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Theorem coeidlem 23270
Description: Lemma for coeid 23271. (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 23227 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
73, 6syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  S  C_  CC )
8 0cnd 9654 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  CC )
98snssd 4108 . . . . . . . . . . . . 13  |-  ( ph  ->  { 0 }  C_  CC )
107, 9unssd 3601 . . . . . . . . . . . 12  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
11 cnex 9638 . . . . . . . . . . . 12  |-  CC  e.  _V
12 ssexg 4542 . . . . . . . . . . . 12  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  CC  e.  _V )  ->  ( S  u.  { 0 } )  e. 
_V )
1310, 11, 12sylancl 675 . . . . . . . . . . 11  |-  ( ph  ->  ( S  u.  {
0 } )  e. 
_V )
14 nn0ex 10899 . . . . . . . . . . 11  |-  NN0  e.  _V
15 elmapg 7503 . . . . . . . . . . 11  |-  ( ( ( S  u.  {
0 } )  e. 
_V  /\  NN0  e.  _V )  ->  ( B  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  <->  B : NN0 --> ( S  u.  { 0 } ) ) )
1613, 14, 15sylancl 675 . . . . . . . . . 10  |-  ( ph  ->  ( B  e.  ( ( S  u.  {
0 } )  ^m  NN0 )  <->  B : NN0 --> ( S  u.  { 0 } ) ) )
175, 16mpbid 215 . . . . . . . . 9  |-  ( ph  ->  B : NN0 --> ( S  u.  { 0 } ) )
1817, 10fssd 5750 . . . . . . . 8  |-  ( ph  ->  B : NN0 --> CC )
19 coeid.6 . . . . . . . 8  |-  ( ph  ->  ( B " ( ZZ>=
`  ( M  + 
1 ) ) )  =  { 0 } )
203, 4, 18, 19, 1coeeq 23260 . . . . . . 7  |-  ( ph  ->  (coeff `  F )  =  B )
212, 20syl5req 2518 . . . . . 6  |-  ( ph  ->  B  =  A )
2221adantr 472 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  B  =  A )
23 fveq1 5878 . . . . . . 7  |-  ( B  =  A  ->  ( B `  k )  =  ( A `  k ) )
2423oveq1d 6323 . . . . . 6  |-  ( B  =  A  ->  (
( B `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2524sumeq2sdv 13847 . . . . 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 472 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  F  e.  (Poly `  S )
)
28 dgrub.2 . . . . . . . . . 10  |-  N  =  (deg `  F )
29 dgrcl 23266 . . . . . . . . . 10  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
3028, 29syl5eqel 2553 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
3127, 30syl 17 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  N  e. 
NN0 )
3231nn0zd 11061 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  e.  ZZ )
334adantr 472 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  M  e. 
NN0 )
3433nn0zd 11061 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  M  e.  ZZ )
3522imaeq1d 5173 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( B
" ( ZZ>= `  ( M  +  1 ) ) )  =  ( A " ( ZZ>= `  ( M  +  1
) ) ) )
3619adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( B
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )
3735, 36eqtr3d 2507 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )
382, 28dgrlb 23269 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  <_  M )
3927, 33, 37, 38syl3anc 1292 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  N  <_  M )
40 eluz2 11188 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  N
)  <->  ( N  e.  ZZ  /\  M  e.  ZZ  /\  N  <_  M ) )
4132, 34, 39, 40syl3anbrc 1214 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  M  e.  ( ZZ>= `  N )
)
42 fzss2 11864 . . . . . 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 11913 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
45 plyssc 23233 . . . . . . . . . . 11  |-  (Poly `  S )  C_  (Poly `  CC )
4645, 3sseldi 3416 . . . . . . . . . 10  |-  ( ph  ->  F  e.  (Poly `  CC ) )
472coef3 23265 . . . . . . . . . 10  |-  ( F  e.  (Poly `  CC )  ->  A : NN0 --> CC )
4846, 47syl 17 . . . . . . . . 9  |-  ( ph  ->  A : NN0 --> CC )
4948adantr 472 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  A : NN0
--> CC )
5049ffvelrnda 6037 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
51 expcl 12328 . . . . . . . 8  |-  ( ( z  e.  CC  /\  k  e.  NN0 )  -> 
( z ^ k
)  e.  CC )
5251adantll 728 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  (
z ^ k )  e.  CC )
5350, 52mulcld 9681 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  (
( A `  k
)  x.  ( z ^ k ) )  e.  CC )
5444, 53sylan2 482 . . . . 5  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  ( z ^ k ) )  e.  CC )
55 eldifn 3545 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  -.  k  e.  ( 0 ... N ) )
5655adantl 473 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  -.  k  e.  ( 0 ... N
) )
57 eldifi 3544 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( 0 ... M
) )
58 elfznn0 11913 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... M )  ->  k  e.  NN0 )
5957, 58syl 17 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  NN0 )
602, 28dgrub 23267 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  N )
61603expia 1233 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  N )
)
6227, 59, 61syl2an 485 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  =/=  0  ->  k  <_  N ) )
63 elfzuz 11822 . . . . . . . . . . . 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 11818 . . . . . . . . . . 11  |-  ( ( k  e.  ( ZZ>= ` 
0 )  /\  N  e.  ZZ )  ->  (
k  e.  ( 0 ... N )  <->  k  <_  N ) )
6664, 32, 65syl2anr 486 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( k  e.  ( 0 ... N
)  <->  k  <_  N
) )
6762, 66sylibrd 242 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  =/=  0  ->  k  e.  ( 0 ... N
) ) )
6867necon1bd 2661 . . . . . . . 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 6323 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  x.  ( z ^ k
) )  =  ( 0  x.  ( z ^ k ) ) )
71 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  z  e.  CC )
7271, 59, 51syl2an 485 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( z ^ k )  e.  CC )
7372mul02d 9849 . . . . . 6  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( 0  x.  ( z ^
k ) )  =  0 )
7470, 73eqtrd 2505 . . . . 5  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( ( 0 ... M )  \  (
0 ... N ) ) )  ->  ( ( A `  k )  x.  ( z ^ k
) )  =  0 )
75 fzfid 12224 . . . . 5  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... M )  e. 
Fin )
7643, 54, 74, 75fsumss 13868 . . . 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 2508 . . 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 4482 . 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 2505 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 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    \ cdif 3387    u. cun 3388    C_ wss 3390   {csn 3959   class class class wbr 4395    |-> cmpt 4454   "cima 4842   -->wf 5585   ` cfv 5589  (class class class)co 6308    ^m cmap 7490   CCcc 9555   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    <_ cle 9694   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   ^cexp 12310   sum_csu 13829  Polycply 23217  coeffccoe 23219  degcdgr 23220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-coe 23223  df-dgr 23224
This theorem is referenced by:  coeid  23271
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