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Theorem coeeu 21578
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 21553 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3340 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 elply2 21549 . . . . . 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 461 . . . . 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 2872 . . . . 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 9366 . . . . . . 7  |-  0  e.  CC
9 snssi 4005 . . . . . . 7  |-  ( 0  e.  CC  ->  { 0 }  C_  CC )
108, 9ax-mp 5 . . . . . 6  |-  { 0 }  C_  CC
11 ssequn2 3517 . . . . . 6  |-  ( { 0 }  C_  CC  <->  ( CC  u.  { 0 } )  =  CC )
1210, 11mpbi 208 . . . . 5  |-  ( CC  u.  { 0 } )  =  CC
1312oveq1i 6090 . . . 4  |-  ( ( CC  u.  { 0 } )  ^m  NN0 )  =  ( CC  ^m 
NN0 )
1413rexeqi 2912 . . 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 2878 . . . 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 1005 . . . . . . 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 1046 . . . . . . 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 1047 . . . . . . 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 1007 . . . . . . 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 1008 . . . . . . 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 1052 . . . . . . 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 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 "
( ZZ>= `  ( m  +  1 ) ) )  =  { 0 } )
24 simp3lr 1053 . . . . . . . 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 6087 . . . . . . . . . . . 12  |-  ( z  =  w  ->  (
z ^ k )  =  ( w ^
k ) )
2625oveq2d 6096 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( a `
 k )  x.  ( w ^ k
) ) )
2726sumeq2sdv 13165 . . . . . . . . . 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 5679 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
a `  k )  =  ( a `  j ) )
29 oveq2 6088 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
w ^ k )  =  ( w ^
j ) )
3028, 29oveq12d 6098 . . . . . . . . . . 11  |-  ( k  =  j  ->  (
( a `  k
)  x.  ( w ^ k ) )  =  ( ( a `
 j )  x.  ( w ^ j
) ) )
3130cbvsumv 13157 . . . . . . . . . 10  |-  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( w ^ k
) )  =  sum_ j  e.  ( 0 ... n ) ( ( a `  j
)  x.  ( w ^ j ) )
3227, 31syl6eq 2481 . . . . . . . . 9  |-  ( z  =  w  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ j  e.  ( 0 ... n ) ( ( a `  j
)  x.  ( w ^ j ) ) )
3332cbvmptv 4371 . . . . . . . 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 2481 . . . . . . 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 1055 . . . . . . . 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 6096 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( b `  k
)  x.  ( z ^ k ) )  =  ( ( b `
 k )  x.  ( w ^ k
) ) )
3736sumeq2sdv 13165 . . . . . . . . . 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 5679 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
b `  k )  =  ( b `  j ) )
3938, 29oveq12d 6098 . . . . . . . . . . 11  |-  ( k  =  j  ->  (
( b `  k
)  x.  ( w ^ k ) )  =  ( ( b `
 j )  x.  ( w ^ j
) ) )
4039cbvsumv 13157 . . . . . . . . . 10  |-  sum_ k  e.  ( 0 ... m
) ( ( b `
 k )  x.  ( w ^ k
) )  =  sum_ j  e.  ( 0 ... m ) ( ( b `  j
)  x.  ( w ^ j ) )
4137, 40syl6eq 2481 . . . . . . . . 9  |-  ( z  =  w  ->  sum_ k  e.  ( 0 ... m
) ( ( b `
 k )  x.  ( z ^ k
) )  =  sum_ j  e.  ( 0 ... m ) ( ( b `  j
)  x.  ( w ^ j ) ) )
4241cbvmptv 4371 . . . . . . . 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 2481 . . . . . . 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 21577 . . . . . 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 1182 . . . . 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 2838 . . . 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 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 5152 . . . . . . 7  |-  ( a  =  b  ->  (
a " ( ZZ>= `  ( n  +  1
) ) )  =  ( b " ( ZZ>=
`  ( n  + 
1 ) ) ) )
5049eqeq1d 2441 . . . . . 6  |-  ( a  =  b  ->  (
( a " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( b " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } ) )
51 fveq1 5678 . . . . . . . . . 10  |-  ( a  =  b  ->  (
a `  k )  =  ( b `  k ) )
5251oveq1d 6095 . . . . . . . . 9  |-  ( a  =  b  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( b `
 k )  x.  ( z ^ k
) ) )
5352sumeq2sdv 13165 . . . . . . . 8  |-  ( a  =  b  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... n ) ( ( b `  k
)  x.  ( z ^ k ) ) )
5453mpteq2dv 4367 . . . . . . 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 2444 . . . . . 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 703 . . . . 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 6087 . . . . . . . . 9  |-  ( n  =  m  ->  (
n  +  1 )  =  ( m  + 
1 ) )
5958fveq2d 5683 . . . . . . . 8  |-  ( n  =  m  ->  ( ZZ>=
`  ( n  + 
1 ) )  =  ( ZZ>= `  ( m  +  1 ) ) )
6059imaeq2d 5157 . . . . . . 7  |-  ( n  =  m  ->  (
b " ( ZZ>= `  ( n  +  1
) ) )  =  ( b " ( ZZ>=
`  ( m  + 
1 ) ) ) )
6160eqeq1d 2441 . . . . . 6  |-  ( n  =  m  ->  (
( b " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( b " ( ZZ>=
`  ( m  + 
1 ) ) )  =  { 0 } ) )
62 oveq2 6088 . . . . . . . . 9  |-  ( n  =  m  ->  (
0 ... n )  =  ( 0 ... m
) )
6362sumeq1d 13162 . . . . . . . 8  |-  ( n  =  m  ->  sum_ k  e.  ( 0 ... n
) ( ( b `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... m ) ( ( b `  k
)  x.  ( z ^ k ) ) )
6463mpteq2dv 4367 . . . . . . 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 2444 . . . . . 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 703 . . . . 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 2938 . . . 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 3142 . 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 657 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 958    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706   E!wreu 2707    u. cun 3314    C_ wss 3316   {csn 3865    e. cmpt 4338   "cima 4830   ` cfv 5406  (class class class)co 6080    ^m cmap 7202   CCcc 9268   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275   NN0cn0 10567   ZZ>=cuz 10849   ...cfz 11424   ^cexp 11849   sum_csu 13147  Polycply 21537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-fz 11425  df-fzo 11533  df-fl 11626  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-rlim 12951  df-sum 13148  df-0p 20990  df-ply 21541
This theorem is referenced by:  coelem  21579  coeeq  21580
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