Theorem plyun0 15489

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 8176	. . . . . . 7 |- 0 e. CC
2		snssi 3818	. . . . . . 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 3380	. . . . 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 3363	. . . . . . . 8 |- ( ( S u. { 0 } ) u. { 0 } ) = ( S u. ( { 0 } u. { 0 } ) )
8		unidm 3349	. . . . . . . . 9 |- ( { 0 } u. { 0 } ) = { 0 }
9	8	uneq2i 3357	. . . . . . . 8 |- ( S u. ( { 0 } u. { 0 } ) ) = ( S u. { 0 } )
10	7, 9	eqtri 2251	. . . . . . 7 |- ( ( S u. { 0 } ) u. { 0 } ) = ( S u. { 0 } )
11	10	oveq1i 6033	. . . . . 6 |- ( ( ( S u. { 0 } ) u. { 0 } ) ^m NN0 ) = ( ( S u. { 0 } ) ^m NN0 )
12	11	rexeqi 2734	. . . . 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 2538	. . . 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 15487	. . 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 15487	. . 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 2227	1 |- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2201   E.wrex 2510   u. cun 3197   C_ wss 3199   {csn 3670   |-> cmpt 4151   ` cfv 5328  (class class class)co 6023   ^m cmap 6822   CCcc 8035   0cc0 8037   x. cmul 8042   NN0cn0 9407   ...cfz 10248   ^cexp 10806   sum_csu 11936  Polycply 15481

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-i2m1 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-map 6824  df-inn 9149  df-n0 9408  df-ply 15483

This theorem is referenced by:  elplyd  15494  ply1term  15496  plyaddlem  15502  plymullem  15503  plycolemc  15511  plycj  15514