Step |
Hyp |
Ref |
Expression |
1 |
|
coeval |
|- ( F e. ( Poly ` S ) -> ( coeff ` F ) = ( 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 ) ) ) ) ) ) |
2 |
|
coeeu |
|- ( F e. ( Poly ` S ) -> E! 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 ) ) ) ) ) |
3 |
|
riotacl2 |
|- ( E! 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 ) ) ) ) -> ( 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 ) ) ) ) ) e. { 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 ) ) ) ) } ) |
4 |
2 3
|
syl |
|- ( F e. ( Poly ` S ) -> ( 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 ) ) ) ) ) e. { 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 ) ) ) ) } ) |
5 |
1 4
|
eqeltrd |
|- ( F e. ( Poly ` S ) -> ( coeff ` F ) e. { 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 ) ) ) ) } ) |
6 |
|
imaeq1 |
|- ( a = ( coeff ` F ) -> ( a " ( ZZ>= ` ( n + 1 ) ) ) = ( ( coeff ` F ) " ( ZZ>= ` ( n + 1 ) ) ) ) |
7 |
6
|
eqeq1d |
|- ( a = ( coeff ` F ) -> ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } <-> ( ( coeff ` F ) " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } ) ) |
8 |
|
fveq1 |
|- ( a = ( coeff ` F ) -> ( a ` k ) = ( ( coeff ` F ) ` k ) ) |
9 |
8
|
oveq1d |
|- ( a = ( coeff ` F ) -> ( ( a ` k ) x. ( z ^ k ) ) = ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) ) |
10 |
9
|
sumeq2sdv |
|- ( a = ( coeff ` F ) -> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... n ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) ) |
11 |
10
|
mpteq2dv |
|- ( a = ( coeff ` F ) -> ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) ) ) |
12 |
11
|
eqeq2d |
|- ( a = ( coeff ` F ) -> ( F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) <-> F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) ) ) ) |
13 |
7 12
|
anbi12d |
|- ( a = ( coeff ` F ) -> ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) <-> ( ( ( coeff ` F ) " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) ) ) ) ) |
14 |
13
|
rexbidv |
|- ( a = ( coeff ` F ) -> ( E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) <-> E. n e. NN0 ( ( ( coeff ` F ) " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) ) ) ) ) |
15 |
14
|
elrab |
|- ( ( coeff ` F ) e. { 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 ) ) ) ) } <-> ( ( coeff ` F ) e. ( CC ^m NN0 ) /\ E. n e. NN0 ( ( ( coeff ` F ) " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) ) ) ) ) |
16 |
5 15
|
sylib |
|- ( F e. ( Poly ` S ) -> ( ( coeff ` F ) e. ( CC ^m NN0 ) /\ E. n e. NN0 ( ( ( coeff ` F ) " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( ( coeff ` F ) ` k ) x. ( z ^ k ) ) ) ) ) ) |