Definition df-shsum 10698

Description: Define subspace sum in SH. See shsumval 10702, shsumval2i 10785, and shsumval3i 10786 for its value.

Assertion

Ref	Expression
df-shsum	|- +H = {<.<.x, y>., z>. | ((x e. SH /\ y e. SH) /\ z = {w e. ~H | E.v e. x E.u e. y w = (v +h u)})}

Distinct variable group:   x,y,z,w,v,u

Detailed syntax breakdown of Definition df-shsum

Step	Hyp	Ref	Expression
1		cph 10224	. 2 class +H
2		vx	. . . . . . 7 set x
3	2	cv 1135	. . . . . 6 class x
4		csh 10221	. . . . . 6 class SH
5	3, 4	wcel 1138	. . . . 5 wff x e. SH
6		vy	. . . . . . 7 set y
7	6	cv 1135	. . . . . 6 class y
8	7, 4	wcel 1138	. . . . 5 wff y e. SH
9	5, 8	wa 239	. . . 4 wff (x e. SH /\ y e. SH)
10		vz	. . . . . 6 set z
11	10	cv 1135	. . . . 5 class z
12		vw	. . . . . . . . . 10 set w
13	12	cv 1135	. . . . . . . . 9 class w
14		vv	. . . . . . . . . . 11 set v
15	14	cv 1135	. . . . . . . . . 10 class v
16		vu	. . . . . . . . . . 11 set u
17	16	cv 1135	. . . . . . . . . 10 class u
18		cva 10213	. . . . . . . . . 10 class +h
19	15, 17, 18	co 4695	. . . . . . . . 9 class (v +h u)
20	13, 19	wceq 1136	. . . . . . . 8 wff w = (v +h u)
21	20, 16, 7	wrex 1940	. . . . . . 7 wff E.u e. y w = (v +h u)
22	21, 14, 3	wrex 1940	. . . . . 6 wff E.v e. x E.u e. y w = (v +h u)
23		chil 10212	. . . . . 6 class ~H
24	22, 12, 23	crab 1942	. . . . 5 class {w e. ~H | E.v e. x E.u e. y w = (v +h u)}
25	11, 24	wceq 1136	. . . 4 wff z = {w e. ~H | E.v e. x E.u e. y w = (v +h u)}
26	9, 25	wa 239	. . 3 wff ((x e. SH /\ y e. SH) /\ z = {w e. ~H | E.v e. x E.u e. y w = (v +h u)})
27	26, 2, 6, 10	copab2 4696	. 2 class {<.<.x, y>., z>. | ((x e. SH /\ y e. SH) /\ z = {w e. ~H | E.v e. x E.u e. y w = (v +h u)})}
28	1, 27	wceq 1136	1 wff +H = {<.<.x, y>., z>. | ((x e. SH /\ y e. SH) /\ z = {w e. ~H | E.v e. x E.u e. y w = (v +h u)})}

This definition is referenced by:  shsumval 10702

Copyright terms: Public domain