Theorem plyun0 15820

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 8312	. . . . . . 7 |- 0 e. CC
2		snssi 3857	. . . . . . 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 3403	. . . . 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 3386	. . . . . . . 8 |- ( ( S u. { 0 } ) u. { 0 } ) = ( S u. ( { 0 } u. { 0 } ) )
8		unidm 3372	. . . . . . . . 9 |- ( { 0 } u. { 0 } ) = { 0 }
9	8	uneq2i 3380	. . . . . . . 8 |- ( S u. ( { 0 } u. { 0 } ) ) = ( S u. { 0 } )
10	7, 9	eqtri 2259	. . . . . . 7 |- ( ( S u. { 0 } ) u. { 0 } ) = ( S u. { 0 } )
11	10	oveq1i 6089	. . . . . 6 |- ( ( ( S u. { 0 } ) u. { 0 } ) ^m NN0 ) = ( ( S u. { 0 } ) ^m NN0 )
12	11	rexeqi 2754	. . . . 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 2557	. . . 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 464	. . 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 15818	. . 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 15818	. . 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 2235	1 |- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   E.wrex 2529   u. cun 3218   C_ wss 3220   {csn 3708   |-> cmpt 4190   ` cfv 5375  (class class class)co 6079   ^m cmap 6916   CCcc 8171   0cc0 8173   x. cmul 8178   NN0cn0 9546   ...cfz 10394   ^cexp 10958   sum_csu 12102  Polycply 15812

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-i2m1 8278

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-map 6918  df-inn 9288  df-n0 9547  df-ply 15814

This theorem is referenced by:  elplyd  15825  ply1term  15827  plyaddlem  15833  plymullem  15834  plycolemc  15842  plycj  15845