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Theorem elply 23147
Description: Definition of a polynomial with coefficients in  S. (Contributed by Mario Carneiro, 17-Jul-2014.)
Assertion
Ref Expression
elply  |-  ( F  e.  (Poly `  S
)  <->  ( S  C_  CC  /\  E. n  e. 
NN0  E. a  e.  ( ( S  u.  {
0 } )  ^m  NN0 ) F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )
Distinct variable groups:    k, a, n, z, S    F, a, n
Allowed substitution hints:    F( z, k)

Proof of Theorem elply
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 plybss 23146 . 2  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
2 plyval 23145 . . . 4  |-  ( S 
C_  CC  ->  (Poly `  S )  =  {
f  |  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) } )
32eleq2d 2492 . . 3  |-  ( S 
C_  CC  ->  ( F  e.  (Poly `  S
)  <->  F  e.  { f  |  E. n  e. 
NN0  E. a  e.  ( ( S  u.  {
0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) } ) )
4 id 22 . . . . . . 7  |-  ( F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )
5 cnex 9627 . . . . . . . 8  |-  CC  e.  _V
65mptex 6151 . . . . . . 7  |-  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) ) )  e. 
_V
74, 6syl6eqel 2515 . . . . . 6  |-  ( F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )  ->  F  e.  _V )
87a1i 11 . . . . 5  |-  ( ( n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )  ->  ( F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )  ->  F  e.  _V ) )
98rexlimivv 2919 . . . 4  |-  ( E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )  ->  F  e.  _V )
10 eqeq1 2426 . . . . 5  |-  ( f  =  F  ->  (
f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) ) )  <->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )
11102rexbidv 2943 . . . 4  |-  ( f  =  F  ->  ( E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) f  =  ( 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 ) F  =  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )
129, 11elab3 3224 . . 3  |-  ( F  e.  { f  |  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) f  =  ( 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 ) F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )
133, 12syl6bb 264 . 2  |-  ( S 
C_  CC  ->  ( F  e.  (Poly `  S
)  <->  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) F  =  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )
141, 13biadan2 646 1  |-  ( F  e.  (Poly `  S
)  <->  ( S  C_  CC  /\  E. n  e. 
NN0  E. a  e.  ( ( S  u.  {
0 } )  ^m  NN0 ) F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2407   E.wrex 2772   _Vcvv 3080    u. cun 3434    C_ wss 3436   {csn 3998    |-> cmpt 4482   ` cfv 5601  (class class class)co 6305    ^m cmap 7483   CCcc 9544   0cc0 9546    x. cmul 9551   NN0cn0 10876   ...cfz 11791   ^cexp 12278   sum_csu 13751  Polycply 23136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-i2m1 9614  ax-1ne0 9615  ax-rrecex 9618  ax-cnre 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-nn 10617  df-n0 10877  df-ply 23140
This theorem is referenced by:  elply2  23148  plyun0  23149  plyf  23150  elplyr  23153  plypf1  23164
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