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Theorem coelem 20146
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 20143 . . 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 20145 . . . 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 6564 . . . 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 2511 . 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 5199 . . . . . 6  |-  ( a  =  (coeff `  F
)  ->  ( a " ( ZZ>= `  (
n  +  1 ) ) )  =  ( (coeff `  F ) " ( ZZ>= `  (
n  +  1 ) ) ) )
76eqeq1d 2445 . . . . 5  |-  ( a  =  (coeff `  F
)  ->  ( (
a " ( ZZ>= `  ( n  +  1
) ) )  =  { 0 }  <->  ( (coeff `  F ) " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } ) )
8 fveq1 5728 . . . . . . . . 9  |-  ( a  =  (coeff `  F
)  ->  ( a `  k )  =  ( (coeff `  F ) `  k ) )
98oveq1d 6097 . . . . . . . 8  |-  ( a  =  (coeff `  F
)  ->  ( (
a `  k )  x.  ( z ^ k
) )  =  ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) ) )
109sumeq2sdv 12499 . . . . . . 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 4297 . . . . . 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 2448 . . . . 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 693 . . . 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 2727 . . 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 3093 . 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 190 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2707   E!wreu 2708   {crab 2710   {csn 3815    e. cmpt 4267   "cima 4882   ` cfv 5455  (class class class)co 6082   iota_crio 6543    ^m cmap 7019   CCcc 8989   0cc0 8991   1c1 8992    + caddc 8994    x. cmul 8996   NN0cn0 10222   ZZ>=cuz 10489   ...cfz 11044   ^cexp 11383   sum_csu 12480  Polycply 20104  coeffccoe 20106
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-oi 7480  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-fz 11045  df-fzo 11137  df-fl 11203  df-seq 11325  df-exp 11384  df-hash 11620  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-clim 12283  df-rlim 12284  df-sum 12481  df-0p 19563  df-ply 20108  df-coe 20110
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