Theorem plyun0 15730

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 8282	. . . . . . 7 |- 0 e. CC
2		snssi 3843	. . . . . . 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 3397	. . . . 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 3380	. . . . . . . 8 |- ( ( S u. { 0 } ) u. { 0 } ) = ( S u. ( { 0 } u. { 0 } ) )
8		unidm 3366	. . . . . . . . 9 |- ( { 0 } u. { 0 } ) = { 0 }
9	8	uneq2i 3374	. . . . . . . 8 |- ( S u. ( { 0 } u. { 0 } ) ) = ( S u. { 0 } )
10	7, 9	eqtri 2255	. . . . . . 7 |- ( ( S u. { 0 } ) u. { 0 } ) = ( S u. { 0 } )
11	10	oveq1i 6068	. . . . . 6 |- ( ( ( S u. { 0 } ) u. { 0 } ) ^m NN0 ) = ( ( S u. { 0 } ) ^m NN0 )
12	11	rexeqi 2748	. . . . 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 2551	. . . 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 15728	. . 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 15728	. . 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 2231	1 |- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2205   E.wrex 2523   u. cun 3212   C_ wss 3214   {csn 3694   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   ^m cmap 6895   CCcc 8141   0cc0 8143   x. cmul 8148   NN0cn0 9516   ...cfz 10364   ^cexp 10927   sum_csu 12066  Polycply 15722

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 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-i2m1 8248

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 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-inn 9258  df-n0 9517  df-ply 15724

This theorem is referenced by:  elplyd  15735  ply1term  15737  plyaddlem  15743  plymullem  15744  plycolemc  15752  plycj  15755