Theorem pm2mpmhmlem2 20624

Description: Lemma 2 for pm2mpmhm 20625. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)

Hypotheses

Ref	Expression
pm2mpmhm.p	|- P = (Poly1 ` R )
pm2mpmhm.c	|- C = ( N Mat P )
pm2mpmhm.a	|- A = ( N Mat R )
pm2mpmhm.q	|- Q = (Poly1 ` A )
pm2mpmhm.t	|- T = ( N pMatToMatPoly R )
pm2mpmhm.b	|- B = ( Base ` C )

Assertion

Ref	Expression
pm2mpmhmlem2	|- ( ( N e. Fin /\ R e. Ring ) -> A. x e. B A. y e. B ( T ` ( x ( .r ` C ) y ) ) = ( ( T ` x ) ( .r ` Q ) ( T ` y ) ) )

Distinct variable groups:   x, B, y   x, N, y   x, R, y

Allowed substitution hints:   A( x, y)   C( x, y)   P( x, y)   Q( x, y)   T( x, y)

Proof of Theorem pm2mpmhmlem2

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

Step	Hyp	Ref	Expression
1		simpll 790	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> N e. Fin )
2		simplr 792	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> R e. Ring )
3		pm2mpmhm.p	. . . . . . . 8 |- P = (Poly1 ` R )
4		pm2mpmhm.c	. . . . . . . 8 |- C = ( N Mat P )
5	3, 4	pmatring 20498	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> C e. Ring )
6	5	adantr 481	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> C e. Ring )
7		simpl 473	. . . . . . 7 |- ( ( x e. B /\ y e. B ) -> x e. B )
8	7	adantl 482	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> x e. B )
9		simpr 477	. . . . . . 7 |- ( ( x e. B /\ y e. B ) -> y e. B )
10	9	adantl 482	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> y e. B )
11		pm2mpmhm.b	. . . . . . 7 |- B = ( Base ` C )
12		eqid 2622	. . . . . . 7 |- ( .r ` C ) = ( .r ` C )
13	11, 12	ringcl 18561	. . . . . 6 |- ( ( C e. Ring /\ x e. B /\ y e. B ) -> ( x ( .r ` C ) y ) e. B )
14	6, 8, 10, 13	syl3anc 1326	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` C ) y ) e. B )
15		eqid 2622	. . . . . 6 |- ( .s ` Q ) = ( .s ` Q )
16		eqid 2622	. . . . . 6 |- (.g ` (mulGrp ` Q ) ) = (.g ` (mulGrp ` Q ) )
17		eqid 2622	. . . . . 6 |- (var1 ` A ) = (var1 ` A )
18		pm2mpmhm.a	. . . . . 6 |- A = ( N Mat R )
19		pm2mpmhm.q	. . . . . 6 |- Q = (Poly1 ` A )
20		pm2mpmhm.t	. . . . . 6 |- T = ( N pMatToMatPoly R )
21	3, 4, 11, 15, 16, 17, 18, 19, 20	pm2mpfval 20601	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ ( x ( .r ` C ) y ) e. B ) -> ( T ` ( x ( .r ` C ) y ) ) = ( Q gsumg ( k e. NN0 |-> ( ( ( x ( .r ` C ) y ) decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
22	1, 2, 14, 21	syl3anc 1326	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( T ` ( x ( .r ` C ) y ) ) = ( Q gsumg ( k e. NN0 |-> ( ( ( x ( .r ` C ) y ) decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
23		simpllr 799	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> R e. Ring )
24		simplr 792	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> ( x e. B /\ y e. B ) )
25		simpr 477	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> k e. NN0 )
26	3, 4, 11, 18	decpmatmul 20577	. . . . . . . 8 |- ( ( R e. Ring /\ ( x e. B /\ y e. B ) /\ k e. NN0 ) -> ( ( x ( .r ` C ) y ) decompPMat k ) = ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) )
27	23, 24, 25, 26	syl3anc 1326	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> ( ( x ( .r ` C ) y ) decompPMat k ) = ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) )
28	27	oveq1d 6665	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> ( ( ( x ( .r ` C ) y ) decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) = ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) )
29	28	mpteq2dva 4744	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( k e. NN0 |-> ( ( ( x ( .r ` C ) y ) decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) = ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) )
30	29	oveq2d 6666	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( Q gsumg ( k e. NN0 |-> ( ( ( x ( .r ` C ) y ) decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) = ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
31		eqid 2622	. . . . . . . 8 |- ( Base ` Q ) = ( Base ` Q )
32	18	matring 20249	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring ) -> A e. Ring )
33	32	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> A e. Ring )
34		eqid 2622	. . . . . . . 8 |- ( Base ` A ) = ( Base ` A )
35		eqid 2622	. . . . . . . 8 |- ( 0g ` A ) = ( 0g ` A )
36		ringcmn 18581	. . . . . . . . . . . 12 |- ( A e. Ring -> A e. CMnd )
37	32, 36	syl 17	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring ) -> A e. CMnd )
38	37	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) -> A e. CMnd )
39		fzfid 12772	. . . . . . . . . 10 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) -> ( 0 ... k ) e. Fin )
40	33	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> A e. Ring )
41		simp-5r 809	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> R e. Ring )
42	8	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> x e. B )
43		elfznn0 12433	. . . . . . . . . . . . . 14 |- ( z e. ( 0 ... k ) -> z e. NN0 )
44	43	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> z e. NN0 )
45	3, 4, 11, 18, 34	decpmatcl 20572	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ x e. B /\ z e. NN0 ) -> ( x decompPMat z ) e. ( Base ` A ) )
46	41, 42, 44, 45	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> ( x decompPMat z ) e. ( Base ` A ) )
47	10	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> y e. B )
48		fznn0sub 12373	. . . . . . . . . . . . . 14 |- ( z e. ( 0 ... k ) -> ( k - z ) e. NN0 )
49	48	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> ( k - z ) e. NN0 )
50	3, 4, 11, 18, 34	decpmatcl 20572	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ y e. B /\ ( k - z ) e. NN0 ) -> ( y decompPMat ( k - z ) ) e. ( Base ` A ) )
51	41, 47, 49, 50	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> ( y decompPMat ( k - z ) ) e. ( Base ` A ) )
52		eqid 2622	. . . . . . . . . . . . 13 |- ( .r ` A ) = ( .r ` A )
53	34, 52	ringcl 18561	. . . . . . . . . . . 12 |- ( ( A e. Ring /\ ( x decompPMat z ) e. ( Base ` A ) /\ ( y decompPMat ( k - z ) ) e. ( Base ` A ) ) -> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) e. ( Base ` A ) )
54	40, 46, 51, 53	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) e. ( Base ` A ) )
55	54	ralrimiva 2966	. . . . . . . . . 10 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) -> A. z e. ( 0 ... k ) ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) e. ( Base ` A ) )
56	34, 38, 39, 55	gsummptcl 18366	. . . . . . . . 9 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) -> ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) e. ( Base ` A ) )
57	56	ralrimiva 2966	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> A. k e. NN0 ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) e. ( Base ` A ) )
58	3, 4, 11, 18, 52, 35	decpmatmulsumfsupp 20578	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( k e. NN0 |-> ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ) finSupp ( 0g ` A ) )
59	58	adantr 481	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( k e. NN0 |-> ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ) finSupp ( 0g ` A ) )
60		simpr 477	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> n e. NN0 )
61	19, 31, 17, 16, 33, 34, 15, 35, 57, 59, 60	gsummoncoe1 19674	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` n ) = [_ n / k ]_ ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) )
62		csbov2g 6691	. . . . . . . . 9 |- ( n e. NN0 -> [_ n / k ]_ ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) = ( A gsumg [_ n / k ]_ ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) )
63		id 22	. . . . . . . . . . 11 |- ( n e. NN0 -> n e. NN0 )
64		oveq2 6658	. . . . . . . . . . . . 13 |- ( k = n -> ( 0 ... k ) = ( 0 ... n ) )
65		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( k = n -> ( k - z ) = ( n - z ) )
66	65	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( k = n -> ( y decompPMat ( k - z ) ) = ( y decompPMat ( n - z ) ) )
67	66	oveq2d 6666	. . . . . . . . . . . . 13 |- ( k = n -> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) = ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( n - z ) ) ) )
68	64, 67	mpteq12dv 4733	. . . . . . . . . . . 12 |- ( k = n -> ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) = ( z e. ( 0 ... n ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( n - z ) ) ) ) )
69	68	adantl 482	. . . . . . . . . . 11 |- ( ( n e. NN0 /\ k = n ) -> ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) = ( z e. ( 0 ... n ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( n - z ) ) ) ) )
70	63, 69	csbied 3560	. . . . . . . . . 10 |- ( n e. NN0 -> [_ n / k ]_ ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) = ( z e. ( 0 ... n ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( n - z ) ) ) ) )
71	70	oveq2d 6666	. . . . . . . . 9 |- ( n e. NN0 -> ( A gsumg [_ n / k ]_ ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) = ( A gsumg ( z e. ( 0 ... n ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( n - z ) ) ) ) ) )
72	62, 71	eqtrd 2656	. . . . . . . 8 |- ( n e. NN0 -> [_ n / k ]_ ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) = ( A gsumg ( z e. ( 0 ... n ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( n - z ) ) ) ) ) )
73	72	adantl 482	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> [_ n / k ]_ ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) = ( A gsumg ( z e. ( 0 ... n ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( n - z ) ) ) ) ) )
74		eqidd 2623	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( r e. NN0 |-> ( A gsumg ( l e. ( 0 ... r ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( r - l ) ) ) ) ) ) = ( r e. NN0 |-> ( A gsumg ( l e. ( 0 ... r ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( r - l ) ) ) ) ) ) )
75		oveq2 6658	. . . . . . . . . . . 12 |- ( r = n -> ( 0 ... r ) = ( 0 ... n ) )
76		oveq1 6657	. . . . . . . . . . . . . 14 |- ( r = n -> ( r - l ) = ( n - l ) )
77	76	fveq2d 6195	. . . . . . . . . . . . 13 |- ( r = n -> ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( r - l ) ) = ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) )
78	77	oveq2d 6666	. . . . . . . . . . . 12 |- ( r = n -> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( r - l ) ) ) = ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) ) )
79	75, 78	mpteq12dv 4733	. . . . . . . . . . 11 |- ( r = n -> ( l e. ( 0 ... r ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( r - l ) ) ) ) = ( l e. ( 0 ... n ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) ) ) )
80	79	oveq2d 6666	. . . . . . . . . 10 |- ( r = n -> ( A gsumg ( l e. ( 0 ... r ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( r - l ) ) ) ) ) = ( A gsumg ( l e. ( 0 ... n ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) ) ) ) )
81	80	adantl 482	. . . . . . . . 9 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ r = n ) -> ( A gsumg ( l e. ( 0 ... r ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( r - l ) ) ) ) ) = ( A gsumg ( l e. ( 0 ... n ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) ) ) ) )
82		ovexd 6680	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( A gsumg ( l e. ( 0 ... n ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) ) ) ) e. _V )
83	74, 81, 60, 82	fvmptd 6288	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( ( r e. NN0 |-> ( A gsumg ( l e. ( 0 ... r ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( r - l ) ) ) ) ) ) ` n ) = ( A gsumg ( l e. ( 0 ... n ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) ) ) ) )
84		eqid 2622	. . . . . . . . . 10 |- ( 0g ` Q ) = ( 0g ` Q )
85	19	ply1ring 19618	. . . . . . . . . . . . 13 |- ( A e. Ring -> Q e. Ring )
86	32, 85	syl 17	. . . . . . . . . . . 12 |- ( ( N e. Fin /\ R e. Ring ) -> Q e. Ring )
87		ringcmn 18581	. . . . . . . . . . . 12 |- ( Q e. Ring -> Q e. CMnd )
88	86, 87	syl 17	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring ) -> Q e. CMnd )
89	88	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> Q e. CMnd )
90		nn0ex 11298	. . . . . . . . . . 11 |- NN0 e. _V
91	90	a1i 11	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> NN0 e. _V )
92	7	anim2i 593	. . . . . . . . . . . . . 14 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( ( N e. Fin /\ R e. Ring ) /\ x e. B ) )
93		df-3an 1039	. . . . . . . . . . . . . 14 |- ( ( N e. Fin /\ R e. Ring /\ x e. B ) <-> ( ( N e. Fin /\ R e. Ring ) /\ x e. B ) )
94	92, 93	sylibr 224	. . . . . . . . . . . . 13 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( N e. Fin /\ R e. Ring /\ x e. B ) )
95	94	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( N e. Fin /\ R e. Ring /\ x e. B ) )
96	3, 4, 11, 15, 16, 17, 18, 19, 31	pm2mpghmlem1 20618	. . . . . . . . . . . 12 |- ( ( ( N e. Fin /\ R e. Ring /\ x e. B ) /\ k e. NN0 ) -> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) e. ( Base ` Q ) )
97	95, 96	sylan 488	. . . . . . . . . . 11 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) -> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) e. ( Base ` Q ) )
98		eqid 2622	. . . . . . . . . . 11 |- ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) = ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) )
99	97, 98	fmptd 6385	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) : NN0 --> ( Base ` Q ) )
100	3, 4, 11, 15, 16, 17, 18, 19	pm2mpghmlem2 20617	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ x e. B ) -> ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) finSupp ( 0g ` Q ) )
101	95, 100	syl 17	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) finSupp ( 0g ` Q ) )
102	31, 84, 89, 91, 99, 101	gsumcl 18316	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. ( Base ` Q ) )
103	9	anim2i 593	. . . . . . . . . . . . . 14 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( ( N e. Fin /\ R e. Ring ) /\ y e. B ) )
104		df-3an 1039	. . . . . . . . . . . . . 14 |- ( ( N e. Fin /\ R e. Ring /\ y e. B ) <-> ( ( N e. Fin /\ R e. Ring ) /\ y e. B ) )
105	103, 104	sylibr 224	. . . . . . . . . . . . 13 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( N e. Fin /\ R e. Ring /\ y e. B ) )
106	105	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( N e. Fin /\ R e. Ring /\ y e. B ) )
107	3, 4, 11, 15, 16, 17, 18, 19, 31	pm2mpghmlem1 20618	. . . . . . . . . . . 12 |- ( ( ( N e. Fin /\ R e. Ring /\ y e. B ) /\ k e. NN0 ) -> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) e. ( Base ` Q ) )
108	106, 107	sylan 488	. . . . . . . . . . 11 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ k e. NN0 ) -> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) e. ( Base ` Q ) )
109		eqid 2622	. . . . . . . . . . 11 |- ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) = ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) )
110	108, 109	fmptd 6385	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) : NN0 --> ( Base ` Q ) )
111	1, 2, 10	3jca 1242	. . . . . . . . . . . 12 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( N e. Fin /\ R e. Ring /\ y e. B ) )
112	111	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( N e. Fin /\ R e. Ring /\ y e. B ) )
113	3, 4, 11, 15, 16, 17, 18, 19	pm2mpghmlem2 20617	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring /\ y e. B ) -> ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) finSupp ( 0g ` Q ) )
114	112, 113	syl 17	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) finSupp ( 0g ` Q ) )
115	31, 84, 89, 91, 110, 114	gsumcl 18316	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. ( Base ` Q ) )
116		eqid 2622	. . . . . . . . . . 11 |- ( .r ` Q ) = ( .r ` Q )
117	19, 116, 52, 31	coe1mul 19640	. . . . . . . . . 10 |- ( ( A e. Ring /\ ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. ( Base ` Q ) /\ ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. ( Base ` Q ) ) -> (coe1 ` ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ) = ( r e. NN0 |-> ( A gsumg ( l e. ( 0 ... r ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( r - l ) ) ) ) ) ) )
118	117	fveq1d 6193	. . . . . . . . 9 |- ( ( A e. Ring /\ ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. ( Base ` Q ) /\ ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. ( Base ` Q ) ) -> ( (coe1 ` ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ) ` n ) = ( ( r e. NN0 |-> ( A gsumg ( l e. ( 0 ... r ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( r - l ) ) ) ) ) ) ` n ) )
119	33, 102, 115, 118	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( (coe1 ` ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ) ` n ) = ( ( r e. NN0 |-> ( A gsumg ( l e. ( 0 ... r ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( r - l ) ) ) ) ) ) ` n ) )
120		oveq2 6658	. . . . . . . . . . . 12 |- ( z = l -> ( x decompPMat z ) = ( x decompPMat l ) )
121		oveq2 6658	. . . . . . . . . . . . 13 |- ( z = l -> ( n - z ) = ( n - l ) )
122	121	oveq2d 6666	. . . . . . . . . . . 12 |- ( z = l -> ( y decompPMat ( n - z ) ) = ( y decompPMat ( n - l ) ) )
123	120, 122	oveq12d 6668	. . . . . . . . . . 11 |- ( z = l -> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( n - z ) ) ) = ( ( x decompPMat l ) ( .r ` A ) ( y decompPMat ( n - l ) ) ) )
124	123	cbvmptv 4750	. . . . . . . . . 10 |- ( z e. ( 0 ... n ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( n - z ) ) ) ) = ( l e. ( 0 ... n ) |-> ( ( x decompPMat l ) ( .r ` A ) ( y decompPMat ( n - l ) ) ) )
125	32	ad3antrrr 766	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> A e. Ring )
126		simp-5r 809	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) /\ k e. NN0 ) -> R e. Ring )
127	8	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) /\ k e. NN0 ) -> x e. B )
128		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) /\ k e. NN0 ) -> k e. NN0 )
129	3, 4, 11, 18, 34	decpmatcl 20572	. . . . . . . . . . . . . . . 16 |- ( ( R e. Ring /\ x e. B /\ k e. NN0 ) -> ( x decompPMat k ) e. ( Base ` A ) )
130	126, 127, 128, 129	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) /\ k e. NN0 ) -> ( x decompPMat k ) e. ( Base ` A ) )
131	130	ralrimiva 2966	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> A. k e. NN0 ( x decompPMat k ) e. ( Base ` A ) )
132	2, 8	jca 554	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( R e. Ring /\ x e. B ) )
133	132	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> ( R e. Ring /\ x e. B ) )
134	3, 4, 11, 18, 35	decpmatfsupp 20574	. . . . . . . . . . . . . . 15 |- ( ( R e. Ring /\ x e. B ) -> ( k e. NN0 |-> ( x decompPMat k ) ) finSupp ( 0g ` A ) )
135	133, 134	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> ( k e. NN0 |-> ( x decompPMat k ) ) finSupp ( 0g ` A ) )
136		elfznn0 12433	. . . . . . . . . . . . . . 15 |- ( l e. ( 0 ... n ) -> l e. NN0 )
137	136	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> l e. NN0 )
138	19, 31, 17, 16, 125, 34, 15, 35, 131, 135, 137	gsummoncoe1 19674	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) = [_ l / k ]_ ( x decompPMat k ) )
139		csbov2g 6691	. . . . . . . . . . . . . . 15 |- ( l e. ( 0 ... n ) -> [_ l / k ]_ ( x decompPMat k ) = ( x decompPMat [_ l / k ]_ k ) )
140		csbvarg 4003	. . . . . . . . . . . . . . . 16 |- ( l e. ( 0 ... n ) -> [_ l / k ]_ k = l )
141	140	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( l e. ( 0 ... n ) -> ( x decompPMat [_ l / k ]_ k ) = ( x decompPMat l ) )
142	139, 141	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( l e. ( 0 ... n ) -> [_ l / k ]_ ( x decompPMat k ) = ( x decompPMat l ) )
143	142	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> [_ l / k ]_ ( x decompPMat k ) = ( x decompPMat l ) )
144	138, 143	eqtr2d 2657	. . . . . . . . . . . 12 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> ( x decompPMat l ) = ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) )
145	10	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) /\ k e. NN0 ) -> y e. B )
146	3, 4, 11, 18, 34	decpmatcl 20572	. . . . . . . . . . . . . . . 16 |- ( ( R e. Ring /\ y e. B /\ k e. NN0 ) -> ( y decompPMat k ) e. ( Base ` A ) )
147	126, 145, 128, 146	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) /\ k e. NN0 ) -> ( y decompPMat k ) e. ( Base ` A ) )
148	147	ralrimiva 2966	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> A. k e. NN0 ( y decompPMat k ) e. ( Base ` A ) )
149	2, 10	jca 554	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( R e. Ring /\ y e. B ) )
150	149	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> ( R e. Ring /\ y e. B ) )
151	3, 4, 11, 18, 35	decpmatfsupp 20574	. . . . . . . . . . . . . . 15 |- ( ( R e. Ring /\ y e. B ) -> ( k e. NN0 |-> ( y decompPMat k ) ) finSupp ( 0g ` A ) )
152	150, 151	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> ( k e. NN0 |-> ( y decompPMat k ) ) finSupp ( 0g ` A ) )
153		fznn0sub 12373	. . . . . . . . . . . . . . 15 |- ( l e. ( 0 ... n ) -> ( n - l ) e. NN0 )
154	153	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> ( n - l ) e. NN0 )
155	19, 31, 17, 16, 125, 34, 15, 35, 148, 152, 154	gsummoncoe1 19674	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) = [_ ( n - l ) / k ]_ ( y decompPMat k ) )
156		ovex 6678	. . . . . . . . . . . . . 14 |- ( n - l ) e. _V
157		csbov2g 6691	. . . . . . . . . . . . . 14 |- ( ( n - l ) e. _V -> [_ ( n - l ) / k ]_ ( y decompPMat k ) = ( y decompPMat [_ ( n - l ) / k ]_ k ) )
158	156, 157	mp1i 13	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> [_ ( n - l ) / k ]_ ( y decompPMat k ) = ( y decompPMat [_ ( n - l ) / k ]_ k ) )
159		csbvarg 4003	. . . . . . . . . . . . . . 15 |- ( ( n - l ) e. _V -> [_ ( n - l ) / k ]_ k = ( n - l ) )
160	156, 159	mp1i 13	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> [_ ( n - l ) / k ]_ k = ( n - l ) )
161	160	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> ( y decompPMat [_ ( n - l ) / k ]_ k ) = ( y decompPMat ( n - l ) ) )
162	155, 158, 161	3eqtrrd 2661	. . . . . . . . . . . 12 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> ( y decompPMat ( n - l ) ) = ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) )
163	144, 162	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) /\ l e. ( 0 ... n ) ) -> ( ( x decompPMat l ) ( .r ` A ) ( y decompPMat ( n - l ) ) ) = ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) ) )
164	163	mpteq2dva 4744	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( l e. ( 0 ... n ) |-> ( ( x decompPMat l ) ( .r ` A ) ( y decompPMat ( n - l ) ) ) ) = ( l e. ( 0 ... n ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) ) ) )
165	124, 164	syl5eq 2668	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( z e. ( 0 ... n ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( n - z ) ) ) ) = ( l e. ( 0 ... n ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) ) ) )
166	165	oveq2d 6666	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( A gsumg ( z e. ( 0 ... n ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( n - z ) ) ) ) ) = ( A gsumg ( l e. ( 0 ... n ) |-> ( ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` l ) ( .r ` A ) ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` ( n - l ) ) ) ) ) )
167	83, 119, 166	3eqtr4rd 2667	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( A gsumg ( z e. ( 0 ... n ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( n - z ) ) ) ) ) = ( (coe1 ` ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ) ` n ) )
168	61, 73, 167	3eqtrd 2660	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ n e. NN0 ) -> ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` n ) = ( (coe1 ` ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ) ` n ) )
169	168	ralrimiva 2966	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> A. n e. NN0 ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` n ) = ( (coe1 ` ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ) ` n ) )
170	32	adantr 481	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> A e. Ring )
171	88	adantr 481	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> Q e. CMnd )
172	90	a1i 11	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> NN0 e. _V )
173	19	ply1lmod 19622	. . . . . . . . . . 11 |- ( A e. Ring -> Q e. LMod )
174	32, 173	syl 17	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring ) -> Q e. LMod )
175	174	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> Q e. LMod )
176	37	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> A e. CMnd )
177		fzfid 12772	. . . . . . . . . . 11 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> ( 0 ... k ) e. Fin )
178	32	ad3antrrr 766	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> A e. Ring )
179		simp-4r 807	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> R e. Ring )
180		simplrl 800	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> x e. B )
181	180	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> x e. B )
182	43	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> z e. NN0 )
183	179, 181, 182, 45	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> ( x decompPMat z ) e. ( Base ` A ) )
184		simplrr 801	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> y e. B )
185	184	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> y e. B )
186	48	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> ( k - z ) e. NN0 )
187	179, 185, 186, 50	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> ( y decompPMat ( k - z ) ) e. ( Base ` A ) )
188	178, 183, 187, 53	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) /\ z e. ( 0 ... k ) ) -> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) e. ( Base ` A ) )
189	188	ralrimiva 2966	. . . . . . . . . . 11 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> A. z e. ( 0 ... k ) ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) e. ( Base ` A ) )
190	34, 176, 177, 189	gsummptcl 18366	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) e. ( Base ` A ) )
191	32	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> A e. Ring )
192	19	ply1sca 19623	. . . . . . . . . . . . 13 |- ( A e. Ring -> A = (Scalar ` Q ) )
193	191, 192	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> A = (Scalar ` Q ) )
194	193	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> (Scalar ` Q ) = A )
195	194	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> ( Base ` (Scalar ` Q ) ) = ( Base ` A ) )
196	190, 195	eleqtrrd 2704	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) e. ( Base ` (Scalar ` Q ) ) )
197		eqid 2622	. . . . . . . . . . 11 |- (mulGrp ` Q ) = (mulGrp ` Q )
198	19, 17, 197, 16, 31	ply1moncl 19641	. . . . . . . . . 10 |- ( ( A e. Ring /\ k e. NN0 ) -> ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) e. ( Base ` Q ) )
199	191, 198	sylancom 701	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) e. ( Base ` Q ) )
200		eqid 2622	. . . . . . . . . 10 |- (Scalar ` Q ) = (Scalar ` Q )
201		eqid 2622	. . . . . . . . . 10 |- ( Base ` (Scalar ` Q ) ) = ( Base ` (Scalar ` Q ) )
202	31, 200, 15, 201	lmodvscl 18880	. . . . . . . . 9 |- ( ( Q e. LMod /\ ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) e. ( Base ` (Scalar ` Q ) ) /\ ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) e. ( Base ` Q ) ) -> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) e. ( Base ` Q ) )
203	175, 196, 199, 202	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) e. ( Base ` Q ) )
204		eqid 2622	. . . . . . . 8 |- ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) = ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) )
205	203, 204	fmptd 6385	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) : NN0 --> ( Base ` Q ) )
206	3, 4, 11, 15, 16, 17, 18, 19, 31, 20	pm2mpmhmlem1 20623	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) finSupp ( 0g ` Q ) )
207	31, 84, 171, 172, 205, 206	gsumcl 18316	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. ( Base ` Q ) )
208	86	adantr 481	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> Q e. Ring )
209	94, 96	sylan 488	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) e. ( Base ` Q ) )
210	209, 98	fmptd 6385	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) : NN0 --> ( Base ` Q ) )
211	94, 100	syl 17	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) finSupp ( 0g ` Q ) )
212	31, 84, 171, 172, 210, 211	gsumcl 18316	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. ( Base ` Q ) )
213	105, 107	sylan 488	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) /\ k e. NN0 ) -> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) e. ( Base ` Q ) )
214	213, 109	fmptd 6385	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) : NN0 --> ( Base ` Q ) )
215	1, 2, 10, 113	syl3anc 1326	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) finSupp ( 0g ` Q ) )
216	31, 84, 171, 172, 214, 215	gsumcl 18316	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. ( Base ` Q ) )
217	31, 116	ringcl 18561	. . . . . . 7 |- ( ( Q e. Ring /\ ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. ( Base ` Q ) /\ ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. ( Base ` Q ) ) -> ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) e. ( Base ` Q ) )
218	208, 212, 216, 217	syl3anc 1326	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) e. ( Base ` Q ) )
219		eqid 2622	. . . . . . 7 |- (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) = (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
220		eqid 2622	. . . . . . 7 |- (coe1 ` ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ) = (coe1 ` ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) )
221	19, 31, 219, 220	ply1coe1eq 19668	. . . . . 6 |- ( ( A e. Ring /\ ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) e. ( Base ` Q ) /\ ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) e. ( Base ` Q ) ) -> ( A. n e. NN0 ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` n ) = ( (coe1 ` ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ) ` n ) <-> ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) = ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ) )
222	170, 207, 218, 221	syl3anc 1326	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( A. n e. NN0 ( (coe1 ` ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ` n ) = ( (coe1 ` ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ) ` n ) <-> ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) = ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) ) )
223	169, 222	mpbid 222	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( Q gsumg ( k e. NN0 |-> ( ( A gsumg ( z e. ( 0 ... k ) |-> ( ( x decompPMat z ) ( .r ` A ) ( y decompPMat ( k - z ) ) ) ) ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) = ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) )
224	22, 30, 223	3eqtrd 2660	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( T ` ( x ( .r ` C ) y ) ) = ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) )
225	3, 4, 11, 15, 16, 17, 18, 19, 20	pm2mpfval 20601	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ x e. B ) -> ( T ` x ) = ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
226	1, 2, 8, 225	syl3anc 1326	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( T ` x ) = ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
227	3, 4, 11, 15, 16, 17, 18, 19, 20	pm2mpfval 20601	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ y e. B ) -> ( T ` y ) = ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
228	1, 2, 10, 227	syl3anc 1326	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( T ` y ) = ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) )
229	226, 228	oveq12d 6668	. . 3 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( ( T ` x ) ( .r ` Q ) ( T ` y ) ) = ( ( Q gsumg ( k e. NN0 |-> ( ( x decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ( .r ` Q ) ( Q gsumg ( k e. NN0 |-> ( ( y decompPMat k ) ( .s ` Q ) ( k (.g ` (mulGrp ` Q ) ) (var1 ` A ) ) ) ) ) ) )
230	224, 229	eqtr4d 2659	. 2 |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( T ` ( x ( .r ` C ) y ) ) = ( ( T ` x ) ( .r ` Q ) ( T ` y ) ) )
231	230	ralrimivva 2971	1 |- ( ( N e. Fin /\ R e. Ring ) -> A. x e. B A. y e. B ( T ` ( x ( .r ` C ) y ) ) = ( ( T ` x ) ( .r ` Q ) ( T ` y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [_csb 3533   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   finSupp cfsupp 8275   0cc0 9936   - cmin 10266   NN0cn0 11292   ...cfz 12326   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   gsum_g cgsu 16101  ._gcmg 17540  CMndccmn 18193  mulGrpcmgp 18489   Ringcrg 18547   LModclmod 18863  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548   Mat cmat 20213   decompPMat cdecpmat 20567   pMatToMatPoly cpm2mp 20597

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-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-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-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-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  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-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-sra 19172  df-rgmod 19173  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-dsmm 20076  df-frlm 20091  df-mamu 20190  df-mat 20214  df-decpmat 20568  df-pm2mp 20598

This theorem is referenced by:  pm2mpmhm  20625