Definition df-marepv 21616

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 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))))

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

Detailed syntax breakdown of Definition df-marepv

Step	Hyp	Ref	Expression
1		cmatrepV 21614	. 2 class matRepV
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3422	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6		vv	. . . 4 setvar 𝑣
7	2	cv 1538	. . . . . 6 class 𝑛
8	3	cv 1538	. . . . . 6 class 𝑟
9		cmat 21464	. . . . . 6 class Mat
10	7, 8, 9	co 7255	. . . . 5 class (𝑛 Mat 𝑟)
11		cbs 16840	. . . . 5 class Base
12	10, 11	cfv 6418	. . . 4 class (Base‘(𝑛 Mat 𝑟))
13	8, 11	cfv 6418	. . . . 5 class (Base‘𝑟)
14		cmap 8573	. . . . 5 class ↑m
15	13, 7, 14	co 7255	. . . 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 1967	. . . . . . 7 wff 𝑗 = 𝑘
20	17	cv 1538	. . . . . . . 8 class 𝑖
21	6	cv 1538	. . . . . . . 8 class 𝑣
22	20, 21	cfv 6418	. . . . . . 7 class (𝑣‘𝑖)
23	18	cv 1538	. . . . . . . 8 class 𝑗
24	5	cv 1538	. . . . . . . 8 class 𝑚
25	20, 23, 24	co 7255	. . . . . . 7 class (𝑖𝑚𝑗)
26	19, 22, 25	cif 4456	. . . . . 6 class if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))
27	17, 18, 7, 7, 26	cmpo 7257	. . . . 5 class (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))
28	16, 7, 27	cmpt 5153	. . . 4 class (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))
29	5, 6, 12, 15, 28	cmpo 7257	. . 3 class (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))
30	2, 3, 4, 4, 29	cmpo 7257	. 2 class (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))))
31	1, 30	wceq 1539	1 wff matRepV = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))))

This definition is referenced by:  marepvfval  21622