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Theorem taylply2 22589
Description: The Taylor polynomial is a polynomial of degree (at most)  N. This version of taylply 22590 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 22586 . . . 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 9574 . . . . . . . . . . . . 13  |-  CC  e.  _V
109a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  CC  e.  _V )
11 elpm2r 7437 . . . . . . . . . . . 12  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : A --> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
1210, 1, 2, 3, 11syl22anc 1229 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
13 dvnbss 22158 . . . . . . . . . . 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 1228 . . . . . . . . . 10  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 N )  C_  dom  F )
15 fdm 5735 . . . . . . . . . . 11  |-  ( F : A --> CC  ->  dom 
F  =  A )
162, 15syl 16 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  A )
1714, 16sseqtrd 3540 . . . . . . . . 9  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 N )  C_  A )
18 recnprss 22135 . . . . . . . . . . 11  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
191, 18syl 16 . . . . . . . . . 10  |-  ( ph  ->  S  C_  CC )
203, 19sstrd 3514 . . . . . . . . 9  |-  ( ph  ->  A  C_  CC )
2117, 20sstrd 3514 . . . . . . . 8  |-  ( ph  ->  dom  ( ( S  Dn F ) `
 N )  C_  CC )
2221, 5sseldd 3505 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
2322adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  CC )  ->  B  e.  CC )
248, 23subcld 9931 . . . . 5  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  -  B )  e.  CC )
25 df-idp 22413 . . . . . . . 8  |-  Xp  =  (  _I  |`  CC )
26 mptresid 5328 . . . . . . . 8  |-  ( x  e.  CC  |->  x )  =  (  _I  |`  CC )
2725, 26eqtr4i 2499 . . . . . . 7  |-  Xp  =  ( x  e.  CC  |->  x )
2827a1i 11 . . . . . 6  |-  ( ph  ->  Xp  =  ( x  e.  CC  |->  x ) )
29 fconstmpt 5043 . . . . . . 7  |-  ( CC 
X.  { B }
)  =  ( x  e.  CC  |->  B )
3029a1i 11 . . . . . 6  |-  ( ph  ->  ( CC  X.  { B } )  =  ( x  e.  CC  |->  B ) )
3110, 8, 23, 28, 30offval2 6541 . . . . 5  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { B }
) )  =  ( x  e.  CC  |->  ( x  -  B ) ) )
32 eqidd 2468 . . . . 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 6292 . . . . . . 7  |-  ( y  =  ( x  -  B )  ->  (
y ^ k )  =  ( ( x  -  B ) ^
k ) )
3433oveq2d 6301 . . . . . 6  |-  ( y  =  ( x  -  B )  ->  (
( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) )  =  ( ( ( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  x.  ( ( x  -  B ) ^ k ) ) )
3534sumeq2sdv 13492 . . . . 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 6055 . . . 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 2511 . . 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 18235 . . . . . . 7  |-  CC  =  ( Base ` fld )
4039subrgss 17242 . . . . . 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 22426 . . . 4  |-  ( ph  ->  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn F ) `  k ) `
 B )  / 
( ! `  k
) )  x.  (
y ^ k ) ) )  e.  (Poly `  D ) )
44 cnfld1 18254 . . . . . . . 8  |-  1  =  ( 1r ` fld )
4544subrg1cl 17249 . . . . . . 7  |-  ( D  e.  (SubRing ` fld )  ->  1  e.  D )
4638, 45syl 16 . . . . . 6  |-  ( ph  ->  1  e.  D )
47 plyid 22433 . . . . . 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 22430 . . . . . 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 17247 . . . . . . 7  |-  ( D  e.  (SubRing ` fld )  ->  D  e.  (SubGrp ` fld ) )
5338, 52syl 16 . . . . . 6  |-  ( ph  ->  D  e.  (SubGrp ` fld )
)
54 cnfldadd 18236 . . . . . . . 8  |-  +  =  ( +g  ` fld )
5554subgcl 16025 . . . . . . 7  |-  ( ( D  e.  (SubGrp ` fld )  /\  u  e.  D  /\  v  e.  D
)  ->  ( u  +  v )  e.  D )
56553expb 1197 . . . . . 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 18237 . . . . . . . 8  |-  x.  =  ( .r ` fld )
5958subrgmcl 17253 . . . . . . 7  |-  ( ( D  e.  (SubRing ` fld )  /\  u  e.  D  /\  v  e.  D )  ->  (
u  x.  v )  e.  D )
60593expb 1197 . . . . . 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 9551 . . . . . . 7  |-  1  e.  CC
63 cnfldneg 18255 . . . . . . 7  |-  ( 1  e.  CC  ->  (
( invg ` fld ) `  1 )  = 
-u 1 )
6462, 63ax-mp 5 . . . . . 6  |-  ( ( invg ` fld ) `  1 )  =  -u 1
65 eqid 2467 . . . . . . . 8  |-  ( invg ` fld )  =  ( invg ` fld )
6665subginvcl 16024 . . . . . . 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 2560 . . . . 5  |-  ( ph  -> 
-u 1  e.  D
)
6948, 51, 57, 61, 68plysub 22443 . . . 4  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { B }
) )  e.  (Poly `  D ) )
7043, 69, 57, 61plyco 22465 . . 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 2555 . 2  |-  ( ph  ->  T  e.  (Poly `  D ) )
7237fveq2d 5870 . . . 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 2467 . . . . 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 2467 . . . . 5  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { B }
) ) )  =  (deg `  ( Xp  oF  -  ( CC  X.  { B }
) ) )
7573, 74, 43, 69dgrco 22498 . . . 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 2467 . . . . . . . . 9  |-  ( Xp  oF  -  ( CC  X.  { B } ) )  =  ( Xp  oF  -  ( CC 
X.  { B }
) )
7776plyremlem 22526 . . . . . . . 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 1009 . . . . . 6  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { B }
) ) )  =  1 )
8079oveq2d 6301 . . . . 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 22457 . . . . . . . 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 10855 . . . . . 6  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  e.  CC )
8483mulid1d 9614 . . . . 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 2508 . . . 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 2512 . . 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 11771 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
9089adantl 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
91 dvnf 22157 . . . . . . 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 1228 . . . . . 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 22160 . . . . . . . 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 1228 . . . . . . 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 3505 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  B  e.  dom  ( ( S  Dn F ) `
 k ) )
9892, 97ffvelrnd 6023 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( S  Dn F ) `  k ) `  B
)  e.  CC )
99 faccl 12332 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
10090, 99syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  e.  NN )
101100nncnd 10553 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  e.  CC )
102100nnne0d 10581 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( ! `  k )  =/=  0 )
10398, 101, 102divcld 10321 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( ( S  Dn F ) `
 k ) `  B )  /  ( ! `  k )
)  e.  CC )
10443, 4, 103, 32dgrle 22467 . . 3  |-  ( ph  ->  (deg `  ( y  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( ( ( S  Dn
F ) `  k
) `  B )  /  ( ! `  k ) )  x.  ( y ^ k
) ) ) )  <_  N )
10586, 104eqbrtrd 4467 . 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   {csn 4027   {cpr 4029   class class class wbr 4447    |-> cmpt 4505    _I cid 4790    X. cxp 4997   `'ccnv 4998   dom cdm 4999    |` cres 5001   "cima 5002    o. ccom 5003   -->wf 5584   ` cfv 5588  (class class class)co 6285    oFcof 6523    ^pm cpm 7422   CCcc 9491   RRcr 9492   0cc0 9493   1c1 9494    + caddc 9496    x. cmul 9498    <_ cle 9630    - cmin 9806   -ucneg 9807    / cdiv 10207   NNcn 10537   NN0cn0 10796   ...cfz 11673   ^cexp 12135   !cfa 12322   sum_csu 13474   invgcminusg 15731  SubGrpcsubg 16009  SubRingcsubrg 17237  ℂfldccnfld 18231    Dncdvn 22095  Polycply 22408   Xpcidp 22409  degcdgr 22411   Tayl ctayl 22574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571  ax-addf 9572  ax-mulf 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-fi 7872  df-sup 7902  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-q 11184  df-rp 11222  df-xneg 11319  df-xadd 11320  df-xmul 11321  df-icc 11537  df-fz 11674  df-fzo 11794  df-fl 11898  df-seq 12077  df-exp 12136  df-fac 12323  df-hash 12375  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-clim 13277  df-rlim 13278  df-sum 13475  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-starv 14573  df-tset 14577  df-ple 14578  df-ds 14580  df-unif 14581  df-rest 14681  df-topn 14682  df-0g 14700  df-gsum 14701  df-topgen 14702  df-mnd 15735  df-grp 15871  df-minusg 15872  df-subg 16012  df-cntz 16169  df-cmn 16615  df-abl 16616  df-mgp 16956  df-ur 16968  df-rng 17014  df-cring 17015  df-subrg 17239  df-psmet 18222  df-xmet 18223  df-met 18224  df-bl 18225  df-mopn 18226  df-fbas 18227  df-fg 18228  df-cnfld 18232  df-top 19206  df-bases 19208  df-topon 19209  df-topsp 19210  df-cld 19326  df-ntr 19327  df-cls 19328  df-nei 19405  df-lp 19443  df-perf 19444  df-cnp 19535  df-haus 19622  df-fil 20174  df-fm 20266  df-flim 20267  df-flf 20268  df-tsms 20452  df-xms 20650  df-ms 20651  df-0p 21904  df-limc 22097  df-dv 22098  df-dvn 22099  df-ply 22412  df-idp 22413  df-coe 22414  df-dgr 22415  df-tayl 22576
This theorem is referenced by:  taylply  22590  taylthlem2  22595
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