Definition df-hvsub 9115

Description: Define vector subtraction. See hvsubvali 9165 for its value and hvsubcli 9166 for its closure.

Assertion

Ref	Expression
df-hvsub	|- -h = {<.<.x, y>., z>. | ((x e. H~ /\ y e. H~) /\ z = (x +h (-u1 .h y)))}

Distinct variable group:   x,y,z

Detailed syntax breakdown of Definition df-hvsub

Step	Hyp	Ref	Expression
1		cmv 9067	. 2 class -h
2		vx	. . . . . . 7 set x
3	2	cv 991	. . . . . 6 class x
4		chil 9063	. . . . . 6 class H~
5	3, 4	wcel 994	. . . . 5 wff x e. H~
6		vy	. . . . . . 7 set y
7	6	cv 991	. . . . . 6 class y
8	7, 4	wcel 994	. . . . 5 wff y e. H~
9	5, 8	wa 221	. . . 4 wff (x e. H~ /\ y e. H~)
10		vz	. . . . . 6 set z
11	10	cv 991	. . . . 5 class z
12		c1 5389	. . . . . . . 8 class 1
13	12	cneg 5447	. . . . . . 7 class -u1
14		csm 9065	. . . . . . 7 class .h
15	13, 7, 14	co 4021	. . . . . 6 class (-u1 .h y)
16		cva 9064	. . . . . 6 class +h
17	3, 15, 16	co 4021	. . . . 5 class (x +h (-u1 .h y))
18	11, 17	wceq 992	. . . 4 wff z = (x +h (-u1 .h y))
19	9, 18	wa 221	. . 3 wff ((x e. H~ /\ y e. H~) /\ z = (x +h (-u1 .h y)))
20	19, 2, 6, 10	copab2 4022	. 2 class {<.<.x, y>., z>. | ((x e. H~ /\ y e. H~) /\ z = (x +h (-u1 .h y)))}
21	1, 20	wceq 992	1 wff -h = {<.<.x, y>., z>. | ((x e. H~ /\ y e. H~) /\ z = (x +h (-u1 .h y)))}

This definition is referenced by:  h2hvs 9121  hvsubopr 9160  hvsubval 9161

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