Definition df-chj 9404

Description: Define Hilbert lattice join. See chjvalt 9451 for its value and chjclt 9458 for its closure law. Note that we define it over all Hilbert space subsets to allow proving more general theorems. Even for general subsets the join belongs to CH; see sshjclt 9456. For an alternate definition see dfchj2 9453.

Assertion

Ref	Expression
df-chj	|- vH = {<.<.x, y>., z>. | ((x (_ H~ /\ y (_ H~) /\ z = (_|_` (_|_` (x u. y))))}

Distinct variable group:   x,y,z

Detailed syntax breakdown of Definition df-chj

Step	Hyp	Ref	Expression
1		chj 8982	. 2 class vH
2		vx	. . . . . . 7 set x
3	2	cv 1098	. . . . . 6 class x
4		chil 8968	. . . . . 6 class H~
5	3, 4	wss 2018	. . . . 5 wff x (_ H~
6		vy	. . . . . . 7 set y
7	6	cv 1098	. . . . . 6 class y
8	7, 4	wss 2018	. . . . 5 wff y (_ H~
9	5, 8	wa 223	. . . 4 wff (x (_ H~ /\ y (_ H~)
10		vz	. . . . . 6 set z
11	10	cv 1098	. . . . 5 class z
12	3, 7	cun 2016	. . . . . . 7 class (x u. y)
13		cort 8979	. . . . . . 7 class _|_
14	12, 13	cfv 3145	. . . . . 6 class (_|_` (x u. y))
15	14, 13	cfv 3145	. . . . 5 class (_|_` (_|_` (x u. y)))
16	11, 15	wceq 1099	. . . 4 wff z = (_|_` (_|_` (x u. y)))
17	9, 16	wa 223	. . 3 wff ((x (_ H~ /\ y (_ H~) /\ z = (_|_` (_|_` (x u. y))))
18	17, 2, 6, 10	copab2 3903	. 2 class {<.<.x, y>., z>. | ((x (_ H~ /\ y (_ H~) /\ z = (_|_` (_|_` (x u. y))))}
19	1, 18	wceq 1099	1 wff vH = {<.<.x, y>., z>. | ((x (_ H~ /\ y (_ H~) /\ z = (_|_` (_|_` (x u. y))))}

This definition is referenced by:  sshjvalt 9449  dfchj2 9453  dfchj3 9454

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