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Metamath Proof Explorer

Theorem cvlatcvr1

Description: An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses cvlatcvr1.j ⊢ ∨ = ( join ‘ 𝐾 )
cvlatcvr1.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
cvlatcvr1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion cvlatcvr1 ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 ↔ 𝑃 𝐶 ( 𝑃 ∨ 𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 cvlatcvr1.j ⊢ ∨ = ( join ‘ 𝐾 )
2 cvlatcvr1.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
3 cvlatcvr1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 simp13 ⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ CvLat )
5 cvlatl ⊢ ( 𝐾 ∈ CvLat → 𝐾 ∈ AtLat )
6 4 5 syl ⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ AtLat )
7 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
8 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
9 7 8 3 atnem0 ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 ↔ ( 𝑃 ( meet ‘ 𝐾 ) 𝑄 ) = ( 0. ‘ 𝐾 ) ) )
10 6 9 syld3an1 ⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 ↔ ( 𝑃 ( meet ‘ 𝐾 ) 𝑄 ) = ( 0. ‘ 𝐾 ) ) )
11 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
12 11 3 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
13 11 1 7 8 2 3 cvlcvrp ⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ 𝐴 ) → ( ( 𝑃 ( meet ‘ 𝐾 ) 𝑄 ) = ( 0. ‘ 𝐾 ) ↔ 𝑃 𝐶 ( 𝑃 ∨ 𝑄 ) ) )
14 12 13 syl3an2 ⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( ( 𝑃 ( meet ‘ 𝐾 ) 𝑄 ) = ( 0. ‘ 𝐾 ) ↔ 𝑃 𝐶 ( 𝑃 ∨ 𝑄 ) ) )
15 10 14 bitrd ⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 ↔ 𝑃 𝐶 ( 𝑃 ∨ 𝑄 ) ) )