Theorem plyun0 14915

Description: The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.)

Assertion

Ref	Expression
plyun0	|- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )

Proof of Theorem plyun0

Dummy variables a f n k z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cn 8013	. . . . . . 7 |- 0 e. CC
2		snssi 3763	. . . . . . 7 |- ( 0 e. CC -> { 0 } C_ CC )
3	1, 2	ax-mp 5	. . . . . 6 |- { 0 } C_ CC
4	3	biantru 302	. . . . 5 |- ( S C_ CC <-> ( S C_ CC /\ { 0 } C_ CC ) )
5		unss 3334	. . . . 5 |- ( ( S C_ CC /\ { 0 } C_ CC ) <-> ( S u. { 0 } ) C_ CC )
6	4, 5	bitr2i 185	. . . 4 |- ( ( S u. { 0 } ) C_ CC <-> S C_ CC )
7		unass 3317	. . . . . . . 8 |- ( ( S u. { 0 } ) u. { 0 } ) = ( S u. ( { 0 } u. { 0 } ) )
8		unidm 3303	. . . . . . . . 9 |- ( { 0 } u. { 0 } ) = { 0 }
9	8	uneq2i 3311	. . . . . . . 8 |- ( S u. ( { 0 } u. { 0 } ) ) = ( S u. { 0 } )
10	7, 9	eqtri 2214	. . . . . . 7 |- ( ( S u. { 0 } ) u. { 0 } ) = ( S u. { 0 } )
11	10	oveq1i 5929	. . . . . 6 |- ( ( ( S u. { 0 } ) u. { 0 } ) ^m NN0 ) = ( ( S u. { 0 } ) ^m NN0 )
12	11	rexeqi 2695	. . . . 5 |- ( E. a e. ( ( ( S u. { 0 } ) u. { 0 } ) ^m NN0 ) f = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) <-> E. a e. ( ( S u. { 0 } ) ^m NN0 ) f = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) )
13	12	rexbii 2501	. . . 4 |- ( E. n e. NN0 E. a e. ( ( ( S u. { 0 } ) 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 ) ) ) )
14	6, 13	anbi12i 460	. . 3 |- ( ( ( S u. { 0 } ) C_ CC /\ E. n e. NN0 E. a e. ( ( ( S u. { 0 } ) u. { 0 } ) ^m NN0 ) f = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) <-> ( 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 ) ) ) ) )
15		elply 14913	. . 3 |- ( f e. (Poly ` ( S u. { 0 } ) ) <-> ( ( S u. { 0 } ) C_ CC /\ E. n e. NN0 E. a e. ( ( ( S u. { 0 } ) u. { 0 } ) ^m NN0 ) f = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
16		elply 14913	. . 3 |- ( 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 ) ) ) ) )
17	14, 15, 16	3bitr4i 212	. 2 |- ( f e. (Poly ` ( S u. { 0 } ) ) <-> f e. (Poly ` S ) )
18	17	eqriv 2190	1 |- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   E.wrex 2473   u. cun 3152   C_ wss 3154   {csn 3619   |-> cmpt 4091   ` cfv 5255  (class class class)co 5919   ^m cmap 6704   CCcc 7872   0cc0 7874   x. cmul 7879   NN0cn0 9243   ...cfz 10077   ^cexp 10612   sum_csu 11499  Polycply 14907

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-i2m1 7979

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-inn 8985  df-n0 9244  df-ply 14909

This theorem is referenced by:  elplyd  14920  ply1term  14922  plyaddlem  14928  plymullem  14929  plycolemc  14936  plycj  14939