Theorem cvlsupr4

Description: Consequence of superposition condition ( P .\/ R ) = ( Q .\/ R ) . (Contributed by NM, 9-Nov-2012)

	Ref	Expression
Hypotheses	cvlsupr2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cvlsupr2.l	⊢ ≤ = ( le ‘ 𝐾 )
	cvlsupr2.j	⊢ ∨ = ( join ‘ 𝐾 )
Assertion	cvlsupr4	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )

Proof

Step	Hyp	Ref	Expression
1		cvlsupr2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2		cvlsupr2.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cvlsupr2.j	⊢ ∨ = ( join ‘ 𝐾 )
4	1 2 3	cvlsupr2	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
5		simp3	⊢ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
6	4 5	biimtrdi	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
7	6	3exp	⊢ ( 𝐾 ∈ CvLat → ( ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 → ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) )
8	7	imp4a	⊢ ( 𝐾 ∈ CvLat → ( ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
9	8	3imp	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )