Definition df-cm 32185

Description: Define the commutes relation (on the Hilbert lattice). Definition of commutes in [Kalmbach] p. 20, who uses the notation xCy for "x commutes with y." See cmbri 32192 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
df-cm	⊢ 𝐶ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))}

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-cm

Step	Hyp	Ref	Expression
1		ccm 31538	. 2 class 𝐶ℋ
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1569	. . . . . 6 class 𝑥
4		cch 31531	. . . . . 6 class Cℋ
5	3, 4	wcel 2145	. . . . 5 wff 𝑥 ∈ Cℋ
6		vy	. . . . . . 7 setvar 𝑦
7	6	cv 1569	. . . . . 6 class 𝑦
8	7, 4	wcel 2145	. . . . 5 wff 𝑦 ∈ Cℋ
9	5, 8	wa 401	. . . 4 wff (𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ )
10	3, 7	cin 3898	. . . . . 6 class (𝑥 ∩ 𝑦)
11		cort 31532	. . . . . . . 8 class ⊥
12	7, 11	cfv 6538	. . . . . . 7 class (⊥‘𝑦)
13	3, 12	cin 3898	. . . . . 6 class (𝑥 ∩ (⊥‘𝑦))
14		chj 31535	. . . . . 6 class ∨ℋ
15	10, 13, 14	co 7420	. . . . 5 class ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦)))
16	3, 15	wceq 1570	. . . 4 wff 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦)))
17	9, 16	wa 401	. . 3 wff ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))
18	17, 2, 6	copab 5167	. 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))}
19	1, 18	wceq 1570	1 wff 𝐶ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))}

This definition is used by:  cmbr  32186