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Definition df-ply 19570
Description: Define the set of polynomials on the complexes 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 19566 . 2  class Poly
2 vx . . 3  set  x
3 cc 8735 . . . 4  class  CC
43cpw 3625 . . 3  class  ~P CC
5 vf . . . . . . . 8  set  f
65cv 1622 . . . . . . 7  class  f
7 vz . . . . . . . 8  set  z
8 cc0 8737 . . . . . . . . . 10  class  0
9 vn . . . . . . . . . . 11  set  n
109cv 1622 . . . . . . . . . 10  class  n
11 cfz 10782 . . . . . . . . . 10  class  ...
128, 10, 11co 5858 . . . . . . . . 9  class  ( 0 ... n )
13 vk . . . . . . . . . . . 12  set  k
1413cv 1622 . . . . . . . . . . 11  class  k
15 va . . . . . . . . . . . 12  set  a
1615cv 1622 . . . . . . . . . . 11  class  a
1714, 16cfv 5255 . . . . . . . . . 10  class  ( a `
 k )
187cv 1622 . . . . . . . . . . 11  class  z
19 cexp 11104 . . . . . . . . . . 11  class  ^
2018, 14, 19co 5858 . . . . . . . . . 10  class  ( z ^ k )
21 cmul 8742 . . . . . . . . . 10  class  x.
2217, 20, 21co 5858 . . . . . . . . 9  class  ( ( a `  k )  x.  ( z ^
k ) )
2312, 22, 13csu 12158 . . . . . . . 8  class  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )
247, 3, 23cmpt 4077 . . . . . . 7  class  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) ) )
256, 24wceq 1623 . . . . . 6  wff  f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )
262cv 1622 . . . . . . . 8  class  x
278csn 3640 . . . . . . . 8  class  { 0 }
2826, 27cun 3150 . . . . . . 7  class  ( x  u.  { 0 } )
29 cn0 9965 . . . . . . 7  class  NN0
30 cmap 6772 . . . . . . 7  class  ^m
3128, 29, 30co 5858 . . . . . 6  class  ( ( x  u.  { 0 } )  ^m  NN0 )
3225, 15, 31wrex 2544 . . . . 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 2544 . . . 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 2269 . . 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 4077 . 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 1623 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 set class
This definition is referenced by:  plyval  19575  plybss  19576
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