Theorem elplyr 15978

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 1028	. 2 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> S C_ CC )
2		simp2 1029	. . 3 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> N e. NN0 )
3		simp3 1030	. . . . 5 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> A : NN0 --> S )
4		ssun1 3392	. . . . 5 |- S C_ ( S u. { 0 } )
5		fss 5549	. . . . 5 |- ( ( A : NN0 --> S /\ S C_ ( S u. { 0 } ) ) -> A : NN0 --> ( S u. { 0 } ) )
6	3, 4, 5	sylancl 417	. . . 4 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> A : NN0 --> ( S u. { 0 } ) )
7		0cnd 8326	. . . . . . . 8 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> 0 e. CC )
8	7	snssd 3860	. . . . . . 7 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> { 0 } C_ CC )
9	1, 8	unssd 3405	. . . . . 6 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> ( S u. { 0 } ) C_ CC )
10		cnex 8310	. . . . . 6 |- CC e. _V
11		ssexg 4274	. . . . . 6 |- ( ( ( S u. { 0 } ) C_ CC /\ CC e. _V ) -> ( S u. { 0 } ) e. _V )
12	9, 10, 11	sylancl 417	. . . . 5 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> ( S u. { 0 } ) e. _V )
13		nn0ex 9581	. . . . 5 |- NN0 e. _V
14		elmapg 6939	. . . . 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 417	. . . 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 2239	. . 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 6096	. . . . . . 7 |- ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
19	18	sumeq1d 12158	. . . . . 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 4224	. . . . 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 2250	. . . 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 5697	. . . . . . . 8 |- ( a = A -> ( a ` k ) = ( A ` k ) )
23	22	oveq1d 6103	. . . . . . 7 |- ( a = A -> ( ( a ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( z ^ k ) ) )
24	23	sumeq2sdv 12162	. . . . . 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 4224	. . . . 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 2250	. . . 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 2945	. . 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 1278	. 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 15972	. 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 421	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 ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   _Vcvv 2821   u. cun 3218   C_ wss 3220   {csn 3709   |-> cmpt 4193   -->wf 5376   ` cfv 5380  (class class class)co 6088   ^m cmap 6926   CCcc 8184   0cc0 8186   x. cmul 8191   NN0cn0 9575   ...cfz 10429   ^cexp 10997   sum_csu 12145  Polycply 15966

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  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 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-addass 8288  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-0id 8294  ax-rnegex 8295  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-ltadd 8302

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-map 6928  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-inn 9315  df-n0 9576  df-z 9657  df-uz 9939  df-fz 10430  df-seqfrec 10907  df-sumdc 12146  df-ply 15968

This theorem is used by:  elplyd  15979