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Theorem coelem 21693
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 21690 . . 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 21692 . . . 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 6065 . . . 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 2516 . 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 5163 . . . . . 6  |-  ( a  =  (coeff `  F
)  ->  ( a " ( ZZ>= `  (
n  +  1 ) ) )  =  ( (coeff `  F ) " ( ZZ>= `  (
n  +  1 ) ) ) )
76eqeq1d 2450 . . . . 5  |-  ( a  =  (coeff `  F
)  ->  ( (
a " ( ZZ>= `  ( n  +  1
) ) )  =  { 0 }  <->  ( (coeff `  F ) " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } ) )
8 fveq1 5689 . . . . . . . . 9  |-  ( a  =  (coeff `  F
)  ->  ( a `  k )  =  ( (coeff `  F ) `  k ) )
98oveq1d 6105 . . . . . . . 8  |-  ( a  =  (coeff `  F
)  ->  ( (
a `  k )  x.  ( z ^ k
) )  =  ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) ) )
109sumeq2sdv 13180 . . . . . . 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 4378 . . . . . 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 2453 . . . . 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 2735 . . 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 3116 . 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 1369    e. wcel 1756   E.wrex 2715   E!wreu 2716   {crab 2718   {csn 3876    e. cmpt 4349   "cima 4842   ` cfv 5417   iota_crio 6050  (class class class)co 6090    ^m cmap 7213   CCcc 9279   0cc0 9281   1c1 9282    + caddc 9284    x. cmul 9286   NN0cn0 10578   ZZ>=cuz 10860   ...cfz 11436   ^cexp 11864   sum_csu 13162  Polycply 21651  coeffccoe 21653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-fz 11437  df-fzo 11548  df-fl 11641  df-seq 11806  df-exp 11865  df-hash 12103  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-clim 12965  df-rlim 12966  df-sum 13163  df-0p 21147  df-ply 21655  df-coe 21657
This theorem is referenced by: (None)
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