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Theorem coeid 23099
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 23057 . . 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 465 . 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 758 . . . . 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 768 . . . . 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 769 . . . . 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 762 . . . . 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 764 . . . . . 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 5872 . . . . . . . . . 10  |-  ( m  =  k  ->  (
a `  m )  =  ( a `  k ) )
11 oveq2 6304 . . . . . . . . . 10  |-  ( m  =  k  ->  (
x ^ m )  =  ( x ^
k ) )
1210, 11oveq12d 6314 . . . . . . . . 9  |-  ( m  =  k  ->  (
( a `  m
)  x.  ( x ^ m ) )  =  ( ( a `
 k )  x.  ( x ^ k
) ) )
1312cbvsumv 13729 . . . . . . . 8  |-  sum_ m  e.  ( 0 ... n
) ( ( a `
 m )  x.  ( x ^ m
) )  =  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( x ^ k ) )
14 oveq1 6303 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x ^ k )  =  ( z ^
k ) )
1514oveq2d 6312 . . . . . . . . 9  |-  ( x  =  z  ->  (
( a `  k
)  x.  ( x ^ k ) )  =  ( ( a `
 k )  x.  ( z ^ k
) ) )
1615sumeq2sdv 13737 . . . . . . . 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 2473 . . . . . . 7  |-  ( x  =  z  ->  sum_ m  e.  ( 0 ... n
) ( ( a `
 m )  x.  ( x ^ m
) )  =  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )
1817cbvmptv 4509 . . . . . 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 2477 . . . . 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 23098 . . . 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 435 . . 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 2922 . 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 370    = wceq 1437    e. wcel 1867   E.wrex 2774    u. cun 3431    C_ wss 3433   {csn 3993    |-> cmpt 4475   "cima 4848   ` cfv 5592  (class class class)co 6296    ^m cmap 7471   CCcc 9526   0cc0 9528   1c1 9529    + caddc 9531    x. cmul 9533   NN0cn0 10858   ZZ>=cuz 11148   ...cfz 11771   ^cexp 12258   sum_csu 13719  Polycply 23045  coeffccoe 23047  degcdgr 23048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-inf 7954  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-fz 11772  df-fzo 11903  df-fl 12014  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519  df-rlim 13520  df-sum 13720  df-0p 22535  df-ply 23049  df-coe 23051  df-dgr 23052
This theorem is referenced by:  coeid2  23100  plyco  23102  0dgrb  23107  coeaddlem  23110  coemullem  23111  coe11  23114  plycn  23122  plycjlem  23137
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