Theorem elplyd 14887

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 5565	. . . . . . 7 |- F/_ k ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j )
2		nfcv 2336	. . . . . . 7 |- F/_ k x.
3		nfcv 2336	. . . . . . 7 |- F/_ k ( z ^ j )
4	1, 2, 3	nfov 5948	. . . . . 6 |- F/_ k ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j ) x. ( z ^ j ) )
5		nfcv 2336	. . . . . 6 |- F/_ j ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) x. ( z ^ k ) )
6		fveq2 5554	. . . . . . 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 5926	. . . . . . 7 |- ( j = k -> ( z ^ j ) = ( z ^ k ) )
8	6, 7	oveq12d 5936	. . . . . 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 11505	. . . . 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 10180	. . . . . . . . 9 |- ( k e. ( 0 ... N ) -> k e. NN0 )
11		iftrue 3562	. . . . . . . . . . 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 2270	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... N ) ) -> if ( k e. ( 0 ... N ) , A , 0 ) e. S )
15		eqid 2193	. . . . . . . . . 10 |- ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) = ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) )
16	15	fvmpt2 5641	. . . . . . . . 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 594	. . . . . . . 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 2226	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) = A )
19	18	oveq1d 5933	. . . . . 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 11511	. . . . 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 2238	. . . 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 4120	. . 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 8012	. . . . . 6 |- ( ph -> 0 e. CC )
25	24	snssd 3763	. . . . 5 |- ( ph -> { 0 } C_ CC )
26	23, 25	unssd 3335	. . . 4 |- ( ph -> ( S u. { 0 } ) C_ CC )
27		elplyd.2	. . . 4 |- ( ph -> N e. NN0 )
28		elun1 3326	. . . . . . . 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 3323	. . . . . . . 8 |- { 0 } C_ ( S u. { 0 } )
32		c0ex 8013	. . . . . . . . 9 |- 0 e. _V
33	32	snss 3753	. . . . . . . 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 9337	. . . . . . . 8 |- ( k e. NN0 -> k e. ZZ )
37	36	adantl 277	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> k e. ZZ )
38		0zd 9329	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> 0 e. ZZ )
39	27	nn0zd 9437	. . . . . . . 8 |- ( ph -> N e. ZZ )
40	39	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> N e. ZZ )
41		fzdcel 10106	. . . . . . 7 |- ( ( k e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> DECID k e. ( 0 ... N ) )
42	37, 38, 40, 41	syl3anc 1249	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> DECID k e. ( 0 ... N ) )
43	30, 35, 42	ifcldadc 3586	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> if ( k e. ( 0 ... N ) , A , 0 ) e. ( S u. { 0 } ) )
44	43	fmpttd 5713	. . . 4 |- ( ph -> ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) : NN0 --> ( S u. { 0 } ) )
45		elplyr 14886	. . . 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 1249	. . 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 2271	. 2 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( A x. ( z ^ k ) ) ) e. (Poly ` ( S u. { 0 } ) ) )
48		plyun0 14882	. 2 |- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )
49	47, 48	eleqtrdi 2286	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 835   = wceq 1364   e. wcel 2164   u. cun 3151   C_ wss 3153   ifcif 3557   {csn 3618   |-> cmpt 4090   -->wf 5250   ` cfv 5254  (class class class)co 5918   CCcc 7870   0cc0 7872   x. cmul 7877   NN0cn0 9240   ZZcz 9317   ...cfz 10074   ^cexp 10609   sum_csu 11496  Polycply 14874

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-map 6704  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-seqfrec 10519  df-sumdc 11497  df-ply 14876

This theorem is referenced by:  ply1term  14889  plyaddlem  14895  plymullem  14896