MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elplyr Structured version   Unicode version

Theorem elplyr 22764
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 994 . 2  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  S  C_  CC )
2 simp2 995 . . 3  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  N  e.  NN0 )
3 simp3 996 . . . . 5  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  A : NN0 --> S )
4 ssun1 3653 . . . . 5  |-  S  C_  ( S  u.  { 0 } )
5 fss 5721 . . . . 5  |-  ( ( A : NN0 --> S  /\  S  C_  ( S  u.  { 0 } ) )  ->  A : NN0 --> ( S  u.  { 0 } ) )
63, 4, 5sylancl 660 . . . 4  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  A : NN0 --> ( S  u.  { 0 } ) )
7 0cnd 9578 . . . . . . . 8  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  0  e.  CC )
87snssd 4161 . . . . . . 7  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  { 0 }  C_  CC )
91, 8unssd 3666 . . . . . 6  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  ( S  u.  { 0 } )  C_  CC )
10 cnex 9562 . . . . . 6  |-  CC  e.  _V
11 ssexg 4583 . . . . . 6  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  CC  e.  _V )  ->  ( S  u.  { 0 } )  e. 
_V )
129, 10, 11sylancl 660 . . . . 5  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  ( S  u.  { 0 } )  e.  _V )
13 nn0ex 10797 . . . . 5  |-  NN0  e.  _V
14 elmapg 7425 . . . . 5  |-  ( ( ( S  u.  {
0 } )  e. 
_V  /\  NN0  e.  _V )  ->  ( A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  <->  A : NN0 --> ( S  u.  { 0 } ) ) )
1512, 13, 14sylancl 660 . . . 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 6278 . . . . . . 7  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
1918sumeq1d 13605 . . . . . 6  |-  ( n  =  N  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( a `  k
)  x.  ( z ^ k ) ) )
2019mpteq2dv 4526 . . . . 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 5847 . . . . . . . 8  |-  ( a  =  A  ->  (
a `  k )  =  ( A `  k ) )
2322oveq1d 6285 . . . . . . 7  |-  ( a  =  A  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2423sumeq2sdv 13608 . . . . . 6  |-  ( a  =  A  ->  sum_ k  e.  ( 0 ... N
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )
2524mpteq2dv 4526 . . . . 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 3218 . . 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 1226 . 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 22758 . 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 662 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 971    = wceq 1398    e. wcel 1823   E.wrex 2805   _Vcvv 3106    u. cun 3459    C_ wss 3461   {csn 4016    |-> cmpt 4497   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   CCcc 9479   0cc0 9481    x. cmul 9486   NN0cn0 10791   ...cfz 11675   ^cexp 12148   sum_csu 13590  Polycply 22747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-seq 12090  df-sum 13591  df-ply 22751
This theorem is referenced by:  elplyd  22765  plypf1  22775  elaa2lem  32255
  Copyright terms: Public domain W3C validator