Theorem atcmp

Description: If two atoms are comparable, they are equal. ( atsseq analog.) (Contributed by NM, 13-Oct-2011)

	Ref	Expression
Hypotheses	atcmp.l	⊢ ≤ = ( le ‘ 𝐾 )
	atcmp.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	atcmp	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄 ) )

Proof

Step	Hyp	Ref	Expression
1		atcmp.l	⊢ ≤ = ( le ‘ 𝐾 )
2		atcmp.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		atlpos	⊢ ( 𝐾 ∈ AtLat → 𝐾 ∈ Poset )
4	3	3ad2ant1	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ Poset )
5		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
6	5 2	atbase	⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
7	6	3ad2ant2	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
8	5 2	atbase	⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
9	8	3ad2ant3	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
10		eqid	⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
11	5 10	atl0cl	⊢ ( 𝐾 ∈ AtLat → ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) )
12	11	3ad2ant1	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) )
13		eqid	⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 )
14	10 13 2	atcvr0	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ) → ( 0. ‘ 𝐾 ) ( ⋖ ‘ 𝐾 ) 𝑃 )
15	14	3adant3	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 0. ‘ 𝐾 ) ( ⋖ ‘ 𝐾 ) 𝑃 )
16	10 13 2	atcvr0	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ) → ( 0. ‘ 𝐾 ) ( ⋖ ‘ 𝐾 ) 𝑄 )
17	16	3adant2	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 0. ‘ 𝐾 ) ( ⋖ ‘ 𝐾 ) 𝑄 )
18	5 1 13	cvrcmp	⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) ∧ ( ( 0. ‘ 𝐾 ) ( ⋖ ‘ 𝐾 ) 𝑃 ∧ ( 0. ‘ 𝐾 ) ( ⋖ ‘ 𝐾 ) 𝑄 ) ) → ( 𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄 ) )
19	4 7 9 12 15 17 18	syl132anc	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄 ) )