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Metamath Proof Explorer

Definition df-md

Description: Define the modular pair relation (on the Hilbert lattice). Definition 1.1 of MaedaMaeda p. 1, who use the notation (x,y)M for "the ordered pair is a modular pair." See mdbr for membership relation. (Contributed by NM, 14-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion df-md 𝑀ℋ = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀ 𝑧 ∈ Cℋ ( 𝑧 ⊆ 𝑦 → ( ( 𝑧 ∨ℋ 𝑥 ) ∩ 𝑦 ) = ( 𝑧 ∨ℋ ( 𝑥 ∩ 𝑦 ) ) ) ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cmd ⊢ 𝑀ℋ
1 vx ⊢ 𝑥
2 vy ⊢ 𝑦
3 1 cv ⊢ 𝑥
4 cch ⊢ Cℋ
5 3 4 wcel ⊢ 𝑥 ∈ Cℋ
6 2 cv ⊢ 𝑦
7 6 4 wcel ⊢ 𝑦 ∈ Cℋ
8 5 7 wa ⊢ ( 𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ )
9 vz ⊢ 𝑧
10 9 cv ⊢ 𝑧
11 10 6 wss ⊢ 𝑧 ⊆ 𝑦
12 chj ⊢ ∨ℋ
13 10 3 12 co ⊢ ( 𝑧 ∨ℋ 𝑥 )
14 13 6 cin ⊢ ( ( 𝑧 ∨ℋ 𝑥 ) ∩ 𝑦 )
15 3 6 cin ⊢ ( 𝑥 ∩ 𝑦 )
16 10 15 12 co ⊢ ( 𝑧 ∨ℋ ( 𝑥 ∩ 𝑦 ) )
17 14 16 wceq ⊢ ( ( 𝑧 ∨ℋ 𝑥 ) ∩ 𝑦 ) = ( 𝑧 ∨ℋ ( 𝑥 ∩ 𝑦 ) )
18 11 17 wi ⊢ ( 𝑧 ⊆ 𝑦 → ( ( 𝑧 ∨ℋ 𝑥 ) ∩ 𝑦 ) = ( 𝑧 ∨ℋ ( 𝑥 ∩ 𝑦 ) ) )
19 18 9 4 wral ⊢ ∀ 𝑧 ∈ Cℋ ( 𝑧 ⊆ 𝑦 → ( ( 𝑧 ∨ℋ 𝑥 ) ∩ 𝑦 ) = ( 𝑧 ∨ℋ ( 𝑥 ∩ 𝑦 ) ) )
20 8 19 wa ⊢ ( ( 𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀ 𝑧 ∈ Cℋ ( 𝑧 ⊆ 𝑦 → ( ( 𝑧 ∨ℋ 𝑥 ) ∩ 𝑦 ) = ( 𝑧 ∨ℋ ( 𝑥 ∩ 𝑦 ) ) ) )
21 20 1 2 copab ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀ 𝑧 ∈ Cℋ ( 𝑧 ⊆ 𝑦 → ( ( 𝑧 ∨ℋ 𝑥 ) ∩ 𝑦 ) = ( 𝑧 ∨ℋ ( 𝑥 ∩ 𝑦 ) ) ) ) }
22 0 21 wceq ⊢ 𝑀ℋ = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀ 𝑧 ∈ Cℋ ( 𝑧 ⊆ 𝑦 → ( ( 𝑧 ∨ℋ 𝑥 ) ∩ 𝑦 ) = ( 𝑧 ∨ℋ ( 𝑥 ∩ 𝑦 ) ) ) ) }