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Theorem elplyr 21805
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 3630 . . . . 5  |-  S  C_  ( S  u.  { 0 } )
5 fss 5678 . . . . 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 9493 . . . . . . . 8  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  0  e.  CC )
87snssd 4129 . . . . . . 7  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  { 0 }  C_  CC )
91, 8unssd 3643 . . . . . 6  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  ( S  u.  { 0 } )  C_  CC )
10 cnex 9477 . . . . . 6  |-  CC  e.  _V
11 ssexg 4549 . . . . . 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 10699 . . . . 5  |-  NN0  e.  _V
14 elmapg 7340 . . . . 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 2455 . . 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 6211 . . . . . . 7  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
1918sumeq1d 13299 . . . . . 6  |-  ( n  =  N  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( a `  k
)  x.  ( z ^ k ) ) )
2019mpteq2dv 4490 . . . . 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 2468 . . . 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 5801 . . . . . . . 8  |-  ( a  =  A  ->  (
a `  k )  =  ( A `  k ) )
2322oveq1d 6218 . . . . . . 7  |-  ( a  =  A  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2423sumeq2sdv 13302 . . . . . 6  |-  ( a  =  A  ->  sum_ k  e.  ( 0 ... N
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )
2524mpteq2dv 4490 . . . . 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 2468 . . . 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 3188 . . 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 1219 . 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 21799 . 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 1370    e. wcel 1758   E.wrex 2800   _Vcvv 3078    u. cun 3437    C_ wss 3439   {csn 3988    |-> cmpt 4461   -->wf 5525   ` cfv 5529  (class class class)co 6203    ^m cmap 7327   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:  elplyd  21806  plypf1  21816
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