |-  ( k  =  0  ->  (
( S  D n F ) `  k
)  =  ( ( S  D n F ) `  0 ) )
2726dmeqd 5064 . . . . . . . 8  |-  ( k  =  0  ->  dom  ( ( S  D n F ) `  k
)  =  dom  (
( S  D n F ) `  0
) )
2827eleq2d 2502 . . . . . . 7  |-  ( k  =  0  ->  ( B  e.  dom  ( ( S  D n F ) `  k )  <-> 
B  e.  dom  (
( S  D n F ) `  0
) ) )
2928rspcv 3040 . . . . . 6  |-  ( 0  e.  ( ( 0 [,] N )  i^i 
ZZ )  ->  ( A. k  e.  (
( 0 [,] N
)  i^i  ZZ ) B  e.  dom  ( ( S  D n F ) `  k )  ->  B  e.  dom  ( ( S  D n F ) `  0
) ) )
3023, 25, 29sylc 58 . . . . 5  |-  ( ph  ->  B  e.  dom  (
( S  D n F ) `  0
) )
31 cnex 9063 . . . . . . . . . 10  |-  CC  e.  _V
3231a1i 11 . . . . . . . . 9  |-  ( ph  ->  CC  e.  _V )
33 taylfval.f . . . . . . . . 9  |-  ( ph  ->  F : A --> CC )
34 elpm2r 7026 . . . . . . . . 9  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
3532, 2, 33, 1, 34syl22anc 1185 . . . . . . . 8  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
36 dvn0 19802 . . . . . . . 8  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
) )  ->  (
( S  D n F ) `  0
)  =  F )
374, 35, 36syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( S  D n F ) `  0
)  =  F )
3837dmeqd 5064 . . . . . 6  |-  ( ph  ->  dom  ( ( S  D n F ) `
 0 )  =  dom  F )
39 fdm 5587 . . . . . . 7  |-  ( F : A --> CC  ->  dom 
F  =  A )
4033, 39syl 16 . . . . . 6  |-  ( ph  ->  dom  F  =  A )
4138, 40eqtrd 2467 . . . . 5  |-  ( ph  ->  dom  ( ( S  D n F ) `
 0 )  =  A )
4230, 41eleqtrd 2511 . . . 4  |-  ( ph  ->  B  e.  A )
435, 42sseldd 3341 . . 3  |-  ( ph  ->  B  e.  CC )
44 cnfldbas 16699 . . . . . . 7  |-  CC  =  ( Base ` fld )
45 cnfld0 16717 . . . . . . 7  |-  0  =  ( 0g ` fld )
46 cnrng 16715 . . . . . . . 8  |-fld  e.  Ring
47 rngmnd 15665 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
4846, 47mp1i 12 . . . . . . 7  |-  ( ph  ->fld  e. 
Mnd )
49 ovex 6098 . . . . . . . . 9  |-  ( 0 [,] N )  e. 
_V
5049inex1 4336 . . . . . . . 8  |-  ( ( 0 [,] N )  i^i  ZZ )  e. 
_V
5150a1i 11 . . . . . . 7  |-  ( ph  ->  ( ( 0 [,] N )  i^i  ZZ )  e.  _V )
522adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  S  e.  { RR ,  CC } )
5335adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  F  e.  ( CC  ^pm  S
) )
54 inss2 3554 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ZZ
55 simpr 448 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )
5654, 55sseldi 3338 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ZZ )
57 inss1 3553 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] N )  i^i  ZZ )  C_  ( 0 [,] N
)
5857, 55sseldi 3338 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  ( 0 [,] N
) )
5915adantr 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  N  e.  RR* )
60 elicc1 10952 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR*  /\  N  e.  RR* )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
616, 59, 60sylancr 645 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  ( 0 [,] N )  <->  ( k  e.  RR*  /\  0  <_ 
k  /\  k  <_  N ) ) )
6258, 61mpbid 202 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
k  e.  RR*  /\  0  <_  k  /\  k  <_  N ) )
6362simp2d 970 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  <_  k )
64 elnn0z 10286 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  <->  ( k  e.  ZZ  /\  0  <_ 
k ) )
6556, 63, 64sylanbrc 646 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  k  e.  NN0 )
66 dvnf 19805 . . . . . . . . . . . 12  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  k  e.  NN0 )  ->  ( ( S  D n F ) `
 k ) : dom  ( ( S  D n F ) `
 k ) --> CC )
6752, 53, 65, 66syl3anc 1184 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( S  D n F ) `  k
) : dom  (
( S  D n F ) `  k
) --> CC )
6867, 24ffvelrnd 5863 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( S  D n F ) `  k
) `  B )  e.  CC )
69 faccl 11568 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
7065, 69syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  NN )
7170nncnd 10008 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  e.  CC )
7270nnne0d 10036 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  ( ! `  k )  =/=  0 )
7368, 71, 72divcld 9782 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
74 0cn 9076 . . . . . . . . . . 11  |-  0  e.  CC
7574a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  0  e.  CC )
7675, 65expcld 11515 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
0 ^ k )  e.  CC )
7773, 76mulcld 9100 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  e.  CC )
78 eqid 2435 . . . . . . . 8  |-  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) )  =  ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) )
7977, 78fmptd 5885 . . . . . . 7  |-  ( ph  ->  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) : ( ( 0 [,] N )  i^i  ZZ )
--> CC )
80 eldifi 3461 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( ( 0 [,] N )  i^i  ZZ )  \  { 0 } )  ->  k  e.  ( ( 0 [,] N
)  i^i  ZZ )
)
8180, 65sylan2 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  e.  NN0 )
82 eldifsni 3920 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( ( 0 [,] N )  i^i  ZZ )  \  { 0 } )  ->  k  =/=  0
)
8382adantl 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  =/=  0
)
84 elnnne0 10227 . . . . . . . . . . . 12  |-  ( k  e.  NN  <->  ( k  e.  NN0  /\  k  =/=  0 ) )
8581, 83, 84sylanbrc 646 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  k  e.  NN )
86850expd 11531 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( 0 ^ k )  =  0 )
8786oveq2d 6089 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) )  =  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  0 ) )
8873mul01d 9257 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( 0 [,] N )  i^i  ZZ ) )  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  0 )  =  0 )
8980, 88sylan2 461 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  0 )  =  0 )
9087, 89eqtrd 2467 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 [,] N )  i^i 
ZZ )  \  {
0 } ) )  ->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) )  =  0 )
9190suppss2 6292 . . . . . . 7  |-  ( ph  ->  ( `' ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) ) " ( _V 
\  { 0 } ) )  C_  { 0 } )
9244, 45, 48, 51, 23, 79, 91gsumpt 15537 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  =  ( ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) ) `  0 ) )
9326fveq1d 5722 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( ( S  D n F ) `  k
) `  B )  =  ( ( ( S  D n F ) `  0 ) `
 B ) )
94 fveq2 5720 . . . . . . . . . . 11  |-  ( k  =  0  ->  ( ! `  k )  =  ( ! ` 
0 ) )
95 fac0 11561 . . . . . . . . . . 11  |-  ( ! `
 0 )  =  1
9694, 95syl6eq 2483 . . . . . . . . . 10  |-  ( k  =  0  ->  ( ! `  k )  =  1 )
9793, 96oveq12d 6091 . . . . . . . . 9  |-  ( k  =  0  ->  (
( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  =  ( ( ( ( S  D n F ) `  0
) `  B )  /  1 ) )
98 oveq2 6081 . . . . . . . . . 10  |-  ( k  =  0  ->  (
0 ^ k )  =  ( 0 ^ 0 ) )
99 exp0 11378 . . . . . . . . . . 11  |-  ( 0  e.  CC  ->  (
0 ^ 0 )  =  1 )
10074, 99ax-mp 8 . . . . . . . . . 10  |-  ( 0 ^ 0 )  =  1
10198, 100syl6eq 2483 . . . . . . . . 9  |-  ( k  =  0  ->  (
0 ^ k )  =  1 )
10297, 101oveq12d 6091 . . . . . . . 8  |-  ( k  =  0  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) )  =  ( ( ( ( ( S  D n F ) `
 0 ) `  B )  /  1
)  x.  1 ) )
103 ovex 6098 . . . . . . . 8  |-  ( ( ( ( ( S  D n F ) `
 0 ) `  B )  /  1
)  x.  1 )  e.  _V
104102, 78, 103fvmpt 5798 . . . . . . 7  |-  ( 0  e.  ( ( 0 [,] N )  i^i 
ZZ )  ->  (
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ` 
0 )  =  ( ( ( ( ( S  D n F ) `  0 ) `
 B )  / 
1 )  x.  1 ) )
10523, 104syl 16 . . . . . 6  |-  ( ph  ->  ( ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ` 
0 )  =  ( ( ( ( ( S  D n F ) `  0 ) `
 B )  / 
1 )  x.  1 ) )
10637fveq1d 5722 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( S  D n F ) `
 0 ) `  B )  =  ( F `  B ) )
107106oveq1d 6088 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( S  D n F ) `  0 ) `
 B )  / 
1 )  =  ( ( F `  B
)  /  1 ) )
10833, 42ffvelrnd 5863 . . . . . . . . . 10  |-  ( ph  ->  ( F `  B
)  e.  CC )
109108div1d 9774 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  /  1
)  =  ( F `
 B ) )
110107, 109eqtrd 2467 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( S  D n F ) `  0 ) `
 B )  / 
1 )  =  ( F `  B ) )
111110oveq1d 6088 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( S  D n F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( ( F `  B
)  x.  1 ) )
112108mulid1d 9097 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  1 )  =  ( F `
 B ) )
113111, 112eqtrd 2467 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  D n F ) `  0
) `  B )  /  1 )  x.  1 )  =  ( F `  B ) )
11492, 105, 1133eqtrd 2471 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  =  ( F `  B ) )
115 rngcmn 15686 . . . . . . 7  |-  (fld  e.  Ring  ->fld  e. CMnd )
11646, 115mp1i 12 . . . . . 6  |-  ( ph  ->fld  e. CMnd
)
117 cnfldtps 18804 . . . . . . 7  |-fld  e.  TopSp
118117a1i 11 . . . . . 6  |-  ( ph  ->fld  e. 
TopSp )
119 snfi 7179 . . . . . . 7  |-  { 0 }  e.  Fin
120 ssfi 7321 . . . . . . 7  |-  ( ( { 0 }  e.  Fin  /\  ( `' ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) ) ) " ( _V  \  { 0 } ) )  C_  { 0 } )  ->  ( `' ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) "
( _V  \  {
0 } ) )  e.  Fin )
121119, 91, 120sylancr 645 . . . . . 6  |-  ( ph  ->  ( `' ( k  e.  ( ( 0 [,] N )  i^i 
ZZ )  |->  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( 0 ^ k ) ) ) " ( _V 
\  { 0 } ) )  e.  Fin )
12244, 45, 116, 118, 51, 79, 121tsmsid 18161 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )  e.  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ) )
123114, 122eqeltrrd 2510 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ) )
12443subidd 9391 . . . . . . . 8  |-  ( ph  ->  ( B  -  B
)  =  0 )
125124oveq1d 6088 . . . . . . 7  |-  ( ph  ->  ( ( B  -  B ) ^ k
)  =  ( 0 ^ k ) )
126125oveq2d 6089 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) )  =  ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
0 ^ k ) ) )
127126mpteq2dv 4288 . . . . 5  |-  ( ph  ->  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) ) )  =  ( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) )
128127oveq2d 6089 . . . 4  |-  ( ph  ->  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) ) ) )  =  (fld tsums 
( k  e.  ( ( 0 [,] N
)  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( 0 ^ k
) ) ) ) )
129123, 128eleqtrrd 2512 . . 3  |-  ( ph  ->  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) ) ) ) )
130 taylfval.t . . . 4  |-  T  =  ( N ( S Tayl 
F ) B )
1312, 33, 1, 8, 24, 130eltayl 20268 . . 3  |-  ( ph  ->  ( B T ( F `  B )  <-> 
( B  e.  CC  /\  ( F `  B
)  e.  (fld tsums  ( k  e.  ( ( 0 [,] N )  i^i  ZZ )  |->  ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( B  -  B ) ^ k
) ) ) ) ) ) )
13243, 129, 131mpbir2and 889 . 2  |-  ( ph  ->  B T ( F `
 B ) )
1332, 33, 1, 8, 24, 130taylf 20269 . . 3  |-  ( ph  ->  T : dom  T --> CC )
134 ffun 5585 . . 3  |-  ( T : dom  T --> CC  ->  Fun 
T )
135 funbrfv2b 5763 . . 3  |-  ( Fun 
T  ->  ( B T ( F `  B )  <->  ( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) ) )
136133, 134, 1353syl 19 . 2  |-  ( ph  ->  ( B T ( F `  B )  <-> 
( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) ) )
137132, 136mpbid 202 1  |-  ( ph  ->  ( B  e.  dom  T  /\  ( T `  B )  =  ( F `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   {csn 3806   {cpr 3807   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   dom cdm 4870   "cima 4873   Fun wfun 5440   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^pm cpm 7011   Fincfn 7101   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    +oocpnf 9109   RR*cxr 9111    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   [,]cicc 10911   ^cexp 11374   !cfa 11558    gsumg cgsu 13716   Mndcmnd 14676  CMndccmn 15404   Ringcrg 15652  ℂfldccnfld 16695   TopSpctps 16953   tsums ctsu 18147    D ncdvn 19743   Tayl ctayl 20261
This theorem is referenced by:  dvntaylp0  20280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-fac 11559  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-rest 13642  df-topn 13643  df-topgen 13659  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cnp 17284  df-haus 17371  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-tsms 18148  df-xms 18342  df-ms 18343  df-limc 19745  df-dv 19746  df-dvn 19747  df-tayl 20263
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