Theorem plyun0 15650

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 8271	. . . . . . 7 |- 0 e. CC
2		snssi 3840	. . . . . . 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 3395	. . . . 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 3378	. . . . . . . 8 |- ( ( S u. { 0 } ) u. { 0 } ) = ( S u. ( { 0 } u. { 0 } ) )
8		unidm 3364	. . . . . . . . 9 |- ( { 0 } u. { 0 } ) = { 0 }
9	8	uneq2i 3372	. . . . . . . 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 6062	. . . . . 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 15648	. . 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 15648	. . 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 3211   C_ wss 3213   {csn 3691   |-> cmpt 4173   ` cfv 5354  (class class class)co 6052   ^m cmap 6884   CCcc 8130   0cc0 8132   x. cmul 8137   NN0cn0 9501   ...cfz 10348   ^cexp 10907   sum_csu 12046  Polycply 15642

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 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-i2m1 8237

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 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-map 6886  df-inn 9243  df-n0 9502  df-ply 15644

This theorem is referenced by:  elplyd  15655  ply1term  15657  plyaddlem  15663  plymullem  15664  plycolemc  15672  plycj  15675