Theorem elplyd 15436

Description: Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)

Hypotheses

Ref	Expression
elplyd.1	|- ( ph -> S C_ CC )
elplyd.2	|- ( ph -> N e. NN0 )
elplyd.3	|- ( ( ph /\ k e. ( 0 ... N ) ) -> A e. S )

Assertion

Ref	Expression
elplyd	|- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( A x. ( z ^ k ) ) ) e. (Poly ` S ) )

Distinct variable groups:   z, A   z, k, N   ph, k, z   S, k, z

Allowed substitution hint:   A( k)

Proof of Theorem elplyd

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nffvmpt1 5643	. . . . . . 7 |- F/_ k ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j )
2		nfcv 2372	. . . . . . 7 |- F/_ k x.
3		nfcv 2372	. . . . . . 7 |- F/_ k ( z ^ j )
4	1, 2, 3	nfov 6040	. . . . . 6 |- F/_ k ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j ) x. ( z ^ j ) )
5		nfcv 2372	. . . . . 6 |- F/_ j ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) x. ( z ^ k ) )
6		fveq2 5632	. . . . . . 7 |- ( j = k -> ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j ) = ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) )
7		oveq2 6018	. . . . . . 7 |- ( j = k -> ( z ^ j ) = ( z ^ k ) )
8	6, 7	oveq12d 6028	. . . . . 6 |- ( j = k -> ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j ) x. ( z ^ j ) ) = ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) x. ( z ^ k ) ) )
9	4, 5, 8	cbvsumi 11894	. . . . 5 |- sum_ j e. ( 0 ... N ) ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j ) x. ( z ^ j ) ) = sum_ k e. ( 0 ... N ) ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) x. ( z ^ k ) )
10		elfznn0 10327	. . . . . . . . 9 |- ( k e. ( 0 ... N ) -> k e. NN0 )
11		iftrue 3607	. . . . . . . . . . 11 |- ( k e. ( 0 ... N ) -> if ( k e. ( 0 ... N ) , A , 0 ) = A )
12	11	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ k e. ( 0 ... N ) ) -> if ( k e. ( 0 ... N ) , A , 0 ) = A )
13		elplyd.3	. . . . . . . . . 10 |- ( ( ph /\ k e. ( 0 ... N ) ) -> A e. S )
14	12, 13	eqeltrd 2306	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... N ) ) -> if ( k e. ( 0 ... N ) , A , 0 ) e. S )
15		eqid 2229	. . . . . . . . . 10 |- ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) = ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) )
16	15	fvmpt2 5723	. . . . . . . . 9 |- ( ( k e. NN0 /\ if ( k e. ( 0 ... N ) , A , 0 ) e. S ) -> ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) = if ( k e. ( 0 ... N ) , A , 0 ) )
17	10, 14, 16	syl2an2 596	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) = if ( k e. ( 0 ... N ) , A , 0 ) )
18	17, 12	eqtrd 2262	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) = A )
19	18	oveq1d 6025	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) x. ( z ^ k ) ) = ( A x. ( z ^ k ) ) )
20	19	sumeq2dv 11900	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... N ) ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... N ) ( A x. ( z ^ k ) ) )
21	9, 20	eqtrid 2274	. . . 4 |- ( ph -> sum_ j e. ( 0 ... N ) ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j ) x. ( z ^ j ) ) = sum_ k e. ( 0 ... N ) ( A x. ( z ^ k ) ) )
22	21	mpteq2dv 4175	. . 3 |- ( ph -> ( z e. CC |-> sum_ j e. ( 0 ... N ) ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j ) x. ( z ^ j ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( A x. ( z ^ k ) ) ) )
23		elplyd.1	. . . . 5 |- ( ph -> S C_ CC )
24		0cnd 8155	. . . . . 6 |- ( ph -> 0 e. CC )
25	24	snssd 3813	. . . . 5 |- ( ph -> { 0 } C_ CC )
26	23, 25	unssd 3380	. . . 4 |- ( ph -> ( S u. { 0 } ) C_ CC )
27		elplyd.2	. . . 4 |- ( ph -> N e. NN0 )
28		elun1 3371	. . . . . . . 8 |- ( A e. S -> A e. ( S u. { 0 } ) )
29	13, 28	syl 14	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... N ) ) -> A e. ( S u. { 0 } ) )
30	29	adantlr 477	. . . . . 6 |- ( ( ( ph /\ k e. NN0 ) /\ k e. ( 0 ... N ) ) -> A e. ( S u. { 0 } ) )
31		ssun2 3368	. . . . . . . 8 |- { 0 } C_ ( S u. { 0 } )
32		c0ex 8156	. . . . . . . . 9 |- 0 e. _V
33	32	snss 3803	. . . . . . . 8 |- ( 0 e. ( S u. { 0 } ) <-> { 0 } C_ ( S u. { 0 } ) )
34	31, 33	mpbir 146	. . . . . . 7 |- 0 e. ( S u. { 0 } )
35	34	a1i 9	. . . . . 6 |- ( ( ( ph /\ k e. NN0 ) /\ -. k e. ( 0 ... N ) ) -> 0 e. ( S u. { 0 } ) )
36		nn0z 9482	. . . . . . . 8 |- ( k e. NN0 -> k e. ZZ )
37	36	adantl 277	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> k e. ZZ )
38		0zd 9474	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> 0 e. ZZ )
39	27	nn0zd 9583	. . . . . . . 8 |- ( ph -> N e. ZZ )
40	39	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> N e. ZZ )
41		fzdcel 10253	. . . . . . 7 |- ( ( k e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> DECID k e. ( 0 ... N ) )
42	37, 38, 40, 41	syl3anc 1271	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> DECID k e. ( 0 ... N ) )
43	30, 35, 42	ifcldadc 3632	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> if ( k e. ( 0 ... N ) , A , 0 ) e. ( S u. { 0 } ) )
44	43	fmpttd 5795	. . . 4 |- ( ph -> ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) : NN0 --> ( S u. { 0 } ) )
45		elplyr 15435	. . . 4 |- ( ( ( S u. { 0 } ) C_ CC /\ N e. NN0 /\ ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) : NN0 --> ( S u. { 0 } ) ) -> ( z e. CC |-> sum_ j e. ( 0 ... N ) ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j ) x. ( z ^ j ) ) ) e. (Poly ` ( S u. { 0 } ) ) )
46	26, 27, 44, 45	syl3anc 1271	. . 3 |- ( ph -> ( z e. CC |-> sum_ j e. ( 0 ... N ) ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j ) x. ( z ^ j ) ) ) e. (Poly ` ( S u. { 0 } ) ) )
47	22, 46	eqeltrrd 2307	. 2 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( A x. ( z ^ k ) ) ) e. (Poly ` ( S u. { 0 } ) ) )
48		plyun0 15431	. 2 |- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )
49	47, 48	eleqtrdi 2322	1 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( A x. ( z ^ k ) ) ) e. (Poly ` S ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 839   = wceq 1395   e. wcel 2200   u. cun 3195   C_ wss 3197   ifcif 3602   {csn 3666   |-> cmpt 4145   -->wf 5317   ` cfv 5321  (class class class)co 6010   CCcc 8013   0cc0 8015   x. cmul 8020   NN0cn0 9385   ZZcz 9462   ...cfz 10221   ^cexp 10777   sum_csu 11885  Polycply 15423

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-map 6810  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-n0 9386  df-z 9463  df-uz 9739  df-fz 10222  df-seqfrec 10687  df-sumdc 11886  df-ply 15425

This theorem is referenced by:  ply1term  15438  plyaddlem  15444  plymullem  15445  plycj  15456  dvply2g  15461