Theorem cvlsupr3

Description: Two equivalent ways of expressing that R is a superposition of P and Q , which can replace the superposition part of ishlat1 , ( x =/= y -> E. z e. A ( z =/= x /\ z =/= y /\ z .<_ ( x .\/ y ) ) ) , with the simpler E. z e. A ( x .\/ z ) = ( y .\/ z ) as shown in ishlat3N . (Contributed by NM, 5-Nov-2012)

	Ref	Expression
Hypotheses	cvlsupr2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cvlsupr2.l	⊢ ≤ = ( le ‘ 𝐾 )
	cvlsupr2.j	⊢ ∨ = ( join ‘ 𝐾 )
Assertion	cvlsupr3	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑃 ≠ 𝑄 → ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 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		df-ne	⊢ ( 𝑃 ≠ 𝑄 ↔ ¬ 𝑃 = 𝑄 )
5	4	imbi1i	⊢ ( ( 𝑃 ≠ 𝑄 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ↔ ( ¬ 𝑃 = 𝑄 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) )
6		oveq1	⊢ ( 𝑃 = 𝑄 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) )
7	6	biantrur	⊢ ( ( ¬ 𝑃 = 𝑄 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ↔ ( ( 𝑃 = 𝑄 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( ¬ 𝑃 = 𝑄 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) )
8		pm4.83	⊢ ( ( ( 𝑃 = 𝑄 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( ¬ 𝑃 = 𝑄 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) ↔ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) )
9	5 7 8	3bitrri	⊢ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑃 ≠ 𝑄 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) )
10	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 ) ) ) )
11	10	3expia	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( 𝑃 ≠ 𝑄 → ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) )
12	11	pm5.74d	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑃 ≠ 𝑄 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ↔ ( 𝑃 ≠ 𝑄 → ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) )
13	9 12	syl5bb	⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑃 ≠ 𝑄 → ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) )