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Theorem elplyd 21645
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 5694 . . . . . . 7  |-  F/_ k
( ( k  e. 
NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `
 j )
2 nfcv 2574 . . . . . . 7  |-  F/_ k  x.
3 nfcv 2574 . . . . . . 7  |-  F/_ k
( z ^ j
)
41, 2, 3nfov 6109 . . . . . 6  |-  F/_ k
( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  j )  x.  ( z ^
j ) )
5 nfcv 2574 . . . . . 6  |-  F/_ j
( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k )  x.  ( z ^
k ) )
6 fveq2 5686 . . . . . . 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 6094 . . . . . . 7  |-  ( j  =  k  ->  (
z ^ j )  =  ( z ^
k ) )
86, 7oveq12d 6104 . . . . . 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 13166 . . . . 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 11473 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
1110adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
12 iftrue 3792 . . . . . . . . . . 11  |-  ( k  e.  ( 0 ... N )  ->  if ( k  e.  ( 0 ... N ) ,  A ,  0 )  =  A )
1312adantl 466 . . . . . . . . . 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 2512 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  if ( k  e.  ( 0 ... N ) ,  A ,  0 )  e.  S )
16 eqid 2438 . . . . . . . . . 10  |-  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )  =  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )
1716fvmpt2 5776 . . . . . . . . 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 661 . . . . . . . 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 2470 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k
)  =  A )
2019oveq1d 6101 . . . . . 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 13172 . . . . 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 2482 . . . 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 4374 . . 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 0cnd 9371 . . . . . 6  |-  ( ph  ->  0  e.  CC )
2625snssd 4013 . . . . 5  |-  ( ph  ->  { 0 }  C_  CC )
2724, 26unssd 3527 . . . 4  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
28 elplyd.2 . . . 4  |-  ( ph  ->  N  e.  NN0 )
29 elun1 3518 . . . . . . . 8  |-  ( A  e.  S  ->  A  e.  ( S  u.  {
0 } ) )
3014, 29syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  A  e.  ( S  u.  {
0 } ) )
3130adantlr 714 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  k  e.  ( 0 ... N
) )  ->  A  e.  ( S  u.  {
0 } ) )
32 ssun2 3515 . . . . . . . 8  |-  { 0 }  C_  ( S  u.  { 0 } )
33 c0ex 9372 . . . . . . . . 9  |-  0  e.  _V
3433snss 3994 . . . . . . . 8  |-  ( 0  e.  ( S  u.  { 0 } )  <->  { 0 }  C_  ( S  u.  { 0 } ) )
3532, 34mpbir 209 . . . . . . 7  |-  0  e.  ( S  u.  {
0 } )
3635a1i 11 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  ( 0 ... N ) )  ->  0  e.  ( S  u.  { 0 } ) )
3731, 36ifclda 3816 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  if (
k  e.  ( 0 ... N ) ,  A ,  0 )  e.  ( S  u.  { 0 } ) )
3837, 16fmptd 5862 . . . 4  |-  ( ph  ->  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) : NN0 --> ( S  u.  { 0 } ) )
39 elplyr 21644 . . . 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 } ) ) )
4027, 28, 38, 39syl3anc 1218 . . 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 } ) ) )
4123, 40eqeltrrd 2513 . 2  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( A  x.  ( z ^ k ) ) )  e.  (Poly `  ( S  u.  { 0 } ) ) )
42 plyun0 21640 . 2  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
4341, 42syl6eleq 2528 1  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( A  x.  ( z ^ k ) ) )  e.  (Poly `  S ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    u. cun 3321    C_ wss 3323   ifcif 3786   {csn 3872    e. cmpt 4345   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274    x. cmul 9279   NN0cn0 10571   ...cfz 11429   ^cexp 11857   sum_csu 13155  Polycply 21627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-seq 11799  df-sum 13156  df-ply 21631
This theorem is referenced by:  ply1term  21647  plyaddlem  21658  plymullem  21659  plycj  21719  dvply2g  21726  elqaalem3  21762  aareccl  21767  taylply2  21808  basellem2  22394
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