Definition df-hvsub 8779

Description: Define vector subtraction. See hvsubval 8829 for its value and hvsubcl 8830 for its closure.

Assertion

Ref	Expression
df-hvsub	⊢ −h = {⟨⟨x, y⟩, z⟩∣((x ∈ ℋ ⋀ y ∈ ℋ ) ⋀ z = (x +h (-1 ·h y)))}

Distinct variable group:   x,y,z

Detailed syntax breakdown of Definition df-hvsub

Step	Hyp	Ref	Expression
1		cmv 8731	. 2 class −h
2		vx	. . . . . . 7 set x
3	2	cv 953	. . . . . 6 class x
4		chil 8727	. . . . . 6 class ℋ
5	3, 4	wcel 956	. . . . 5 wff x ∈ ℋ
6		vy	. . . . . . 7 set y
7	6	cv 953	. . . . . 6 class y
8	7, 4	wcel 956	. . . . 5 wff y ∈ ℋ
9	5, 8	wa 223	. . . 4 wff (x ∈ ℋ ⋀ y ∈ ℋ )
10		vz	. . . . . 6 set z
11	10	cv 953	. . . . 5 class z
12		c1 5215	. . . . . . . 8 class 1
13	12	cneg 5273	. . . . . . 7 class -1
14		csm 8729	. . . . . . 7 class ·h
15	13, 7, 14	co 3954	. . . . . 6 class (-1 ·h y)
16		cva 8728	. . . . . 6 class +h
17	3, 15, 16	co 3954	. . . . 5 class (x +h (-1 ·h y))
18	11, 17	wceq 954	. . . 4 wff z = (x +h (-1 ·h y))
19	9, 18	wa 223	. . 3 wff ((x ∈ ℋ ⋀ y ∈ ℋ ) ⋀ z = (x +h (-1 ·h y)))
20	19, 2, 6, 10	copab2 3955	. 2 class {⟨⟨x, y⟩, z⟩∣((x ∈ ℋ ⋀ y ∈ ℋ ) ⋀ z = (x +h (-1 ·h y)))}
21	1, 20	wceq 954	1 wff −h = {⟨⟨x, y⟩, z⟩∣((x ∈ ℋ ⋀ y ∈ ℋ ) ⋀ z = (x +h (-1 ·h y)))}

This definition is referenced by:  h2hvs 8785  hvsubopr 8824  hvsubvalt 8825

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