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Theorem elplyr 21644
Description: Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
elplyr  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )  e.  (Poly `  S ) )
Distinct variable groups:    z, k, A    k, N, z    S, k, z

Proof of Theorem elplyr
Dummy variables  a  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 988 . 2  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  S  C_  CC )
2 simp2 989 . . 3  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  N  e.  NN0 )
3 simp3 990 . . . . 5  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  A : NN0 --> S )
4 ssun1 3514 . . . . 5  |-  S  C_  ( S  u.  { 0 } )
5 fss 5562 . . . . 5  |-  ( ( A : NN0 --> S  /\  S  C_  ( S  u.  { 0 } ) )  ->  A : NN0 --> ( S  u.  { 0 } ) )
63, 4, 5sylancl 662 . . . 4  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  A : NN0 --> ( S  u.  { 0 } ) )
7 0cnd 9371 . . . . . . . 8  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  0  e.  CC )
87snssd 4013 . . . . . . 7  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  { 0 }  C_  CC )
91, 8unssd 3527 . . . . . 6  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  ( S  u.  { 0 } )  C_  CC )
10 cnex 9355 . . . . . 6  |-  CC  e.  _V
11 ssexg 4433 . . . . . 6  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  CC  e.  _V )  ->  ( S  u.  { 0 } )  e. 
_V )
129, 10, 11sylancl 662 . . . . 5  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  ( S  u.  { 0 } )  e.  _V )
13 nn0ex 10577 . . . . 5  |-  NN0  e.  _V
14 elmapg 7219 . . . . 5  |-  ( ( ( S  u.  {
0 } )  e. 
_V  /\  NN0  e.  _V )  ->  ( A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  <->  A : NN0 --> ( S  u.  { 0 } ) ) )
1512, 13, 14sylancl 662 . . . 4  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  ( A  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) 
<->  A : NN0 --> ( S  u.  { 0 } ) ) )
166, 15mpbird 232 . . 3  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )
17 eqidd 2439 . . 3  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
18 oveq2 6094 . . . . . . 7  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
1918sumeq1d 13170 . . . . . 6  |-  ( n  =  N  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( a `  k
)  x.  ( z ^ k ) ) )
2019mpteq2dv 4374 . . . . 5  |-  ( n  =  N  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( a `
 k )  x.  ( z ^ k
) ) ) )
2120eqeq2d 2449 . . . 4  |-  ( n  =  N  ->  (
( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) ) )  <->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )
22 fveq1 5685 . . . . . . . 8  |-  ( a  =  A  ->  (
a `  k )  =  ( A `  k ) )
2322oveq1d 6101 . . . . . . 7  |-  ( a  =  A  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2423sumeq2sdv 13173 . . . . . 6  |-  ( a  =  A  ->  sum_ k  e.  ( 0 ... N
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )
2524mpteq2dv 4374 . . . . 5  |-  ( a  =  A  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( a `  k
)  x.  ( z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
2625eqeq2d 2449 . . . 4  |-  ( a  =  A  ->  (
( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( a `
 k )  x.  ( z ^ k
) ) )  <->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) ) ) )
2721, 26rspc2ev 3076 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  /\  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )  ->  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )
282, 16, 17, 27syl3anc 1218 . 2  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )
29 elply 21638 . 2  |-  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )  e.  (Poly `  S )  <->  ( S  C_  CC  /\  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )
301, 28, 29sylanbrc 664 1  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )  e.  (Poly `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2711   _Vcvv 2967    u. cun 3321    C_ wss 3323   {csn 3872    e. cmpt 4345   -->wf 5409   ` cfv 5413  (class class class)co 6086    ^m cmap 7206   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:  elplyd  21645  plypf1  21655
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