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Definition df-coe 22350
Description: Define the coefficient function for a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
Assertion
Ref Expression
df-coe  |- coeff  =  ( f  e.  (Poly `  CC )  |->  ( iota_ a  e.  ( CC  ^m  NN0 ) E. n  e. 
NN0  ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) ) ) )
Distinct variable group:    f, a, k, n, z

Detailed syntax breakdown of Definition df-coe
StepHypRef Expression
1 ccoe 22346 . 2  class coeff
2 vf . . 3  setvar  f
3 cc 9490 . . . 4  class  CC
4 cply 22344 . . . 4  class Poly
53, 4cfv 5588 . . 3  class  (Poly `  CC )
6 va . . . . . . . . 9  setvar  a
76cv 1378 . . . . . . . 8  class  a
8 vn . . . . . . . . . . 11  setvar  n
98cv 1378 . . . . . . . . . 10  class  n
10 c1 9493 . . . . . . . . . 10  class  1
11 caddc 9495 . . . . . . . . . 10  class  +
129, 10, 11co 6284 . . . . . . . . 9  class  ( n  +  1 )
13 cuz 11082 . . . . . . . . 9  class  ZZ>=
1412, 13cfv 5588 . . . . . . . 8  class  ( ZZ>= `  ( n  +  1
) )
157, 14cima 5002 . . . . . . 7  class  ( a
" ( ZZ>= `  (
n  +  1 ) ) )
16 cc0 9492 . . . . . . . 8  class  0
1716csn 4027 . . . . . . 7  class  { 0 }
1815, 17wceq 1379 . . . . . 6  wff  ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }
192cv 1378 . . . . . . 7  class  f
20 vz . . . . . . . 8  setvar  z
21 cfz 11672 . . . . . . . . . 10  class  ...
2216, 9, 21co 6284 . . . . . . . . 9  class  ( 0 ... n )
23 vk . . . . . . . . . . . 12  setvar  k
2423cv 1378 . . . . . . . . . . 11  class  k
2524, 7cfv 5588 . . . . . . . . . 10  class  ( a `
 k )
2620cv 1378 . . . . . . . . . . 11  class  z
27 cexp 12134 . . . . . . . . . . 11  class  ^
2826, 24, 27co 6284 . . . . . . . . . 10  class  ( z ^ k )
29 cmul 9497 . . . . . . . . . 10  class  x.
3025, 28, 29co 6284 . . . . . . . . 9  class  ( ( a `  k )  x.  ( z ^
k ) )
3122, 30, 23csu 13471 . . . . . . . 8  class  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )
3220, 3, 31cmpt 4505 . . . . . . 7  class  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) ) )
3319, 32wceq 1379 . . . . . 6  wff  f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )
3418, 33wa 369 . . . . 5  wff  ( ( a " ( ZZ>= `  ( n  +  1
) ) )  =  { 0 }  /\  f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )
35 cn0 10795 . . . . 5  class  NN0
3634, 8, 35wrex 2815 . . . 4  wff  E. n  e.  NN0  ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )
37 cmap 7420 . . . . 5  class  ^m
383, 35, 37co 6284 . . . 4  class  ( CC 
^m  NN0 )
3936, 6, 38crio 6244 . . 3  class  ( iota_ a  e.  ( CC  ^m  NN0 ) E. n  e. 
NN0  ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) ) )
402, 5, 39cmpt 4505 . 2  class  ( f  e.  (Poly `  CC )  |->  ( iota_ a  e.  ( CC  ^m  NN0 ) E. n  e.  NN0  ( ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) ) )
411, 40wceq 1379 1  wff coeff  =  ( f  e.  (Poly `  CC )  |->  ( iota_ a  e.  ( CC  ^m  NN0 ) E. n  e. 
NN0  ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  coeval  22383
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