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Theorem elplyd 23235
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 5887 . . . . . . 7  |-  F/_ k
( ( k  e. 
NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `
 j )
2 nfcv 2612 . . . . . . 7  |-  F/_ k  x.
3 nfcv 2612 . . . . . . 7  |-  F/_ k
( z ^ j
)
41, 2, 3nfov 6334 . . . . . 6  |-  F/_ k
( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  j )  x.  ( z ^
j ) )
5 nfcv 2612 . . . . . 6  |-  F/_ j
( ( ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k )  x.  ( z ^
k ) )
6 fveq2 5879 . . . . . . 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 6316 . . . . . . 7  |-  ( j  =  k  ->  (
z ^ j )  =  ( z ^
k ) )
86, 7oveq12d 6326 . . . . . 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 13840 . . . . 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 11913 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
1110adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
12 iftrue 3878 . . . . . . . . . . 11  |-  ( k  e.  ( 0 ... N )  ->  if ( k  e.  ( 0 ... N ) ,  A ,  0 )  =  A )
1312adantl 473 . . . . . . . . . 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 2549 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  if ( k  e.  ( 0 ... N ) ,  A ,  0 )  e.  S )
16 eqid 2471 . . . . . . . . . 10  |-  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )  =  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) )
1716fvmpt2 5972 . . . . . . . . 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 673 . . . . . . . 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 2505 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) `  k
)  =  A )
2019oveq1d 6323 . . . . . 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 13846 . . . . 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 2517 . . . 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 4483 . . 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 9654 . . . . . 6  |-  ( ph  ->  0  e.  CC )
2625snssd 4108 . . . . 5  |-  ( ph  ->  { 0 }  C_  CC )
2724, 26unssd 3601 . . . 4  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
28 elplyd.2 . . . 4  |-  ( ph  ->  N  e.  NN0 )
29 elun1 3592 . . . . . . . 8  |-  ( A  e.  S  ->  A  e.  ( S  u.  {
0 } ) )
3014, 29syl 17 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  A  e.  ( S  u.  {
0 } ) )
3130adantlr 729 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  k  e.  ( 0 ... N
) )  ->  A  e.  ( S  u.  {
0 } ) )
32 ssun2 3589 . . . . . . . 8  |-  { 0 }  C_  ( S  u.  { 0 } )
33 c0ex 9655 . . . . . . . . 9  |-  0  e.  _V
3433snss 4087 . . . . . . . 8  |-  ( 0  e.  ( S  u.  { 0 } )  <->  { 0 }  C_  ( S  u.  { 0 } ) )
3532, 34mpbir 214 . . . . . . 7  |-  0  e.  ( S  u.  {
0 } )
3635a1i 11 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  e.  ( 0 ... N ) )  ->  0  e.  ( S  u.  { 0 } ) )
3731, 36ifclda 3904 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  if (
k  e.  ( 0 ... N ) ,  A ,  0 )  e.  ( S  u.  { 0 } ) )
3837, 16fmptd 6061 . . . 4  |-  ( ph  ->  ( k  e.  NN0  |->  if ( k  e.  ( 0 ... N ) ,  A ,  0 ) ) : NN0 --> ( S  u.  { 0 } ) )
39 elplyr 23234 . . . 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 1292 . . 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 2550 . 2  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( A  x.  ( z ^ k ) ) )  e.  (Poly `  ( S  u.  { 0 } ) ) )
42 plyun0 23230 . 2  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
4341, 42syl6eleq 2559 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 376    = wceq 1452    e. wcel 1904    u. cun 3388    C_ wss 3390   ifcif 3872   {csn 3959    |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557    x. cmul 9562   NN0cn0 10893   ...cfz 11810   ^cexp 12310   sum_csu 13829  Polycply 23217
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-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
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-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-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-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-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-seq 12252  df-sum 13830  df-ply 23221
This theorem is referenced by:  ply1term  23237  plyaddlem  23248  plymullem  23249  plycj  23310  dvply2g  23317  elqaalem3  23353  elqaalem3OLD  23356  aareccl  23361  taylply2  23402  basellem2  24087  aacllem  41046
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