This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cvr1

Description: A Hilbert lattice has the covering property. Proposition 1(ii) in Kalmbach p. 140 (and its converse). ( chcv1 analog.) (Contributed by NM, 17-Nov-2011)

Ref Expression
Hypotheses cvr1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cvr1.l ⊢ ≤ = ( le ‘ 𝐾 )
cvr1.j ⊢ ∨ = ( join ‘ 𝐾 )
cvr1.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
cvr1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion cvr1 ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( ¬ 𝑃 ≤ 𝑋 ↔ 𝑋 𝐶 ( 𝑋 ∨ 𝑃 ) ) )

Proof

Step Hyp Ref Expression
1 cvr1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cvr1.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cvr1.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cvr1.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
5 cvr1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 hlomcmcv ⊢ ( 𝐾 ∈ HL → ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) )
7 1 2 3 4 5 cvlcvr1 ⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( ¬ 𝑃 ≤ 𝑋 ↔ 𝑋 𝐶 ( 𝑋 ∨ 𝑃 ) ) )
8 6 7 syl3an1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( ¬ 𝑃 ≤ 𝑋 ↔ 𝑋 𝐶 ( 𝑋 ∨ 𝑃 ) ) )