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Metamath Proof Explorer

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 bitrid ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑃 ≠ 𝑄 → ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) )