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Theorem coeidlem 20156
Description: Lemma for coeid 20157. (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 20113 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
73, 6syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  S  C_  CC )
8 0cn 9084 . . . . . . . . . . . . . . 15  |-  0  e.  CC
98a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  CC )
109snssd 3943 . . . . . . . . . . . . 13  |-  ( ph  ->  { 0 }  C_  CC )
117, 10unssd 3523 . . . . . . . . . . . 12  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
12 cnex 9071 . . . . . . . . . . . 12  |-  CC  e.  _V
13 ssexg 4349 . . . . . . . . . . . 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 10227 . . . . . . . . . . 11  |-  NN0  e.  _V
16 elmapg 7031 . . . . . . . . . . 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 5599 . . . . . . . . 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 20146 . . . . . . 7  |-  ( ph  ->  (coeff `  F )  =  B )
232, 22syl5req 2481 . . . . . 6  |-  ( ph  ->  B  =  A )
2423adantr 452 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  B  =  A )
25 fveq1 5727 . . . . . . 7  |-  ( B  =  A  ->  ( B `  k )  =  ( A `  k ) )
2625oveq1d 6096 . . . . . 6  |-  ( B  =  A  ->  (
( B `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2726sumeq2sdv 12498 . . . . 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 20152 . . . . . . . . . 10  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
3230, 31syl5eqel 2520 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
3329, 32syl 16 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  N  e. 
NN0 )
3433nn0zd 10373 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  e.  ZZ )
354adantr 452 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  M  e. 
NN0 )
3635nn0zd 10373 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  M  e.  ZZ )
3724imaeq1d 5202 . . . . . . . . 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 2470 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )
402, 30dgrlb 20155 . . . . . . . 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 10494 . . . . . . 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 11092 . . . . . 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 11083 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
47 plyssc 20119 . . . . . . . . . . 11  |-  (Poly `  S )  C_  (Poly `  CC )
4847, 3sseldi 3346 . . . . . . . . . 10  |-  ( ph  ->  F  e.  (Poly `  CC ) )
492coef3 20151 . . . . . . . . . 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 5870 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
53 expcl 11399 . . . . . . . 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 9108 . . . . . 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 3470 . . . . . . . . 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 3469 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( 0 ... M
) )
60 elfznn0 11083 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... M )  ->  k  e.  NN0 )
6159, 60syl 16 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  NN0 )
622, 30dgrub 20153 . . . . . . . . . . . 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 11055 . . . . . . . . . . . 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 11051 . . . . . . . . . . 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 2672 . . . . . . . 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 6096 . . . . . 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 9264 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( 0  x.  ( z ^
k ) )  =  0 )
7672, 75eqtrd 2468 . . . . 5  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  x.  ( z ^ k
) )  =  0 )
77 fzfid 11312 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... M )  e. 
Fin )
7845, 56, 76, 77fsumss 12519 . . . 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 2471 . . 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 4295 . 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 2468 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 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    \ cdif 3317    u. cun 3318    C_ wss 3320   {csn 3814   class class class wbr 4212    e. cmpt 4266   "cima 4881   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    <_ cle 9121   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043   ^cexp 11382   sum_csu 12479  Polycply 20103  coeffccoe 20105  degcdgr 20106
This theorem is referenced by:  coeid  20157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107  df-coe 20109  df-dgr 20110
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