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Theorem coeidlem 20109
Description: Lemma for coeid 20110. (Contributed by Mario Carneiro, 22-Jul-2014.)
Hypotheses
Ref Expression
dgrub.1  |-  A  =  (coeff `  F )
dgrub.2  |-  N  =  (deg `  F )
coeid.3  |-  ( ph  ->  F  e.  (Poly `  S ) )
coeid.4  |-  ( ph  ->  M  e.  NN0 )
coeid.5  |-  ( ph  ->  B  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )
coeid.6  |-  ( ph  ->  ( B " ( ZZ>=
`  ( M  + 
1 ) ) )  =  { 0 } )
coeid.7  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... M
) ( ( B `
 k )  x.  ( z ^ k
) ) ) )
Assertion
Ref Expression
coeidlem  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
Distinct variable groups:    z, k, A    k, F    ph, k, z    S, k, z    B, k, z    k, M, z   
k, N, z
Allowed substitution hint:    F( z)

Proof of Theorem coeidlem
StepHypRef Expression
1 coeid.7 . 2  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... M
) ( ( B `
 k )  x.  ( z ^ k
) ) ) )
2 dgrub.1 . . . . . . 7  |-  A  =  (coeff `  F )
3 coeid.3 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  S ) )
4 coeid.4 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
5 coeid.5 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )
6 plybss 20066 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
73, 6syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  S  C_  CC )
8 0cn 9040 . . . . . . . . . . . . . . 15  |-  0  e.  CC
98a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  CC )
109snssd 3903 . . . . . . . . . . . . 13  |-  ( ph  ->  { 0 }  C_  CC )
117, 10unssd 3483 . . . . . . . . . . . 12  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
12 cnex 9027 . . . . . . . . . . . 12  |-  CC  e.  _V
13 ssexg 4309 . . . . . . . . . . . 12  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  CC  e.  _V )  ->  ( S  u.  { 0 } )  e. 
_V )
1411, 12, 13sylancl 644 . . . . . . . . . . 11  |-  ( ph  ->  ( S  u.  {
0 } )  e. 
_V )
15 nn0ex 10183 . . . . . . . . . . 11  |-  NN0  e.  _V
16 elmapg 6990 . . . . . . . . . . 11  |-  ( ( ( S  u.  {
0 } )  e. 
_V  /\  NN0  e.  _V )  ->  ( B  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  <->  B : NN0 --> ( S  u.  { 0 } ) ) )
1714, 15, 16sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  ( B  e.  ( ( S  u.  {
0 } )  ^m  NN0 )  <->  B : NN0 --> ( S  u.  { 0 } ) ) )
185, 17mpbid 202 . . . . . . . . 9  |-  ( ph  ->  B : NN0 --> ( S  u.  { 0 } ) )
19 fss 5558 . . . . . . . . 9  |-  ( ( B : NN0 --> ( S  u.  { 0 } )  /\  ( S  u.  { 0 } )  C_  CC )  ->  B : NN0 --> CC )
2018, 11, 19syl2anc 643 . . . . . . . 8  |-  ( ph  ->  B : NN0 --> CC )
21 coeid.6 . . . . . . . 8  |-  ( ph  ->  ( B " ( ZZ>=
`  ( M  + 
1 ) ) )  =  { 0 } )
223, 4, 20, 21, 1coeeq 20099 . . . . . . 7  |-  ( ph  ->  (coeff `  F )  =  B )
232, 22syl5req 2449 . . . . . 6  |-  ( ph  ->  B  =  A )
2423adantr 452 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  B  =  A )
25 fveq1 5686 . . . . . . 7  |-  ( B  =  A  ->  ( B `  k )  =  ( A `  k ) )
2625oveq1d 6055 . . . . . 6  |-  ( B  =  A  ->  (
( B `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2726sumeq2sdv 12453 . . . . 5  |-  ( B  =  A  ->  sum_ k  e.  ( 0 ... M
) ( ( B `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( z ^ k ) ) )
2824, 27syl 16 . . . 4  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... M
) ( ( B `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( z ^ k ) ) )
293adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  F  e.  (Poly `  S )
)
30 dgrub.2 . . . . . . . . . 10  |-  N  =  (deg `  F )
31 dgrcl 20105 . . . . . . . . . 10  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
3230, 31syl5eqel 2488 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
3329, 32syl 16 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  N  e. 
NN0 )
3433nn0zd 10329 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  e.  ZZ )
354adantr 452 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  M  e. 
NN0 )
3635nn0zd 10329 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  M  e.  ZZ )
3724imaeq1d 5161 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( B
" ( ZZ>= `  ( M  +  1 ) ) )  =  ( A " ( ZZ>= `  ( M  +  1
) ) ) )
3821adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( B
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )
3937, 38eqtr3d 2438 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )
402, 30dgrlb 20108 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  <_  M )
4129, 35, 39, 40syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  <_  M )
42 eluz2 10450 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  N
)  <->  ( N  e.  ZZ  /\  M  e.  ZZ  /\  N  <_  M ) )
4334, 36, 41, 42syl3anbrc 1138 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  M  e.  ( ZZ>= `  N )
)
44 fzss2 11048 . . . . . 6  |-  ( M  e.  ( ZZ>= `  N
)  ->  ( 0 ... N )  C_  ( 0 ... M
) )
4543, 44syl 16 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... N )  C_  ( 0 ... M
) )
46 elfznn0 11039 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
47 plyssc 20072 . . . . . . . . . . 11  |-  (Poly `  S )  C_  (Poly `  CC )
4847, 3sseldi 3306 . . . . . . . . . 10  |-  ( ph  ->  F  e.  (Poly `  CC ) )
492coef3 20104 . . . . . . . . . 10  |-  ( F  e.  (Poly `  CC )  ->  A : NN0 --> CC )
5048, 49syl 16 . . . . . . . . 9  |-  ( ph  ->  A : NN0 --> CC )
5150adantr 452 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  A : NN0
--> CC )
5251ffvelrnda 5829 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
53 expcl 11354 . . . . . . . 8  |-  ( ( z  e.  CC  /\  k  e.  NN0 )  -> 
( z ^ k
)  e.  CC )
5453adantll 695 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  (
z ^ k )  e.  CC )
5552, 54mulcld 9064 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  (
( A `  k
)  x.  ( z ^ k ) )  e.  CC )
5646, 55sylan2 461 . . . . 5  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  ( z ^ k ) )  e.  CC )
57 eldifn 3430 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  -.  k  e.  ( 0 ... N ) )
5857adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  -.  k  e.  ( 0 ... N
) )
59 eldifi 3429 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( 0 ... M
) )
60 elfznn0 11039 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... M )  ->  k  e.  NN0 )
6159, 60syl 16 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  NN0 )
622, 30dgrub 20106 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  N )
63623expia 1155 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  N )
)
6429, 61, 63syl2an 464 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  =/=  0  ->  k  <_  N ) )
65 elfzuz 11011 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... M )  ->  k  e.  ( ZZ>= `  0 )
)
6659, 65syl 16 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( ZZ>= `  0 )
)
67 elfz5 11007 . . . . . . . . . . 11  |-  ( ( k  e.  ( ZZ>= ` 
0 )  /\  N  e.  ZZ )  ->  (
k  e.  ( 0 ... N )  <->  k  <_  N ) )
6866, 34, 67syl2anr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( k  e.  ( 0 ... N
)  <->  k  <_  N
) )
6964, 68sylibrd 226 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  =/=  0  ->  k  e.  ( 0 ... N
) ) )
7069necon1bd 2635 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( -.  k  e.  ( 0 ... N )  -> 
( A `  k
)  =  0 ) )
7158, 70mpd 15 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( A `  k )  =  0 )
7271oveq1d 6055 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  x.  ( z ^ k
) )  =  ( 0  x.  ( z ^ k ) ) )
73 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  z  e.  CC )
7473, 61, 53syl2an 464 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( z ^ k )  e.  CC )
7574mul02d 9220 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( 0  x.  ( z ^
k ) )  =  0 )
7672, 75eqtrd 2436 . . . . 5  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  x.  ( z ^ k
) )  =  0 )
77 fzfid 11267 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... M )  e. 
Fin )
7845, 56, 76, 77fsumss 12474 . . . 4  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( z ^ k ) ) )
7928, 78eqtr4d 2439 . . 3  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... M
) ( ( B `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )
8079mpteq2dva 4255 . 2  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... M ) ( ( B `  k
)  x.  ( z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
811, 80eqtrd 2436 1  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916    \ cdif 3277    u. cun 3278    C_ wss 3280   {csn 3774   class class class wbr 4172    e. cmpt 4226   "cima 4840   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    <_ cle 9077   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   ^cexp 11337   sum_csu 12434  Polycply 20056  coeffccoe 20058  degcdgr 20059
This theorem is referenced by:  coeid  20110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-coe 20062  df-dgr 20063
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