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Metamath Proof Explorer

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 ) )