Theorem m2cpminvid2 20560

Description: The transformation applied to the inverse transformation of a constant polynomial matrix over the ring R results in the matrix itself. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.)

Hypotheses

Ref	Expression
m2cpminvid2.s	|- S = ( N ConstPolyMat R )
m2cpminvid2.i	|- I = ( N cPolyMatToMat R )
m2cpminvid2.t	|- T = ( N matToPolyMat R )

Assertion

Ref	Expression
m2cpminvid2	|- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( T ` ( I ` M ) ) = M )

Proof of Theorem m2cpminvid2

Dummy variables i j x y n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		m2cpminvid2.i	. . . 4 |- I = ( N cPolyMatToMat R )
2		m2cpminvid2.s	. . . 4 |- S = ( N ConstPolyMat R )
3	1, 2	cpm2mval 20555	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( I ` M ) = ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) )
4	3	fveq2d 6195	. 2 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( T ` ( I ` M ) ) = ( T ` ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) ) )
5		simp1 1061	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> N e. Fin )
6		simp2 1062	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> R e. Ring )
7		eqid 2622	. . . . 5 |- ( N Mat R ) = ( N Mat R )
8		eqid 2622	. . . . 5 |- ( Base ` R ) = ( Base ` R )
9		eqid 2622	. . . . 5 |- ( Base ` ( N Mat R ) ) = ( Base ` ( N Mat R ) )
10		eqid 2622	. . . . . . 7 |- ( N Mat (Poly1 ` R ) ) = ( N Mat (Poly1 ` R ) )
11		eqid 2622	. . . . . . 7 |- ( Base ` (Poly1 ` R ) ) = ( Base ` (Poly1 ` R ) )
12		eqid 2622	. . . . . . 7 |- ( Base ` ( N Mat (Poly1 ` R ) ) ) = ( Base ` ( N Mat (Poly1 ` R ) ) )
13		simp2 1062	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ x e. N /\ y e. N ) -> x e. N )
14		simp3 1063	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ x e. N /\ y e. N ) -> y e. N )
15		eqid 2622	. . . . . . . . 9 |- (Poly1 ` R ) = (Poly1 ` R )
16	2, 15, 10, 12	cpmatpmat 20515	. . . . . . . 8 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> M e. ( Base ` ( N Mat (Poly1 ` R ) ) ) )
17	16	3ad2ant1 1082	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ x e. N /\ y e. N ) -> M e. ( Base ` ( N Mat (Poly1 ` R ) ) ) )
18	10, 11, 12, 13, 14, 17	matecld 20232	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ x e. N /\ y e. N ) -> ( x M y ) e. ( Base ` (Poly1 ` R ) ) )
19		0nn0 11307	. . . . . 6 |- 0 e. NN0
20		eqid 2622	. . . . . . 7 |- (coe1 ` ( x M y ) ) = (coe1 ` ( x M y ) )
21	20, 11, 15, 8	coe1fvalcl 19582	. . . . . 6 |- ( ( ( x M y ) e. ( Base ` (Poly1 ` R ) ) /\ 0 e. NN0 ) -> ( (coe1 ` ( x M y ) ) ` 0 ) e. ( Base ` R ) )
22	18, 19, 21	sylancl 694	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ x e. N /\ y e. N ) -> ( (coe1 ` ( x M y ) ) ` 0 ) e. ( Base ` R ) )
23	7, 8, 9, 5, 6, 22	matbas2d 20229	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) e. ( Base ` ( N Mat R ) ) )
24		m2cpminvid2.t	. . . . 5 |- T = ( N matToPolyMat R )
25		eqid 2622	. . . . 5 |- (algSc ` (Poly1 ` R ) ) = (algSc ` (Poly1 ` R ) )
26	24, 7, 9, 15, 25	mat2pmatval 20529	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) e. ( Base ` ( N Mat R ) ) ) -> ( T ` ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) ) = ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( i ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) j ) ) ) )
27	5, 6, 23, 26	syl3anc 1326	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( T ` ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) ) = ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( i ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) j ) ) ) )
28		eqidd 2623	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ i e. N /\ j e. N ) -> ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) = ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) )
29		oveq12 6659	. . . . . . . . 9 |- ( ( x = i /\ y = j ) -> ( x M y ) = ( i M j ) )
30	29	fveq2d 6195	. . . . . . . 8 |- ( ( x = i /\ y = j ) -> (coe1 ` ( x M y ) ) = (coe1 ` ( i M j ) ) )
31	30	fveq1d 6193	. . . . . . 7 |- ( ( x = i /\ y = j ) -> ( (coe1 ` ( x M y ) ) ` 0 ) = ( (coe1 ` ( i M j ) ) ` 0 ) )
32	31	adantl 482	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ i e. N /\ j e. N ) /\ ( x = i /\ y = j ) ) -> ( (coe1 ` ( x M y ) ) ` 0 ) = ( (coe1 ` ( i M j ) ) ` 0 ) )
33		simp2 1062	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ i e. N /\ j e. N ) -> i e. N )
34		simp3 1063	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ i e. N /\ j e. N ) -> j e. N )
35		fvexd 6203	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ i e. N /\ j e. N ) -> ( (coe1 ` ( i M j ) ) ` 0 ) e. _V )
36	28, 32, 33, 34, 35	ovmpt2d 6788	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ i e. N /\ j e. N ) -> ( i ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) j ) = ( (coe1 ` ( i M j ) ) ` 0 ) )
37	36	fveq2d 6195	. . . 4 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ i e. N /\ j e. N ) -> ( (algSc ` (Poly1 ` R ) ) ` ( i ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) j ) ) = ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) )
38	37	mpt2eq3dva 6719	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( i ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) j ) ) ) = ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) )
39	27, 38	eqtrd 2656	. 2 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( T ` ( x e. N , y e. N |-> ( (coe1 ` ( x M y ) ) ` 0 ) ) ) = ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) )
40	2, 15	m2cpminvid2lem 20559	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> A. n e. NN0 ( (coe1 ` ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) )
41		simpl2 1065	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> R e. Ring )
42		simprl 794	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> x e. N )
43		simprr 796	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> y e. N )
44	16	adantr 481	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> M e. ( Base ` ( N Mat (Poly1 ` R ) ) ) )
45	10, 11, 12, 42, 43, 44	matecld 20232	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( x M y ) e. ( Base ` (Poly1 ` R ) ) )
46	45, 19, 21	sylancl 694	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( (coe1 ` ( x M y ) ) ` 0 ) e. ( Base ` R ) )
47	15, 25, 8, 11	ply1sclcl 19656	. . . . . . . 8 |- ( ( R e. Ring /\ ( (coe1 ` ( x M y ) ) ` 0 ) e. ( Base ` R ) ) -> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) e. ( Base ` (Poly1 ` R ) ) )
48	41, 46, 47	syl2anc 693	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) e. ( Base ` (Poly1 ` R ) ) )
49		eqid 2622	. . . . . . . . 9 |- (coe1 ` ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) = (coe1 ` ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) )
50	15, 11, 49, 20	ply1coe1eq 19668	. . . . . . . 8 |- ( ( R e. Ring /\ ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) e. ( Base ` (Poly1 ` R ) ) /\ ( x M y ) e. ( Base ` (Poly1 ` R ) ) ) -> ( A. n e. NN0 ( (coe1 ` ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) <-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) = ( x M y ) ) )
51	50	bicomd 213	. . . . . . 7 |- ( ( R e. Ring /\ ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) e. ( Base ` (Poly1 ` R ) ) /\ ( x M y ) e. ( Base ` (Poly1 ` R ) ) ) -> ( ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) = ( x M y ) <-> A. n e. NN0 ( (coe1 ` ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) ) )
52	41, 48, 45, 51	syl3anc 1326	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) = ( x M y ) <-> A. n e. NN0 ( (coe1 ` ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) ) ` n ) = ( (coe1 ` ( x M y ) ) ` n ) ) )
53	40, 52	mpbird 247	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) = ( x M y ) )
54	53	ralrimivva 2971	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> A. x e. N A. y e. N ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) = ( x M y ) )
55		eqidd 2623	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ x e. N ) /\ y e. N ) -> ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) = ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) )
56		oveq12 6659	. . . . . . . . . . . 12 |- ( ( i = x /\ j = y ) -> ( i M j ) = ( x M y ) )
57	56	fveq2d 6195	. . . . . . . . . . 11 |- ( ( i = x /\ j = y ) -> (coe1 ` ( i M j ) ) = (coe1 ` ( x M y ) ) )
58	57	fveq1d 6193	. . . . . . . . . 10 |- ( ( i = x /\ j = y ) -> ( (coe1 ` ( i M j ) ) ` 0 ) = ( (coe1 ` ( x M y ) ) ` 0 ) )
59	58	fveq2d 6195	. . . . . . . . 9 |- ( ( i = x /\ j = y ) -> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) = ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) )
60	59	adantl 482	. . . . . . . 8 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ x e. N ) /\ y e. N ) /\ ( i = x /\ j = y ) ) -> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) = ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) )
61		simplr 792	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ x e. N ) /\ y e. N ) -> x e. N )
62		simpr 477	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ x e. N ) /\ y e. N ) -> y e. N )
63		fvexd 6203	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ x e. N ) /\ y e. N ) -> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) e. _V )
64	55, 60, 61, 62, 63	ovmpt2d 6788	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ x e. N ) /\ y e. N ) -> ( x ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) y ) = ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) )
65	64	eqeq1d 2624	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ x e. N ) /\ y e. N ) -> ( ( x ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) y ) = ( x M y ) <-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) = ( x M y ) ) )
66	65	anasss 679	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ ( x e. N /\ y e. N ) ) -> ( ( x ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) y ) = ( x M y ) <-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) = ( x M y ) ) )
67	66	2ralbidva 2988	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( A. x e. N A. y e. N ( x ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) y ) = ( x M y ) <-> A. x e. N A. y e. N ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( x M y ) ) ` 0 ) ) = ( x M y ) ) )
68	54, 67	mpbird 247	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> A. x e. N A. y e. N ( x ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) y ) = ( x M y ) )
69		fvexd 6203	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> (Poly1 ` R ) e. _V )
70	6	3ad2ant1 1082	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ i e. N /\ j e. N ) -> R e. Ring )
71	16	3ad2ant1 1082	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ i e. N /\ j e. N ) -> M e. ( Base ` ( N Mat (Poly1 ` R ) ) ) )
72	10, 11, 12, 33, 34, 71	matecld 20232	. . . . . . 7 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ i e. N /\ j e. N ) -> ( i M j ) e. ( Base ` (Poly1 ` R ) ) )
73		eqid 2622	. . . . . . . 8 |- (coe1 ` ( i M j ) ) = (coe1 ` ( i M j ) )
74	73, 11, 15, 8	coe1fvalcl 19582	. . . . . . 7 |- ( ( ( i M j ) e. ( Base ` (Poly1 ` R ) ) /\ 0 e. NN0 ) -> ( (coe1 ` ( i M j ) ) ` 0 ) e. ( Base ` R ) )
75	72, 19, 74	sylancl 694	. . . . . 6 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ i e. N /\ j e. N ) -> ( (coe1 ` ( i M j ) ) ` 0 ) e. ( Base ` R ) )
76	15, 25, 8, 11	ply1sclcl 19656	. . . . . 6 |- ( ( R e. Ring /\ ( (coe1 ` ( i M j ) ) ` 0 ) e. ( Base ` R ) ) -> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) e. ( Base ` (Poly1 ` R ) ) )
77	70, 75, 76	syl2anc 693	. . . . 5 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. S ) /\ i e. N /\ j e. N ) -> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) e. ( Base ` (Poly1 ` R ) ) )
78	10, 11, 12, 5, 69, 77	matbas2d 20229	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) e. ( Base ` ( N Mat (Poly1 ` R ) ) ) )
79	10, 12	eqmat 20230	. . . 4 |- ( ( ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) e. ( Base ` ( N Mat (Poly1 ` R ) ) ) /\ M e. ( Base ` ( N Mat (Poly1 ` R ) ) ) ) -> ( ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) = M <-> A. x e. N A. y e. N ( x ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) y ) = ( x M y ) ) )
80	78, 16, 79	syl2anc 693	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) = M <-> A. x e. N A. y e. N ( x ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) y ) = ( x M y ) ) )
81	68, 80	mpbird 247	. 2 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( i e. N , j e. N |-> ( (algSc ` (Poly1 ` R ) ) ` ( (coe1 ` ( i M j ) ) ` 0 ) ) ) = M )
82	4, 39, 81	3eqtrd 2660	1 |- ( ( N e. Fin /\ R e. Ring /\ M e. S ) -> ( T ` ( I ` M ) ) = M )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   0cc0 9936   NN0cn0 11292   Basecbs 15857   Ringcrg 18547  algSccascl 19311  Poly₁cpl1 19547  coe₁cco1 19548   Mat cmat 20213   ConstPolyMat ccpmat 20508   matToPolyMat cmat2pmat 20509   cPolyMatToMat ccpmat2mat 20510

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-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-dsmm 20076  df-frlm 20091  df-mat 20214  df-cpmat 20511  df-mat2pmat 20512  df-cpmat2mat 20513

This theorem is referenced by:  m2cpmfo  20561  m2cpminv  20565  cpmadumatpoly  20688  cayhamlem4  20693