Definition df-hvsub 9033

Description: Define vector subtraction. See hvsubval 9039 for its value and hvsubcl 9040 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 8972	. 2 class -h
2		vx	. . . . . . 7 set x
3	2	cv 1098	. . . . . 6 class x
4		chil 8968	. . . . . 6 class H~
5	3, 4	wcel 1105	. . . . 5 wff x e. H~
6		vy	. . . . . . 7 set y
7	6	cv 1098	. . . . . 6 class y
8	7, 4	wcel 1105	. . . . 5 wff y e. H~
9	5, 8	wa 223	. . . 4 wff (x e. H~ /\ y e. H~)
10		vz	. . . . . 6 set z
11	10	cv 1098	. . . . 5 class z
12		c1 5158	. . . . . . . 8 class 1
13	12	cneg 5216	. . . . . . 7 class -u1
14		csm 8970	. . . . . . 7 class .h
15	13, 7, 14	co 3902	. . . . . 6 class (-u1 .h y)
16		cva 8969	. . . . . 6 class +h
17	3, 15, 16	co 3902	. . . . 5 class (x +h (-u1 .h y))
18	11, 17	wceq 1099	. . . 4 wff z = (x +h (-u1 .h y))
19	9, 18	wa 223	. . 3 wff ((x e. H~ /\ y e. H~) /\ z = (x +h (-u1 .h y)))
20	19, 2, 6, 10	copab2 3903	. 2 class {<.<.x, y>., z>. | ((x e. H~ /\ y e. H~) /\ z = (x +h (-u1 .h y)))}
21	1, 20	wceq 1099	1 wff -h = {<.<.x, y>., z>. | ((x e. H~ /\ y e. H~) /\ z = (x +h (-u1 .h y)))}

This definition is referenced by:  hvsubopr 9034  hvsubvalt 9035  hhvs 9186

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