Theorem elplyr 15060

Description: Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Assertion

Ref	Expression
elplyr	|- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) e. (Poly ` S ) )

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

Proof of Theorem elplyr

Dummy variables a n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 999	. 2 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> S C_ CC )
2		simp2 1000	. . 3 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> N e. NN0 )
3		simp3 1001	. . . . 5 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> A : NN0 --> S )
4		ssun1 3327	. . . . 5 |- S C_ ( S u. { 0 } )
5		fss 5422	. . . . 5 |- ( ( A : NN0 --> S /\ S C_ ( S u. { 0 } ) ) -> A : NN0 --> ( S u. { 0 } ) )
6	3, 4, 5	sylancl 413	. . . 4 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> A : NN0 --> ( S u. { 0 } ) )
7		0cnd 8036	. . . . . . . 8 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> 0 e. CC )
8	7	snssd 3768	. . . . . . 7 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> { 0 } C_ CC )
9	1, 8	unssd 3340	. . . . . 6 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> ( S u. { 0 } ) C_ CC )
10		cnex 8020	. . . . . 6 |- CC e. _V
11		ssexg 4173	. . . . . 6 |- ( ( ( S u. { 0 } ) C_ CC /\ CC e. _V ) -> ( S u. { 0 } ) e. _V )
12	9, 10, 11	sylancl 413	. . . . 5 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> ( S u. { 0 } ) e. _V )
13		nn0ex 9272	. . . . 5 |- NN0 e. _V
14		elmapg 6729	. . . . 5 |- ( ( ( S u. { 0 } ) e. _V /\ NN0 e. _V ) -> ( A e. ( ( S u. { 0 } ) ^m NN0 ) <-> A : NN0 --> ( S u. { 0 } ) ) )
15	12, 13, 14	sylancl 413	. . . 4 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> ( A e. ( ( S u. { 0 } ) ^m NN0 ) <-> A : NN0 --> ( S u. { 0 } ) ) )
16	6, 15	mpbird 167	. . 3 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> A e. ( ( S u. { 0 } ) ^m NN0 ) )
17		eqidd 2197	. . 3 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )
18		oveq2 5933	. . . . . . 7 |- ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
19	18	sumeq1d 11548	. . . . . 6 |- ( n = N -> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... N ) ( ( a ` k ) x. ( z ^ k ) ) )
20	19	mpteq2dv 4125	. . . . 5 |- ( n = N -> ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( a ` k ) x. ( z ^ k ) ) ) )
21	20	eqeq2d 2208	. . . 4 |- ( n = N -> ( ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) <-> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
22		fveq1 5560	. . . . . . . 8 |- ( a = A -> ( a ` k ) = ( A ` k ) )
23	22	oveq1d 5940	. . . . . . 7 |- ( a = A -> ( ( a ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( z ^ k ) ) )
24	23	sumeq2sdv 11552	. . . . . 6 |- ( a = A -> sum_ k e. ( 0 ... N ) ( ( a ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) )
25	24	mpteq2dv 4125	. . . . 5 |- ( a = A -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( a ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )
26	25	eqeq2d 2208	. . . 4 |- ( a = A -> ( ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( a ` k ) x. ( z ^ k ) ) ) <-> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) ) )
27	21, 26	rspc2ev 2883	. . 3 |- ( ( N e. NN0 /\ A e. ( ( S u. { 0 } ) ^m NN0 ) /\ ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) = ( 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 ) ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) )
28	2, 16, 17, 27	syl3anc 1249	. 2 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> E. n e. NN0 E. a e. ( ( S u. { 0 } ) ^m NN0 ) ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) )
29		elply 15054	. 2 |- ( ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) e. (Poly ` S ) <-> ( S C_ CC /\ E. n e. NN0 E. a e. ( ( S u. { 0 } ) ^m NN0 ) ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
30	1, 28, 29	sylanbrc 417	1 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) e. (Poly ` S ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   E.wrex 2476   _Vcvv 2763   u. cun 3155   C_ wss 3157   {csn 3623   |-> cmpt 4095   -->wf 5255   ` cfv 5259  (class class class)co 5925   ^m cmap 6716   CCcc 7894   0cc0 7896   x. cmul 7901   NN0cn0 9266   ...cfz 10100   ^cexp 10647   sum_csu 11535  Polycply 15048

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101  df-seqfrec 10557  df-sumdc 11536  df-ply 15050

This theorem is referenced by:  elplyd  15061