Theorem plycj 24033

Description: The double conjugation of a polynomial is a polynomial. (The single conjugation is not because our definition of polynomial includes only holomorphic functions, i.e. no dependence on ( * ` z ) independently of z.) (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
plycj.1	|- N = (deg ` F )
plycj.2	|- G = ( ( * o. F ) o. * )
plycj.3	|- ( ( ph /\ x e. S ) -> ( * ` x ) e. S )
plycj.4	|- ( ph -> F e. (Poly ` S ) )

Assertion

Ref	Expression
plycj	|- ( ph -> G e. (Poly ` S ) )

Distinct variable groups:   x, F   x, N   ph, x   x, S

Allowed substitution hint:   G( x)

Proof of Theorem plycj

Dummy variables k z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plycj.4	. . . 4 |- ( ph -> F e. (Poly ` S ) )
2		plycj.1	. . . . 5 |- N = (deg ` F )
3		plycj.2	. . . . 5 |- G = ( ( * o. F ) o. * )
4		eqid 2622	. . . . 5 |- (coeff ` F ) = (coeff ` F )
5	2, 3, 4	plycjlem 24032	. . . 4 |- ( F e. (Poly ` S ) -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( * o. (coeff ` F ) ) ` k ) x. ( z ^ k ) ) ) )
6	1, 5	syl 17	. . 3 |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( * o. (coeff ` F ) ) ` k ) x. ( z ^ k ) ) ) )
7		plybss 23950	. . . . . 6 |- ( F e. (Poly ` S ) -> S C_ CC )
8	1, 7	syl 17	. . . . 5 |- ( ph -> S C_ CC )
9		0cnd 10033	. . . . . 6 |- ( ph -> 0 e. CC )
10	9	snssd 4340	. . . . 5 |- ( ph -> { 0 } C_ CC )
11	8, 10	unssd 3789	. . . 4 |- ( ph -> ( S u. { 0 } ) C_ CC )
12		dgrcl 23989	. . . . . 6 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
13	1, 12	syl 17	. . . . 5 |- ( ph -> (deg ` F ) e. NN0 )
14	2, 13	syl5eqel 2705	. . . 4 |- ( ph -> N e. NN0 )
15	4	coef 23986	. . . . . . 7 |- ( F e. (Poly ` S ) -> (coeff ` F ) : NN0 --> ( S u. { 0 } ) )
16	1, 15	syl 17	. . . . . 6 |- ( ph -> (coeff ` F ) : NN0 --> ( S u. { 0 } ) )
17		elfznn0 12433	. . . . . 6 |- ( k e. ( 0 ... N ) -> k e. NN0 )
18		fvco3 6275	. . . . . 6 |- ( ( (coeff ` F ) : NN0 --> ( S u. { 0 } ) /\ k e. NN0 ) -> ( ( * o. (coeff ` F ) ) ` k ) = ( * ` ( (coeff ` F ) ` k ) ) )
19	16, 17, 18	syl2an 494	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( * o. (coeff ` F ) ) ` k ) = ( * ` ( (coeff ` F ) ` k ) ) )
20		ffvelrn 6357	. . . . . . 7 |- ( ( (coeff ` F ) : NN0 --> ( S u. { 0 } ) /\ k e. NN0 ) -> ( (coeff ` F ) ` k ) e. ( S u. { 0 } ) )
21	16, 17, 20	syl2an 494	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( (coeff ` F ) ` k ) e. ( S u. { 0 } ) )
22		plycj.3	. . . . . . . . . . 11 |- ( ( ph /\ x e. S ) -> ( * ` x ) e. S )
23	22	ralrimiva 2966	. . . . . . . . . 10 |- ( ph -> A. x e. S ( * ` x ) e. S )
24		fveq2 6191	. . . . . . . . . . . 12 |- ( x = ( (coeff ` F ) ` k ) -> ( * ` x ) = ( * ` ( (coeff ` F ) ` k ) ) )
25	24	eleq1d 2686	. . . . . . . . . . 11 |- ( x = ( (coeff ` F ) ` k ) -> ( ( * ` x ) e. S <-> ( * ` ( (coeff ` F ) ` k ) ) e. S ) )
26	25	rspccv 3306	. . . . . . . . . 10 |- ( A. x e. S ( * ` x ) e. S -> ( ( (coeff ` F ) ` k ) e. S -> ( * ` ( (coeff ` F ) ` k ) ) e. S ) )
27	23, 26	syl 17	. . . . . . . . 9 |- ( ph -> ( ( (coeff ` F ) ` k ) e. S -> ( * ` ( (coeff ` F ) ` k ) ) e. S ) )
28		elsni 4194	. . . . . . . . . . . . 13 |- ( ( (coeff ` F ) ` k ) e. { 0 } -> ( (coeff ` F ) ` k ) = 0 )
29	28	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( (coeff ` F ) ` k ) e. { 0 } -> ( * ` ( (coeff ` F ) ` k ) ) = ( * ` 0 ) )
30		cj0 13898	. . . . . . . . . . . 12 |- ( * ` 0 ) = 0
31	29, 30	syl6eq 2672	. . . . . . . . . . 11 |- ( ( (coeff ` F ) ` k ) e. { 0 } -> ( * ` ( (coeff ` F ) ` k ) ) = 0 )
32		fvex 6201	. . . . . . . . . . . 12 |- ( * ` ( (coeff ` F ) ` k ) ) e. _V
33	32	elsn 4192	. . . . . . . . . . 11 |- ( ( * ` ( (coeff ` F ) ` k ) ) e. { 0 } <-> ( * ` ( (coeff ` F ) ` k ) ) = 0 )
34	31, 33	sylibr 224	. . . . . . . . . 10 |- ( ( (coeff ` F ) ` k ) e. { 0 } -> ( * ` ( (coeff ` F ) ` k ) ) e. { 0 } )
35	34	a1i 11	. . . . . . . . 9 |- ( ph -> ( ( (coeff ` F ) ` k ) e. { 0 } -> ( * ` ( (coeff ` F ) ` k ) ) e. { 0 } ) )
36	27, 35	orim12d 883	. . . . . . . 8 |- ( ph -> ( ( ( (coeff ` F ) ` k ) e. S \/ ( (coeff ` F ) ` k ) e. { 0 } ) -> ( ( * ` ( (coeff ` F ) ` k ) ) e. S \/ ( * ` ( (coeff ` F ) ` k ) ) e. { 0 } ) ) )
37		elun 3753	. . . . . . . 8 |- ( ( (coeff ` F ) ` k ) e. ( S u. { 0 } ) <-> ( ( (coeff ` F ) ` k ) e. S \/ ( (coeff ` F ) ` k ) e. { 0 } ) )
38		elun 3753	. . . . . . . 8 |- ( ( * ` ( (coeff ` F ) ` k ) ) e. ( S u. { 0 } ) <-> ( ( * ` ( (coeff ` F ) ` k ) ) e. S \/ ( * ` ( (coeff ` F ) ` k ) ) e. { 0 } ) )
39	36, 37, 38	3imtr4g 285	. . . . . . 7 |- ( ph -> ( ( (coeff ` F ) ` k ) e. ( S u. { 0 } ) -> ( * ` ( (coeff ` F ) ` k ) ) e. ( S u. { 0 } ) ) )
40	39	adantr 481	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( (coeff ` F ) ` k ) e. ( S u. { 0 } ) -> ( * ` ( (coeff ` F ) ` k ) ) e. ( S u. { 0 } ) ) )
41	21, 40	mpd 15	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( * ` ( (coeff ` F ) ` k ) ) e. ( S u. { 0 } ) )
42	19, 41	eqeltrd 2701	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( * o. (coeff ` F ) ) ` k ) e. ( S u. { 0 } ) )
43	11, 14, 42	elplyd 23958	. . 3 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( * o. (coeff ` F ) ) ` k ) x. ( z ^ k ) ) ) e. (Poly ` ( S u. { 0 } ) ) )
44	6, 43	eqeltrd 2701	. 2 |- ( ph -> G e. (Poly ` ( S u. { 0 } ) ) )
45		plyun0 23953	. 2 |- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )
46	44, 45	syl6eleq 2711	1 |- ( ph -> G e. (Poly ` S ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   u. cun 3572   C_ wss 3574   {csn 4177   |-> cmpt 4729   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   NN0cn0 11292   ...cfz 12326   ^cexp 12860   *ccj 13836   sum_csu 14416  Polycply 23940  coeffccoe 23942  degcdgr 23943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  coecj  24034