Metamath Proof Explorer


Theorem hlatlej1

Description: A join's first argument is less than or equal to the join. Special case of latlej1 to show an atom is on a line. (Contributed by NM, 15-May-2013)

Ref Expression
Hypotheses hlatlej.l = ( le ‘ 𝐾 )
hlatlej.j = ( join ‘ 𝐾 )
hlatlej.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion hlatlej1 ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → 𝑃 ( 𝑃 𝑄 ) )

Proof

Step Hyp Ref Expression
1 hlatlej.l = ( le ‘ 𝐾 )
2 hlatlej.j = ( join ‘ 𝐾 )
3 hlatlej.a 𝐴 = ( Atoms ‘ 𝐾 )
4 hllat ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
5 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
6 5 3 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
7 5 3 atbase ( 𝑄𝐴𝑄 ∈ ( Base ‘ 𝐾 ) )
8 5 1 2 latlej1 ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → 𝑃 ( 𝑃 𝑄 ) )
9 4 6 7 8 syl3an ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → 𝑃 ( 𝑃 𝑄 ) )