Theorem elplyd 15468

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 5650	. . . . . . 7 |- F/_ k ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j )
2		nfcv 2374	. . . . . . 7 |- F/_ k x.
3		nfcv 2374	. . . . . . 7 |- F/_ k ( z ^ j )
4	1, 2, 3	nfov 6048	. . . . . 6 |- F/_ k ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` j ) x. ( z ^ j ) )
5		nfcv 2374	. . . . . 6 |- F/_ j ( ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) x. ( z ^ k ) )
6		fveq2 5639	. . . . . . 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 6026	. . . . . . 7 |- ( j = k -> ( z ^ j ) = ( z ^ k ) )
8	6, 7	oveq12d 6036	. . . . . 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 11924	. . . . 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 10349	. . . . . . . . 9 |- ( k e. ( 0 ... N ) -> k e. NN0 )
11		iftrue 3610	. . . . . . . . . . 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 2308	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... N ) ) -> if ( k e. ( 0 ... N ) , A , 0 ) e. S )
15		eqid 2231	. . . . . . . . . 10 |- ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) = ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) )
16	15	fvmpt2 5730	. . . . . . . . 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 598	. . . . . . . 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 2264	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) ` k ) = A )
19	18	oveq1d 6033	. . . . . 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 11930	. . . . 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 2276	. . . 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 4180	. . 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 8172	. . . . . 6 |- ( ph -> 0 e. CC )
25	24	snssd 3818	. . . . 5 |- ( ph -> { 0 } C_ CC )
26	23, 25	unssd 3383	. . . 4 |- ( ph -> ( S u. { 0 } ) C_ CC )
27		elplyd.2	. . . 4 |- ( ph -> N e. NN0 )
28		elun1 3374	. . . . . . . 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 3371	. . . . . . . 8 |- { 0 } C_ ( S u. { 0 } )
32		c0ex 8173	. . . . . . . . 9 |- 0 e. _V
33	32	snss 3808	. . . . . . . 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 9499	. . . . . . . 8 |- ( k e. NN0 -> k e. ZZ )
37	36	adantl 277	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> k e. ZZ )
38		0zd 9491	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> 0 e. ZZ )
39	27	nn0zd 9600	. . . . . . . 8 |- ( ph -> N e. ZZ )
40	39	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> N e. ZZ )
41		fzdcel 10275	. . . . . . 7 |- ( ( k e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> DECID k e. ( 0 ... N ) )
42	37, 38, 40, 41	syl3anc 1273	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> DECID k e. ( 0 ... N ) )
43	30, 35, 42	ifcldadc 3635	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> if ( k e. ( 0 ... N ) , A , 0 ) e. ( S u. { 0 } ) )
44	43	fmpttd 5802	. . . 4 |- ( ph -> ( k e. NN0 |-> if ( k e. ( 0 ... N ) , A , 0 ) ) : NN0 --> ( S u. { 0 } ) )
45		elplyr 15467	. . . 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 1273	. . 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 2309	. 2 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( A x. ( z ^ k ) ) ) e. (Poly ` ( S u. { 0 } ) ) )
48		plyun0 15463	. 2 |- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )
49	47, 48	eleqtrdi 2324	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 841   = wceq 1397   e. wcel 2202   u. cun 3198   C_ wss 3200   ifcif 3605   {csn 3669   |-> cmpt 4150   -->wf 5322   ` cfv 5326  (class class class)co 6018   CCcc 8030   0cc0 8032   x. cmul 8037   NN0cn0 9402   ZZcz 9479   ...cfz 10243   ^cexp 10801   sum_csu 11915  Polycply 15455

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-map 6819  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-seqfrec 10711  df-sumdc 11916  df-ply 15457

This theorem is referenced by:  ply1term  15470  plyaddlem  15476  plymullem  15477  plycj  15488  dvply2g  15493