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Metamath Proof Explorer

Theorem cvlsupr8

Description: Consequence of superposition condition ( P .\/ R ) = ( Q .\/ R ) . (Contributed by NM, 24-Nov-2012)

Ref Expression
Hypotheses cvlsupr5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cvlsupr5.j ⊢ ∨ = ( join ‘ 𝐾 )
Assertion cvlsupr8 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 cvlsupr5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2 cvlsupr5.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cvllat ⊢ ( 𝐾 ∈ CvLat → 𝐾 ∈ Lat )
4 3 3ad2ant1 ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → 𝐾 ∈ Lat )
5 simp22 ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → 𝑄 ∈ 𝐴 )
6 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
7 6 1 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
8 5 7 syl ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
9 simp23 ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → 𝑅 ∈ 𝐴 )
10 6 1 atbase ⊢ ( 𝑅 ∈ 𝐴 → 𝑅 ∈ ( Base ‘ 𝐾 ) )
11 9 10 syl ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
12 6 2 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑄 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑄 ) )
13 4 8 11 12 syl3anc ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → ( 𝑄 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑄 ) )
14 simp3r ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) )
15 1 2 cvlsupr7 ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) )
16 13 14 15 3eqtr4rd ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) )