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Metamath Proof Explorer

Theorem cvrp

Description: A Hilbert lattice satisfies the covering property of Definition 7.4 of MaedaMaeda p. 31 and its converse. ( cvp analog.) (Contributed by NM, 18-Nov-2011)

Ref Expression
Hypotheses cvrp.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cvrp.j ⊢ ∨ = ( join ‘ 𝐾 )
cvrp.m ⊢ ∧ = ( meet ‘ 𝐾 )
cvrp.z ⊢ 0 = ( 0. ‘ 𝐾 )
cvrp.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
cvrp.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion cvrp ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝑋 ∧ 𝑃 ) = 0 ↔ 𝑋 𝐶 ( 𝑋 ∨ 𝑃 ) ) )

Proof

Step Hyp Ref Expression
1 cvrp.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cvrp.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cvrp.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cvrp.z ⊢ 0 = ( 0. ‘ 𝐾 )
5 cvrp.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
6 cvrp.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
7 hlomcmcv ⊢ ( 𝐾 ∈ HL → ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) )
8 1 2 3 4 5 6 cvlcvrp ⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝑋 ∧ 𝑃 ) = 0 ↔ 𝑋 𝐶 ( 𝑋 ∨ 𝑃 ) ) )
9 7 8 syl3an1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝑋 ∧ 𝑃 ) = 0 ↔ 𝑋 𝐶 ( 𝑋 ∨ 𝑃 ) ) )