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Theorem taylply2 22929
Description: The Taylor polynomial is a polynomial of degree (at most)  N. This version of taylply 22930 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 22926 . . . 4  |-  ( ph  ->  T  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( ( ( ( S  Dn F ) `  k ) `  B
)  /  ( ! `
 k ) )  x.  ( ( x  -  B ) ^
k ) ) ) )
8 simpr 459 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
9 cnex 9562 . . . . . . . . . . . . 13  |-  CC  e.  _V
109a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  CC  e.  _V )
11 elpm2r 7429 . . . . . . . . . . . 12  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
1210, 1, 2, 3, 11syl22anc 1227 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
13 dvnbss 22497 . . . . . . . . . . 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 1226 . . . . . . . . . 10  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 N )  C_  dom  F )
15 fdm 5717 . . . . . . . . . . 11  |-  ( F : A --> CC  ->  dom 
F  =  A )
162, 15syl 16 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  A )
1714, 16sseqtrd 3525 . . . . . . . . 9  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 N )  C_  A )
18 recnprss 22474 . . . . . . . . . . 11  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
191, 18syl 16 . . . . . . . . . 10  |-  ( ph  ->  S  C_  CC )
203, 19sstrd 3499 . . . . . . . . 9  |-  ( ph  ->  A  C_  CC )
2117, 20sstrd 3499 . . . . . . . 8  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 N )  C_  CC )
2221, 5sseldd 3490 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
2322adantr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  CC )
248, 23subcld 9922 . . . . 5  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  -  B )  e.  CC )
25 df-idp 22752 . . . . . . . 8  |-  Xp  =  (  _I  |`  CC )
26 mptresid 5316 . . . . . . . 8  |-  ( x  e.  CC  |->  x )  =  (  _I  |`  CC )
2725, 26eqtr4i 2486 . . . . . . 7  |-  Xp  =  ( x  e.  CC  |->  x )
2827a1i 11 . . . . . 6  |-  ( ph  ->  Xp  =  ( x  e.  CC  |->  x ) )
29 fconstmpt 5032 . . . . . . 7  |-  ( CC 
X.  { B }
)  =  ( x  e.  CC  |->  B )
3029a1i 11 . . . . . 6  |-  ( ph  ->  ( CC  X.  { B } )  =  ( x  e.  CC  |->  B ) )
3110, 8, 23, 28, 30offval2 6529 . . . . 5  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { B }
) )  =  ( x  e.  CC  |->  ( x  -  B ) ) )
32 eqidd 2455 . . . . 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 6277 . . . . . . 7  |-  ( y  =  ( x  -  B )  ->  (
y ^ k )  =  ( ( x  -  B ) ^
k ) )
3433oveq2d 6286 . . . . . 6  |-  ( y  =  ( x  -  B )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) )  =  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) )
3534sumeq2sdv 13608 . . . . 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 6040 . . . 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 2498 . . 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 18619 . . . . . . 7  |-  CC  =  ( Base ` fld )
4039subrgss 17625 . . . . . 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 22765 . . . 4  |-  ( ph  ->  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  e.  (Poly `  D ) )
44 cnfld1 18638 . . . . . . . 8  |-  1  =  ( 1r ` fld )
4544subrg1cl 17632 . . . . . . 7  |-  ( D  e.  (SubRing ` fld )  ->  1  e.  D )
4638, 45syl 16 . . . . . 6  |-  ( ph  ->  1  e.  D )
47 plyid 22772 . . . . . 6  |-  ( ( D  C_  CC  /\  1  e.  D )  ->  Xp  e.  (Poly `  D
) )
4841, 46, 47syl2anc 659 . . . . 5  |-  ( ph  ->  Xp  e.  (Poly `  D ) )
49 taylply2.2 . . . . . 6  |-  ( ph  ->  B  e.  D )
50 plyconst 22769 . . . . . 6  |-  ( ( D  C_  CC  /\  B  e.  D )  ->  ( CC  X.  { B }
)  e.  (Poly `  D ) )
5141, 49, 50syl2anc 659 . . . . 5  |-  ( ph  ->  ( CC  X.  { B } )  e.  (Poly `  D ) )
52 subrgsubg 17630 . . . . . . 7  |-  ( D  e.  (SubRing ` fld )  ->  D  e.  (SubGrp ` fld ) )
5338, 52syl 16 . . . . . 6  |-  ( ph  ->  D  e.  (SubGrp ` fld )
)
54 cnfldadd 18620 . . . . . . . 8  |-  +  =  ( +g  ` fld )
5554subgcl 16410 . . . . . . 7  |-  ( ( D  e.  (SubGrp ` fld )  /\  u  e.  D  /\  v  e.  D
)  ->  ( u  +  v )  e.  D )
56553expb 1195 . . . . . 6  |-  ( ( D  e.  (SubGrp ` fld )  /\  ( u  e.  D  /\  v  e.  D
) )  ->  (
u  +  v )  e.  D )
5753, 56sylan 469 . . . . 5  |-  ( (
ph  /\  ( u  e.  D  /\  v  e.  D ) )  -> 
( u  +  v )  e.  D )
58 cnfldmul 18621 . . . . . . . 8  |-  x.  =  ( .r ` fld )
5958subrgmcl 17636 . . . . . . 7  |-  ( ( D  e.  (SubRing ` fld )  /\  u  e.  D  /\  v  e.  D )  ->  (
u  x.  v )  e.  D )
60593expb 1195 . . . . . 6  |-  ( ( D  e.  (SubRing ` fld )  /\  (
u  e.  D  /\  v  e.  D )
)  ->  ( u  x.  v )  e.  D
)
6138, 60sylan 469 . . . . 5  |-  ( (
ph  /\  ( u  e.  D  /\  v  e.  D ) )  -> 
( u  x.  v
)  e.  D )
62 ax-1cn 9539 . . . . . . 7  |-  1  e.  CC
63 cnfldneg 18639 . . . . . . 7  |-  ( 1  e.  CC  ->  (
( invg ` fld ) `  1 )  = 
-u 1 )
6462, 63ax-mp 5 . . . . . 6  |-  ( ( invg ` fld ) `  1 )  =  -u 1
65 eqid 2454 . . . . . . . 8  |-  ( invg ` fld )  =  ( invg ` fld )
6665subginvcl 16409 . . . . . . 7  |-  ( ( D  e.  (SubGrp ` fld )  /\  1  e.  D
)  ->  ( ( invg ` fld ) `  1 )  e.  D )
6753, 46, 66syl2anc 659 . . . . . 6  |-  ( ph  ->  ( ( invg ` fld ) `  1 )  e.  D )
6864, 67syl5eqelr 2547 . . . . 5  |-  ( ph  -> 
-u 1  e.  D
)
6948, 51, 57, 61, 68plysub 22782 . . . 4  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { B }
) )  e.  (Poly `  D ) )
7043, 69, 57, 61plyco 22804 . . 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 2542 . 2  |-  ( ph  ->  T  e.  (Poly `  D ) )
7237fveq2d 5852 . . . 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 2454 . . . . 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 2454 . . . . 5  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { B }
) ) )  =  (deg `  ( Xp  oF  -  ( CC  X.  { B }
) ) )
7573, 74, 43, 69dgrco 22838 . . . 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 2454 . . . . . . . . 9  |-  ( Xp  oF  -  ( CC  X.  { B } ) )  =  ( Xp  oF  -  ( CC 
X.  { B }
) )
7776plyremlem 22866 . . . . . . . 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 1007 . . . . . 6  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { B }
) ) )  =  1 )
8079oveq2d 6286 . . . . 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 22796 . . . . . . . 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 10850 . . . . . 6  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  e.  CC )
8483mulid1d 9602 . . . . 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 2495 . . . 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 2499 . . 3  |-  ( ph  ->  (deg `  T )  =  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) ) )
871adantr 463 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  S  e.  { RR ,  CC } )
8812adantr 463 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  F  e.  ( CC  ^pm  S
) )
89 elfznn0 11775 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
9089adantl 464 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
91 dvnf 22496 . . . . . . 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 1226 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( S  Dn
F ) `  k
) : dom  (
( S  Dn
F ) `  k
) --> CC )
93 simpr 459 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  ( 0 ... N
) )
94 dvn2bss 22499 . . . . . . . 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 1226 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  dom  ( ( S  Dn F ) `  N )  C_  dom  ( ( S  Dn F ) `  k ) )
965adantr 463 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  B  e.  dom  ( ( S  Dn F ) `
 N ) )
9795, 96sseldd 3490 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  B  e.  dom  ( ( S  Dn F ) `
 k ) )
9892, 97ffvelrnd 6008 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( S  Dn F ) `  k ) `  B
)  e.  CC )
99 faccl 12345 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
10090, 99syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  e.  NN )
101100nncnd 10547 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  e.  CC )
102100nnne0d 10576 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  =/=  0 )
10398, 101, 102divcld 10316 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
10443, 4, 103, 32dgrle 22806 . . 3  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  <_  N )
10586, 104eqbrtrd 4459 . 2  |-  ( ph  ->  (deg `  T )  <_  N )
10671, 105jca 530 1  |-  ( ph  ->  ( T  e.  (Poly `  D )  /\  (deg `  T )  <_  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   {csn 4016   {cpr 4018   class class class wbr 4439    |-> cmpt 4497    _I cid 4779    X. cxp 4986   `'ccnv 4987   dom cdm 4988    |` cres 4990   "cima 4991    o. ccom 4992   -->wf 5566   ` cfv 5570  (class class class)co 6270    oFcof 6511    ^pm cpm 7413   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    <_ cle 9618    - cmin 9796   -ucneg 9797    / cdiv 10202   NNcn 10531   NN0cn0 10791   ...cfz 11675   ^cexp 12148   !cfa 12335   sum_csu 13590   invgcminusg 16253  SubGrpcsubg 16394  SubRingcsubrg 17620  ℂfldccnfld 18615    Dncdvn 22434  Polycply 22747   Xpcidp 22748  degcdgr 22750   Tayl ctayl 22914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-fac 12336  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-rlim 13394  df-sum 13591  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-subg 16397  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-subrg 17622  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cnp 19896  df-haus 19983  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-tsms 20791  df-xms 20989  df-ms 20990  df-0p 22243  df-limc 22436  df-dv 22437  df-dvn 22438  df-ply 22751  df-idp 22752  df-coe 22753  df-dgr 22754  df-tayl 22916
This theorem is referenced by:  taylply  22930  taylthlem2  22935
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