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Theorem coeidlem 22926
Description: Lemma for coeid 22927. (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 22883 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
73, 6syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  S  C_  CC )
8 0cnd 9619 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  CC )
98snssd 4117 . . . . . . . . . . . . 13  |-  ( ph  ->  { 0 }  C_  CC )
107, 9unssd 3619 . . . . . . . . . . . 12  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
11 cnex 9603 . . . . . . . . . . . 12  |-  CC  e.  _V
12 ssexg 4540 . . . . . . . . . . . 12  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  CC  e.  _V )  ->  ( S  u.  { 0 } )  e. 
_V )
1310, 11, 12sylancl 660 . . . . . . . . . . 11  |-  ( ph  ->  ( S  u.  {
0 } )  e. 
_V )
14 nn0ex 10842 . . . . . . . . . . 11  |-  NN0  e.  _V
15 elmapg 7470 . . . . . . . . . . 11  |-  ( ( ( S  u.  {
0 } )  e. 
_V  /\  NN0  e.  _V )  ->  ( B  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  <->  B : NN0 --> ( S  u.  { 0 } ) ) )
1613, 14, 15sylancl 660 . . . . . . . . . 10  |-  ( ph  ->  ( B  e.  ( ( S  u.  {
0 } )  ^m  NN0 )  <->  B : NN0 --> ( S  u.  { 0 } ) ) )
175, 16mpbid 210 . . . . . . . . 9  |-  ( ph  ->  B : NN0 --> ( S  u.  { 0 } ) )
1817, 10fssd 5723 . . . . . . . 8  |-  ( ph  ->  B : NN0 --> CC )
19 coeid.6 . . . . . . . 8  |-  ( ph  ->  ( B " ( ZZ>=
`  ( M  + 
1 ) ) )  =  { 0 } )
203, 4, 18, 19, 1coeeq 22916 . . . . . . 7  |-  ( ph  ->  (coeff `  F )  =  B )
212, 20syl5req 2456 . . . . . 6  |-  ( ph  ->  B  =  A )
2221adantr 463 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  B  =  A )
23 fveq1 5848 . . . . . . 7  |-  ( B  =  A  ->  ( B `  k )  =  ( A `  k ) )
2423oveq1d 6293 . . . . . 6  |-  ( B  =  A  ->  (
( B `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2524sumeq2sdv 13675 . . . . 5  |-  ( B  =  A  ->  sum_ k  e.  ( 0 ... M
) ( ( B `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( z ^ k ) ) )
2622, 25syl 17 . . . 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 ) ) )
273adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  F  e.  (Poly `  S )
)
28 dgrub.2 . . . . . . . . . 10  |-  N  =  (deg `  F )
29 dgrcl 22922 . . . . . . . . . 10  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
3028, 29syl5eqel 2494 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
3127, 30syl 17 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  N  e. 
NN0 )
3231nn0zd 11006 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  e.  ZZ )
334adantr 463 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  M  e. 
NN0 )
3433nn0zd 11006 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  M  e.  ZZ )
3522imaeq1d 5156 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( B
" ( ZZ>= `  ( M  +  1 ) ) )  =  ( A " ( ZZ>= `  ( M  +  1
) ) ) )
3619adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( B
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )
3735, 36eqtr3d 2445 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )
382, 28dgrlb 22925 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  <_  M )
3927, 33, 37, 38syl3anc 1230 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  <_  M )
40 eluz2 11133 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  N
)  <->  ( N  e.  ZZ  /\  M  e.  ZZ  /\  N  <_  M ) )
4132, 34, 39, 40syl3anbrc 1181 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  M  e.  ( ZZ>= `  N )
)
42 fzss2 11778 . . . . . 6  |-  ( M  e.  ( ZZ>= `  N
)  ->  ( 0 ... N )  C_  ( 0 ... M
) )
4341, 42syl 17 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... N )  C_  ( 0 ... M
) )
44 elfznn0 11826 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
45 plyssc 22889 . . . . . . . . . . 11  |-  (Poly `  S )  C_  (Poly `  CC )
4645, 3sseldi 3440 . . . . . . . . . 10  |-  ( ph  ->  F  e.  (Poly `  CC ) )
472coef3 22921 . . . . . . . . . 10  |-  ( F  e.  (Poly `  CC )  ->  A : NN0 --> CC )
4846, 47syl 17 . . . . . . . . 9  |-  ( ph  ->  A : NN0 --> CC )
4948adantr 463 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  A : NN0
--> CC )
5049ffvelrnda 6009 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
51 expcl 12228 . . . . . . . 8  |-  ( ( z  e.  CC  /\  k  e.  NN0 )  -> 
( z ^ k
)  e.  CC )
5251adantll 712 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  (
z ^ k )  e.  CC )
5350, 52mulcld 9646 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  (
( A `  k
)  x.  ( z ^ k ) )  e.  CC )
5444, 53sylan2 472 . . . . 5  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  ( z ^ k ) )  e.  CC )
55 eldifn 3566 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  -.  k  e.  ( 0 ... N ) )
5655adantl 464 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  -.  k  e.  ( 0 ... N
) )
57 eldifi 3565 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( 0 ... M
) )
58 elfznn0 11826 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... M )  ->  k  e.  NN0 )
5957, 58syl 17 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  NN0 )
602, 28dgrub 22923 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  N )
61603expia 1199 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  N )
)
6227, 59, 61syl2an 475 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  =/=  0  ->  k  <_  N ) )
63 elfzuz 11738 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... M )  ->  k  e.  ( ZZ>= `  0 )
)
6457, 63syl 17 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( ZZ>= `  0 )
)
65 elfz5 11734 . . . . . . . . . . 11  |-  ( ( k  e.  ( ZZ>= ` 
0 )  /\  N  e.  ZZ )  ->  (
k  e.  ( 0 ... N )  <->  k  <_  N ) )
6664, 32, 65syl2anr 476 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( k  e.  ( 0 ... N
)  <->  k  <_  N
) )
6762, 66sylibrd 234 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  =/=  0  ->  k  e.  ( 0 ... N
) ) )
6867necon1bd 2621 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( -.  k  e.  ( 0 ... N )  -> 
( A `  k
)  =  0 ) )
6956, 68mpd 15 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( A `  k )  =  0 )
7069oveq1d 6293 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  x.  ( z ^ k
) )  =  ( 0  x.  ( z ^ k ) ) )
71 simpr 459 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  z  e.  CC )
7271, 59, 51syl2an 475 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( z ^ k )  e.  CC )
7372mul02d 9812 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( 0  x.  ( z ^
k ) )  =  0 )
7470, 73eqtrd 2443 . . . . 5  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  x.  ( z ^ k
) )  =  0 )
75 fzfid 12124 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... M )  e. 
Fin )
7643, 54, 74, 75fsumss 13696 . . . 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 ) ) )
7726, 76eqtr4d 2446 . . 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 ) ) )
7877mpteq2dva 4481 . 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
) ) ) )
791, 78eqtrd 2443 1  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3059    \ cdif 3411    u. cun 3412    C_ wss 3414   {csn 3972   class class class wbr 4395    |-> cmpt 4453   "cima 4826   -->wf 5565   ` cfv 5569  (class class class)co 6278    ^m cmap 7457   CCcc 9520   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527    <_ cle 9659   NN0cn0 10836   ZZcz 10905   ZZ>=cuz 11127   ...cfz 11726   ^cexp 12210   sum_csu 13657  Polycply 22873  coeffccoe 22875  degcdgr 22876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-rlim 13461  df-sum 13658  df-0p 22369  df-ply 22877  df-coe 22879  df-dgr 22880
This theorem is referenced by:  coeid  22927
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