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Theorem elplyr 19599
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 955 . 2  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  S  C_  CC )
2 simp2 956 . . 3  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  N  e.  NN0 )
3 simp3 957 . . . . 5  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  A : NN0 --> S )
4 ssun1 3351 . . . . 5  |-  S  C_  ( S  u.  { 0 } )
5 fss 5413 . . . . 5  |-  ( ( A : NN0 --> S  /\  S  C_  ( S  u.  { 0 } ) )  ->  A : NN0 --> ( S  u.  { 0 } ) )
63, 4, 5sylancl 643 . . . 4  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  A : NN0 --> ( S  u.  { 0 } ) )
7 0cn 8847 . . . . . . . . 9  |-  0  e.  CC
87a1i 10 . . . . . . . 8  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  0  e.  CC )
98snssd 3776 . . . . . . 7  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  { 0 }  C_  CC )
101, 9unssd 3364 . . . . . 6  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  ( S  u.  { 0 } )  C_  CC )
11 cnex 8834 . . . . . 6  |-  CC  e.  _V
12 ssexg 4176 . . . . . 6  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  CC  e.  _V )  ->  ( S  u.  { 0 } )  e. 
_V )
1310, 11, 12sylancl 643 . . . . 5  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  ( S  u.  { 0 } )  e.  _V )
14 nn0ex 9987 . . . . 5  |-  NN0  e.  _V
15 elmapg 6801 . . . . 5  |-  ( ( ( S  u.  {
0 } )  e. 
_V  /\  NN0  e.  _V )  ->  ( A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  <->  A : NN0 --> ( S  u.  { 0 } ) ) )
1613, 14, 15sylancl 643 . . . 4  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  ( A  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) 
<->  A : NN0 --> ( S  u.  { 0 } ) ) )
176, 16mpbird 223 . . 3  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )
18 eqidd 2297 . . 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
) ) ) )
19 oveq2 5882 . . . . . . 7  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
2019sumeq1d 12190 . . . . . 6  |-  ( n  =  N  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( a `  k
)  x.  ( z ^ k ) ) )
2120mpteq2dv 4123 . . . . 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
) ) ) )
2221eqeq2d 2307 . . . 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 ) ) ) ) )
23 fveq1 5540 . . . . . . . 8  |-  ( a  =  A  ->  (
a `  k )  =  ( A `  k ) )
2423oveq1d 5889 . . . . . . 7  |-  ( a  =  A  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2524sumeq2sdv 12193 . . . . . 6  |-  ( a  =  A  ->  sum_ k  e.  ( 0 ... N
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )
2625mpteq2dv 4123 . . . . 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
) ) ) )
2726eqeq2d 2307 . . . 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 ) ) ) ) )
2822, 27rspc2ev 2905 . . 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 ) ) ) )
292, 17, 18, 28syl3anc 1182 . 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 ) ) ) )
30 elply 19593 . 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 ) ) ) ) )
311, 29, 30sylanbrc 645 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    u. cun 3163    C_ wss 3165   {csn 3653    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751   0cc0 8753    x. cmul 8758   NN0cn0 9981   ...cfz 10798   ^cexp 11120   sum_csu 12174  Polycply 19582
This theorem is referenced by:  elplyd  19600  plypf1  19610
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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