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Metamath Proof Explorer

Theorem cvlsupr6

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

Ref Expression
Hypotheses cvlsupr5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cvlsupr5.j ⊢ ∨ = ( join ‘ 𝐾 )
Assertion cvlsupr6 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → 𝑅 ≠ 𝑄 )

Proof

Step Hyp Ref Expression
1 cvlsupr5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2 cvlsupr5.j ⊢ ∨ = ( join ‘ 𝐾 )
3 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
4 1 3 2 cvlsupr2 ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) ) )
5 simp2 ⊢ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ≠ 𝑄 )
6 4 5 biimtrdi ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) → 𝑅 ≠ 𝑄 ) )
7 6 3exp ⊢ ( 𝐾 ∈ CvLat → ( ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 → ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) → 𝑅 ≠ 𝑄 ) ) ) )
8 7 imp4a ⊢ ( 𝐾 ∈ CvLat → ( ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) → 𝑅 ≠ 𝑄 ) ) )
9 8 3imp ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → 𝑅 ≠ 𝑄 )