Theorem hlexchb1

Description: A Hilbert lattice has the exchange property. (Contributed by NM, 16-Nov-2011)

	Ref	Expression
Hypotheses	hlsuprexch.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	hlsuprexch.l	⊢ ≤ = ( le ‘ 𝐾 )
	hlsuprexch.j	⊢ ∨ = ( join ‘ 𝐾 )
	hlsuprexch.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	hlexchb1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ 𝑃 ≤ 𝑋 ) → ( 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ↔ ( 𝑋 ∨ 𝑃 ) = ( 𝑋 ∨ 𝑄 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hlsuprexch.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		hlsuprexch.l	⊢ ≤ = ( le ‘ 𝐾 )
3		hlsuprexch.j	⊢ ∨ = ( join ‘ 𝐾 )
4		hlsuprexch.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		hlcvl	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CvLat )
6	1 2 3 4	cvlexchb1	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ 𝑃 ≤ 𝑋 ) → ( 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ↔ ( 𝑋 ∨ 𝑃 ) = ( 𝑋 ∨ 𝑄 ) ) )
7	5 6	syl3an1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ 𝑃 ≤ 𝑋 ) → ( 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ↔ ( 𝑋 ∨ 𝑃 ) = ( 𝑋 ∨ 𝑄 ) ) )