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Theorem coeeq 21707
Description: If  A satisfies the properties of the coefficient function, it must be equal to the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
coeeq.1  |-  ( ph  ->  F  e.  (Poly `  S ) )
coeeq.2  |-  ( ph  ->  N  e.  NN0 )
coeeq.3  |-  ( ph  ->  A : NN0 --> CC )
coeeq.4  |-  ( ph  ->  ( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } )
coeeq.5  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
Assertion
Ref Expression
coeeq  |-  ( ph  ->  (coeff `  F )  =  A )
Distinct variable groups:    z, k, A    k, N, z
Allowed substitution hints:    ph( z, k)    S( z, k)    F( z, k)

Proof of Theorem coeeq
Dummy variables  a  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coeeq.1 . . 3  |-  ( ph  ->  F  e.  (Poly `  S ) )
2 coeval 21703 . . 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 ) ) ) ) ) )
31, 2syl 16 . 2  |-  ( ph  ->  (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 ) ) ) ) ) )
4 coeeq.2 . . . 4  |-  ( ph  ->  N  e.  NN0 )
5 coeeq.4 . . . 4  |-  ( ph  ->  ( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } )
6 coeeq.5 . . . 4  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
7 oveq1 6110 . . . . . . . . 9  |-  ( n  =  N  ->  (
n  +  1 )  =  ( N  + 
1 ) )
87fveq2d 5707 . . . . . . . 8  |-  ( n  =  N  ->  ( ZZ>=
`  ( n  + 
1 ) )  =  ( ZZ>= `  ( N  +  1 ) ) )
98imaeq2d 5181 . . . . . . 7  |-  ( n  =  N  ->  ( A " ( ZZ>= `  (
n  +  1 ) ) )  =  ( A " ( ZZ>= `  ( N  +  1
) ) ) )
109eqeq1d 2451 . . . . . 6  |-  ( n  =  N  ->  (
( A " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } ) )
11 oveq2 6111 . . . . . . . . 9  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
1211sumeq1d 13190 . . . . . . . 8  |-  ( n  =  N  ->  sum_ k  e.  ( 0 ... n
) ( ( A `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )
1312mpteq2dv 4391 . . . . . . 7  |-  ( n  =  N  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( A `  k
)  x.  ( z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
1413eqeq2d 2454 . . . . . 6  |-  ( n  =  N  ->  ( 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 ) ) ) ) )
1510, 14anbi12d 710 . . . . 5  |-  ( n  =  N  ->  (
( ( 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 ) ) ) ) ) )
1615rspcev 3085 . . . 4  |-  ( ( 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 ) ) ) ) )
174, 5, 6, 16syl12anc 1216 . . 3  |-  ( ph  ->  E. n  e.  NN0  ( ( A "
( ZZ>= `  ( n  +  1 ) ) )  =  { 0 }  /\  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( A `  k
)  x.  ( z ^ k ) ) ) ) )
18 coeeq.3 . . . . 5  |-  ( ph  ->  A : NN0 --> CC )
19 cnex 9375 . . . . . 6  |-  CC  e.  _V
20 nn0ex 10597 . . . . . 6  |-  NN0  e.  _V
2119, 20elmap 7253 . . . . 5  |-  ( A  e.  ( CC  ^m  NN0 )  <->  A : NN0 --> CC )
2218, 21sylibr 212 . . . 4  |-  ( ph  ->  A  e.  ( CC 
^m  NN0 ) )
23 coeeu 21705 . . . . 5  |-  ( 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 ) ) ) ) )
241, 23syl 16 . . . 4  |-  ( ph  ->  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 ) ) ) ) )
25 imaeq1 5176 . . . . . . . 8  |-  ( a  =  A  ->  (
a " ( ZZ>= `  ( n  +  1
) ) )  =  ( A " ( ZZ>=
`  ( n  + 
1 ) ) ) )
2625eqeq1d 2451 . . . . . . 7  |-  ( a  =  A  ->  (
( a " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( A " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } ) )
27 fveq1 5702 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a `  k )  =  ( A `  k ) )
2827oveq1d 6118 . . . . . . . . . 10  |-  ( a  =  A  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2928sumeq2sdv 13193 . . . . . . . . 9  |-  ( a  =  A  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... n ) ( ( A `  k
)  x.  ( z ^ k ) ) )
3029mpteq2dv 4391 . . . . . . . 8  |-  ( a  =  A  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
3130eqeq2d 2454 . . . . . . 7  |-  ( a  =  A  ->  ( 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 ) ) ) ) )
3226, 31anbi12d 710 . . . . . 6  |-  ( a  =  A  ->  (
( ( 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 ) ) ) ) ) )
3332rexbidv 2748 . . . . 5  |-  ( a  =  A  ->  ( 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 ) ) ) ) ) )
3433riota2 6087 . . . 4  |-  ( ( A  e.  ( CC 
^m  NN0 )  /\  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 ) ) ) ) )  ->  ( 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 ) ) ) ) )  =  A ) )
3522, 24, 34syl2anc 661 . . 3  |-  ( ph  ->  ( 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 ) ) ) ) )  =  A ) )
3617, 35mpbid 210 . 2  |-  ( ph  ->  ( 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 ) ) ) ) )  =  A )
373, 36eqtrd 2475 1  |-  ( ph  ->  (coeff `  F )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2728   E!wreu 2729   {csn 3889    e. cmpt 4362   "cima 4855   -->wf 5426   ` cfv 5430   iota_crio 6063  (class class class)co 6103    ^m cmap 7226   CCcc 9292   0cc0 9294   1c1 9295    + caddc 9297    x. cmul 9299   NN0cn0 10591   ZZ>=cuz 10873   ...cfz 11449   ^cexp 11877   sum_csu 13175  Polycply 21664  coeffccoe 21666
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-oi 7736  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-fz 11450  df-fzo 11561  df-fl 11654  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-rlim 12979  df-sum 13176  df-0p 21160  df-ply 21668  df-coe 21670
This theorem is referenced by:  dgrlem  21709  coeidlem  21717  coeeq2  21722  dgreq  21724  coeaddlem  21728  coemullem  21729  coe1termlem  21737  coecj  21757  basellem2  22431
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