Definition df-uvc 22069

Description: ((𝑅 unitVec 𝐼)‘𝑗) is the unit vector in (𝑅 freeLMod 𝐼) along the 𝑗 axis. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
df-uvc	⊢ unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)))))

Distinct variable group:   𝑖,𝑟,𝑗,𝑘

Detailed syntax breakdown of Definition df-uvc

Step	Hyp	Ref	Expression
1		cuvc 22068	. 2 class unitVec
2		vr	. . 3 setvar 𝑟
3		vi	. . 3 setvar 𝑖
4		cvv 3451	. . 3 class V
5		vj	. . . 4 setvar 𝑗
6	3	cv 1569	. . . 4 class 𝑖
7		vk	. . . . 5 setvar 𝑘
8	7, 5	weq 1995	. . . . . 6 wff 𝑘 = 𝑗
9	2	cv 1569	. . . . . . 7 class 𝑟
10		cur 20387	. . . . . . 7 class 1r
11	9, 10	cfv 6531	. . . . . 6 class (1r‘𝑟)
12		c0g 17590	. . . . . . 7 class 0g
13	9, 12	cfv 6531	. . . . . 6 class (0g‘𝑟)
14	8, 11, 13	cif 4482	. . . . 5 class if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟))
15	7, 6, 14	cmpt 5186	. . . 4 class (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)))
16	5, 6, 15	cmpt 5186	. . 3 class (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟))))
17	2, 3, 4, 4, 16	cmpo 7414	. 2 class (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)))))
18	1, 17	wceq 1570	1 wff unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)))))

This definition is used by:  uvcfval  22070