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Theorem elplyd 21806
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 5810 . . . . . . 7  |-  F/_ k
( ( k  e. 
NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `
 j )
2 nfcv 2616 . . . . . . 7  |-  F/_ k  x.
3 nfcv 2616 . . . . . . 7  |-  F/_ k
( z ^ j
)
41, 2, 3nfov 6226 . . . . . 6  |-  F/_ k
( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  j )  x.  ( z ^
j ) )
5 nfcv 2616 . . . . . 6  |-  F/_ j
( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k )  x.  ( z ^
k ) )
6 fveq2 5802 . . . . . . 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 6211 . . . . . . 7  |-  ( j  =  k  ->  (
z ^ j )  =  ( z ^
k ) )
86, 7oveq12d 6221 . . . . . 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 13295 . . . . 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 11601 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
1110adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
12 iftrue 3908 . . . . . . . . . . 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 2542 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  if ( k  e.  ( 0 ... N ) ,  A ,  0 )  e.  S )
16 eqid 2454 . . . . . . . . . 10  |-  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )  =  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )
1716fvmpt2 5893 . . . . . . . . 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 2495 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k
)  =  A )
2019oveq1d 6218 . . . . . 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 13301 . . . . 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 2507 . . . 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 4490 . . 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 9493 . . . . . 6  |-  ( ph  ->  0  e.  CC )
2625snssd 4129 . . . . 5  |-  ( ph  ->  { 0 }  C_  CC )
2724, 26unssd 3643 . . . 4  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
28 elplyd.2 . . . 4  |-  ( ph  ->  N  e.  NN0 )
29 elun1 3634 . . . . . . . 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 3631 . . . . . . . 8  |-  { 0 }  C_  ( S  u.  { 0 } )
33 c0ex 9494 . . . . . . . . 9  |-  0  e.  _V
3433snss 4110 . . . . . . . 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 3932 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  if (
k  e.  ( 0 ... N ) ,  A ,  0 )  e.  ( S  u.  { 0 } ) )
3837, 16fmptd 5979 . . . 4  |-  ( ph  ->  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) : NN0 --> ( S  u.  { 0 } ) )
39 elplyr 21805 . . . 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 1219 . . 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 2543 . 2  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( A  x.  ( z ^ k ) ) )  e.  (Poly `  ( S  u.  { 0 } ) ) )
42 plyun0 21801 . 2  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
4341, 42syl6eleq 2552 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 1370    e. wcel 1758    u. cun 3437    C_ wss 3439   ifcif 3902   {csn 3988    |-> cmpt 4461   -->wf 5525   ` cfv 5529  (class class class)co 6203   CCcc 9394   0cc0 9396    x. cmul 9401   NN0cn0 10693   ...cfz 11557   ^cexp 11985   sum_csu 13284  Polycply 21788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-seq 11927  df-sum 13285  df-ply 21792
This theorem is referenced by:  ply1term  21808  plyaddlem  21819  plymullem  21820  plycj  21880  dvply2g  21887  elqaalem3  21923  aareccl  21928  taylply2  21969  basellem2  22555
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