Theorem elply 15730

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 15729	. 2 |- ( F e. (Poly ` S ) -> S C_ CC )
2		plyval 15728	. . . 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 2304	. . 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 8268	. . . . . . . 8 |- CC e. _V
6	5	mptex 5918	. . . . . . 7 |- ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) e. _V
7	4, 6	eqeltrdi 2325	. . . . . 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 2668	. . . 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 2241	. . . . 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 2569	. . . 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 2972	. . 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 617	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 1398   e. wcel 2205   {cab 2220   E.wrex 2523   _Vcvv 2815   u. cun 3212   C_ wss 3214   {csn 3695   |-> cmpt 4177   ` cfv 5358  (class class class)co 6059   ^m cmap 6896   CCcc 8142   0cc0 8144   x. cmul 8149   NN0cn0 9517   ...cfz 10365   ^cexp 10928   sum_csu 12068  Polycply 15724

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-i2m1 8249

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-map 6898  df-inn 9259  df-n0 9518  df-ply 15726

This theorem is referenced by:  elply2  15731  plyun0  15732  plyf  15733  elplyr  15736  plycj  15757  plycn  15758  plyrecj  15759