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Theorem coeid 19620
Description: Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
Hypotheses
Ref Expression
dgrub.1  |-  A  =  (coeff `  F )
dgrub.2  |-  N  =  (deg `  F )
Assertion
Ref Expression
coeid  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
Distinct variable groups:    z, k, A    k, F    S, k,
z    k, N, z    z, F

Proof of Theorem coeid
Dummy variables  a  n  x  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elply2 19578 . . 3  |-  ( F  e.  (Poly `  S
)  <->  ( S  C_  CC  /\  E. n  e. 
NN0  E. a  e.  ( ( S  u.  {
0 } )  ^m  NN0 ) ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( x  e.  CC  |->  sum_ m  e.  ( 0 ... n ) ( ( a `  m )  x.  (
x ^ m ) ) ) ) ) )
21simprbi 450 . 2  |-  ( F  e.  (Poly `  S
)  ->  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( x  e.  CC  |->  sum_ m  e.  ( 0 ... n ) ( ( a `  m )  x.  (
x ^ m ) ) ) ) )
3 dgrub.1 . . . . 5  |-  A  =  (coeff `  F )
4 dgrub.2 . . . . 5  |-  N  =  (deg `  F )
5 simpll 730 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  ( ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  F  =  ( x  e.  CC  |->  sum_
m  e.  ( 0 ... n ) ( ( a `  m
)  x.  ( x ^ m ) ) ) ) )  ->  F  e.  (Poly `  S
) )
6 simplrl 736 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  ( ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  F  =  ( x  e.  CC  |->  sum_
m  e.  ( 0 ... n ) ( ( a `  m
)  x.  ( x ^ m ) ) ) ) )  ->  n  e.  NN0 )
7 simplrr 737 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  ( ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  F  =  ( x  e.  CC  |->  sum_
m  e.  ( 0 ... n ) ( ( a `  m
)  x.  ( x ^ m ) ) ) ) )  -> 
a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )
8 simprl 732 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  ( ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  F  =  ( x  e.  CC  |->  sum_
m  e.  ( 0 ... n ) ( ( a `  m
)  x.  ( x ^ m ) ) ) ) )  -> 
( a " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } )
9 simprr 733 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  ( ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  F  =  ( x  e.  CC  |->  sum_
m  e.  ( 0 ... n ) ( ( a `  m
)  x.  ( x ^ m ) ) ) ) )  ->  F  =  ( x  e.  CC  |->  sum_ m  e.  ( 0 ... n ) ( ( a `  m )  x.  (
x ^ m ) ) ) )
10 fveq2 5525 . . . . . . . . . 10  |-  ( m  =  k  ->  (
a `  m )  =  ( a `  k ) )
11 oveq2 5866 . . . . . . . . . 10  |-  ( m  =  k  ->  (
x ^ m )  =  ( x ^
k ) )
1210, 11oveq12d 5876 . . . . . . . . 9  |-  ( m  =  k  ->  (
( a `  m
)  x.  ( x ^ m ) )  =  ( ( a `
 k )  x.  ( x ^ k
) ) )
1312cbvsumv 12169 . . . . . . . 8  |-  sum_ m  e.  ( 0 ... n
) ( ( a `
 m )  x.  ( x ^ m
) )  =  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( x ^ k ) )
14 oveq1 5865 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x ^ k )  =  ( z ^
k ) )
1514oveq2d 5874 . . . . . . . . 9  |-  ( x  =  z  ->  (
( a `  k
)  x.  ( x ^ k ) )  =  ( ( a `
 k )  x.  ( z ^ k
) ) )
1615sumeq2sdv 12177 . . . . . . . 8  |-  ( x  =  z  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( x ^ k
) )  =  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )
1713, 16syl5eq 2327 . . . . . . 7  |-  ( x  =  z  ->  sum_ m  e.  ( 0 ... n
) ( ( a `
 m )  x.  ( x ^ m
) )  =  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )
1817cbvmptv 4111 . . . . . 6  |-  ( x  e.  CC  |->  sum_ m  e.  ( 0 ... n
) ( ( a `
 m )  x.  ( x ^ m
) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )
199, 18syl6eq 2331 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  ( ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  F  =  ( x  e.  CC  |->  sum_
m  e.  ( 0 ... n ) ( ( a `  m
)  x.  ( x ^ m ) ) ) ) )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )
203, 4, 5, 6, 7, 8, 19coeidlem 19619 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  ( ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  F  =  ( x  e.  CC  |->  sum_
m  e.  ( 0 ... n ) ( ( a `  m
)  x.  ( x ^ m ) ) ) ) )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) ) )
2120ex 423 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( x  e.  CC  |->  sum_ m  e.  ( 0 ... n ) ( ( a `  m )  x.  (
x ^ m ) ) ) )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) ) ) )
2221rexlimdvva 2674 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( x  e.  CC  |->  sum_ m  e.  ( 0 ... n ) ( ( a `  m )  x.  (
x ^ m ) ) ) )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) ) ) )
232, 22mpd 14 1  |-  ( F  e.  (Poly `  S
)  ->  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 358    = wceq 1623    e. wcel 1684   E.wrex 2544    u. cun 3150    C_ wss 3152   {csn 3640    e. cmpt 4077   "cima 4692   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   NN0cn0 9965   ZZ>=cuz 10230   ...cfz 10782   ^cexp 11104   sum_csu 12158  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  coeid2  19621  plyco  19623  0dgrb  19628  coeaddlem  19630  coemullem  19631  coe11  19634  plycn  19642  plycjlem  19657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573
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