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Metamath Proof Explorer

Theorem atnem0

Description: The meet of distinct atoms is zero. ( atnemeq0 analog.) (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses atnem0.m ⊢ ∧ = ( meet ‘ 𝐾 )
atnem0.z ⊢ 0 = ( 0. ‘ 𝐾 )
atnem0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion atnem0 ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 ↔ ( 𝑃 ∧ 𝑄 ) = 0 ) )

Proof

Step Hyp Ref Expression
1 atnem0.m ⊢ ∧ = ( meet ‘ 𝐾 )
2 atnem0.z ⊢ 0 = ( 0. ‘ 𝐾 )
3 atnem0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
5 4 3 atncmp ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑄 ↔ 𝑃 ≠ 𝑄 ) )
6 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
7 6 3 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
8 6 4 1 2 3 atnle ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑄 ↔ ( 𝑃 ∧ 𝑄 ) = 0 ) )
9 7 8 syl3an3 ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( ¬ 𝑃 ( le ‘ 𝐾 ) 𝑄 ↔ ( 𝑃 ∧ 𝑄 ) = 0 ) )
10 5 9 bitr3d ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 ↔ ( 𝑃 ∧ 𝑄 ) = 0 ) )