Theorem elplyr 15592

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 1024	. 2 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> S C_ CC )
2		simp2 1025	. . 3 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> N e. NN0 )
3		simp3 1026	. . . . 5 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> A : NN0 --> S )
4		ssun1 3381	. . . . 5 |- S C_ ( S u. { 0 } )
5		fss 5520	. . . . 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 8263	. . . . . . . 8 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> 0 e. CC )
8	7	snssd 3838	. . . . . . 7 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> { 0 } C_ CC )
9	1, 8	unssd 3394	. . . . . 6 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> ( S u. { 0 } ) C_ CC )
10		cnex 8247	. . . . . 6 |- CC e. _V
11		ssexg 4248	. . . . . 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 9498	. . . . 5 |- NN0 e. _V
14		elmapg 6894	. . . . 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 2233	. . 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 6057	. . . . . . 7 |- ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
19	18	sumeq1d 12044	. . . . . 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 4200	. . . . 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 2244	. . . 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 5668	. . . . . . . 8 |- ( a = A -> ( a ` k ) = ( A ` k ) )
23	22	oveq1d 6064	. . . . . . 7 |- ( a = A -> ( ( a ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( z ^ k ) ) )
24	23	sumeq2sdv 12048	. . . . . 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 4200	. . . . 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 2244	. . . 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 2935	. . 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 1274	. 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 15586	. 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 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   _Vcvv 2812   u. cun 3208   C_ wss 3210   {csn 3688   |-> cmpt 4170   -->wf 5347   ` cfv 5351  (class class class)co 6049   ^m cmap 6881   CCcc 8121   0cc0 8123   x. cmul 8128   NN0cn0 9492   ...cfz 10338   ^cexp 10896   sum_csu 12031  Polycply 15580

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-map 6883  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-inn 9234  df-n0 9493  df-z 9574  df-uz 9850  df-fz 10339  df-seqfrec 10806  df-sumdc 12032  df-ply 15582

This theorem is referenced by:  elplyd  15593