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Theorem coeid 20149
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 20107 . . 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 451 . 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 731 . . . . 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 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 ) ) ) ) )  ->  n  e.  NN0 )
7 simplrr 738 . . . . 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 733 . . . . 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 734 . . . . . 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 5720 . . . . . . . . . 10  |-  ( m  =  k  ->  (
a `  m )  =  ( a `  k ) )
11 oveq2 6081 . . . . . . . . . 10  |-  ( m  =  k  ->  (
x ^ m )  =  ( x ^
k ) )
1210, 11oveq12d 6091 . . . . . . . . 9  |-  ( m  =  k  ->  (
( a `  m
)  x.  ( x ^ m ) )  =  ( ( a `
 k )  x.  ( x ^ k
) ) )
1312cbvsumv 12482 . . . . . . . 8  |-  sum_ m  e.  ( 0 ... n
) ( ( a `
 m )  x.  ( x ^ m
) )  =  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( x ^ k ) )
14 oveq1 6080 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x ^ k )  =  ( z ^
k ) )
1514oveq2d 6089 . . . . . . . . 9  |-  ( x  =  z  ->  (
( a `  k
)  x.  ( x ^ k ) )  =  ( ( a `
 k )  x.  ( z ^ k
) ) )
1615sumeq2sdv 12490 . . . . . . . 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 2479 . . . . . . 7  |-  ( x  =  z  ->  sum_ m  e.  ( 0 ... n
) ( ( a `
 m )  x.  ( x ^ m
) )  =  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )
1817cbvmptv 4292 . . . . . 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 2483 . . . . 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 20148 . . . 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 424 . . 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 2829 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698    u. cun 3310    C_ wss 3312   {csn 3806    e. cmpt 4258   "cima 4873   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987   NN0cn0 10213   ZZ>=cuz 10480   ...cfz 11035   ^cexp 11374   sum_csu 12471  Polycply 20095  coeffccoe 20097  degcdgr 20098
This theorem is referenced by:  coeid2  20150  plyco  20152  0dgrb  20157  coeaddlem  20159  coemullem  20160  coe11  20163  plycn  20171  plycjlem  20186
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-coe 20101  df-dgr 20102
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