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 ) ) )