Definition df-shsum 5275

Description: Define subspace sum in SH. See shsumvalt 5279, shsumval2 5361, and shsumval3 5362 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 +v u)})}

Distinct variable group(s):   x,y,z,w,v,u

Detailed syntax breakdown of Definition df-shsum

Step	Hyp	Ref	Expression
1		cph 4970	. 2 class +H
2		vx	. . . . . . 7 set x
3	2	cv 1089	. . . . . 6 class x
4		csh 4967	. . . . . 6 class SH
5	3, 4	wcel 1092	. . . . 5 wff x e. SH
6		vy	. . . . . . 7 set y
7	6	cv 1089	. . . . . 6 class y
8	7, 4	wcel 1092	. . . . 5 wff y e. SH
9	5, 8	wa 196	. . . 4 wff (x e. SH /\ y e. SH)
10		vz	. . . . . 6 set z
11	10	cv 1089	. . . . 5 class z
12		vw	. . . . . . . . . 10 set w
13	12	cv 1089	. . . . . . . . 9 class w
14		vv	. . . . . . . . . . 11 set v
15	14	cv 1089	. . . . . . . . . 10 class v
16		vu	. . . . . . . . . . 11 set u
17	16	cv 1089	. . . . . . . . . 10 class u
18		cva 4959	. . . . . . . . . 10 class +v
19	15, 17, 18	co 3001	. . . . . . . . 9 class (v +v u)
20	13, 19	wceq 1091	. . . . . . . 8 wff w = (v +v u)
21	20, 16, 7	wrex 1202	. . . . . . 7 wff E.u e. y w = (v +v u)
22	21, 14, 3	wrex 1202	. . . . . 6 wff E.v e. x E.u e. y w = (v +v u)
23		chil 4958	. . . . . 6 class H~
24	22, 12, 23	crab 1204	. . . . 5 class {w e. H~ | E.v e. x E.u e. y w = (v +v u)}
25	11, 24	wceq 1091	. . . 4 wff z = {w e. H~ | E.v e. x E.u e. y w = (v +v u)}
26	9, 25	wa 196	. . 3 wff ((x e. SH /\ y e. SH) /\ z = {w e. H~ | E.v e. x E.u e. y w = (v +v u)})
27	26, 2, 6, 10	copab2 3002	. 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 +v u)})}
28	1, 27	wceq 1091	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 +v u)})}

This definition is referenced by:  shsumvalt 5279

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