Definition df-subr 44428

Description: Define the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)

Assertion

Ref	Expression
df-subr	⊢ -𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣))))

Distinct variable group:   𝑥,𝑣,𝑦

Detailed syntax breakdown of Definition df-subr

Step	Hyp	Ref	Expression
1		cminusr 44422	. 2 class -𝑟
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cvv 3464	. . 3 class V
5		vv	. . . 4 setvar 𝑣
6		cr 11137	. . . 4 class ℝ
7	5	cv 1538	. . . . . 6 class 𝑣
8	2	cv 1538	. . . . . 6 class 𝑥
9	7, 8	cfv 6542	. . . . 5 class (𝑥‘𝑣)
10	3	cv 1538	. . . . . 6 class 𝑦
11	7, 10	cfv 6542	. . . . 5 class (𝑦‘𝑣)
12		cmin 11475	. . . . 5 class −
13	9, 11, 12	co 7414	. . . 4 class ((𝑥‘𝑣) − (𝑦‘𝑣))
14	5, 6, 13	cmpt 5207	. . 3 class (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣)))
15	2, 3, 4, 4, 14	cmpo 7416	. 2 class (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣))))
16	1, 15	wceq 1539	1 wff -𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣))))

This definition is referenced by:  subrval  44431