Theorem hlsupr

Description: A Hilbert lattice has the superposition property. Theorem 13.2 in Crawley p. 107. (Contributed by NM, 30-Jan-2012)

	Ref	Expression
Hypotheses	hlsupr.l	⊢ ≤ = ( le ‘ 𝐾 )
	hlsupr.j	⊢ ∨ = ( join ‘ 𝐾 )
	hlsupr.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	hlsupr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hlsupr.l	⊢ ≤ = ( le ‘ 𝐾 )
2		hlsupr.j	⊢ ∨ = ( join ‘ 𝐾 )
3		hlsupr.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
5	4 1 2 3	hlsuprexch	⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( ( 𝑃 ≠ 𝑄 → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ∀ 𝑟 ∈ ( Base ‘ 𝐾 ) ( ( ¬ 𝑃 ≤ 𝑟 ∧ 𝑃 ≤ ( 𝑟 ∨ 𝑄 ) ) → 𝑄 ≤ ( 𝑟 ∨ 𝑃 ) ) ) )
6	5	simpld	⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
7	6	imp	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )