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Theorem elplyr 20120
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 957 . 2  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  S  C_  CC )
2 simp2 958 . . 3  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  N  e.  NN0 )
3 simp3 959 . . . . 5  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  A : NN0 --> S )
4 ssun1 3510 . . . . 5  |-  S  C_  ( S  u.  { 0 } )
5 fss 5599 . . . . 5  |-  ( ( A : NN0 --> S  /\  S  C_  ( S  u.  { 0 } ) )  ->  A : NN0 --> ( S  u.  { 0 } ) )
63, 4, 5sylancl 644 . . . 4  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  A : NN0 --> ( S  u.  { 0 } ) )
7 0cn 9084 . . . . . . . . 9  |-  0  e.  CC
87a1i 11 . . . . . . . 8  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  0  e.  CC )
98snssd 3943 . . . . . . 7  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  { 0 }  C_  CC )
101, 9unssd 3523 . . . . . 6  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  ( S  u.  { 0 } )  C_  CC )
11 cnex 9071 . . . . . 6  |-  CC  e.  _V
12 ssexg 4349 . . . . . 6  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  CC  e.  _V )  ->  ( S  u.  { 0 } )  e. 
_V )
1310, 11, 12sylancl 644 . . . . 5  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  ( S  u.  { 0 } )  e.  _V )
14 nn0ex 10227 . . . . 5  |-  NN0  e.  _V
15 elmapg 7031 . . . . 5  |-  ( ( ( S  u.  {
0 } )  e. 
_V  /\  NN0  e.  _V )  ->  ( A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  <->  A : NN0 --> ( S  u.  { 0 } ) ) )
1613, 14, 15sylancl 644 . . . 4  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  ( A  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) 
<->  A : NN0 --> ( S  u.  { 0 } ) ) )
176, 16mpbird 224 . . 3  |-  ( ( S  C_  CC  /\  N  e.  NN0  /\  A : NN0
--> S )  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )
18 eqidd 2437 . . 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 6089 . . . . . . 7  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
2019sumeq1d 12495 . . . . . 6  |-  ( n  =  N  ->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( a `  k
)  x.  ( z ^ k ) ) )
2120mpteq2dv 4296 . . . . 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 2447 . . . 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 5727 . . . . . . . 8  |-  ( a  =  A  ->  (
a `  k )  =  ( A `  k ) )
2423oveq1d 6096 . . . . . . 7  |-  ( a  =  A  ->  (
( a `  k
)  x.  ( z ^ k ) )  =  ( ( A `
 k )  x.  ( z ^ k
) ) )
2524sumeq2sdv 12498 . . . . . 6  |-  ( a  =  A  ->  sum_ k  e.  ( 0 ... N
) ( ( a `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) )
2625mpteq2dv 4296 . . . . 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 2447 . . . 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 3060 . . 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 1184 . 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 20114 . 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 646 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 177    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706   _Vcvv 2956    u. cun 3318    C_ wss 3320   {csn 3814    e. cmpt 4266   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   CCcc 8988   0cc0 8990    x. cmul 8995   NN0cn0 10221   ...cfz 11043   ^cexp 11382   sum_csu 12479  Polycply 20103
This theorem is referenced by:  elplyd  20121  plypf1  20131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-seq 11324  df-sum 12480  df-ply 20107
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