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Theorem coeidlem 21590
Description: Lemma for coeid 21591. (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 21547 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
73, 6syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  S  C_  CC )
8 0cnd 9367 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  CC )
98snssd 4006 . . . . . . . . . . . . 13  |-  ( ph  ->  { 0 }  C_  CC )
107, 9unssd 3520 . . . . . . . . . . . 12  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
11 cnex 9351 . . . . . . . . . . . 12  |-  CC  e.  _V
12 ssexg 4426 . . . . . . . . . . . 12  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  CC  e.  _V )  ->  ( S  u.  { 0 } )  e. 
_V )
1310, 11, 12sylancl 655 . . . . . . . . . . 11  |-  ( ph  ->  ( S  u.  {
0 } )  e. 
_V )
14 nn0ex 10573 . . . . . . . . . . 11  |-  NN0  e.  _V
15 elmapg 7215 . . . . . . . . . . 11  |-  ( ( ( S  u.  {
0 } )  e. 
_V  /\  NN0  e.  _V )  ->  ( B  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  <->  B : NN0 --> ( S  u.  { 0 } ) ) )
1613, 14, 15sylancl 655 . . . . . . . . . 10  |-  ( ph  ->  ( B  e.  ( ( S  u.  {
0 } )  ^m  NN0 )  <->  B : NN0 --> ( S  u.  { 0 } ) ) )
175, 16mpbid 210 . . . . . . . . 9  |-  ( ph  ->  B : NN0 --> ( S  u.  { 0 } ) )
18 fss 5555 . . . . . . . . 9  |-  ( ( B : NN0 --> ( S  u.  { 0 } )  /\  ( S  u.  { 0 } )  C_  CC )  ->  B : NN0 --> CC )
1917, 10, 18syl2anc 654 . . . . . . . 8  |-  ( ph  ->  B : NN0 --> CC )
20 coeid.6 . . . . . . . 8  |-  ( ph  ->  ( B " ( ZZ>=
`  ( M  + 
1 ) ) )  =  { 0 } )
213, 4, 19, 20, 1coeeq 21580 . . . . . . 7  |-  ( ph  ->  (coeff `  F )  =  B )
222, 21syl5req 2478 . . . . . 6  |-  ( ph  ->  B  =  A )
2322adantr 462 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  B  =  A )
24 fveq1 5678 . . . . . . 7  |-  ( B  =  A  ->  ( B `  k )  =  ( A `  k ) )
2524oveq1d 6095 . . . . . 6  |-  ( B  =  A  ->  (
( B `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2625sumeq2sdv 13165 . . . . 5  |-  ( B  =  A  ->  sum_ k  e.  ( 0 ... M
) ( ( B `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( z ^ k ) ) )
2723, 26syl 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 ) ) )
283adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  F  e.  (Poly `  S )
)
29 dgrub.2 . . . . . . . . . 10  |-  N  =  (deg `  F )
30 dgrcl 21586 . . . . . . . . . 10  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
3129, 30syl5eqel 2517 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
3228, 31syl 16 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  N  e. 
NN0 )
3332nn0zd 10733 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  e.  ZZ )
344adantr 462 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  M  e. 
NN0 )
3534nn0zd 10733 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  M  e.  ZZ )
3623imaeq1d 5156 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( B
" ( ZZ>= `  ( M  +  1 ) ) )  =  ( A " ( ZZ>= `  ( M  +  1
) ) ) )
3720adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( B
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )
3836, 37eqtr3d 2467 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )
392, 29dgrlb 21589 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  <_  M )
4028, 34, 38, 39syl3anc 1211 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  <_  M )
41 eluz2 10855 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  N
)  <->  ( N  e.  ZZ  /\  M  e.  ZZ  /\  N  <_  M ) )
4233, 35, 40, 41syl3anbrc 1165 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  M  e.  ( ZZ>= `  N )
)
43 fzss2 11485 . . . . . 6  |-  ( M  e.  ( ZZ>= `  N
)  ->  ( 0 ... N )  C_  ( 0 ... M
) )
4442, 43syl 16 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... N )  C_  ( 0 ... M
) )
45 elfznn0 11468 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
46 plyssc 21553 . . . . . . . . . . 11  |-  (Poly `  S )  C_  (Poly `  CC )
4746, 3sseldi 3342 . . . . . . . . . 10  |-  ( ph  ->  F  e.  (Poly `  CC ) )
482coef3 21585 . . . . . . . . . 10  |-  ( F  e.  (Poly `  CC )  ->  A : NN0 --> CC )
4947, 48syl 16 . . . . . . . . 9  |-  ( ph  ->  A : NN0 --> CC )
5049adantr 462 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  A : NN0
--> CC )
5150ffvelrnda 5831 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
52 expcl 11867 . . . . . . . 8  |-  ( ( z  e.  CC  /\  k  e.  NN0 )  -> 
( z ^ k
)  e.  CC )
5352adantll 706 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  (
z ^ k )  e.  CC )
5451, 53mulcld 9394 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  (
( A `  k
)  x.  ( z ^ k ) )  e.  CC )
5545, 54sylan2 471 . . . . 5  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  ( z ^ k ) )  e.  CC )
56 eldifn 3467 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  -.  k  e.  ( 0 ... N ) )
5756adantl 463 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  -.  k  e.  ( 0 ... N
) )
58 eldifi 3466 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( 0 ... M
) )
59 elfznn0 11468 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... M )  ->  k  e.  NN0 )
6058, 59syl 16 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  NN0 )
612, 29dgrub 21587 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  N )
62613expia 1182 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  N )
)
6328, 60, 62syl2an 474 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  =/=  0  ->  k  <_  N ) )
64 elfzuz 11436 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... M )  ->  k  e.  ( ZZ>= `  0 )
)
6558, 64syl 16 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( ZZ>= `  0 )
)
66 elfz5 11432 . . . . . . . . . . 11  |-  ( ( k  e.  ( ZZ>= ` 
0 )  /\  N  e.  ZZ )  ->  (
k  e.  ( 0 ... N )  <->  k  <_  N ) )
6765, 33, 66syl2anr 475 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( k  e.  ( 0 ... N
)  <->  k  <_  N
) )
6863, 67sylibrd 234 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  =/=  0  ->  k  e.  ( 0 ... N
) ) )
6968necon1bd 2669 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( -.  k  e.  ( 0 ... N )  -> 
( A `  k
)  =  0 ) )
7057, 69mpd 15 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( A `  k )  =  0 )
7170oveq1d 6095 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  x.  ( z ^ k
) )  =  ( 0  x.  ( z ^ k ) ) )
72 simpr 458 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  z  e.  CC )
7372, 60, 52syl2an 474 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( z ^ k )  e.  CC )
7473mul02d 9555 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( 0  x.  ( z ^
k ) )  =  0 )
7571, 74eqtrd 2465 . . . . 5  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  x.  ( z ^ k
) )  =  0 )
76 fzfid 11779 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... M )  e. 
Fin )
7744, 55, 75, 76fsumss 13186 . . . 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 ) ) )
7827, 77eqtr4d 2468 . . 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 ) ) )
7978mpteq2dva 4366 . 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
) ) ) )
801, 79eqtrd 2465 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 369    = wceq 1362    e. wcel 1755    =/= wne 2596   _Vcvv 2962    \ cdif 3313    u. cun 3314    C_ wss 3316   {csn 3865   class class class wbr 4280    e. cmpt 4338   "cima 4830   -->wf 5402   ` cfv 5406  (class class class)co 6080    ^m cmap 7202   CCcc 9268   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    <_ cle 9407   NN0cn0 10567   ZZcz 10634   ZZ>=cuz 10849   ...cfz 11424   ^cexp 11849   sum_csu 13147  Polycply 21537  coeffccoe 21539  degcdgr 21540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-fz 11425  df-fzo 11533  df-fl 11626  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-rlim 12951  df-sum 13148  df-0p 20990  df-ply 21541  df-coe 21543  df-dgr 21544
This theorem is referenced by:  coeid  21591
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