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Theorem taylply2 20276
Description: The Taylor polynomial is a polynomial of degree (at most)  N. This version of taylply 20277 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 20273 . . . 4  |-  ( ph  ->  T  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) ) )
8 simpr 448 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
9 cnex 9063 . . . . . . . . . . . . 13  |-  CC  e.  _V
109a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  CC  e.  _V )
11 elpm2r 7026 . . . . . . . . . . . 12  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
1210, 1, 2, 3, 11syl22anc 1185 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
13 dvnbss 19806 . . . . . . . . . . 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 1184 . . . . . . . . . 10  |-  ( ph  ->  dom  ( ( S  D n F ) `
 N )  C_  dom  F )
15 fdm 5587 . . . . . . . . . . 11  |-  ( F : A --> CC  ->  dom 
F  =  A )
162, 15syl 16 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  A )
1714, 16sseqtrd 3376 . . . . . . . . 9  |-  ( ph  ->  dom  ( ( S  D n F ) `
 N )  C_  A )
18 recnprss 19783 . . . . . . . . . . 11  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
191, 18syl 16 . . . . . . . . . 10  |-  ( ph  ->  S  C_  CC )
203, 19sstrd 3350 . . . . . . . . 9  |-  ( ph  ->  A  C_  CC )
2117, 20sstrd 3350 . . . . . . . 8  |-  ( ph  ->  dom  ( ( S  D n F ) `
 N )  C_  CC )
2221, 5sseldd 3341 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
2322adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  CC )
248, 23subcld 9403 . . . . 5  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  -  B )  e.  CC )
25 df-idp 20100 . . . . . . . 8  |-  X p  =  (  _I  |`  CC )
26 mptresid 5187 . . . . . . . 8  |-  ( x  e.  CC  |->  x )  =  (  _I  |`  CC )
2725, 26eqtr4i 2458 . . . . . . 7  |-  X p  =  ( x  e.  CC  |->  x )
2827a1i 11 . . . . . 6  |-  ( ph  ->  X p  =  ( x  e.  CC  |->  x ) )
29 fconstmpt 4913 . . . . . . 7  |-  ( CC 
X.  { B }
)  =  ( x  e.  CC  |->  B )
3029a1i 11 . . . . . 6  |-  ( ph  ->  ( CC  X.  { B } )  =  ( x  e.  CC  |->  B ) )
3110, 8, 23, 28, 30offval2 6314 . . . . 5  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { B } ) )  =  ( x  e.  CC  |->  ( x  -  B ) ) )
32 eqidd 2436 . . . . 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 6080 . . . . . . 7  |-  ( y  =  ( x  -  B )  ->  (
y ^ k )  =  ( ( x  -  B ) ^
k ) )
3433oveq2d 6089 . . . . . 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 12490 . . . . 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 5893 . . . 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 2470 . . 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 16699 . . . . . . 7  |-  CC  =  ( Base ` fld )
4039subrgss 15861 . . . . . 6  |-  ( D  e.  (SubRing ` fld )  ->  D  C_  CC )
4138, 40syl 16 . . . . 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 20113 . . . 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 16718 . . . . . . . 8  |-  1  =  ( 1r ` fld )
4544subrg1cl 15868 . . . . . . 7  |-  ( D  e.  (SubRing ` fld )  ->  1  e.  D )
4638, 45syl 16 . . . . . 6  |-  ( ph  ->  1  e.  D )
47 plyid 20120 . . . . . 6  |-  ( ( D  C_  CC  /\  1  e.  D )  ->  X p  e.  (Poly `  D
) )
4841, 46, 47syl2anc 643 . . . . 5  |-  ( ph  ->  X p  e.  (Poly `  D ) )
49 taylply2.2 . . . . . 6  |-  ( ph  ->  B  e.  D )
50 plyconst 20117 . . . . . 6  |-  ( ( D  C_  CC  /\  B  e.  D )  ->  ( CC  X.  { B }
)  e.  (Poly `  D ) )
5141, 49, 50syl2anc 643 . . . . 5  |-  ( ph  ->  ( CC  X.  { B } )  e.  (Poly `  D ) )
52 subrgsubg 15866 . . . . . . 7  |-  ( D  e.  (SubRing ` fld )  ->  D  e.  (SubGrp ` fld ) )
5338, 52syl 16 . . . . . 6  |-  ( ph  ->  D  e.  (SubGrp ` fld )
)
54 cnfldadd 16700 . . . . . . . 8  |-  +  =  ( +g  ` fld )
5554subgcl 14946 . . . . . . 7  |-  ( ( D  e.  (SubGrp ` fld )  /\  u  e.  D  /\  v  e.  D
)  ->  ( u  +  v )  e.  D )
56553expb 1154 . . . . . 6  |-  ( ( D  e.  (SubGrp ` fld )  /\  ( u  e.  D  /\  v  e.  D
) )  ->  (
u  +  v )  e.  D )
5753, 56sylan 458 . . . . 5  |-  ( (
ph  /\  ( u  e.  D  /\  v  e.  D ) )  -> 
( u  +  v )  e.  D )
58 cnfldmul 16701 . . . . . . . 8  |-  x.  =  ( .r ` fld )
5958subrgmcl 15872 . . . . . . 7  |-  ( ( D  e.  (SubRing ` fld )  /\  u  e.  D  /\  v  e.  D )  ->  (
u  x.  v )  e.  D )
60593expb 1154 . . . . . 6  |-  ( ( D  e.  (SubRing ` fld )  /\  (
u  e.  D  /\  v  e.  D )
)  ->  ( u  x.  v )  e.  D
)
6138, 60sylan 458 . . . . 5  |-  ( (
ph  /\  ( u  e.  D  /\  v  e.  D ) )  -> 
( u  x.  v
)  e.  D )
62 ax-1cn 9040 . . . . . . 7  |-  1  e.  CC
63 cnfldneg 16719 . . . . . . 7  |-  ( 1  e.  CC  ->  (
( inv g ` fld ) `  1 )  = 
-u 1 )
6462, 63ax-mp 8 . . . . . 6  |-  ( ( inv g ` fld ) `  1 )  =  -u 1
65 eqid 2435 . . . . . . . 8  |-  ( inv g ` fld )  =  ( inv g ` fld )
6665subginvcl 14945 . . . . . . 7  |-  ( ( D  e.  (SubGrp ` fld )  /\  1  e.  D
)  ->  ( ( inv g ` fld ) `  1 )  e.  D )
6753, 46, 66syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( inv g ` fld ) `  1 )  e.  D )
6864, 67syl5eqelr 2520 . . . . 5  |-  ( ph  -> 
-u 1  e.  D
)
6948, 51, 57, 61, 68plysub 20130 . . . 4  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { B } ) )  e.  (Poly `  D ) )
7043, 69, 57, 61plyco 20152 . . 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 2509 . 2  |-  ( ph  ->  T  e.  (Poly `  D ) )
7237fveq2d 5724 . . . 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 2435 . . . . 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 2435 . . . . 5  |-  (deg `  ( X p  o F  -  ( CC  X.  { B } ) ) )  =  (deg
`  ( X p  o F  -  ( CC  X.  { B }
) ) )
7573, 74, 43, 69dgrco 20185 . . . 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 2435 . . . . . . . . 9  |-  ( X p  o F  -  ( CC  X.  { B } ) )  =  ( X p  o F  -  ( CC  X.  { B } ) )
7776plyremlem 20213 . . . . . . . 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 16 . . . . . . 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 970 . . . . . 6  |-  ( ph  ->  (deg `  ( X p  o F  -  ( CC  X.  { B }
) ) )  =  1 )
8079oveq2d 6089 . . . . 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 20144 . . . . . . . 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 16 . . . . . . 7  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  e.  NN0 )
8382nn0cnd 10268 . . . . . 6  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  e.  CC )
8483mulid1d 9097 . . . . 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 2467 . . . 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 2471 . . 3  |-  ( ph  ->  (deg `  T )  =  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) ) )
871adantr 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  S  e.  { RR ,  CC } )
8812adantr 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  F  e.  ( CC  ^pm  S
) )
89 elfznn0 11075 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
9089adantl 453 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
91 dvnf 19805 . . . . . . 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 1184 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( S  D n F ) `  k
) : dom  (
( S  D n F ) `  k
) --> CC )
93 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  ( 0 ... N
) )
94 dvn2bss 19808 . . . . . . . 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 1184 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  dom  ( ( S  D n F ) `  N
)  C_  dom  ( ( S  D n F ) `  k ) )
965adantr 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  B  e.  dom  ( ( S  D n F ) `
 N ) )
9795, 96sseldd 3341 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  B  e.  dom  ( ( S  D n F ) `
 k ) )
9892, 97ffvelrnd 5863 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( S  D n F ) `  k
) `  B )  e.  CC )
99 faccl 11568 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
10090, 99syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  e.  NN )
101100nncnd 10008 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  e.  CC )
102100nnne0d 10036 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  =/=  0 )
10398, 101, 102divcld 9782 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( ( S  D n F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
10443, 4, 103, 32dgrle 20154 . . 3  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  D n F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  <_  N )
10586, 104eqbrtrd 4224 . 2  |-  ( ph  ->  (deg `  T )  <_  N )
10671, 105jca 519 1  |-  ( ph  ->  ( T  e.  (Poly `  D )  /\  (deg `  T )  <_  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   {csn 3806   {cpr 3807   class class class wbr 4204    e. cmpt 4258    _I cid 4485    X. cxp 4868   `'ccnv 4869   dom cdm 4870    |` cres 4872   "cima 4873    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295    ^pm cpm 7011   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    <_ cle 9113    - cmin 9283   -ucneg 9284    / cdiv 9669   NNcn 9992   NN0cn0 10213   ...cfz 11035   ^cexp 11374   !cfa 11558   sum_csu 12471   inv gcminusg 14678  SubGrpcsubg 14930  SubRingcsubrg 15856  ℂfldccnfld 16695    D ncdvn 19743  Polycply 20095   X pcidp 20096  degcdgr 20098   Tayl ctayl 20261
This theorem is referenced by:  taylply  20277  taylthlem2  20282
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-of 6297  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-fl 11194  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-clim 12274  df-rlim 12275  df-sum 12472  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-mnd 14682  df-grp 14804  df-minusg 14805  df-subg 14933  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-subrg 15858  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-0p 19554  df-limc 19745  df-dv 19746  df-dvn 19747  df-ply 20099  df-idp 20100  df-coe 20101  df-dgr 20102  df-tayl 20263
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