MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coeeq Structured version   Unicode version

Theorem coeeq 20146
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 20142 . . 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 6088 . . . . . . . . 9  |-  ( n  =  N  ->  (
n  +  1 )  =  ( N  + 
1 ) )
87fveq2d 5732 . . . . . . . 8  |-  ( n  =  N  ->  ( ZZ>=
`  ( n  + 
1 ) )  =  ( ZZ>= `  ( N  +  1 ) ) )
98imaeq2d 5203 . . . . . . 7  |-  ( n  =  N  ->  ( A " ( ZZ>= `  (
n  +  1 ) ) )  =  ( A " ( ZZ>= `  ( N  +  1
) ) ) )
109eqeq1d 2444 . . . . . 6  |-  ( n  =  N  ->  (
( A " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } ) )
11 oveq2 6089 . . . . . . . . 9  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
1211sumeq1d 12495 . . . . . . . 8  |-  ( n  =  N  ->  sum_ k  e.  ( 0 ... n
) ( ( A `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )
1312mpteq2dv 4296 . . . . . . 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 2447 . . . . . 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 692 . . . . 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 3052 . . . 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 1182 . . 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 9071 . . . . . 6  |-  CC  e.  _V
20 nn0ex 10227 . . . . . 6  |-  NN0  e.  _V
2119, 20elmap 7042 . . . . 5  |-  ( A  e.  ( CC  ^m  NN0 )  <->  A : NN0 --> CC )
2218, 21sylibr 204 . . . 4  |-  ( ph  ->  A  e.  ( CC 
^m  NN0 ) )
23 coeeu 20144 . . . . 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 5198 . . . . . . . 8  |-  ( a  =  A  ->  (
a " ( ZZ>= `  ( n  +  1
) ) )  =  ( A " ( ZZ>=
`  ( n  + 
1 ) ) ) )
2625eqeq1d 2444 . . . . . . 7  |-  ( a  =  A  ->  (
( a " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( A " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } ) )
27 fveq1 5727 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a `  k )  =  ( A `  k ) )
2827oveq1d 6096 . . . . . . . . . 10  |-  ( a  =  A  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2928sumeq2sdv 12498 . . . . . . . . 9  |-  ( a  =  A  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... n ) ( ( A `  k
)  x.  ( z ^ k ) ) )
3029mpteq2dv 4296 . . . . . . . 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 2447 . . . . . . 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 692 . . . . . 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 2726 . . . . 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 6572 . . . 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 643 . . 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 202 . 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 2468 1  |-  ( ph  ->  (coeff `  F )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   E!wreu 2707   {csn 3814    e. cmpt 4266   "cima 4881   -->wf 5450   ` cfv 5454  (class class class)co 6081   iota_crio 6542    ^m cmap 7018   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995   NN0cn0 10221   ZZ>=cuz 10488   ...cfz 11043   ^cexp 11382   sum_csu 12479  Polycply 20103  coeffccoe 20105
This theorem is referenced by:  dgrlem  20148  coeidlem  20156  coeeq2  20161  dgreq  20163  coeaddlem  20167  coemullem  20168  coe1termlem  20176  coecj  20196  basellem2  20864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107  df-coe 20109
  Copyright terms: Public domain W3C validator