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 ) ) ) ) }