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Theorem coeeq 22790
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 22786 . . 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 6277 . . . . . . . . 9  |-  ( n  =  N  ->  (
n  +  1 )  =  ( N  + 
1 ) )
87fveq2d 5852 . . . . . . . 8  |-  ( n  =  N  ->  ( ZZ>=
`  ( n  + 
1 ) )  =  ( ZZ>= `  ( N  +  1 ) ) )
98imaeq2d 5325 . . . . . . 7  |-  ( n  =  N  ->  ( A " ( ZZ>= `  (
n  +  1 ) ) )  =  ( A " ( ZZ>= `  ( N  +  1
) ) ) )
109eqeq1d 2456 . . . . . 6  |-  ( n  =  N  ->  (
( A " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } ) )
11 oveq2 6278 . . . . . . . . 9  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
1211sumeq1d 13605 . . . . . . . 8  |-  ( n  =  N  ->  sum_ k  e.  ( 0 ... n
) ( ( A `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )
1312mpteq2dv 4526 . . . . . . 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 2468 . . . . . 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 708 . . . . 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 3207 . . . 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 1224 . . 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 9562 . . . . . 6  |-  CC  e.  _V
20 nn0ex 10797 . . . . . 6  |-  NN0  e.  _V
2119, 20elmap 7440 . . . . 5  |-  ( A  e.  ( CC  ^m  NN0 )  <->  A : NN0 --> CC )
2218, 21sylibr 212 . . . 4  |-  ( ph  ->  A  e.  ( CC 
^m  NN0 ) )
23 coeeu 22788 . . . . 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 5320 . . . . . . . 8  |-  ( a  =  A  ->  (
a " ( ZZ>= `  ( n  +  1
) ) )  =  ( A " ( ZZ>=
`  ( n  + 
1 ) ) ) )
2625eqeq1d 2456 . . . . . . 7  |-  ( a  =  A  ->  (
( a " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( A " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } ) )
27 fveq1 5847 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a `  k )  =  ( A `  k ) )
2827oveq1d 6285 . . . . . . . . . 10  |-  ( a  =  A  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2928sumeq2sdv 13608 . . . . . . . . 9  |-  ( a  =  A  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... n ) ( ( A `  k
)  x.  ( z ^ k ) ) )
3029mpteq2dv 4526 . . . . . . . 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 2468 . . . . . . 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 708 . . . . . 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 2965 . . . . 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 6254 . . . 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 659 . . 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 2495 1  |-  ( ph  ->  (coeff `  F )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805   E!wreu 2806   {csn 4016    |-> cmpt 4497   "cima 4991   -->wf 5566   ` cfv 5570   iota_crio 6231  (class class class)co 6270    ^m cmap 7412   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   NN0cn0 10791   ZZ>=cuz 11082   ...cfz 11675   ^cexp 12148   sum_csu 13590  Polycply 22747  coeffccoe 22749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-rlim 13394  df-sum 13591  df-0p 22243  df-ply 22751  df-coe 22753
This theorem is referenced by:  dgrlem  22792  coeidlem  22800  coeeq2  22805  dgreq  22807  coeaddlem  22812  coemullem  22813  coe1termlem  22821  coecj  22841  basellem2  23553  aacllem  33604
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