Definition df-marepv 20365

Description: Function replacing a column of a matrix by a vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 26-Feb-2019.)

Assertion

Ref	Expression
df-marepv	|- matRepV = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) , v e. ( ( Base ` r ) ^m n ) |-> ( k e. n |-> ( i e. n , j e. n |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) )

Distinct variable group:   i, j, k, m, n, r, v

Detailed syntax breakdown of Definition df-marepv

Step	Hyp	Ref	Expression
1		cmatrepV 20363	. 2 class matRepV
2		vn	. . 3 setvar n
3		vr	. . 3 setvar r
4		cvv 3200	. . 3 class _V
5		vm	. . . 4 setvar m
6		vv	. . . 4 setvar v
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	8, 11	cfv 5888	. . . . 5 class ( Base ` r )
14		cmap 7857	. . . . 5 class ^m
15	13, 7, 14	co 6650	. . . 4 class ( ( Base ` r ) ^m n )
16		vk	. . . . 5 setvar k
17		vi	. . . . . 6 setvar i
18		vj	. . . . . 6 setvar j
19	18, 16	weq 1874	. . . . . . 7 wff j = k
20	17	cv 1482	. . . . . . . 8 class i
21	6	cv 1482	. . . . . . . 8 class v
22	20, 21	cfv 5888	. . . . . . 7 class ( v ` i )
23	18	cv 1482	. . . . . . . 8 class j
24	5	cv 1482	. . . . . . . 8 class m
25	20, 23, 24	co 6650	. . . . . . 7 class ( i m j )
26	19, 22, 25	cif 4086	. . . . . 6 class if ( j = k , ( v ` i ) , ( i m j ) )
27	17, 18, 7, 7, 26	cmpt2 6652	. . . . 5 class ( i e. n , j e. n |-> if ( j = k , ( v ` i ) , ( i m j ) ) )
28	16, 7, 27	cmpt 4729	. . . 4 class ( k e. n |-> ( i e. n , j e. n |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) )
29	5, 6, 12, 15, 28	cmpt2 6652	. . 3 class ( m e. ( Base ` ( n Mat r ) ) , v e. ( ( Base ` r ) ^m n ) |-> ( k e. n |-> ( i e. n , j e. n |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) )
30	2, 3, 4, 4, 29	cmpt2 6652	. 2 class ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) , v e. ( ( Base ` r ) ^m n ) |-> ( k e. n |-> ( i e. n , j e. n |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) )
31	1, 30	wceq 1483	1 wff matRepV = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) , v e. ( ( Base ` r ) ^m n ) |-> ( k e. n |-> ( i e. n , j e. n |-> if ( j = k , ( v ` i ) , ( i m j ) ) ) ) ) )

This definition is referenced by:  marepvfval  20371