Theorem ply1coe 19666

Description: Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 7-Oct-2019.)

Hypotheses

Ref	Expression
ply1coe.p	|- P = (Poly1 ` R )
ply1coe.x	|- X = (var1 ` R )
ply1coe.b	|- B = ( Base ` P )
ply1coe.n	|- .x. = ( .s ` P )
ply1coe.m	|- M = (mulGrp ` P )
ply1coe.e	|- .^ = (.g ` M )
ply1coe.a	|- A = (coe1 ` K )

Assertion

Ref	Expression
ply1coe	|- ( ( R e. Ring /\ K e. B ) -> K = ( P gsumg ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) )

Distinct variable groups:   A, k   B, k   k, K   k, X   .^ , k   R, k   .x. , k   P, k

Allowed substitution hint:   M( k)

Proof of Theorem ply1coe

Dummy variables a b c x d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( 1o mPoly R ) = ( 1o mPoly R )
2		psr1baslem 19555	. . 3 |- ( NN0 ^m 1o ) = { d e. ( NN0 ^m 1o ) | ( `' d " NN ) e. Fin }
3		eqid 2622	. . 3 |- ( 0g ` R ) = ( 0g ` R )
4		eqid 2622	. . 3 |- ( 1r ` R ) = ( 1r ` R )
5		1onn 7719	. . . 4 |- 1o e. om
6	5	a1i 11	. . 3 |- ( ( R e. Ring /\ K e. B ) -> 1o e. om )
7		ply1coe.p	. . . 4 |- P = (Poly1 ` R )
8		eqid 2622	. . . 4 |- (PwSer1 ` R ) = (PwSer1 ` R )
9		ply1coe.b	. . . 4 |- B = ( Base ` P )
10	7, 8, 9	ply1bas 19565	. . 3 |- B = ( Base ` ( 1o mPoly R ) )
11		ply1coe.n	. . . 4 |- .x. = ( .s ` P )
12	7, 1, 11	ply1vsca 19596	. . 3 |- .x. = ( .s ` ( 1o mPoly R ) )
13		simpl 473	. . 3 |- ( ( R e. Ring /\ K e. B ) -> R e. Ring )
14		simpr 477	. . 3 |- ( ( R e. Ring /\ K e. B ) -> K e. B )
15	1, 2, 3, 4, 6, 10, 12, 13, 14	mplcoe1 19465	. 2 |- ( ( R e. Ring /\ K e. B ) -> K = ( ( 1o mPoly R ) gsumg ( a e. ( NN0 ^m 1o ) |-> ( ( K ` a ) .x. ( b e. ( NN0 ^m 1o ) |-> if ( b = a , ( 1r ` R ) , ( 0g ` R ) ) ) ) ) ) )
16		ply1coe.a	. . . . . . 7 |- A = (coe1 ` K )
17	16	fvcoe1 19577	. . . . . 6 |- ( ( K e. B /\ a e. ( NN0 ^m 1o ) ) -> ( K ` a ) = ( A ` ( a ` (/) ) ) )
18	17	adantll 750	. . . . 5 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( K ` a ) = ( A ` ( a ` (/) ) ) )
19	5	a1i 11	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> 1o e. om )
20		eqid 2622	. . . . . . 7 |- (mulGrp ` ( 1o mPoly R ) ) = (mulGrp ` ( 1o mPoly R ) )
21		eqid 2622	. . . . . . 7 |- (.g ` (mulGrp ` ( 1o mPoly R ) ) ) = (.g ` (mulGrp ` ( 1o mPoly R ) ) )
22		eqid 2622	. . . . . . 7 |- ( 1o mVar R ) = ( 1o mVar R )
23		simpll 790	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> R e. Ring )
24		simpr 477	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> a e. ( NN0 ^m 1o ) )
25		eqidd 2623	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) )
26		0ex 4790	. . . . . . . . . . 11 |- (/) e. _V
27		fveq2 6191	. . . . . . . . . . . . 13 |- ( b = (/) -> ( ( 1o mVar R ) ` b ) = ( ( 1o mVar R ) ` (/) ) )
28	27	oveq1d 6665	. . . . . . . . . . . 12 |- ( b = (/) -> ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) )
29	27	oveq2d 6666	. . . . . . . . . . . 12 |- ( b = (/) -> ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) )
30	28, 29	eqeq12d 2637	. . . . . . . . . . 11 |- ( b = (/) -> ( ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) )
31	26, 30	ralsn 4222	. . . . . . . . . 10 |- ( A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) )
32	25, 31	sylibr 224	. . . . . . . . 9 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) )
33		fveq2 6191	. . . . . . . . . . . . 13 |- ( x = (/) -> ( ( 1o mVar R ) ` x ) = ( ( 1o mVar R ) ` (/) ) )
34	33	oveq2d 6666	. . . . . . . . . . . 12 |- ( x = (/) -> ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) )
35	33	oveq1d 6665	. . . . . . . . . . . 12 |- ( x = (/) -> ( ( ( 1o mVar R ) ` x ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) )
36	34, 35	eqeq12d 2637	. . . . . . . . . . 11 |- ( x = (/) -> ( ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) ) )
37	36	ralbidv 2986	. . . . . . . . . 10 |- ( x = (/) -> ( A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) ) )
38	26, 37	ralsn 4222	. . . . . . . . 9 |- ( A. x e. { (/) } A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) = ( ( ( 1o mVar R ) ` (/) ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) )
39	32, 38	sylibr 224	. . . . . . . 8 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> A. x e. { (/) } A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) )
40		df1o2 7572	. . . . . . . . 9 |- 1o = { (/) }
41	40	raleqi 3142	. . . . . . . . 9 |- ( A. b e. 1o ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) )
42	40, 41	raleqbii 2990	. . . . . . . 8 |- ( A. x e. 1o A. b e. 1o ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) <-> A. x e. { (/) } A. b e. { (/) } ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) )
43	39, 42	sylibr 224	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> A. x e. 1o A. b e. 1o ( ( ( 1o mVar R ) ` b ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` x ) ) = ( ( ( 1o mVar R ) ` x ) ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` b ) ) )
44	1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 43	mplcoe5 19468	. . . . . 6 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( b e. ( NN0 ^m 1o ) |-> if ( b = a , ( 1r ` R ) , ( 0g ` R ) ) ) = ( (mulGrp ` ( 1o mPoly R ) ) gsumg ( c e. 1o |-> ( ( a ` c ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) ) )
45		mpteq1 4737	. . . . . . . . 9 |- ( 1o = { (/) } -> ( c e. 1o |-> ( ( a ` c ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) = ( c e. { (/) } |-> ( ( a ` c ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) )
46	40, 45	ax-mp 5	. . . . . . . 8 |- ( c e. 1o |-> ( ( a ` c ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) = ( c e. { (/) } |-> ( ( a ` c ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) )
47	46	oveq2i 6661	. . . . . . 7 |- ( (mulGrp ` ( 1o mPoly R ) ) gsumg ( c e. 1o |-> ( ( a ` c ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) ) = ( (mulGrp ` ( 1o mPoly R ) ) gsumg ( c e. { (/) } |-> ( ( a ` c ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) )
48	1	mplring 19452	. . . . . . . . . . 11 |- ( ( 1o e. om /\ R e. Ring ) -> ( 1o mPoly R ) e. Ring )
49	5, 48	mpan 706	. . . . . . . . . 10 |- ( R e. Ring -> ( 1o mPoly R ) e. Ring )
50	20	ringmgp 18553	. . . . . . . . . 10 |- ( ( 1o mPoly R ) e. Ring -> (mulGrp ` ( 1o mPoly R ) ) e. Mnd )
51	49, 50	syl 17	. . . . . . . . 9 |- ( R e. Ring -> (mulGrp ` ( 1o mPoly R ) ) e. Mnd )
52	51	ad2antrr 762	. . . . . . . 8 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> (mulGrp ` ( 1o mPoly R ) ) e. Mnd )
53	26	a1i 11	. . . . . . . 8 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> (/) e. _V )
54		ply1coe.e	. . . . . . . . . . . 12 |- .^ = (.g ` M )
55	20, 10	mgpbas 18495	. . . . . . . . . . . . 13 |- B = ( Base ` (mulGrp ` ( 1o mPoly R ) ) )
56	55	a1i 11	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ K e. B ) -> B = ( Base ` (mulGrp ` ( 1o mPoly R ) ) ) )
57		ply1coe.m	. . . . . . . . . . . . . 14 |- M = (mulGrp ` P )
58	57, 9	mgpbas 18495	. . . . . . . . . . . . 13 |- B = ( Base ` M )
59	58	a1i 11	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ K e. B ) -> B = ( Base ` M ) )
60		ssv 3625	. . . . . . . . . . . . 13 |- B C_ _V
61	60	a1i 11	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ K e. B ) -> B C_ _V )
62		ovexd 6680	. . . . . . . . . . . 12 |- ( ( ( R e. Ring /\ K e. B ) /\ ( a e. _V /\ b e. _V ) ) -> ( a ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) b ) e. _V )
63		eqid 2622	. . . . . . . . . . . . . . . . 17 |- ( .r ` P ) = ( .r ` P )
64	7, 1, 63	ply1mulr 19597	. . . . . . . . . . . . . . . 16 |- ( .r ` P ) = ( .r ` ( 1o mPoly R ) )
65	20, 64	mgpplusg 18493	. . . . . . . . . . . . . . 15 |- ( .r ` P ) = ( +g ` (mulGrp ` ( 1o mPoly R ) ) )
66	57, 63	mgpplusg 18493	. . . . . . . . . . . . . . 15 |- ( .r ` P ) = ( +g ` M )
67	65, 66	eqtr3i 2646	. . . . . . . . . . . . . 14 |- ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) = ( +g ` M )
68	67	oveqi 6663	. . . . . . . . . . . . 13 |- ( a ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) b ) = ( a ( +g ` M ) b )
69	68	a1i 11	. . . . . . . . . . . 12 |- ( ( ( R e. Ring /\ K e. B ) /\ ( a e. _V /\ b e. _V ) ) -> ( a ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) b ) = ( a ( +g ` M ) b ) )
70	21, 54, 56, 59, 61, 62, 69	mulgpropd 17584	. . . . . . . . . . 11 |- ( ( R e. Ring /\ K e. B ) -> (.g ` (mulGrp ` ( 1o mPoly R ) ) ) = .^ )
71	70	oveqd 6667	. . . . . . . . . 10 |- ( ( R e. Ring /\ K e. B ) -> ( ( a ` (/) ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) X ) = ( ( a ` (/) ) .^ X ) )
72	71	adantr 481	. . . . . . . . 9 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( ( a ` (/) ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) X ) = ( ( a ` (/) ) .^ X ) )
73	7	ply1ring 19618	. . . . . . . . . . . 12 |- ( R e. Ring -> P e. Ring )
74	57	ringmgp 18553	. . . . . . . . . . . 12 |- ( P e. Ring -> M e. Mnd )
75	73, 74	syl 17	. . . . . . . . . . 11 |- ( R e. Ring -> M e. Mnd )
76	75	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> M e. Mnd )
77		elmapi 7879	. . . . . . . . . . . 12 |- ( a e. ( NN0 ^m 1o ) -> a : 1o --> NN0 )
78		0lt1o 7584	. . . . . . . . . . . 12 |- (/) e. 1o
79		ffvelrn 6357	. . . . . . . . . . . 12 |- ( ( a : 1o --> NN0 /\ (/) e. 1o ) -> ( a ` (/) ) e. NN0 )
80	77, 78, 79	sylancl 694	. . . . . . . . . . 11 |- ( a e. ( NN0 ^m 1o ) -> ( a ` (/) ) e. NN0 )
81	80	adantl 482	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( a ` (/) ) e. NN0 )
82		ply1coe.x	. . . . . . . . . . . 12 |- X = (var1 ` R )
83	82, 7, 9	vr1cl 19587	. . . . . . . . . . 11 |- ( R e. Ring -> X e. B )
84	83	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> X e. B )
85	58, 54	mulgnn0cl 17558	. . . . . . . . . 10 |- ( ( M e. Mnd /\ ( a ` (/) ) e. NN0 /\ X e. B ) -> ( ( a ` (/) ) .^ X ) e. B )
86	76, 81, 84, 85	syl3anc 1326	. . . . . . . . 9 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( ( a ` (/) ) .^ X ) e. B )
87	72, 86	eqeltrd 2701	. . . . . . . 8 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( ( a ` (/) ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) X ) e. B )
88		fveq2 6191	. . . . . . . . . 10 |- ( c = (/) -> ( a ` c ) = ( a ` (/) ) )
89		fveq2 6191	. . . . . . . . . . 11 |- ( c = (/) -> ( ( 1o mVar R ) ` c ) = ( ( 1o mVar R ) ` (/) ) )
90	82	vr1val 19562	. . . . . . . . . . 11 |- X = ( ( 1o mVar R ) ` (/) )
91	89, 90	syl6eqr 2674	. . . . . . . . . 10 |- ( c = (/) -> ( ( 1o mVar R ) ` c ) = X )
92	88, 91	oveq12d 6668	. . . . . . . . 9 |- ( c = (/) -> ( ( a ` c ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) = ( ( a ` (/) ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) X ) )
93	55, 92	gsumsn 18354	. . . . . . . 8 |- ( ( (mulGrp ` ( 1o mPoly R ) ) e. Mnd /\ (/) e. _V /\ ( ( a ` (/) ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) X ) e. B ) -> ( (mulGrp ` ( 1o mPoly R ) ) gsumg ( c e. { (/) } |-> ( ( a ` c ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) ) = ( ( a ` (/) ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) X ) )
94	52, 53, 87, 93	syl3anc 1326	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( (mulGrp ` ( 1o mPoly R ) ) gsumg ( c e. { (/) } |-> ( ( a ` c ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) ) = ( ( a ` (/) ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) X ) )
95	47, 94	syl5eq 2668	. . . . . 6 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( (mulGrp ` ( 1o mPoly R ) ) gsumg ( c e. 1o |-> ( ( a ` c ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` c ) ) ) ) = ( ( a ` (/) ) (.g ` (mulGrp ` ( 1o mPoly R ) ) ) X ) )
96	44, 95, 72	3eqtrd 2660	. . . . 5 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( b e. ( NN0 ^m 1o ) |-> if ( b = a , ( 1r ` R ) , ( 0g ` R ) ) ) = ( ( a ` (/) ) .^ X ) )
97	18, 96	oveq12d 6668	. . . 4 |- ( ( ( R e. Ring /\ K e. B ) /\ a e. ( NN0 ^m 1o ) ) -> ( ( K ` a ) .x. ( b e. ( NN0 ^m 1o ) |-> if ( b = a , ( 1r ` R ) , ( 0g ` R ) ) ) ) = ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) )
98	97	mpteq2dva 4744	. . 3 |- ( ( R e. Ring /\ K e. B ) -> ( a e. ( NN0 ^m 1o ) |-> ( ( K ` a ) .x. ( b e. ( NN0 ^m 1o ) |-> if ( b = a , ( 1r ` R ) , ( 0g ` R ) ) ) ) ) = ( a e. ( NN0 ^m 1o ) |-> ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) ) )
99	98	oveq2d 6666	. 2 |- ( ( R e. Ring /\ K e. B ) -> ( ( 1o mPoly R ) gsumg ( a e. ( NN0 ^m 1o ) |-> ( ( K ` a ) .x. ( b e. ( NN0 ^m 1o ) |-> if ( b = a , ( 1r ` R ) , ( 0g ` R ) ) ) ) ) ) = ( ( 1o mPoly R ) gsumg ( a e. ( NN0 ^m 1o ) |-> ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) ) ) )
100		nn0ex 11298	. . . . . 6 |- NN0 e. _V
101	100	mptex 6486	. . . . 5 |- ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) e. _V
102	101	a1i 11	. . . 4 |- ( ( R e. Ring /\ K e. B ) -> ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) e. _V )
103		fvex 6201	. . . . . 6 |- (Poly1 ` R ) e. _V
104	7, 103	eqeltri 2697	. . . . 5 |- P e. _V
105	104	a1i 11	. . . 4 |- ( ( R e. Ring /\ K e. B ) -> P e. _V )
106		ovexd 6680	. . . 4 |- ( ( R e. Ring /\ K e. B ) -> ( 1o mPoly R ) e. _V )
107	9, 10	eqtr3i 2646	. . . . 5 |- ( Base ` P ) = ( Base ` ( 1o mPoly R ) )
108	107	a1i 11	. . . 4 |- ( ( R e. Ring /\ K e. B ) -> ( Base ` P ) = ( Base ` ( 1o mPoly R ) ) )
109		eqid 2622	. . . . . 6 |- ( +g ` P ) = ( +g ` P )
110	7, 1, 109	ply1plusg 19595	. . . . 5 |- ( +g ` P ) = ( +g ` ( 1o mPoly R ) )
111	110	a1i 11	. . . 4 |- ( ( R e. Ring /\ K e. B ) -> ( +g ` P ) = ( +g ` ( 1o mPoly R ) ) )
112	102, 105, 106, 108, 111	gsumpropd 17272	. . 3 |- ( ( R e. Ring /\ K e. B ) -> ( P gsumg ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) = ( ( 1o mPoly R ) gsumg ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) )
113		eqid 2622	. . . . 5 |- ( 0g ` P ) = ( 0g ` P )
114	1, 7, 113	ply1mpl0 19625	. . . 4 |- ( 0g ` P ) = ( 0g ` ( 1o mPoly R ) )
115	1	mpllmod 19451	. . . . . 6 |- ( ( 1o e. om /\ R e. Ring ) -> ( 1o mPoly R ) e. LMod )
116	5, 13, 115	sylancr 695	. . . . 5 |- ( ( R e. Ring /\ K e. B ) -> ( 1o mPoly R ) e. LMod )
117		lmodcmn 18911	. . . . 5 |- ( ( 1o mPoly R ) e. LMod -> ( 1o mPoly R ) e. CMnd )
118	116, 117	syl 17	. . . 4 |- ( ( R e. Ring /\ K e. B ) -> ( 1o mPoly R ) e. CMnd )
119	100	a1i 11	. . . 4 |- ( ( R e. Ring /\ K e. B ) -> NN0 e. _V )
120	7	ply1lmod 19622	. . . . . . 7 |- ( R e. Ring -> P e. LMod )
121	120	ad2antrr 762	. . . . . 6 |- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> P e. LMod )
122		eqid 2622	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
123	16, 9, 7, 122	coe1f 19581	. . . . . . . . 9 |- ( K e. B -> A : NN0 --> ( Base ` R ) )
124	123	adantl 482	. . . . . . . 8 |- ( ( R e. Ring /\ K e. B ) -> A : NN0 --> ( Base ` R ) )
125	124	ffvelrnda 6359	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> ( A ` k ) e. ( Base ` R ) )
126	7	ply1sca 19623	. . . . . . . . . 10 |- ( R e. Ring -> R = (Scalar ` P ) )
127	126	eqcomd 2628	. . . . . . . . 9 |- ( R e. Ring -> (Scalar ` P ) = R )
128	127	ad2antrr 762	. . . . . . . 8 |- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> (Scalar ` P ) = R )
129	128	fveq2d 6195	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> ( Base ` (Scalar ` P ) ) = ( Base ` R ) )
130	125, 129	eleqtrrd 2704	. . . . . 6 |- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> ( A ` k ) e. ( Base ` (Scalar ` P ) ) )
131	75	ad2antrr 762	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> M e. Mnd )
132		simpr 477	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> k e. NN0 )
133	83	ad2antrr 762	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> X e. B )
134	58, 54	mulgnn0cl 17558	. . . . . . 7 |- ( ( M e. Mnd /\ k e. NN0 /\ X e. B ) -> ( k .^ X ) e. B )
135	131, 132, 133, 134	syl3anc 1326	. . . . . 6 |- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> ( k .^ X ) e. B )
136		eqid 2622	. . . . . . 7 |- (Scalar ` P ) = (Scalar ` P )
137		eqid 2622	. . . . . . 7 |- ( Base ` (Scalar ` P ) ) = ( Base ` (Scalar ` P ) )
138	9, 136, 11, 137	lmodvscl 18880	. . . . . 6 |- ( ( P e. LMod /\ ( A ` k ) e. ( Base ` (Scalar ` P ) ) /\ ( k .^ X ) e. B ) -> ( ( A ` k ) .x. ( k .^ X ) ) e. B )
139	121, 130, 135, 138	syl3anc 1326	. . . . 5 |- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> ( ( A ` k ) .x. ( k .^ X ) ) e. B )
140		eqid 2622	. . . . 5 |- ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) = ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) )
141	139, 140	fmptd 6385	. . . 4 |- ( ( R e. Ring /\ K e. B ) -> ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) : NN0 --> B )
142	7, 82, 9, 11, 57, 54, 16	ply1coefsupp 19665	. . . 4 |- ( ( R e. Ring /\ K e. B ) -> ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) finSupp ( 0g ` P ) )
143		eqid 2622	. . . . . 6 |- ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) = ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) )
144	40, 100, 26, 143	mapsnf1o2 7905	. . . . 5 |- ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) : ( NN0 ^m 1o ) -1-1-onto-> NN0
145	144	a1i 11	. . . 4 |- ( ( R e. Ring /\ K e. B ) -> ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) : ( NN0 ^m 1o ) -1-1-onto-> NN0 )
146	10, 114, 118, 119, 141, 142, 145	gsumf1o 18317	. . 3 |- ( ( R e. Ring /\ K e. B ) -> ( ( 1o mPoly R ) gsumg ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) = ( ( 1o mPoly R ) gsumg ( ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) o. ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ) ) )
147		eqidd 2623	. . . . 5 |- ( ( R e. Ring /\ K e. B ) -> ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) = ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) )
148		eqidd 2623	. . . . 5 |- ( ( R e. Ring /\ K e. B ) -> ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) = ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) )
149		fveq2 6191	. . . . . 6 |- ( k = ( a ` (/) ) -> ( A ` k ) = ( A ` ( a ` (/) ) ) )
150		oveq1 6657	. . . . . 6 |- ( k = ( a ` (/) ) -> ( k .^ X ) = ( ( a ` (/) ) .^ X ) )
151	149, 150	oveq12d 6668	. . . . 5 |- ( k = ( a ` (/) ) -> ( ( A ` k ) .x. ( k .^ X ) ) = ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) )
152	81, 147, 148, 151	fmptco 6396	. . . 4 |- ( ( R e. Ring /\ K e. B ) -> ( ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) o. ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ) = ( a e. ( NN0 ^m 1o ) |-> ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) ) )
153	152	oveq2d 6666	. . 3 |- ( ( R e. Ring /\ K e. B ) -> ( ( 1o mPoly R ) gsumg ( ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) o. ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ) ) = ( ( 1o mPoly R ) gsumg ( a e. ( NN0 ^m 1o ) |-> ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) ) ) )
154	112, 146, 153	3eqtrrd 2661	. 2 |- ( ( R e. Ring /\ K e. B ) -> ( ( 1o mPoly R ) gsumg ( a e. ( NN0 ^m 1o ) |-> ( ( A ` ( a ` (/) ) ) .x. ( ( a ` (/) ) .^ X ) ) ) ) = ( P gsumg ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) )
155	15, 99, 154	3eqtrd 2660	1 |- ( ( R e. Ring /\ K e. B ) -> K = ( P gsumg ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   |-> cmpt 4729   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   omcom 7065   1oc1o 7553   ^m cmap 7857   NN0cn0 11292   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  ._gcmg 17540  CMndccmn 18193  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547   LModclmod 18863   mVar cmvr 19352   mPoly cmpl 19353  PwSer₁cps1 19545  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548

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

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-iin 4523  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-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  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-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-ple 15961  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553

This theorem is referenced by:  eqcoe1ply1eq  19667  pmatcollpw1lem2  20580  mp2pm2mp  20616  plypf1  23968