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Theorem coeval 22789
Description: Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Assertion
Ref Expression
coeval  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
)  =  ( iota_ 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 coeval
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 plyssc 22766 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3485 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 eqeq1 2458 . . . . . 6  |-  ( f  =  F  ->  (
f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) ) )  <->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )
43anbi2d 701 . . . . 5  |-  ( f  =  F  ->  (
( ( a "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )  <->  ( (
a " ( ZZ>= `  ( n  +  1
) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) ) ) )
54rexbidv 2965 . . . 4  |-  ( f  =  F  ->  ( 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  ( ( a
" ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) ) ) )
65riotabidv 6234 . . 3  |-  ( f  =  F  ->  ( iota_ 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 ) ) ) ) )  =  ( iota_ 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 ) ) ) ) ) )
7 df-coe 22756 . . 3  |- coeff  =  ( f  e.  (Poly `  CC )  |->  ( iota_ 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 ) ) ) ) ) )
8 riotaex 6236 . . 3  |-  ( iota_ 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.  _V
96, 7, 8fvmpt 5931 . 2  |-  ( F  e.  (Poly `  CC )  ->  (coeff `  F
)  =  ( iota_ 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 ) ) ) ) ) )
102, 9syl 16 1  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
)  =  ( iota_ 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 367    = wceq 1398    e. wcel 1823   E.wrex 2805   {csn 4016    |-> cmpt 4497   "cima 4991   ` cfv 5570   iota_crio 6231  (class class class)co 6270    ^m cmap 7412   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   NN0cn0 10791   ZZ>=cuz 11082   ...cfz 11675   ^cexp 12151   sum_csu 13593  Polycply 22750  coeffccoe 22752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rrecex 9553  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-map 7414  df-nn 10532  df-n0 10792  df-ply 22754  df-coe 22756
This theorem is referenced by:  coelem  22792  coeeq  22793
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