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Definition df-ply 21788
Description: Define the set of polynomials on the complex numbers with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
Assertion
Ref Expression
df-ply  |- Poly  =  ( x  e.  ~P CC  |->  { f  |  E. n  e.  NN0  E. a  e.  ( ( x  u. 
{ 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) } )
Distinct variable group:    f, a, k, n, x, z

Detailed syntax breakdown of Definition df-ply
StepHypRef Expression
1 cply 21784 . 2  class Poly
2 vx . . 3  setvar  x
3 cc 9390 . . . 4  class  CC
43cpw 3967 . . 3  class  ~P CC
5 vf . . . . . . . 8  setvar  f
65cv 1369 . . . . . . 7  class  f
7 vz . . . . . . . 8  setvar  z
8 cc0 9392 . . . . . . . . . 10  class  0
9 vn . . . . . . . . . . 11  setvar  n
109cv 1369 . . . . . . . . . 10  class  n
11 cfz 11553 . . . . . . . . . 10  class  ...
128, 10, 11co 6199 . . . . . . . . 9  class  ( 0 ... n )
13 vk . . . . . . . . . . . 12  setvar  k
1413cv 1369 . . . . . . . . . . 11  class  k
15 va . . . . . . . . . . . 12  setvar  a
1615cv 1369 . . . . . . . . . . 11  class  a
1714, 16cfv 5525 . . . . . . . . . 10  class  ( a `
 k )
187cv 1369 . . . . . . . . . . 11  class  z
19 cexp 11981 . . . . . . . . . . 11  class  ^
2018, 14, 19co 6199 . . . . . . . . . 10  class  ( z ^ k )
21 cmul 9397 . . . . . . . . . 10  class  x.
2217, 20, 21co 6199 . . . . . . . . 9  class  ( ( a `  k )  x.  ( z ^
k ) )
2312, 22, 13csu 13280 . . . . . . . 8  class  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )
247, 3, 23cmpt 4457 . . . . . . 7  class  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) ) )
256, 24wceq 1370 . . . . . 6  wff  f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )
262cv 1369 . . . . . . . 8  class  x
278csn 3984 . . . . . . . 8  class  { 0 }
2826, 27cun 3433 . . . . . . 7  class  ( x  u.  { 0 } )
29 cn0 10689 . . . . . . 7  class  NN0
30 cmap 7323 . . . . . . 7  class  ^m
3128, 29, 30co 6199 . . . . . 6  class  ( ( x  u.  { 0 } )  ^m  NN0 )
3225, 15, 31wrex 2799 . . . . 5  wff  E. a  e.  ( ( x  u. 
{ 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )
3332, 9, 29wrex 2799 . . . 4  wff  E. n  e.  NN0  E. a  e.  ( ( x  u. 
{ 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )
3433, 5cab 2439 . . 3  class  { f  |  E. n  e. 
NN0  E. a  e.  ( ( x  u.  {
0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) }
352, 4, 34cmpt 4457 . 2  class  ( x  e.  ~P CC  |->  { f  |  E. n  e.  NN0  E. a  e.  ( ( x  u. 
{ 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) } )
361, 35wceq 1370 1  wff Poly  =  ( x  e.  ~P CC  |->  { f  |  E. n  e.  NN0  E. a  e.  ( ( x  u. 
{ 0 } )  ^m  NN0 ) 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:  plyval  21793  plybss  21794
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