Theorem elplyr 15467

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 1023	. 2 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> S C_ CC )
2		simp2 1024	. . 3 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> N e. NN0 )
3		simp3 1025	. . . . 5 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> A : NN0 --> S )
4		ssun1 3370	. . . . 5 |- S C_ ( S u. { 0 } )
5		fss 5494	. . . . 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 8172	. . . . . . . 8 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> 0 e. CC )
8	7	snssd 3818	. . . . . . 7 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> { 0 } C_ CC )
9	1, 8	unssd 3383	. . . . . 6 |- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> ( S u. { 0 } ) C_ CC )
10		cnex 8156	. . . . . 6 |- CC e. _V
11		ssexg 4228	. . . . . 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 9408	. . . . 5 |- NN0 e. _V
14		elmapg 6830	. . . . 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 2232	. . 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 6026	. . . . . . 7 |- ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
19	18	sumeq1d 11928	. . . . . 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 4180	. . . . 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 2243	. . . 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 5638	. . . . . . . 8 |- ( a = A -> ( a ` k ) = ( A ` k ) )
23	22	oveq1d 6033	. . . . . . 7 |- ( a = A -> ( ( a ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( z ^ k ) ) )
24	23	sumeq2sdv 11932	. . . . . 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 4180	. . . . 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 2243	. . . 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 2925	. . 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 1273	. 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 15461	. 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 1004   = wceq 1397   e. wcel 2202   E.wrex 2511   _Vcvv 2802   u. cun 3198   C_ wss 3200   {csn 3669   |-> cmpt 4150   -->wf 5322   ` cfv 5326  (class class class)co 6018   ^m cmap 6817   CCcc 8030   0cc0 8032   x. cmul 8037   NN0cn0 9402   ...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:  elplyd  15468