MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coeeu Structured version   Unicode version

Theorem coeeu 21696
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 21671 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3355 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 elply2 21667 . . . . . 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 2885 . . . . 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 9381 . . . . . . 7  |-  0  e.  CC
9 snssi 4020 . . . . . . 7  |-  ( 0  e.  CC  ->  { 0 }  C_  CC )
108, 9ax-mp 5 . . . . . 6  |-  { 0 }  C_  CC
11 ssequn2 3532 . . . . . 6  |-  ( { 0 }  C_  CC  <->  ( CC  u.  { 0 } )  =  CC )
1210, 11mpbi 208 . . . . 5  |-  ( CC  u.  { 0 } )  =  CC
1312oveq1i 6104 . . . 4  |-  ( ( CC  u.  { 0 } )  ^m  NN0 )  =  ( CC  ^m 
NN0 )
1413rexeqi 2925 . . 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 2891 . . . 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 1012 . . . . . . 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 1053 . . . . . . 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 1054 . . . . . . 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 1014 . . . . . . 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 1015 . . . . . . 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 1059 . . . . . . 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 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 ) ) ) ) ) )  ->  ( b "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 } )
24 simp3lr 1060 . . . . . . . 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 6101 . . . . . . . . . . . 12  |-  ( z  =  w  ->  (
z ^ k )  =  ( w ^
k ) )
2625oveq2d 6110 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( a `
 k )  x.  ( w ^ k
) ) )
2726sumeq2sdv 13184 . . . . . . . . . 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 5694 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
a `  k )  =  ( a `  j ) )
29 oveq2 6102 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
w ^ k )  =  ( w ^
j ) )
3028, 29oveq12d 6112 . . . . . . . . . . 11  |-  ( k  =  j  ->  (
( a `  k
)  x.  ( w ^ k ) )  =  ( ( a `
 j )  x.  ( w ^ j
) ) )
3130cbvsumv 13176 . . . . . . . . . 10  |-  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( w ^ k
) )  =  sum_ j  e.  ( 0 ... n ) ( ( a `  j
)  x.  ( w ^ j ) )
3227, 31syl6eq 2491 . . . . . . . . 9  |-  ( z  =  w  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ j  e.  ( 0 ... n ) ( ( a `  j
)  x.  ( w ^ j ) ) )
3332cbvmptv 4386 . . . . . . . 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 2491 . . . . . . 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 1062 . . . . . . . 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 6110 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( b `  k
)  x.  ( z ^ k ) )  =  ( ( b `
 k )  x.  ( w ^ k
) ) )
3736sumeq2sdv 13184 . . . . . . . . . 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 5694 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
b `  k )  =  ( b `  j ) )
3938, 29oveq12d 6112 . . . . . . . . . . 11  |-  ( k  =  j  ->  (
( b `  k
)  x.  ( w ^ k ) )  =  ( ( b `
 j )  x.  ( w ^ j
) ) )
4039cbvsumv 13176 . . . . . . . . . 10  |-  sum_ k  e.  ( 0 ... m
) ( ( b `
 k )  x.  ( w ^ k
) )  =  sum_ j  e.  ( 0 ... m ) ( ( b `  j
)  x.  ( w ^ j ) )
4137, 40syl6eq 2491 . . . . . . . . 9  |-  ( z  =  w  ->  sum_ k  e.  ( 0 ... m
) ( ( b `
 k )  x.  ( z ^ k
) )  =  sum_ j  e.  ( 0 ... m ) ( ( b `  j
)  x.  ( w ^ j ) ) )
4241cbvmptv 4386 . . . . . . . 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 2491 . . . . . . 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 21695 . . . . . 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 1189 . . . . 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 2851 . . . 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 2811 . 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 5167 . . . . . . 7  |-  ( a  =  b  ->  (
a " ( ZZ>= `  ( n  +  1
) ) )  =  ( b " ( ZZ>=
`  ( n  + 
1 ) ) ) )
5049eqeq1d 2451 . . . . . 6  |-  ( a  =  b  ->  (
( a " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( b " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } ) )
51 fveq1 5693 . . . . . . . . . 10  |-  ( a  =  b  ->  (
a `  k )  =  ( b `  k ) )
5251oveq1d 6109 . . . . . . . . 9  |-  ( a  =  b  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( b `
 k )  x.  ( z ^ k
) ) )
5352sumeq2sdv 13184 . . . . . . . 8  |-  ( a  =  b  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... n ) ( ( b `  k
)  x.  ( z ^ k ) ) )
5453mpteq2dv 4382 . . . . . . 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 2454 . . . . . 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 2739 . . . 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 6101 . . . . . . . . 9  |-  ( n  =  m  ->  (
n  +  1 )  =  ( m  + 
1 ) )
5958fveq2d 5698 . . . . . . . 8  |-  ( n  =  m  ->  ( ZZ>=
`  ( n  + 
1 ) )  =  ( ZZ>= `  ( m  +  1 ) ) )
6059imaeq2d 5172 . . . . . . 7  |-  ( n  =  m  ->  (
b " ( ZZ>= `  ( n  +  1
) ) )  =  ( b " ( ZZ>=
`  ( m  + 
1 ) ) ) )
6160eqeq1d 2451 . . . . . 6  |-  ( n  =  m  ->  (
( b " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( b " ( ZZ>=
`  ( m  + 
1 ) ) )  =  { 0 } ) )
62 oveq2 6102 . . . . . . . . 9  |-  ( n  =  m  ->  (
0 ... n )  =  ( 0 ... m
) )
6362sumeq1d 13181 . . . . . . . 8  |-  ( n  =  m  ->  sum_ k  e.  ( 0 ... n
) ( ( b `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) )
6463mpteq2dv 4382 . . . . . . 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 2454 . . . . . 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 2951 . . . 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 3156 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2718   E.wrex 2719   E!wreu 2720    u. cun 3329    C_ wss 3331   {csn 3880    e. cmpt 4353   "cima 4846   ` cfv 5421  (class class class)co 6094    ^m cmap 7217   CCcc 9283   0cc0 9285   1c1 9286    + caddc 9288    x. cmul 9290   NN0cn0 10582   ZZ>=cuz 10864   ...cfz 11440   ^cexp 11868   sum_csu 13166  Polycply 21655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-oi 7727  df-card 8112  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-rp 10995  df-fz 11441  df-fzo 11552  df-fl 11645  df-seq 11810  df-exp 11869  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-rlim 12970  df-sum 13167  df-0p 21151  df-ply 21659
This theorem is referenced by:  coelem  21697  coeeq  21698
  Copyright terms: Public domain W3C validator