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Definition df-cm 27826
 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 27833 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
df-cm 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))))}
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-cm
StepHypRef Expression
1 ccm 27177 . 2 class 𝐶
2 vx . . . . . . 7 setvar 𝑥
32cv 1474 . . . . . 6 class 𝑥
4 cch 27170 . . . . . 6 class C
53, 4wcel 1977 . . . . 5 wff 𝑥C
6 vy . . . . . . 7 setvar 𝑦
76cv 1474 . . . . . 6 class 𝑦
87, 4wcel 1977 . . . . 5 wff 𝑦C
95, 8wa 383 . . . 4 wff (𝑥C𝑦C )
103, 7cin 3539 . . . . . 6 class (𝑥𝑦)
11 cort 27171 . . . . . . . 8 class
127, 11cfv 5804 . . . . . . 7 class (⊥‘𝑦)
133, 12cin 3539 . . . . . 6 class (𝑥 ∩ (⊥‘𝑦))
14 chj 27174 . . . . . 6 class
1510, 13, 14co 6549 . . . . 5 class ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦)))
163, 15wceq 1475 . . . 4 wff 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦)))
179, 16wa 383 . . 3 wff ((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))))
1817, 2, 6copab 4642 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))))}
191, 18wceq 1475 1 wff 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))))}
 Colors of variables: wff setvar class This definition is referenced by:  cmbr  27827
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