Theorem elply 15416

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 15415	. 2 |- ( F e. (Poly ` S ) -> S C_ CC )
2		plyval 15414	. . . 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 2299	. . 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 8131	. . . . . . . 8 |- CC e. _V
6	5	mptex 5869	. . . . . . 7 |- ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) e. _V
7	4, 6	eqeltrdi 2320	. . . . . 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 2654	. . . 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 2236	. . . . 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 2555	. . . 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 2955	. . 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 1395   e. wcel 2200   {cab 2215   E.wrex 2509   _Vcvv 2799   u. cun 3195   C_ wss 3197   {csn 3666   |-> cmpt 4145   ` cfv 5318  (class class class)co 6007   ^m cmap 6803   CCcc 8005   0cc0 8007   x. cmul 8012   NN0cn0 9377   ...cfz 10212   ^cexp 10768   sum_csu 11872  Polycply 15410

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-i2m1 8112

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-map 6805  df-inn 9119  df-n0 9378  df-ply 15412

This theorem is referenced by:  elply2  15417  plyun0  15418  plyf  15419  elplyr  15422  plycj  15443  plycn  15444  plyrecj  15445