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Theorem elplyd 19600
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 nfmpt1 4125 . . . . . . . 8  |-  F/_ k
( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )
2 nfcv 2432 . . . . . . . 8  |-  F/_ k
j
31, 2nffv 5548 . . . . . . 7  |-  F/_ k
( ( k  e. 
NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `
 j )
4 nfcv 2432 . . . . . . 7  |-  F/_ k  x.
5 nfcv 2432 . . . . . . 7  |-  F/_ k
( z ^ j
)
63, 4, 5nfov 5897 . . . . . 6  |-  F/_ k
( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  j )  x.  ( z ^
j ) )
7 nfcv 2432 . . . . . 6  |-  F/_ j
( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k )  x.  ( z ^
k ) )
8 fveq2 5541 . . . . . . 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 ) )
9 oveq2 5882 . . . . . . 7  |-  ( j  =  k  ->  (
z ^ j )  =  ( z ^
k ) )
108, 9oveq12d 5892 . . . . . 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
) ) )
116, 7, 10cbvsumi 12186 . . . . 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 ) )
12 elfznn0 10838 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
1312adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
14 iftrue 3584 . . . . . . . . . . 11  |-  ( k  e.  ( 0 ... N )  ->  if ( k  e.  ( 0 ... N ) ,  A ,  0 )  =  A )
1514adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  if ( k  e.  ( 0 ... N ) ,  A ,  0 )  =  A )
16 elplyd.3 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  A  e.  S )
1715, 16eqeltrd 2370 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  if ( k  e.  ( 0 ... N ) ,  A ,  0 )  e.  S )
18 eqid 2296 . . . . . . . . . 10  |-  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )  =  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )
1918fvmpt2 5624 . . . . . . . . 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 ) )
2013, 17, 19syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k
)  =  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )
2120, 15eqtrd 2328 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k
)  =  A )
2221oveq1d 5889 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( ( k  e. 
NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `
 k )  x.  ( z ^ k
) )  =  ( A  x.  ( z ^ k ) ) )
2322sumeq2dv 12192 . . . . 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 ) ) )
2411, 23syl5eq 2340 . . . 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 ) ) )
2524mpteq2dv 4123 . . 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 ) ) ) )
26 elplyd.1 . . . . 5  |-  ( ph  ->  S  C_  CC )
27 0cn 8847 . . . . . . 7  |-  0  e.  CC
2827a1i 10 . . . . . 6  |-  ( ph  ->  0  e.  CC )
2928snssd 3776 . . . . 5  |-  ( ph  ->  { 0 }  C_  CC )
3026, 29unssd 3364 . . . 4  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
31 elplyd.2 . . . 4  |-  ( ph  ->  N  e.  NN0 )
32 elun1 3355 . . . . . . . 8  |-  ( A  e.  S  ->  A  e.  ( S  u.  {
0 } ) )
3316, 32syl 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  A  e.  ( S  u.  {
0 } ) )
3433adantlr 695 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  k  e.  ( 0 ... N
) )  ->  A  e.  ( S  u.  {
0 } ) )
35 ssun2 3352 . . . . . . . 8  |-  { 0 }  C_  ( S  u.  { 0 } )
36 c0ex 8848 . . . . . . . . 9  |-  0  e.  _V
3736snss 3761 . . . . . . . 8  |-  ( 0  e.  ( S  u.  { 0 } )  <->  { 0 }  C_  ( S  u.  { 0 } ) )
3835, 37mpbir 200 . . . . . . 7  |-  0  e.  ( S  u.  {
0 } )
3938a1i 10 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  ( 0 ... N ) )  ->  0  e.  ( S  u.  { 0 } ) )
4034, 39ifclda 3605 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  if (
k  e.  ( 0 ... N ) ,  A ,  0 )  e.  ( S  u.  { 0 } ) )
4140, 18fmptd 5700 . . . 4  |-  ( ph  ->  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) : NN0 --> ( S  u.  { 0 } ) )
42 elplyr 19599 . . . 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 } ) ) )
4330, 31, 41, 42syl3anc 1182 . . 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 } ) ) )
4425, 43eqeltrrd 2371 . 2  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( A  x.  ( z ^ k ) ) )  e.  (Poly `  ( S  u.  { 0 } ) ) )
45 plyun0 19595 . 2  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
4644, 45syl6eleq 2386 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 358    = wceq 1632    e. wcel 1696    u. cun 3163    C_ wss 3165   ifcif 3578   {csn 3653    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753    x. cmul 8758   NN0cn0 9981   ...cfz 10798   ^cexp 11120   sum_csu 12174  Polycply 19582
This theorem is referenced by:  ply1term  19602  plyaddlem  19613  plymullem  19614  plycj  19674  dvply2g  19681  elqaalem3  19717  aareccl  19722  taylply2  19763  basellem2  20335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-sum 12175  df-ply 19586
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