Theorem elply 15373

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:   S, a, n   k, a, z, n   F, a, n

Allowed substitution hints:   S( z, k)   F( z, k)

Proof of Theorem elply

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		plybss 15372	. 2 |- ( F e. (Poly ` S ) -> S C_ CC )
2		plyval 15371	. . . 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 ) ) ) } )
3	2	eleq2d 2279	. . 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 19	. . . . . . 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 8091	. . . . . . . 8 |- CC e. _V
6	5	mptex 5838	. . . . . . 7 |- ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) e. _V
7	4, 6	eqeltrdi 2300	. . . . . 6 |- ( F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) -> F e. _V )
8	7	a1i 9	. . . . 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 ) )
9	8	rexlimivv 2634	. . . 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 2216	. . . . 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 ) ) ) ) )
11	10	2rexbidv 2535	. . . 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 ) ) ) ) )
12	9, 11	elab3 2935	. . 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 ) ) ) )
13	3, 12	bitrdi 196	. 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 ) ) ) ) )
14	1, 13	biadanii 615	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 ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1375   e. wcel 2180   {cab 2195   E.wrex 2489   _Vcvv 2779   u. cun 3175   C_ wss 3177   {csn 3646   |-> cmpt 4124   ` cfv 5294  (class class class)co 5974   ^m cmap 6765   CCcc 7965   0cc0 7967   x. cmul 7972   NN0cn0 9337   ...cfz 10172   ^cexp 10727   sum_csu 11830  Polycply 15367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-i2m1 8072

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-map 6767  df-inn 9079  df-n0 9338  df-ply 15369

This theorem is referenced by:  elply2  15374  plyun0  15375  plyf  15376  elplyr  15379  plycj  15400  plycn  15401  plyrecj  15402