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Theorem coeeq 23181
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 23177 . . 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 17 . 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 6297 . . . . . . . . 9  |-  ( n  =  N  ->  (
n  +  1 )  =  ( N  + 
1 ) )
87fveq2d 5869 . . . . . . . 8  |-  ( n  =  N  ->  ( ZZ>=
`  ( n  + 
1 ) )  =  ( ZZ>= `  ( N  +  1 ) ) )
98imaeq2d 5168 . . . . . . 7  |-  ( n  =  N  ->  ( A " ( ZZ>= `  (
n  +  1 ) ) )  =  ( A " ( ZZ>= `  ( N  +  1
) ) ) )
109eqeq1d 2453 . . . . . 6  |-  ( n  =  N  ->  (
( A " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } ) )
11 oveq2 6298 . . . . . . . . 9  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
1211sumeq1d 13767 . . . . . . . 8  |-  ( n  =  N  ->  sum_ k  e.  ( 0 ... n
) ( ( A `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )
1312mpteq2dv 4490 . . . . . . 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 2461 . . . . . 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 717 . . . . 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 3150 . . . 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 1266 . . 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 9620 . . . . . 6  |-  CC  e.  _V
20 nn0ex 10875 . . . . . 6  |-  NN0  e.  _V
2119, 20elmap 7500 . . . . 5  |-  ( A  e.  ( CC  ^m  NN0 )  <->  A : NN0 --> CC )
2218, 21sylibr 216 . . . 4  |-  ( ph  ->  A  e.  ( CC 
^m  NN0 ) )
23 coeeu 23179 . . . . 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 17 . . . 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 5163 . . . . . . . 8  |-  ( a  =  A  ->  (
a " ( ZZ>= `  ( n  +  1
) ) )  =  ( A " ( ZZ>=
`  ( n  + 
1 ) ) ) )
2625eqeq1d 2453 . . . . . . 7  |-  ( a  =  A  ->  (
( a " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( A " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } ) )
27 fveq1 5864 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a `  k )  =  ( A `  k ) )
2827oveq1d 6305 . . . . . . . . . 10  |-  ( a  =  A  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2928sumeq2sdv 13770 . . . . . . . . 9  |-  ( a  =  A  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... n ) ( ( A `  k
)  x.  ( z ^ k ) ) )
3029mpteq2dv 4490 . . . . . . . 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 2461 . . . . . . 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 717 . . . . . 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 2901 . . . . 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 6274 . . . 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 667 . . 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 214 . 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 2485 1  |-  ( ph  ->  (coeff `  F )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   E.wrex 2738   E!wreu 2739   {csn 3968    |-> cmpt 4461   "cima 4837   -->wf 5578   ` cfv 5582   iota_crio 6251  (class class class)co 6290    ^m cmap 7472   CCcc 9537   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   NN0cn0 10869   ZZ>=cuz 11159   ...cfz 11784   ^cexp 12272   sum_csu 13752  Polycply 23138  coeffccoe 23140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753  df-0p 22628  df-ply 23142  df-coe 23144
This theorem is referenced by:  dgrlem  23183  coeidlem  23191  coeeq2  23196  dgreq  23198  coeaddlem  23203  coemullem  23204  coe1termlem  23212  coecj  23232  basellem2  24008  aacllem  40593
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