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Theorem taylply2 19763
Description: The Taylor polynomial is a polynomial of degree (at most)  N. This version of taylply 19764 shows that the coefficients of  T are in a subring of the complexes. (Contributed by Mario Carneiro, 1-Jan-2017.)
Hypotheses
Ref Expression
taylpfval.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
taylpfval.f  |-  ( ph  ->  F : A --> CC )
taylpfval.a  |-  ( ph  ->  A  C_  S )
taylpfval.n  |-  ( ph  ->  N  e.  NN0 )
taylpfval.b  |-  ( ph  ->  B  e.  dom  (
( S  D n F ) `  N
) )
taylpfval.t  |-  T  =  ( N ( S Tayl 
F ) B )
taylply2.1  |-  ( ph  ->  D  e.  (SubRing ` fld ) )
taylply2.2  |-  ( ph  ->  B  e.  D )
taylply2.3  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  D )
Assertion
Ref Expression
taylply2  |-  ( ph  ->  ( T  e.  (Poly `  D )  /\  (deg `  T )  <_  N
) )
Distinct variable groups:    B, k    k, F    k, N    ph, k    D, k    S, k
Allowed substitution hints:    A( k)    T( k)

Proof of Theorem taylply2
Dummy variables  u  v  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 taylpfval.s . . . . 5  |-  ( ph  ->  S  e.  { RR ,  CC } )
2 taylpfval.f . . . . 5  |-  ( ph  ->  F : A --> CC )
3 taylpfval.a . . . . 5  |-  ( ph  ->  A  C_  S )
4 taylpfval.n . . . . 5  |-  ( ph  ->  N  e.  NN0 )
5 taylpfval.b . . . . 5  |-  ( ph  ->  B  e.  dom  (
( S  D n F ) `  N
) )
6 taylpfval.t . . . . 5  |-  T  =  ( N ( S Tayl 
F ) B )
71, 2, 3, 4, 5, 6taylpfval 19760 . . . 4  |-  ( ph  ->  T  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) )
8 simpr 447 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
9 cnex 8834 . . . . . . . . . . . . 13  |-  CC  e.  _V
109a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  CC  e.  _V )
11 elpm2r 6804 . . . . . . . . . . . 12  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
1210, 1, 2, 3, 11syl22anc 1183 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
13 dvnbss 19293 . . . . . . . . . . 11  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  dom  ( ( S  D n F ) `  N ) 
C_  dom  F )
141, 12, 4, 13syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  dom  ( ( S  D n F ) `
 N )  C_  dom  F )
15 fdm 5409 . . . . . . . . . . 11  |-  ( F : A --> CC  ->  dom 
F  =  A )
162, 15syl 15 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  A )
1714, 16sseqtrd 3227 . . . . . . . . 9  |-  ( ph  ->  dom  ( ( S  D n F ) `
 N )  C_  A )
18 recnprss 19270 . . . . . . . . . . 11  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
191, 18syl 15 . . . . . . . . . 10  |-  ( ph  ->  S  C_  CC )
203, 19sstrd 3202 . . . . . . . . 9  |-  ( ph  ->  A  C_  CC )
2117, 20sstrd 3202 . . . . . . . 8  |-  ( ph  ->  dom  ( ( S  D n F ) `
 N )  C_  CC )
2221, 5sseldd 3194 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
2322adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  CC )
248, 23subcld 9173 . . . . 5  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  -  B )  e.  CC )
25 df-idp 19587 . . . . . . . 8  |-  X p  =  (  _I  |`  CC )
26 mptresid 5020 . . . . . . . 8  |-  ( x  e.  CC  |->  x )  =  (  _I  |`  CC )
2725, 26eqtr4i 2319 . . . . . . 7  |-  X p  =  ( x  e.  CC  |->  x )
2827a1i 10 . . . . . 6  |-  ( ph  ->  X p  =  ( x  e.  CC  |->  x ) )
29 fconstmpt 4748 . . . . . . 7  |-  ( CC 
X.  { B }
)  =  ( x  e.  CC  |->  B )
3029a1i 10 . . . . . 6  |-  ( ph  ->  ( CC  X.  { B } )  =  ( x  e.  CC  |->  B ) )
3110, 8, 23, 28, 30offval2 6111 . . . . 5  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { B } ) )  =  ( x  e.  CC  |->  ( x  -  B ) ) )
32 eqidd 2297 . . . . 5  |-  ( ph  ->  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  =  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )
33 oveq1 5881 . . . . . . 7  |-  ( y  =  ( x  -  B )  ->  (
y ^ k )  =  ( ( x  -  B ) ^
k ) )
3433oveq2d 5890 . . . . . 6  |-  ( y  =  ( x  -  B )  ->  (
( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) )  =  ( ( ( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) )
3534sumeq2sdv 12193 . . . . 5  |-  ( y  =  ( x  -  B )  ->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) )
3624, 31, 32, 35fmptco 5707 . . . 4  |-  ( ph  ->  ( ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) )  o.  ( X p  o F  -  ( CC  X.  { B } ) ) )  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) )
377, 36eqtr4d 2331 . . 3  |-  ( ph  ->  T  =  ( ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  o.  (
X p  o F  -  ( CC  X.  { B } ) ) ) )
38 taylply2.1 . . . . . 6  |-  ( ph  ->  D  e.  (SubRing ` fld ) )
39 cnfldbas 16399 . . . . . . 7  |-  CC  =  ( Base ` fld )
4039subrgss 15562 . . . . . 6  |-  ( D  e.  (SubRing ` fld )  ->  D  C_  CC )
4138, 40syl 15 . . . . 5  |-  ( ph  ->  D  C_  CC )
42 taylply2.3 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  D )
4341, 4, 42elplyd 19600 . . . 4  |-  ( ph  ->  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  e.  (Poly `  D ) )
44 cnfld1 16415 . . . . . . . 8  |-  1  =  ( 1r ` fld )
4544subrg1cl 15569 . . . . . . 7  |-  ( D  e.  (SubRing ` fld )  ->  1  e.  D )
4638, 45syl 15 . . . . . 6  |-  ( ph  ->  1  e.  D )
47 plyid 19607 . . . . . 6  |-  ( ( D  C_  CC  /\  1  e.  D )  ->  X p  e.  (Poly `  D
) )
4841, 46, 47syl2anc 642 . . . . 5  |-  ( ph  ->  X p  e.  (Poly `  D ) )
49 taylply2.2 . . . . . 6  |-  ( ph  ->  B  e.  D )
50 plyconst 19604 . . . . . 6  |-  ( ( D  C_  CC  /\  B  e.  D )  ->  ( CC  X.  { B }
)  e.  (Poly `  D ) )
5141, 49, 50syl2anc 642 . . . . 5  |-  ( ph  ->  ( CC  X.  { B } )  e.  (Poly `  D ) )
52 subrgsubg 15567 . . . . . . 7  |-  ( D  e.  (SubRing ` fld )  ->  D  e.  (SubGrp ` fld ) )
5338, 52syl 15 . . . . . 6  |-  ( ph  ->  D  e.  (SubGrp ` fld )
)
54 cnfldadd 16400 . . . . . . . 8  |-  +  =  ( +g  ` fld )
5554subgcl 14647 . . . . . . 7  |-  ( ( D  e.  (SubGrp ` fld )  /\  u  e.  D  /\  v  e.  D
)  ->  ( u  +  v )  e.  D )
56553expb 1152 . . . . . 6  |-  ( ( D  e.  (SubGrp ` fld )  /\  ( u  e.  D  /\  v  e.  D
) )  ->  (
u  +  v )  e.  D )
5753, 56sylan 457 . . . . 5  |-  ( (
ph  /\  ( u  e.  D  /\  v  e.  D ) )  -> 
( u  +  v )  e.  D )
58 cnfldmul 16401 . . . . . . . 8  |-  x.  =  ( .r ` fld )
5958subrgmcl 15573 . . . . . . 7  |-  ( ( D  e.  (SubRing ` fld )  /\  u  e.  D  /\  v  e.  D )  ->  (
u  x.  v )  e.  D )
60593expb 1152 . . . . . 6  |-  ( ( D  e.  (SubRing ` fld )  /\  (
u  e.  D  /\  v  e.  D )
)  ->  ( u  x.  v )  e.  D
)
6138, 60sylan 457 . . . . 5  |-  ( (
ph  /\  ( u  e.  D  /\  v  e.  D ) )  -> 
( u  x.  v
)  e.  D )
62 ax-1cn 8811 . . . . . . 7  |-  1  e.  CC
63 cnfldneg 16416 . . . . . . 7  |-  ( 1  e.  CC  ->  (
( inv g ` fld ) `  1 )  = 
-u 1 )
6462, 63ax-mp 8 . . . . . 6  |-  ( ( inv g ` fld ) `  1 )  =  -u 1
65 eqid 2296 . . . . . . . 8  |-  ( inv g ` fld )  =  ( inv g ` fld )
6665subginvcl 14646 . . . . . . 7  |-  ( ( D  e.  (SubGrp ` fld )  /\  1  e.  D
)  ->  ( ( inv g ` fld ) `  1 )  e.  D )
6753, 46, 66syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( inv g ` fld ) `  1 )  e.  D )
6864, 67syl5eqelr 2381 . . . . 5  |-  ( ph  -> 
-u 1  e.  D
)
6948, 51, 57, 61, 68plysub 19617 . . . 4  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { B } ) )  e.  (Poly `  D ) )
7043, 69, 57, 61plyco 19639 . . 3  |-  ( ph  ->  ( ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) )  o.  ( X p  o F  -  ( CC  X.  { B } ) ) )  e.  (Poly `  D ) )
7137, 70eqeltrd 2370 . 2  |-  ( ph  ->  T  e.  (Poly `  D ) )
7237fveq2d 5545 . . . 4  |-  ( ph  ->  (deg `  T )  =  (deg `  ( (
y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  o.  (
X p  o F  -  ( CC  X.  { B } ) ) ) ) )
73 eqid 2296 . . . . 5  |-  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )  =  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )
74 eqid 2296 . . . . 5  |-  (deg `  ( X p  o F  -  ( CC  X.  { B } ) ) )  =  (deg
`  ( X p  o F  -  ( CC  X.  { B }
) ) )
7573, 74, 43, 69dgrco 19672 . . . 4  |-  ( ph  ->  (deg `  ( (
y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  o.  (
X p  o F  -  ( CC  X.  { B } ) ) ) )  =  ( (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  x.  (deg `  (
X p  o F  -  ( CC  X.  { B } ) ) ) ) )
76 eqid 2296 . . . . . . . . 9  |-  ( X p  o F  -  ( CC  X.  { B } ) )  =  ( X p  o F  -  ( CC  X.  { B } ) )
7776plyremlem 19700 . . . . . . . 8  |-  ( B  e.  CC  ->  (
( X p  o F  -  ( CC  X.  { B } ) )  e.  (Poly `  CC )  /\  (deg `  ( X p  o F  -  ( CC  X.  { B } ) ) )  =  1  /\  ( `' ( X p  o F  -  ( CC  X.  { B } ) )
" { 0 } )  =  { B } ) )
7822, 77syl 15 . . . . . . 7  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { B }
) )  e.  (Poly `  CC )  /\  (deg `  ( X p  o F  -  ( CC  X.  { B } ) ) )  =  1  /\  ( `' ( X p  o F  -  ( CC  X.  { B } ) )
" { 0 } )  =  { B } ) )
7978simp2d 968 . . . . . 6  |-  ( ph  ->  (deg `  ( X p  o F  -  ( CC  X.  { B }
) ) )  =  1 )
8079oveq2d 5890 . . . . 5  |-  ( ph  ->  ( (deg `  (
y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )  x.  (deg `  ( X p  o F  -  ( CC  X.  { B }
) ) ) )  =  ( (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )  x.  1 ) )
81 dgrcl 19631 . . . . . . . 8  |-  ( ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  e.  (Poly `  D )  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )  e. 
NN0 )
8243, 81syl 15 . . . . . . 7  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  e.  NN0 )
8382nn0cnd 10036 . . . . . 6  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  e.  CC )
8483mulid1d 8868 . . . . 5  |-  ( ph  ->  ( (deg `  (
y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )  x.  1 )  =  (deg
`  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) ) )
8580, 84eqtrd 2328 . . . 4  |-  ( ph  ->  ( (deg `  (
y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )  x.  (deg `  ( X p  o F  -  ( CC  X.  { B }
) ) ) )  =  (deg `  (
y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) ) )
8672, 75, 853eqtrd 2332 . . 3  |-  ( ph  ->  (deg `  T )  =  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) ) )
871adantr 451 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  S  e.  { RR ,  CC } )
8812adantr 451 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  F  e.  ( CC  ^pm  S
) )
89 elfznn0 10838 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
9089adantl 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
91 dvnf 19292 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  k  e.  NN0 )  ->  ( ( S  D n F ) `
 k ) : dom  ( ( S  D n F ) `
 k ) --> CC )
9287, 88, 90, 91syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( S  D n F ) `  k
) : dom  (
( S  D n F ) `  k
) --> CC )
93 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  ( 0 ... N
) )
94 dvn2bss 19295 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  k  e.  ( 0 ... N
) )  ->  dom  ( ( S  D n F ) `  N
)  C_  dom  ( ( S  D n F ) `  k ) )
9587, 88, 93, 94syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  dom  ( ( S  D n F ) `  N
)  C_  dom  ( ( S  D n F ) `  k ) )
965adantr 451 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  B  e.  dom  ( ( S  D n F ) `
 N ) )
9795, 96sseldd 3194 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  B  e.  dom  ( ( S  D n F ) `
 k ) )
9892, 97ffvelrnd 5682 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( S  D n F ) `  k
) `  B )  e.  CC )
99 faccl 11314 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
10090, 99syl 15 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  e.  NN )
101100nncnd 9778 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  e.  CC )
102100nnne0d 9806 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  =/=  0 )
10398, 101, 102divcld 9552 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
10443, 4, 103, 32dgrle 19641 . . 3  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  <_  N )
10586, 104eqbrtrd 4059 . 2  |-  ( ph  ->  (deg `  T )  <_  N )
10671, 105jca 518 1  |-  ( ph  ->  ( T  e.  (Poly `  D )  /\  (deg `  T )  <_  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   {csn 3653   {cpr 3654   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703   `'ccnv 4704   dom cdm 4705    |` cres 4707   "cima 4708    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092    ^pm cpm 6789   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   NN0cn0 9981   ...cfz 10798   ^cexp 11120   !cfa 11304   sum_csu 12174   inv gcminusg 14379  SubGrpcsubg 14631  SubRingcsubrg 15557  ℂfldccnfld 16393    D ncdvn 19230  Polycply 19582   X pcidp 19583  degcdgr 19585   Tayl ctayl 19748
This theorem is referenced by:  taylply  19764  taylthlem2  19769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-0g 13420  df-gsum 13421  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-subrg 15559  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cnp 16974  df-haus 17059  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-tsms 17825  df-xms 17901  df-ms 17902  df-0p 19041  df-limc 19232  df-dv 19233  df-dvn 19234  df-ply 19586  df-idp 19587  df-coe 19588  df-dgr 19589  df-tayl 19750
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