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Theorem coelem 22748
Description: Lemma for properties of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
coelem  |-  ( F  e.  (Poly `  S
)  ->  ( (coeff `  F )  e.  ( CC  ^m  NN0 )  /\  E. n  e.  NN0  ( ( (coeff `  F ) " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) ) ) ) ) )
Distinct variable groups:    z, k    n, F    S, n    k, n, z, F
Allowed substitution hints:    S( z, k)

Proof of Theorem coelem
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 coeval 22745 . . 3  |-  ( 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 ) ) ) ) ) )
2 coeeu 22747 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  E! 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 ) ) ) ) )
3 riotacl2 6271 . . . 4  |-  ( E! 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 ) ) ) ) )  e. 
{ 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 ) ) ) ) } )
42, 3syl 16 . . 3  |-  ( F  e.  (Poly `  S
)  ->  ( 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.  { 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 ) ) ) ) } )
51, 4eqeltrd 2545 . 2  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
)  e.  { 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 ) ) ) ) } )
6 imaeq1 5342 . . . . . 6  |-  ( a  =  (coeff `  F
)  ->  ( a " ( ZZ>= `  (
n  +  1 ) ) )  =  ( (coeff `  F ) " ( ZZ>= `  (
n  +  1 ) ) ) )
76eqeq1d 2459 . . . . 5  |-  ( a  =  (coeff `  F
)  ->  ( (
a " ( ZZ>= `  ( n  +  1
) ) )  =  { 0 }  <->  ( (coeff `  F ) " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } ) )
8 fveq1 5871 . . . . . . . . 9  |-  ( a  =  (coeff `  F
)  ->  ( a `  k )  =  ( (coeff `  F ) `  k ) )
98oveq1d 6311 . . . . . . . 8  |-  ( a  =  (coeff `  F
)  ->  ( (
a `  k )  x.  ( z ^ k
) )  =  ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) ) )
109sumeq2sdv 13537 . . . . . . 7  |-  ( a  =  (coeff `  F
)  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... n ) ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) ) )
1110mpteq2dv 4544 . . . . . 6  |-  ( a  =  (coeff `  F
)  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) ) ) )
1211eqeq2d 2471 . . . . 5  |-  ( a  =  (coeff `  F
)  ->  ( F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )  <->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) ) ) ) )
137, 12anbi12d 710 . . . 4  |-  ( a  =  (coeff `  F
)  ->  ( (
( a " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )  <->  ( (
(coeff `  F ) " ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( (coeff `  F ) `  k
)  x.  ( z ^ k ) ) ) ) ) )
1413rexbidv 2968 . . 3  |-  ( a  =  (coeff `  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  ( ( (coeff `  F ) " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) ) ) ) ) )
1514elrab 3257 . 2  |-  ( (coeff `  F )  e.  {
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 ) ) ) ) }  <-> 
( (coeff `  F
)  e.  ( CC 
^m  NN0 )  /\  E. n  e.  NN0  ( ( (coeff `  F ) " ( ZZ>= `  (
n  +  1 ) ) )  =  {
0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( (coeff `  F ) `  k
)  x.  ( z ^ k ) ) ) ) ) )
165, 15sylib 196 1  |-  ( F  e.  (Poly `  S
)  ->  ( (coeff `  F )  e.  ( CC  ^m  NN0 )  /\  E. n  e.  NN0  ( ( (coeff `  F ) " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   E!wreu 2809   {crab 2811   {csn 4032    |-> cmpt 4515   "cima 5011   ` cfv 5594   iota_crio 6257  (class class class)co 6296    ^m cmap 7438   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   NN0cn0 10816   ZZ>=cuz 11106   ...cfz 11697   ^cexp 12168   sum_csu 13519  Polycply 22706  coeffccoe 22708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11821  df-fl 11931  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-rlim 13323  df-sum 13520  df-0p 22202  df-ply 22710  df-coe 22712
This theorem is referenced by: (None)
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