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 ) )