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Theorem coeeu 22748
Description: Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
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 ) ) ) ) )
Distinct variable groups:    z, k    n, a, F    S, a, n    k, a, z, n
Allowed substitution hints:    S( z, k)    F( z, k)

Proof of Theorem coeeu
Dummy variables  b 
j  m  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyssc 22723 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3495 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 elply2 22719 . . . . . 6  |-  ( F  e.  (Poly `  CC ) 
<->  ( CC  C_  CC  /\ 
E. n  e.  NN0  E. a  e.  ( ( CC  u.  { 0 } )  ^m  NN0 ) ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) ) ) )
43simprbi 464 . . . . 5  |-  ( F  e.  (Poly `  CC )  ->  E. n  e.  NN0  E. a  e.  ( ( CC  u.  { 0 } )  ^m  NN0 ) ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) ) )
5 rexcom 3019 . . . . 5  |-  ( E. n  e.  NN0  E. a  e.  ( ( CC  u.  { 0 } )  ^m  NN0 ) ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  <->  E. a  e.  ( ( CC  u.  { 0 } )  ^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 ) ) ) ) )
64, 5sylib 196 . . . 4  |-  ( F  e.  (Poly `  CC )  ->  E. a  e.  ( ( CC  u.  {
0 } )  ^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 ) ) ) ) )
72, 6syl 16 . . 3  |-  ( F  e.  (Poly `  S
)  ->  E. a  e.  ( ( CC  u.  { 0 } )  ^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 ) ) ) ) )
8 0cn 9605 . . . . . . 7  |-  0  e.  CC
9 snssi 4176 . . . . . . 7  |-  ( 0  e.  CC  ->  { 0 }  C_  CC )
108, 9ax-mp 5 . . . . . 6  |-  { 0 }  C_  CC
11 ssequn2 3673 . . . . . 6  |-  ( { 0 }  C_  CC  <->  ( CC  u.  { 0 } )  =  CC )
1210, 11mpbi 208 . . . . 5  |-  ( CC  u.  { 0 } )  =  CC
1312oveq1i 6306 . . . 4  |-  ( ( CC  u.  { 0 } )  ^m  NN0 )  =  ( CC  ^m 
NN0 )
1413rexeqi 3059 . . 3  |-  ( E. a  e.  ( ( CC  u.  { 0 } )  ^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 ) ) ) ) )
157, 14sylib 196 . 2  |-  ( 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 ) ) ) ) )
16 reeanv 3025 . . . 4  |-  ( E. n  e.  NN0  E. m  e.  NN0  ( ( ( a " ( ZZ>= `  ( n  +  1
) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) )  <->  ( E. n  e.  NN0  ( ( a " ( ZZ>= `  ( n  +  1
) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  E. m  e.  NN0  (
( b " ( ZZ>=
`  ( m  + 
1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )
17 simp1l 1020 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )  /\  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )  ->  F  e.  (Poly `  S ) )
18 simp1rl 1061 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )  /\  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )  ->  a  e.  ( CC  ^m  NN0 )
)
19 simp1rr 1062 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )  /\  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )  ->  b  e.  ( CC  ^m  NN0 )
)
20 simp2l 1022 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )  /\  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )  ->  n  e.  NN0 )
21 simp2r 1023 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )  /\  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )  ->  m  e.  NN0 )
22 simp3ll 1067 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )  /\  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )  ->  ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 } )
23 simp3rl 1069 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )  /\  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )  ->  ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 } )
24 simp3lr 1068 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )  /\  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )
25 oveq1 6303 . . . . . . . . . . . 12  |-  ( z  =  w  ->  (
z ^ k )  =  ( w ^
k ) )
2625oveq2d 6312 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( a `
 k )  x.  ( w ^ k
) ) )
2726sumeq2sdv 13538 . . . . . . . . . 10  |-  ( z  =  w  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( w ^ k ) ) )
28 fveq2 5872 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
a `  k )  =  ( a `  j ) )
29 oveq2 6304 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
w ^ k )  =  ( w ^
j ) )
3028, 29oveq12d 6314 . . . . . . . . . . 11  |-  ( k  =  j  ->  (
( a `  k
)  x.  ( w ^ k ) )  =  ( ( a `
 j )  x.  ( w ^ j
) ) )
3130cbvsumv 13530 . . . . . . . . . 10  |-  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( w ^ k
) )  =  sum_ j  e.  ( 0 ... n ) ( ( a `  j
)  x.  ( w ^ j ) )
3227, 31syl6eq 2514 . . . . . . . . 9  |-  ( z  =  w  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ j  e.  ( 0 ... n ) ( ( a `  j
)  x.  ( w ^ j ) ) )
3332cbvmptv 4548 . . . . . . . 8  |-  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) ) )  =  ( w  e.  CC  |->  sum_ j  e.  ( 0 ... n ) ( ( a `  j
)  x.  ( w ^ j ) ) )
3424, 33syl6eq 2514 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )  /\  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )  ->  F  =  ( w  e.  CC  |->  sum_ j  e.  ( 0 ... n ) ( ( a `  j
)  x.  ( w ^ j ) ) ) )
35 simp3rr 1070 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )  /\  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) )
3625oveq2d 6312 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( b `  k
)  x.  ( z ^ k ) )  =  ( ( b `
 k )  x.  ( w ^ k
) ) )
3736sumeq2sdv 13538 . . . . . . . . . 10  |-  ( z  =  w  ->  sum_ k  e.  ( 0 ... m
) ( ( b `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( w ^ k ) ) )
38 fveq2 5872 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
b `  k )  =  ( b `  j ) )
3938, 29oveq12d 6314 . . . . . . . . . . 11  |-  ( k  =  j  ->  (
( b `  k
)  x.  ( w ^ k ) )  =  ( ( b `
 j )  x.  ( w ^ j
) ) )
4039cbvsumv 13530 . . . . . . . . . 10  |-  sum_ k  e.  ( 0 ... m
) ( ( b `
 k )  x.  ( w ^ k
) )  =  sum_ j  e.  ( 0 ... m ) ( ( b `  j
)  x.  ( w ^ j ) )
4137, 40syl6eq 2514 . . . . . . . . 9  |-  ( z  =  w  ->  sum_ k  e.  ( 0 ... m
) ( ( b `
 k )  x.  ( z ^ k
) )  =  sum_ j  e.  ( 0 ... m ) ( ( b `  j
)  x.  ( w ^ j ) ) )
4241cbvmptv 4548 . . . . . . . 8  |-  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m
) ( ( b `
 k )  x.  ( z ^ k
) ) )  =  ( w  e.  CC  |->  sum_ j  e.  ( 0 ... m ) ( ( b `  j
)  x.  ( w ^ j ) ) )
4335, 42syl6eq 2514 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )  /\  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )  ->  F  =  ( w  e.  CC  |->  sum_ j  e.  ( 0 ... m ) ( ( b `  j
)  x.  ( w ^ j ) ) ) )
4417, 18, 19, 20, 21, 22, 23, 34, 43coeeulem 22747 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )  /\  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) ) )  ->  a  =  b )
45443expia 1198 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  /\  ( n  e.  NN0  /\  m  e.  NN0 )
)  ->  ( (
( ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )  /\  (
( b " ( ZZ>=
`  ( m  + 
1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) )  -> 
a  =  b ) )
4645rexlimdvva 2956 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  e.  ( CC  ^m  NN0 ) ) )  -> 
( E. n  e. 
NN0  E. m  e.  NN0  ( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  /\  ( ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) )  -> 
a  =  b ) )
4716, 46syl5bir 218 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  (
a  e.  ( CC 
^m  NN0 )  /\  b  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. m  e.  NN0  (
( b " ( ZZ>=
`  ( m  + 
1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) )  -> 
a  =  b ) )
4847ralrimivva 2878 . 2  |-  ( F  e.  (Poly `  S
)  ->  A. a  e.  ( CC  ^m  NN0 ) A. b  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. m  e.  NN0  (
( b " ( ZZ>=
`  ( m  + 
1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) )  -> 
a  =  b ) )
49 imaeq1 5342 . . . . . . 7  |-  ( a  =  b  ->  (
a " ( ZZ>= `  ( n  +  1
) ) )  =  ( b " ( ZZ>=
`  ( n  + 
1 ) ) ) )
5049eqeq1d 2459 . . . . . 6  |-  ( a  =  b  ->  (
( a " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( b " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } ) )
51 fveq1 5871 . . . . . . . . . 10  |-  ( a  =  b  ->  (
a `  k )  =  ( b `  k ) )
5251oveq1d 6311 . . . . . . . . 9  |-  ( a  =  b  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( b `
 k )  x.  ( z ^ k
) ) )
5352sumeq2sdv 13538 . . . . . . . 8  |-  ( a  =  b  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... n ) ( ( b `  k
)  x.  ( z ^ k ) ) )
5453mpteq2dv 4544 . . . . . . 7  |-  ( a  =  b  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n
) ( ( b `
 k )  x.  ( z ^ k
) ) ) )
5554eqeq2d 2471 . . . . . 6  |-  ( a  =  b  ->  ( F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )  <->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) )
5650, 55anbi12d 710 . . . . 5  |-  ( a  =  b  ->  (
( ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )  <->  ( (
b " ( ZZ>= `  ( n  +  1
) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( b `  k )  x.  (
z ^ k ) ) ) ) ) )
5756rexbidv 2968 . . . 4  |-  ( a  =  b  ->  ( 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  ( ( b
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( b `  k )  x.  (
z ^ k ) ) ) ) ) )
58 oveq1 6303 . . . . . . . . 9  |-  ( n  =  m  ->  (
n  +  1 )  =  ( m  + 
1 ) )
5958fveq2d 5876 . . . . . . . 8  |-  ( n  =  m  ->  ( ZZ>=
`  ( n  + 
1 ) )  =  ( ZZ>= `  ( m  +  1 ) ) )
6059imaeq2d 5347 . . . . . . 7  |-  ( n  =  m  ->  (
b " ( ZZ>= `  ( n  +  1
) ) )  =  ( b " ( ZZ>=
`  ( m  + 
1 ) ) ) )
6160eqeq1d 2459 . . . . . 6  |-  ( n  =  m  ->  (
( b " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( b " ( ZZ>=
`  ( m  + 
1 ) ) )  =  { 0 } ) )
62 oveq2 6304 . . . . . . . . 9  |-  ( n  =  m  ->  (
0 ... n )  =  ( 0 ... m
) )
6362sumeq1d 13535 . . . . . . . 8  |-  ( n  =  m  ->  sum_ k  e.  ( 0 ... n
) ( ( b `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) )
6463mpteq2dv 4544 . . . . . . 7  |-  ( n  =  m  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( b `  k
)  x.  ( z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m
) ( ( b `
 k )  x.  ( z ^ k
) ) ) )
6564eqeq2d 2471 . . . . . 6  |-  ( n  =  m  ->  ( F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( b `  k )  x.  (
z ^ k ) ) )  <->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) ) ) )
6661, 65anbi12d 710 . . . . 5  |-  ( n  =  m  ->  (
( ( b "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( b `  k
)  x.  ( z ^ k ) ) ) )  <->  ( (
b " ( ZZ>= `  ( m  +  1
) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k )  x.  (
z ^ k ) ) ) ) ) )
6766cbvrexv 3085 . . . 4  |-  ( E. n  e.  NN0  (
( b " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( b `  k
)  x.  ( z ^ k ) ) ) )  <->  E. m  e.  NN0  ( ( b
" ( ZZ>= `  (
m  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k )  x.  (
z ^ k ) ) ) ) )
6857, 67syl6bb 261 . . 3  |-  ( a  =  b  ->  ( E. n  e.  NN0  ( ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )  <->  E. m  e.  NN0  ( ( b
" ( ZZ>= `  (
m  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k )  x.  (
z ^ k ) ) ) ) ) )
6968reu4 3293 . 2  |-  ( 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 ) ) ) )  <->  ( 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 ) ) ) )  /\  A. a  e.  ( CC 
^m  NN0 ) A. b  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. m  e.  NN0  ( ( b " ( ZZ>= `  ( m  +  1
) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... m ) ( ( b `  k )  x.  (
z ^ k ) ) ) ) )  ->  a  =  b ) ) )
7015, 48, 69sylanbrc 664 1  |-  ( 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 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   E!wreu 2809    u. cun 3469    C_ wss 3471   {csn 4032    |-> cmpt 4515   "cima 5011   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   NN0cn0 10816   ZZ>=cuz 11106   ...cfz 11697   ^cexp 12169   sum_csu 13520  Polycply 22707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-0p 22203  df-ply 22711
This theorem is referenced by:  coelem  22749  coeeq  22750
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