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Theorem taylply2 21855
Description: The Taylor polynomial is a polynomial of degree (at most)  N. This version of taylply 21856 shows that the coefficients of  T are in a subring of the complex numbers. (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  Dn
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  Dn 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  Dn
F ) `  N
) )
6 taylpfval.t . . . . 5  |-  T  =  ( N ( S Tayl 
F ) B )
71, 2, 3, 4, 5, 6taylpfval 21852 . . . 4  |-  ( ph  ->  T  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) ) )
8 simpr 461 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
9 cnex 9384 . . . . . . . . . . . . 13  |-  CC  e.  _V
109a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  CC  e.  _V )
11 elpm2r 7251 . . . . . . . . . . . 12  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
1210, 1, 2, 3, 11syl22anc 1219 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
13 dvnbss 21424 . . . . . . . . . . 11  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  N  e.  NN0 )  ->  dom  ( ( S  Dn F ) `  N ) 
C_  dom  F )
141, 12, 4, 13syl3anc 1218 . . . . . . . . . 10  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 N )  C_  dom  F )
15 fdm 5584 . . . . . . . . . . 11  |-  ( F : A --> CC  ->  dom 
F  =  A )
162, 15syl 16 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  A )
1714, 16sseqtrd 3413 . . . . . . . . 9  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 N )  C_  A )
18 recnprss 21401 . . . . . . . . . . 11  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
191, 18syl 16 . . . . . . . . . 10  |-  ( ph  ->  S  C_  CC )
203, 19sstrd 3387 . . . . . . . . 9  |-  ( ph  ->  A  C_  CC )
2117, 20sstrd 3387 . . . . . . . 8  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 N )  C_  CC )
2221, 5sseldd 3378 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
2322adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  CC )
248, 23subcld 9740 . . . . 5  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  -  B )  e.  CC )
25 df-idp 21679 . . . . . . . 8  |-  Xp  =  (  _I  |`  CC )
26 mptresid 5181 . . . . . . . 8  |-  ( x  e.  CC  |->  x )  =  (  _I  |`  CC )
2725, 26eqtr4i 2466 . . . . . . 7  |-  Xp  =  ( x  e.  CC  |->  x )
2827a1i 11 . . . . . 6  |-  ( ph  ->  Xp  =  ( x  e.  CC  |->  x ) )
29 fconstmpt 4903 . . . . . . 7  |-  ( CC 
X.  { B }
)  =  ( x  e.  CC  |->  B )
3029a1i 11 . . . . . 6  |-  ( ph  ->  ( CC  X.  { B } )  =  ( x  e.  CC  |->  B ) )
3110, 8, 23, 28, 30offval2 6357 . . . . 5  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { B }
) )  =  ( x  e.  CC  |->  ( x  -  B ) ) )
32 eqidd 2444 . . . . 5  |-  ( ph  ->  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  =  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )
33 oveq1 6119 . . . . . . 7  |-  ( y  =  ( x  -  B )  ->  (
y ^ k )  =  ( ( x  -  B ) ^
k ) )
3433oveq2d 6128 . . . . . 6  |-  ( y  =  ( x  -  B )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) )  =  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) )
3534sumeq2sdv 13202 . . . . 5  |-  ( y  =  ( x  -  B )  ->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( y ^
k ) )  = 
sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( ( x  -  B ) ^ k
) ) )
3624, 31, 32, 35fmptco 5897 . . . 4  |-  ( ph  ->  ( ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) )  o.  ( Xp  oF  -  ( CC 
X.  { B }
) ) )  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
( x  -  B
) ^ k ) ) ) )
377, 36eqtr4d 2478 . . 3  |-  ( ph  ->  T  =  ( ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  o.  (
Xp  oF  -  ( CC  X.  { B } ) ) ) )
38 taylply2.1 . . . . . 6  |-  ( ph  ->  D  e.  (SubRing ` fld ) )
39 cnfldbas 17844 . . . . . . 7  |-  CC  =  ( Base ` fld )
4039subrgss 16888 . . . . . 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  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  D )
4341, 4, 42elplyd 21692 . . . 4  |-  ( ph  ->  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  e.  (Poly `  D ) )
44 cnfld1 17863 . . . . . . . 8  |-  1  =  ( 1r ` fld )
4544subrg1cl 16895 . . . . . . 7  |-  ( D  e.  (SubRing ` fld )  ->  1  e.  D )
4638, 45syl 16 . . . . . 6  |-  ( ph  ->  1  e.  D )
47 plyid 21699 . . . . . 6  |-  ( ( D  C_  CC  /\  1  e.  D )  ->  Xp  e.  (Poly `  D
) )
4841, 46, 47syl2anc 661 . . . . 5  |-  ( ph  ->  Xp  e.  (Poly `  D ) )
49 taylply2.2 . . . . . 6  |-  ( ph  ->  B  e.  D )
50 plyconst 21696 . . . . . 6  |-  ( ( D  C_  CC  /\  B  e.  D )  ->  ( CC  X.  { B }
)  e.  (Poly `  D ) )
5141, 49, 50syl2anc 661 . . . . 5  |-  ( ph  ->  ( CC  X.  { B } )  e.  (Poly `  D ) )
52 subrgsubg 16893 . . . . . . 7  |-  ( D  e.  (SubRing ` fld )  ->  D  e.  (SubGrp ` fld ) )
5338, 52syl 16 . . . . . 6  |-  ( ph  ->  D  e.  (SubGrp ` fld )
)
54 cnfldadd 17845 . . . . . . . 8  |-  +  =  ( +g  ` fld )
5554subgcl 15712 . . . . . . 7  |-  ( ( D  e.  (SubGrp ` fld )  /\  u  e.  D  /\  v  e.  D
)  ->  ( u  +  v )  e.  D )
56553expb 1188 . . . . . 6  |-  ( ( D  e.  (SubGrp ` fld )  /\  ( u  e.  D  /\  v  e.  D
) )  ->  (
u  +  v )  e.  D )
5753, 56sylan 471 . . . . 5  |-  ( (
ph  /\  ( u  e.  D  /\  v  e.  D ) )  -> 
( u  +  v )  e.  D )
58 cnfldmul 17846 . . . . . . . 8  |-  x.  =  ( .r ` fld )
5958subrgmcl 16899 . . . . . . 7  |-  ( ( D  e.  (SubRing ` fld )  /\  u  e.  D  /\  v  e.  D )  ->  (
u  x.  v )  e.  D )
60593expb 1188 . . . . . 6  |-  ( ( D  e.  (SubRing ` fld )  /\  (
u  e.  D  /\  v  e.  D )
)  ->  ( u  x.  v )  e.  D
)
6138, 60sylan 471 . . . . 5  |-  ( (
ph  /\  ( u  e.  D  /\  v  e.  D ) )  -> 
( u  x.  v
)  e.  D )
62 ax-1cn 9361 . . . . . . 7  |-  1  e.  CC
63 cnfldneg 17864 . . . . . . 7  |-  ( 1  e.  CC  ->  (
( invg ` fld ) `  1 )  = 
-u 1 )
6462, 63ax-mp 5 . . . . . 6  |-  ( ( invg ` fld ) `  1 )  =  -u 1
65 eqid 2443 . . . . . . . 8  |-  ( invg ` fld )  =  ( invg ` fld )
6665subginvcl 15711 . . . . . . 7  |-  ( ( D  e.  (SubGrp ` fld )  /\  1  e.  D
)  ->  ( ( invg ` fld ) `  1 )  e.  D )
6753, 46, 66syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( invg ` fld ) `  1 )  e.  D )
6864, 67syl5eqelr 2528 . . . . 5  |-  ( ph  -> 
-u 1  e.  D
)
6948, 51, 57, 61, 68plysub 21709 . . . 4  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { B }
) )  e.  (Poly `  D ) )
7043, 69, 57, 61plyco 21731 . . 3  |-  ( ph  ->  ( ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) )  o.  ( Xp  oF  -  ( CC 
X.  { B }
) ) )  e.  (Poly `  D )
)
7137, 70eqeltrd 2517 . 2  |-  ( ph  ->  T  e.  (Poly `  D ) )
7237fveq2d 5716 . . . 4  |-  ( ph  ->  (deg `  T )  =  (deg `  ( (
y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  o.  (
Xp  oF  -  ( CC  X.  { B } ) ) ) ) )
73 eqid 2443 . . . . 5  |-  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )  =  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )
74 eqid 2443 . . . . 5  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { B }
) ) )  =  (deg `  ( Xp  oF  -  ( CC  X.  { B }
) ) )
7573, 74, 43, 69dgrco 21764 . . . 4  |-  ( ph  ->  (deg `  ( (
y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  o.  (
Xp  oF  -  ( CC  X.  { B } ) ) ) )  =  ( (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  x.  (deg `  (
Xp  oF  -  ( CC  X.  { B } ) ) ) ) )
76 eqid 2443 . . . . . . . . 9  |-  ( Xp  oF  -  ( CC  X.  { B } ) )  =  ( Xp  oF  -  ( CC 
X.  { B }
) )
7776plyremlem 21792 . . . . . . . 8  |-  ( B  e.  CC  ->  (
( Xp  oF  -  ( CC 
X.  { B }
) )  e.  (Poly `  CC )  /\  (deg `  ( Xp  oF  -  ( CC 
X.  { B }
) ) )  =  1  /\  ( `' ( Xp  oF  -  ( CC 
X.  { B }
) ) " {
0 } )  =  { B } ) )
7822, 77syl 16 . . . . . . 7  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { B }
) )  e.  (Poly `  CC )  /\  (deg `  ( Xp  oF  -  ( CC 
X.  { B }
) ) )  =  1  /\  ( `' ( Xp  oF  -  ( CC 
X.  { B }
) ) " {
0 } )  =  { B } ) )
7978simp2d 1001 . . . . . 6  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { B }
) ) )  =  1 )
8079oveq2d 6128 . . . . 5  |-  ( ph  ->  ( (deg `  (
y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )  x.  (deg `  ( Xp  oF  -  ( CC  X.  { B }
) ) ) )  =  ( (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )  x.  1 ) )
81 dgrcl 21723 . . . . . . . 8  |-  ( ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  e.  (Poly `  D )  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )  e. 
NN0 )
8243, 81syl 16 . . . . . . 7  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  e.  NN0 )
8382nn0cnd 10659 . . . . . 6  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  e.  CC )
8483mulid1d 9424 . . . . 5  |-  ( ph  ->  ( (deg `  (
y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )  x.  1 )  =  (deg
`  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) ) )
8580, 84eqtrd 2475 . . . 4  |-  ( ph  ->  ( (deg `  (
y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) )  x.  (deg `  ( Xp  oF  -  ( CC  X.  { B }
) ) ) )  =  (deg `  (
y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) ) ) )
8672, 75, 853eqtrd 2479 . . 3  |-  ( ph  ->  (deg `  T )  =  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) ) )
871adantr 465 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  S  e.  { RR ,  CC } )
8812adantr 465 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  F  e.  ( CC  ^pm  S
) )
89 elfznn0 11502 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
9089adantl 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
91 dvnf 21423 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  k  e.  NN0 )  ->  ( ( S  Dn F ) `
 k ) : dom  ( ( S  Dn F ) `
 k ) --> CC )
9287, 88, 90, 91syl3anc 1218 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( S  Dn
F ) `  k
) : dom  (
( S  Dn
F ) `  k
) --> CC )
93 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  ( 0 ... N
) )
94 dvn2bss 21426 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
)  /\  k  e.  ( 0 ... N
) )  ->  dom  ( ( S  Dn F ) `  N )  C_  dom  ( ( S  Dn F ) `  k ) )
9587, 88, 93, 94syl3anc 1218 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  dom  ( ( S  Dn F ) `  N )  C_  dom  ( ( S  Dn F ) `  k ) )
965adantr 465 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  B  e.  dom  ( ( S  Dn F ) `
 N ) )
9795, 96sseldd 3378 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  B  e.  dom  ( ( S  Dn F ) `
 k ) )
9892, 97ffvelrnd 5865 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( S  Dn F ) `  k ) `  B
)  e.  CC )
99 faccl 12082 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
10090, 99syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  e.  NN )
101100nncnd 10359 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  e.  CC )
102100nnne0d 10387 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  =/=  0 )
10398, 101, 102divcld 10128 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
10443, 4, 103, 32dgrle 21733 . . 3  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  <_  N )
10586, 104eqbrtrd 4333 . 2  |-  ( ph  ->  (deg `  T )  <_  N )
10671, 105jca 532 1  |-  ( ph  ->  ( T  e.  (Poly `  D )  /\  (deg `  T )  <_  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2993    C_ wss 3349   {csn 3898   {cpr 3900   class class class wbr 4313    e. cmpt 4371    _I cid 4652    X. cxp 4859   `'ccnv 4860   dom cdm 4861    |` cres 4863   "cima 4864    o. ccom 4865   -->wf 5435   ` cfv 5439  (class class class)co 6112    oFcof 6339    ^pm cpm 7236   CCcc 9301   RRcr 9302   0cc0 9303   1c1 9304    + caddc 9306    x. cmul 9308    <_ cle 9440    - cmin 9616   -ucneg 9617    / cdiv 10014   NNcn 10343   NN0cn0 10600   ...cfz 11458   ^cexp 11886   !cfa 12072   sum_csu 13184   invgcminusg 15432  SubGrpcsubg 15696  SubRingcsubrg 16883  ℂfldccnfld 17840    Dncdvn 21361  Polycply 21674   Xpcidp 21675  degcdgr 21677   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-fal 1375  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-of 6341  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-fl 11663  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-clim 12987  df-rlim 12988  df-sum 13185  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-mnd 15436  df-grp 15566  df-minusg 15567  df-subg 15699  df-cntz 15856  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-cring 16670  df-subrg 16885  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-0p 21170  df-limc 21363  df-dv 21364  df-dvn 21365  df-ply 21678  df-idp 21679  df-coe 21680  df-dgr 21681  df-tayl 21842
This theorem is referenced by:  taylply  21856  taylthlem2  21861
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