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Theorem elplyd 20113
Description: Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)
Hypotheses
Ref Expression
elplyd.1  |-  ( ph  ->  S  C_  CC )
elplyd.2  |-  ( ph  ->  N  e.  NN0 )
elplyd.3  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  A  e.  S )
Assertion
Ref Expression
elplyd  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( A  x.  ( z ^ k ) ) )  e.  (Poly `  S ) )
Distinct variable groups:    z, A    z, k, N    ph, k, z    S, k, z
Allowed substitution hint:    A( k)

Proof of Theorem elplyd
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 nffvmpt1 5728 . . . . . . 7  |-  F/_ k
( ( k  e. 
NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `
 j )
2 nfcv 2571 . . . . . . 7  |-  F/_ k  x.
3 nfcv 2571 . . . . . . 7  |-  F/_ k
( z ^ j
)
41, 2, 3nfov 6096 . . . . . 6  |-  F/_ k
( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  j )  x.  ( z ^
j ) )
5 nfcv 2571 . . . . . 6  |-  F/_ j
( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k )  x.  ( z ^
k ) )
6 fveq2 5720 . . . . . . 7  |-  ( j  =  k  ->  (
( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  j
)  =  ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k ) )
7 oveq2 6081 . . . . . . 7  |-  ( j  =  k  ->  (
z ^ j )  =  ( z ^
k ) )
86, 7oveq12d 6091 . . . . . 6  |-  ( j  =  k  ->  (
( ( k  e. 
NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `
 j )  x.  ( z ^ j
) )  =  ( ( ( k  e. 
NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `
 k )  x.  ( z ^ k
) ) )
94, 5, 8cbvsumi 12483 . . . . 5  |-  sum_ j  e.  ( 0 ... N
) ( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  j )  x.  ( z ^
j ) )  = 
sum_ k  e.  ( 0 ... N ) ( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k )  x.  ( z ^
k ) )
10 elfznn0 11075 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
1110adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
12 iftrue 3737 . . . . . . . . . . 11  |-  ( k  e.  ( 0 ... N )  ->  if ( k  e.  ( 0 ... N ) ,  A ,  0 )  =  A )
1312adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  if ( k  e.  ( 0 ... N ) ,  A ,  0 )  =  A )
14 elplyd.3 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  A  e.  S )
1513, 14eqeltrd 2509 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  if ( k  e.  ( 0 ... N ) ,  A ,  0 )  e.  S )
16 eqid 2435 . . . . . . . . . 10  |-  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )  =  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )
1716fvmpt2 5804 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  if ( k  e.  ( 0 ... N ) ,  A ,  0 )  e.  S )  ->  ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k )  =  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )
1811, 15, 17syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k
)  =  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )
1918, 13eqtrd 2467 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k
)  =  A )
2019oveq1d 6088 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( k  e. 
NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `
 k )  x.  ( z ^ k
) )  =  ( A  x.  ( z ^ k ) ) )
2120sumeq2dv 12489 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... N ) ( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k )  x.  ( z ^
k ) )  = 
sum_ k  e.  ( 0 ... N ) ( A  x.  (
z ^ k ) ) )
229, 21syl5eq 2479 . . . 4  |-  ( ph  -> 
sum_ j  e.  ( 0 ... N ) ( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  j )  x.  ( z ^
j ) )  = 
sum_ k  e.  ( 0 ... N ) ( A  x.  (
z ^ k ) ) )
2322mpteq2dv 4288 . . 3  |-  ( ph  ->  ( z  e.  CC  |->  sum_ j  e.  ( 0 ... N ) ( ( ( k  e. 
NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `
 j )  x.  ( z ^ j
) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( A  x.  ( z ^ k ) ) ) )
24 elplyd.1 . . . . 5  |-  ( ph  ->  S  C_  CC )
25 0cn 9076 . . . . . . 7  |-  0  e.  CC
2625a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  CC )
2726snssd 3935 . . . . 5  |-  ( ph  ->  { 0 }  C_  CC )
2824, 27unssd 3515 . . . 4  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
29 elplyd.2 . . . 4  |-  ( ph  ->  N  e.  NN0 )
30 elun1 3506 . . . . . . . 8  |-  ( A  e.  S  ->  A  e.  ( S  u.  {
0 } ) )
3114, 30syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  A  e.  ( S  u.  {
0 } ) )
3231adantlr 696 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  k  e.  ( 0 ... N
) )  ->  A  e.  ( S  u.  {
0 } ) )
33 ssun2 3503 . . . . . . . 8  |-  { 0 }  C_  ( S  u.  { 0 } )
34 c0ex 9077 . . . . . . . . 9  |-  0  e.  _V
3534snss 3918 . . . . . . . 8  |-  ( 0  e.  ( S  u.  { 0 } )  <->  { 0 }  C_  ( S  u.  { 0 } ) )
3633, 35mpbir 201 . . . . . . 7  |-  0  e.  ( S  u.  {
0 } )
3736a1i 11 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  ( 0 ... N ) )  ->  0  e.  ( S  u.  { 0 } ) )
3832, 37ifclda 3758 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  if (
k  e.  ( 0 ... N ) ,  A ,  0 )  e.  ( S  u.  { 0 } ) )
3938, 16fmptd 5885 . . . 4  |-  ( ph  ->  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) : NN0 --> ( S  u.  { 0 } ) )
40 elplyr 20112 . . . 4  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  N  e.  NN0  /\  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) : NN0 --> ( S  u.  { 0 } ) )  -> 
( z  e.  CC  |->  sum_ j  e.  ( 0 ... N ) ( ( ( k  e. 
NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `
 j )  x.  ( z ^ j
) ) )  e.  (Poly `  ( S  u.  { 0 } ) ) )
4128, 29, 39, 40syl3anc 1184 . . 3  |-  ( ph  ->  ( z  e.  CC  |->  sum_ j  e.  ( 0 ... N ) ( ( ( k  e. 
NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `
 j )  x.  ( z ^ j
) ) )  e.  (Poly `  ( S  u.  { 0 } ) ) )
4223, 41eqeltrrd 2510 . 2  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( A  x.  ( z ^ k ) ) )  e.  (Poly `  ( S  u.  { 0 } ) ) )
43 plyun0 20108 . 2  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
4442, 43syl6eleq 2525 1  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( A  x.  ( z ^ k ) ) )  e.  (Poly `  S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    u. cun 3310    C_ wss 3312   ifcif 3731   {csn 3806    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982    x. cmul 8987   NN0cn0 10213   ...cfz 11035   ^cexp 11374   sum_csu 12471  Polycply 20095
This theorem is referenced by:  ply1term  20115  plyaddlem  20126  plymullem  20127  plycj  20187  dvply2g  20194  elqaalem3  20230  aareccl  20235  taylply2  20276  basellem2  20856
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-seq 11316  df-sum 12472  df-ply 20099
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