Theorem chcoeffeqlem 20690

Description: Lemma for chcoeffeq 20691. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)

Hypotheses

Ref	Expression
chcoeffeq.a	|- A = ( N Mat R )
chcoeffeq.b	|- B = ( Base ` A )
chcoeffeq.p	|- P = (Poly1 ` R )
chcoeffeq.y	|- Y = ( N Mat P )
chcoeffeq.r	|- .X. = ( .r ` Y )
chcoeffeq.s	|- .- = ( -g ` Y )
chcoeffeq.0	|- .0. = ( 0g ` Y )
chcoeffeq.t	|- T = ( N matToPolyMat R )
chcoeffeq.c	|- C = ( N CharPlyMat R )
chcoeffeq.k	|- K = ( C ` M )
chcoeffeq.g	|- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )
chcoeffeq.w	|- W = ( Base ` Y )
chcoeffeq.1	|- .1. = ( 1r ` A )
chcoeffeq.m	|- .* = ( .s ` A )
chcoeffeq.u	|- U = ( N cPolyMatToMat R )

Assertion

Ref

Expression

chcoeffeqlem

|- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) )

Distinct variable groups:   A, n   B, n   n, G   n, K   n, M   n, N   R, n   U, n   n, Y   .1. , n   .* , n   n, b   n, s

Allowed substitution hints:   A( s, b)   B( s, b)   C( n, s, b)   P( n, s, b)   R( s, b)   T( n, s, b)   .X. ( n, s, b)   U( s, b)   .1. ( s, b)   G( s, b)   .* ( s, b)   K( s, b)   M( s, b)   .- ( n, s, b)   N( s, b)   W( n, s, b)   Y( s, b)   .0. ( n, s, b)

Proof of Theorem chcoeffeqlem

Dummy variable l is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- (Poly1 ` A ) = (Poly1 ` A )
2		eqid 2622	. . . . 5 |- (var1 ` A ) = (var1 ` A )
3		eqid 2622	. . . . 5 |- (.g ` (mulGrp ` (Poly1 ` A ) ) ) = (.g ` (mulGrp ` (Poly1 ` A ) ) )
4		crngring 18558	. . . . . . . 8 |- ( R e. CRing -> R e. Ring )
5		chcoeffeq.a	. . . . . . . . 9 |- A = ( N Mat R )
6	5	matring 20249	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring ) -> A e. Ring )
7	4, 6	sylan2 491	. . . . . . 7 |- ( ( N e. Fin /\ R e. CRing ) -> A e. Ring )
8	7	3adant3 1081	. . . . . 6 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> A e. Ring )
9	8	adantr 481	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> A e. Ring )
10		chcoeffeq.b	. . . . 5 |- B = ( Base ` A )
11		eqid 2622	. . . . 5 |- ( .s ` (Poly1 ` A ) ) = ( .s ` (Poly1 ` A ) )
12		eqid 2622	. . . . 5 |- ( 0g ` A ) = ( 0g ` A )
13		chcoeffeq.p	. . . . . . . 8 |- P = (Poly1 ` R )
14		chcoeffeq.y	. . . . . . . 8 |- Y = ( N Mat P )
15		chcoeffeq.t	. . . . . . . 8 |- T = ( N matToPolyMat R )
16		chcoeffeq.r	. . . . . . . 8 |- .X. = ( .r ` Y )
17		chcoeffeq.s	. . . . . . . 8 |- .- = ( -g ` Y )
18		chcoeffeq.0	. . . . . . . 8 |- .0. = ( 0g ` Y )
19		chcoeffeq.g	. . . . . . . 8 |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )
20		eqid 2622	. . . . . . . 8 |- ( N ConstPolyMat R ) = ( N ConstPolyMat R )
21		eqid 2622	. . . . . . . 8 |- ( .s ` Y ) = ( .s ` Y )
22		eqid 2622	. . . . . . . 8 |- ( 1r ` Y ) = ( 1r ` Y )
23		eqid 2622	. . . . . . . 8 |- (var1 ` R ) = (var1 ` R )
24		eqid 2622	. . . . . . . 8 |- ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) = ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) )
25		eqid 2622	. . . . . . . 8 |- ( N maAdju P ) = ( N maAdju P )
26		chcoeffeq.w	. . . . . . . 8 |- W = ( Base ` Y )
27		chcoeffeq.u	. . . . . . . 8 |- U = ( N cPolyMatToMat R )
28	5, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 1, 2, 11, 3, 27	cpmadumatpolylem1 20686	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ s e. NN ) /\ b e. ( B ^m ( 0 ... s ) ) ) -> ( U o. G ) e. ( B ^m NN0 ) )
29	28	anasss 679	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( U o. G ) e. ( B ^m NN0 ) )
30	5, 10, 13, 14, 16, 17, 18, 15, 19, 20	chfacfisfcpmat 20660	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> G : NN0 --> ( N ConstPolyMat R ) )
31	4, 30	syl3anl2 1375	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> G : NN0 --> ( N ConstPolyMat R ) )
32	31	adantr 481	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( U o. G ) e. ( B ^m NN0 ) ) -> G : NN0 --> ( N ConstPolyMat R ) )
33		fvco3 6275	. . . . . . . . . 10 |- ( ( G : NN0 --> ( N ConstPolyMat R ) /\ l e. NN0 ) -> ( ( U o. G ) ` l ) = ( U ` ( G ` l ) ) )
34	33	eqcomd 2628	. . . . . . . . 9 |- ( ( G : NN0 --> ( N ConstPolyMat R ) /\ l e. NN0 ) -> ( U ` ( G ` l ) ) = ( ( U o. G ) ` l ) )
35	32, 34	sylan 488	. . . . . . . 8 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( U o. G ) e. ( B ^m NN0 ) ) /\ l e. NN0 ) -> ( U ` ( G ` l ) ) = ( ( U o. G ) ` l ) )
36		elmapi 7879	. . . . . . . . . 10 |- ( ( U o. G ) e. ( B ^m NN0 ) -> ( U o. G ) : NN0 --> B )
37	36	adantl 482	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( U o. G ) e. ( B ^m NN0 ) ) -> ( U o. G ) : NN0 --> B )
38	37	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( U o. G ) e. ( B ^m NN0 ) ) /\ l e. NN0 ) -> ( ( U o. G ) ` l ) e. B )
39	35, 38	eqeltrd 2701	. . . . . . 7 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( U o. G ) e. ( B ^m NN0 ) ) /\ l e. NN0 ) -> ( U ` ( G ` l ) ) e. B )
40	39	ralrimiva 2966	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( U o. G ) e. ( B ^m NN0 ) ) -> A. l e. NN0 ( U ` ( G ` l ) ) e. B )
41	29, 40	mpdan 702	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> A. l e. NN0 ( U ` ( G ` l ) ) e. B )
42	4	anim2i 593	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. CRing ) -> ( N e. Fin /\ R e. Ring ) )
43	42	3adant3 1081	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( N e. Fin /\ R e. Ring ) )
44	43	adantr 481	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( N e. Fin /\ R e. Ring ) )
45	5, 10, 20, 27	cpm2mf 20557	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring ) -> U : ( N ConstPolyMat R ) --> B )
46	44, 45	syl 17	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> U : ( N ConstPolyMat R ) --> B )
47		fcompt 6400	. . . . . . 7 |- ( ( U : ( N ConstPolyMat R ) --> B /\ G : NN0 --> ( N ConstPolyMat R ) ) -> ( U o. G ) = ( l e. NN0 |-> ( U ` ( G ` l ) ) ) )
48	46, 31, 47	syl2anc 693	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( U o. G ) = ( l e. NN0 |-> ( U ` ( G ` l ) ) ) )
49	5, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 1, 2, 11, 3, 27	cpmadumatpolylem2 20687	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ s e. NN ) /\ b e. ( B ^m ( 0 ... s ) ) ) -> ( U o. G ) finSupp ( 0g ` A ) )
50	49	anasss 679	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( U o. G ) finSupp ( 0g ` A ) )
51	48, 50	eqbrtrrd 4677	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( l e. NN0 |-> ( U ` ( G ` l ) ) ) finSupp ( 0g ` A ) )
52		simpll1 1100	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ l e. NN0 ) -> N e. Fin )
53	4	3ad2ant2 1083	. . . . . . . 8 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> R e. Ring )
54	53	ad2antrr 762	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ l e. NN0 ) -> R e. Ring )
55		chcoeffeq.k	. . . . . . . . . 10 |- K = ( C ` M )
56		chcoeffeq.c	. . . . . . . . . . 11 |- C = ( N CharPlyMat R )
57		eqid 2622	. . . . . . . . . . 11 |- ( Base ` P ) = ( Base ` P )
58	56, 5, 10, 13, 57	chpmatply1 20637	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( C ` M ) e. ( Base ` P ) )
59	55, 58	syl5eqel 2705	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> K e. ( Base ` P ) )
60		eqid 2622	. . . . . . . . . 10 |- (coe1 ` K ) = (coe1 ` K )
61		eqid 2622	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
62	60, 57, 13, 61	coe1fvalcl 19582	. . . . . . . . 9 |- ( ( K e. ( Base ` P ) /\ l e. NN0 ) -> ( (coe1 ` K ) ` l ) e. ( Base ` R ) )
63	59, 62	sylan 488	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ l e. NN0 ) -> ( (coe1 ` K ) ` l ) e. ( Base ` R ) )
64	63	adantlr 751	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ l e. NN0 ) -> ( (coe1 ` K ) ` l ) e. ( Base ` R ) )
65		chcoeffeq.1	. . . . . . . . . 10 |- .1. = ( 1r ` A )
66	10, 65	ringidcl 18568	. . . . . . . . 9 |- ( A e. Ring -> .1. e. B )
67	8, 66	syl 17	. . . . . . . 8 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> .1. e. B )
68	67	ad2antrr 762	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ l e. NN0 ) -> .1. e. B )
69		chcoeffeq.m	. . . . . . . 8 |- .* = ( .s ` A )
70	61, 5, 10, 69	matvscl 20237	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( ( (coe1 ` K ) ` l ) e. ( Base ` R ) /\ .1. e. B ) ) -> ( ( (coe1 ` K ) ` l ) .* .1. ) e. B )
71	52, 54, 64, 68, 70	syl22anc 1327	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ l e. NN0 ) -> ( ( (coe1 ` K ) ` l ) .* .1. ) e. B )
72	71	ralrimiva 2966	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> A. l e. NN0 ( ( (coe1 ` K ) ` l ) .* .1. ) e. B )
73		nn0ex 11298	. . . . . . 7 |- NN0 e. _V
74	73	a1i 11	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> NN0 e. _V )
75	5	matlmod 20235	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring ) -> A e. LMod )
76	4, 75	sylan2 491	. . . . . . . 8 |- ( ( N e. Fin /\ R e. CRing ) -> A e. LMod )
77	76	3adant3 1081	. . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> A e. LMod )
78	77	adantr 481	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> A e. LMod )
79		eqidd 2623	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> (Scalar ` A ) = (Scalar ` A ) )
80		fvexd 6203	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ l e. NN0 ) -> ( (coe1 ` K ) ` l ) e. _V )
81		eqid 2622	. . . . . 6 |- ( 0g ` (Scalar ` A ) ) = ( 0g ` (Scalar ` A ) )
82	5	matsca2 20226	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. CRing ) -> R = (Scalar ` A ) )
83	82	3adant3 1081	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> R = (Scalar ` A ) )
84	83, 53	eqeltrrd 2702	. . . . . . . 8 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> (Scalar ` A ) e. Ring )
85	83	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> (Scalar ` A ) = R )
86	85	fveq2d 6195	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> (Poly1 ` (Scalar ` A ) ) = (Poly1 ` R ) )
87	86, 13	syl6eqr 2674	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> (Poly1 ` (Scalar ` A ) ) = P )
88	87	fveq2d 6195	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( Base ` (Poly1 ` (Scalar ` A ) ) ) = ( Base ` P ) )
89	59, 88	eleqtrrd 2704	. . . . . . . 8 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> K e. ( Base ` (Poly1 ` (Scalar ` A ) ) ) )
90		eqid 2622	. . . . . . . . 9 |- (Poly1 ` (Scalar ` A ) ) = (Poly1 ` (Scalar ` A ) )
91		eqid 2622	. . . . . . . . 9 |- ( Base ` (Poly1 ` (Scalar ` A ) ) ) = ( Base ` (Poly1 ` (Scalar ` A ) ) )
92	90, 91, 81	mptcoe1fsupp 19585	. . . . . . . 8 |- ( ( (Scalar ` A ) e. Ring /\ K e. ( Base ` (Poly1 ` (Scalar ` A ) ) ) ) -> ( l e. NN0 |-> ( (coe1 ` K ) ` l ) ) finSupp ( 0g ` (Scalar ` A ) ) )
93	84, 89, 92	syl2anc 693	. . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( l e. NN0 |-> ( (coe1 ` K ) ` l ) ) finSupp ( 0g ` (Scalar ` A ) ) )
94	93	adantr 481	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( l e. NN0 |-> ( (coe1 ` K ) ` l ) ) finSupp ( 0g ` (Scalar ` A ) ) )
95	74, 78, 79, 10, 80, 68, 12, 81, 69, 94	mptscmfsupp0 18928	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( l e. NN0 |-> ( ( (coe1 ` K ) ` l ) .* .1. ) ) finSupp ( 0g ` A ) )
96		fveq2 6191	. . . . . . . . . 10 |- ( n = l -> ( G ` n ) = ( G ` l ) )
97	96	fveq2d 6195	. . . . . . . . 9 |- ( n = l -> ( U ` ( G ` n ) ) = ( U ` ( G ` l ) ) )
98		oveq1 6657	. . . . . . . . 9 |- ( n = l -> ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) = ( l (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) )
99	97, 98	oveq12d 6668	. . . . . . . 8 |- ( n = l -> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) = ( ( U ` ( G ` l ) ) ( .s ` (Poly1 ` A ) ) ( l (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) )
100	99	cbvmptv 4750	. . . . . . 7 |- ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) = ( l e. NN0 |-> ( ( U ` ( G ` l ) ) ( .s ` (Poly1 ` A ) ) ( l (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) )
101	100	oveq2i 6661	. . . . . 6 |- ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( l e. NN0 |-> ( ( U ` ( G ` l ) ) ( .s ` (Poly1 ` A ) ) ( l (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) )
102	101	a1i 11	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( l e. NN0 |-> ( ( U ` ( G ` l ) ) ( .s ` (Poly1 ` A ) ) ( l (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) )
103		fveq2 6191	. . . . . . . . . 10 |- ( n = l -> ( (coe1 ` K ) ` n ) = ( (coe1 ` K ) ` l ) )
104	103	oveq1d 6665	. . . . . . . . 9 |- ( n = l -> ( ( (coe1 ` K ) ` n ) .* .1. ) = ( ( (coe1 ` K ) ` l ) .* .1. ) )
105	104, 98	oveq12d 6668	. . . . . . . 8 |- ( n = l -> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) = ( ( ( (coe1 ` K ) ` l ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( l (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) )
106	105	cbvmptv 4750	. . . . . . 7 |- ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) = ( l e. NN0 |-> ( ( ( (coe1 ` K ) ` l ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( l (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) )
107	106	oveq2i 6661	. . . . . 6 |- ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( l e. NN0 |-> ( ( ( (coe1 ` K ) ` l ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( l (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) )
108	107	a1i 11	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( l e. NN0 |-> ( ( ( (coe1 ` K ) ` l ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( l (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) )
109	1, 2, 3, 9, 10, 11, 12, 41, 51, 72, 95, 102, 108	gsumply1eq 19675	. . . 4 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) <-> A. l e. NN0 ( U ` ( G ` l ) ) = ( ( (coe1 ` K ) ` l ) .* .1. ) ) )
110	109	biimpa 501	. . 3 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) -> A. l e. NN0 ( U ` ( G ` l ) ) = ( ( (coe1 ` K ) ` l ) .* .1. ) )
111	97, 104	eqeq12d 2637	. . . 4 |- ( n = l -> ( ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) <-> ( U ` ( G ` l ) ) = ( ( (coe1 ` K ) ` l ) .* .1. ) ) )
112	111	cbvralv 3171	. . 3 |- ( A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) <-> A. l e. NN0 ( U ` ( G ` l ) ) = ( ( (coe1 ` K ) ` l ) .* .1. ) )
113	110, 112	sylibr 224	. 2 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) )
114	113	ex 450	1 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   finSupp cfsupp 8275   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   NNcn 11020   NN0cn0 11292   ...cfz 12326   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   gsum_g cgsu 16101   -gcsg 17424  ._gcmg 17540  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547   CRingccrg 18548   LModclmod 18863  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548   Mat cmat 20213   maAdju cmadu 20438   ConstPolyMat ccpmat 20508   matToPolyMat cmat2pmat 20509   cPolyMatToMat ccpmat2mat 20510   CharPlyMat cchpmat 20631

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-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  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-ot 4186  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-tpos 7352  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-sup 8348  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  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-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  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-gim 17701  df-cntz 17750  df-oppg 17776  df-symg 17798  df-pmtr 17862  df-psgn 17911  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-dsmm 20076  df-frlm 20091  df-mamu 20190  df-mat 20214  df-mdet 20391  df-cpmat 20511  df-mat2pmat 20512  df-cpmat2mat 20513  df-chpmat 20632

This theorem is referenced by:  chcoeffeq  20691