Definition df-mat2pmat 20512

Description: Transformation of a matrix (over a ring) into a matrix over the corresponding polynomial ring. (Contributed by AV, 31-Jul-2019.)

Assertion

Ref	Expression
df-mat2pmat	|- matToPolyMat = ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( x e. n , y e. n |-> ( (algSc ` (Poly1 ` r ) ) ` ( x m y ) ) ) ) )

Distinct variable group:   m, n, r, x, y

Detailed syntax breakdown of Definition df-mat2pmat

Step	Hyp	Ref	Expression
1		cmat2pmat 20509	. 2 class matToPolyMat
2		vn	. . 3 setvar n
3		vr	. . 3 setvar r
4		cfn 7955	. . 3 class Fin
5		cvv 3200	. . 3 class _V
6		vm	. . . 4 setvar m
7	2	cv 1482	. . . . . 6 class n
8	3	cv 1482	. . . . . 6 class r
9		cmat 20213	. . . . . 6 class Mat
10	7, 8, 9	co 6650	. . . . 5 class ( n Mat r )
11		cbs 15857	. . . . 5 class Base
12	10, 11	cfv 5888	. . . 4 class ( Base ` ( n Mat r ) )
13		vx	. . . . 5 setvar x
14		vy	. . . . 5 setvar y
15	13	cv 1482	. . . . . . 7 class x
16	14	cv 1482	. . . . . . 7 class y
17	6	cv 1482	. . . . . . 7 class m
18	15, 16, 17	co 6650	. . . . . 6 class ( x m y )
19		cpl1 19547	. . . . . . . 8 class Poly1
20	8, 19	cfv 5888	. . . . . . 7 class (Poly1 ` r )
21		cascl 19311	. . . . . . 7 class algSc
22	20, 21	cfv 5888	. . . . . 6 class (algSc ` (Poly1 ` r ) )
23	18, 22	cfv 5888	. . . . 5 class ( (algSc ` (Poly1 ` r ) ) ` ( x m y ) )
24	13, 14, 7, 7, 23	cmpt2 6652	. . . 4 class ( x e. n , y e. n |-> ( (algSc ` (Poly1 ` r ) ) ` ( x m y ) ) )
25	6, 12, 24	cmpt 4729	. . 3 class ( m e. ( Base ` ( n Mat r ) ) |-> ( x e. n , y e. n |-> ( (algSc ` (Poly1 ` r ) ) ` ( x m y ) ) ) )
26	2, 3, 4, 5, 25	cmpt2 6652	. 2 class ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( x e. n , y e. n |-> ( (algSc ` (Poly1 ` r ) ) ` ( x m y ) ) ) ) )
27	1, 26	wceq 1483	1 wff matToPolyMat = ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( x e. n , y e. n |-> ( (algSc ` (Poly1 ` r ) ) ` ( x m y ) ) ) ) )

This definition is referenced by:  mat2pmatfval  20528