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Theorem coeeq 19625
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 19621 . . 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 15 . 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 5881 . . . . . . . . 9  |-  ( n  =  N  ->  (
n  +  1 )  =  ( N  + 
1 ) )
87fveq2d 5545 . . . . . . . 8  |-  ( n  =  N  ->  ( ZZ>=
`  ( n  + 
1 ) )  =  ( ZZ>= `  ( N  +  1 ) ) )
98imaeq2d 5028 . . . . . . 7  |-  ( n  =  N  ->  ( A " ( ZZ>= `  (
n  +  1 ) ) )  =  ( A " ( ZZ>= `  ( N  +  1
) ) ) )
109eqeq1d 2304 . . . . . 6  |-  ( n  =  N  ->  (
( A " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } ) )
11 oveq2 5882 . . . . . . . . 9  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
1211sumeq1d 12190 . . . . . . . 8  |-  ( n  =  N  ->  sum_ k  e.  ( 0 ... n
) ( ( A `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )
1312mpteq2dv 4123 . . . . . . 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 2307 . . . . . 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 691 . . . . 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 2897 . . . 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 1180 . . 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 8834 . . . . . 6  |-  CC  e.  _V
20 nn0ex 9987 . . . . . 6  |-  NN0  e.  _V
2119, 20elmap 6812 . . . . 5  |-  ( A  e.  ( CC  ^m  NN0 )  <->  A : NN0 --> CC )
2218, 21sylibr 203 . . . 4  |-  ( ph  ->  A  e.  ( CC 
^m  NN0 ) )
23 coeeu 19623 . . . . 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 15 . . . 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 5023 . . . . . . . 8  |-  ( a  =  A  ->  (
a " ( ZZ>= `  ( n  +  1
) ) )  =  ( A " ( ZZ>=
`  ( n  + 
1 ) ) ) )
2625eqeq1d 2304 . . . . . . 7  |-  ( a  =  A  ->  (
( a " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 }  <-> 
( A " ( ZZ>=
`  ( n  + 
1 ) ) )  =  { 0 } ) )
27 fveq1 5540 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a `  k )  =  ( A `  k ) )
2827oveq1d 5889 . . . . . . . . . 10  |-  ( a  =  A  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2928sumeq2sdv 12193 . . . . . . . . 9  |-  ( a  =  A  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... n ) ( ( A `  k
)  x.  ( z ^ k ) ) )
3029mpteq2dv 4123 . . . . . . . 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 2307 . . . . . . 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 691 . . . . . 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 2577 . . . . 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 6343 . . . 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 642 . . 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 201 . 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 2328 1  |-  ( ph  ->  (coeff `  F )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   E!wreu 2558   {csn 3653    e. cmpt 4093   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   iota_crio 6313    ^m cmap 6788   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   NN0cn0 9981   ZZ>=cuz 10246   ...cfz 10798   ^cexp 11120   sum_csu 12174  Polycply 19582  coeffccoe 19584
This theorem is referenced by:  dgrlem  19627  coeidlem  19635  coeeq2  19640  dgreq  19642  coeaddlem  19646  coemullem  19647  coe1termlem  19655  coecj  19675  basellem2  20335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588
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