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Theorem coeid 22462
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 22420 . . 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 464 . 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 753 . . . . 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 759 . . . . 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 760 . . . . 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 755 . . . . 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 756 . . . . . 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 5866 . . . . . . . . . 10  |-  ( m  =  k  ->  (
a `  m )  =  ( a `  k ) )
11 oveq2 6293 . . . . . . . . . 10  |-  ( m  =  k  ->  (
x ^ m )  =  ( x ^
k ) )
1210, 11oveq12d 6303 . . . . . . . . 9  |-  ( m  =  k  ->  (
( a `  m
)  x.  ( x ^ m ) )  =  ( ( a `
 k )  x.  ( x ^ k
) ) )
1312cbvsumv 13484 . . . . . . . 8  |-  sum_ m  e.  ( 0 ... n
) ( ( a `
 m )  x.  ( x ^ m
) )  =  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( x ^ k ) )
14 oveq1 6292 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x ^ k )  =  ( z ^
k ) )
1514oveq2d 6301 . . . . . . . . 9  |-  ( x  =  z  ->  (
( a `  k
)  x.  ( x ^ k ) )  =  ( ( a `
 k )  x.  ( z ^ k
) ) )
1615sumeq2sdv 13492 . . . . . . . 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 2520 . . . . . . 7  |-  ( x  =  z  ->  sum_ m  e.  ( 0 ... n
) ( ( a `
 m )  x.  ( x ^ m
) )  =  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )
1817cbvmptv 4538 . . . . . 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 2524 . . . . 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 22461 . . . 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 434 . . 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 2962 . 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 15 1  |-  ( F  e.  (Poly `  S
)  ->  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    = wceq 1379    e. wcel 1767   E.wrex 2815    u. cun 3474    C_ wss 3476   {csn 4027    |-> cmpt 4505   "cima 5002   ` cfv 5588  (class class class)co 6285    ^m cmap 7421   CCcc 9491   0cc0 9493   1c1 9494    + caddc 9496    x. cmul 9498   NN0cn0 10796   ZZ>=cuz 11083   ...cfz 11673   ^cexp 12135   sum_csu 13474  Polycply 22408  coeffccoe 22410  degcdgr 22411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571  ax-addf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fz 11674  df-fzo 11794  df-fl 11898  df-seq 12077  df-exp 12136  df-hash 12375  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-clim 13277  df-rlim 13278  df-sum 13475  df-0p 21904  df-ply 22412  df-coe 22414  df-dgr 22415
This theorem is referenced by:  coeid2  22463  plyco  22465  0dgrb  22470  coeaddlem  22472  coemullem  22473  coe11  22476  plycn  22484  plycjlem  22499
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