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Theorem coeeu 20144
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 20119 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3344 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 elply2 20115 . . . . . 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 451 . . . . 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 2869 . . . . 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 189 . . . 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 9084 . . . . . . 7  |-  0  e.  CC
9 snssi 3942 . . . . . . 7  |-  ( 0  e.  CC  ->  { 0 }  C_  CC )
108, 9ax-mp 8 . . . . . 6  |-  { 0 }  C_  CC
11 ssequn2 3520 . . . . . 6  |-  ( { 0 }  C_  CC  <->  ( CC  u.  { 0 } )  =  CC )
1210, 11mpbi 200 . . . . 5  |-  ( CC  u.  { 0 } )  =  CC
1312oveq1i 6091 . . . 4  |-  ( ( CC  u.  { 0 } )  ^m  NN0 )  =  ( CC  ^m 
NN0 )
1413rexeqi 2909 . . 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 189 . 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 2875 . . . 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 981 . . . . . . 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 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 ) ) ) ) ) )  ->  a  e.  ( CC  ^m  NN0 )
)
19 simp1rr 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 ) ) ) ) ) )  ->  b  e.  ( CC  ^m  NN0 )
)
20 simp2l 983 . . . . . . 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 984 . . . . . . 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 1028 . . . . . . 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 1030 . . . . . . 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 1029 . . . . . . . 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 6088 . . . . . . . . . . . 12  |-  ( z  =  w  ->  (
z ^ k )  =  ( w ^
k ) )
2625oveq2d 6097 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( a `
 k )  x.  ( w ^ k
) ) )
2726sumeq2sdv 12498 . . . . . . . . . 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 5728 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
a `  k )  =  ( a `  j ) )
29 oveq2 6089 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
w ^ k )  =  ( w ^
j ) )
3028, 29oveq12d 6099 . . . . . . . . . . 11  |-  ( k  =  j  ->  (
( a `  k
)  x.  ( w ^ k ) )  =  ( ( a `
 j )  x.  ( w ^ j
) ) )
3130cbvsumv 12490 . . . . . . . . . 10  |-  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( w ^ k
) )  =  sum_ j  e.  ( 0 ... n ) ( ( a `  j
)  x.  ( w ^ j ) )
3227, 31syl6eq 2484 . . . . . . . . 9  |-  ( z  =  w  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ j  e.  ( 0 ... n ) ( ( a `  j
)  x.  ( w ^ j ) ) )
3332cbvmptv 4300 . . . . . . . 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 2484 . . . . . . 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 1031 . . . . . . . 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 6097 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( b `  k
)  x.  ( z ^ k ) )  =  ( ( b `
 k )  x.  ( w ^ k
) ) )
3736sumeq2sdv 12498 . . . . . . . . . 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 5728 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
b `  k )  =  ( b `  j ) )
3938, 29oveq12d 6099 . . . . . . . . . . 11  |-  ( k  =  j  ->  (
( b `  k
)  x.  ( w ^ k ) )  =  ( ( b `
 j )  x.  ( w ^ j
) ) )
4039cbvsumv 12490 . . . . . . . . . 10  |-  sum_ k  e.  ( 0 ... m
) ( ( b `
 k )  x.  ( w ^ k
) )  =  sum_ j  e.  ( 0 ... m ) ( ( b `  j
)  x.  ( w ^ j ) )
4137, 40syl6eq 2484 . . . . . . . . 9  |-  ( z  =  w  ->  sum_ k  e.  ( 0 ... m
) ( ( b `
 k )  x.  ( z ^ k
) )  =  sum_ j  e.  ( 0 ... m ) ( ( b `  j
)  x.  ( w ^ j ) ) )
4241cbvmptv 4300 . . . . . . . 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 2484 . . . . . . 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 20143 . . . . . 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 1155 . . . . 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 2837 . . . 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 210 . . 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 2798 . 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 5198 . . . . . . 7  |-  ( a  =  b  ->  (
a " ( ZZ>= `  ( n  +  1
) ) )  =  ( b " ( ZZ>=
`  ( n  + 
1 ) ) ) )
5049eqeq1d 2444 . . . . . 6  |-  ( a  =  b  ->  (
( a " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( b " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } ) )
51 fveq1 5727 . . . . . . . . . 10  |-  ( a  =  b  ->  (
a `  k )  =  ( b `  k ) )
5251oveq1d 6096 . . . . . . . . 9  |-  ( a  =  b  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( b `
 k )  x.  ( z ^ k
) ) )
5352sumeq2sdv 12498 . . . . . . . 8  |-  ( a  =  b  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... n ) ( ( b `  k
)  x.  ( z ^ k ) ) )
5453mpteq2dv 4296 . . . . . . 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 2447 . . . . . 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 692 . . . . 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 2726 . . . 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 6088 . . . . . . . . 9  |-  ( n  =  m  ->  (
n  +  1 )  =  ( m  + 
1 ) )
5958fveq2d 5732 . . . . . . . 8  |-  ( n  =  m  ->  ( ZZ>=
`  ( n  + 
1 ) )  =  ( ZZ>= `  ( m  +  1 ) ) )
6059imaeq2d 5203 . . . . . . 7  |-  ( n  =  m  ->  (
b " ( ZZ>= `  ( n  +  1
) ) )  =  ( b " ( ZZ>=
`  ( m  + 
1 ) ) ) )
6160eqeq1d 2444 . . . . . 6  |-  ( n  =  m  ->  (
( b " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( b " ( ZZ>=
`  ( m  + 
1 ) ) )  =  { 0 } ) )
62 oveq2 6089 . . . . . . . . 9  |-  ( n  =  m  ->  (
0 ... n )  =  ( 0 ... m
) )
6362sumeq1d 12495 . . . . . . . 8  |-  ( n  =  m  ->  sum_ k  e.  ( 0 ... n
) ( ( b `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) )
6463mpteq2dv 4296 . . . . . . 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 2447 . . . . . 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 692 . . . . 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 2933 . . . 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 253 . . 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 3128 . 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 646 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   E!wreu 2707    u. cun 3318    C_ wss 3320   {csn 3814    e. cmpt 4266   "cima 4881   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995   NN0cn0 10221   ZZ>=cuz 10488   ...cfz 11043   ^cexp 11382   sum_csu 12479  Polycply 20103
This theorem is referenced by:  coelem  20145  coeeq  20146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107
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