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 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) |
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 | ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
7 | 5 3 | atbase | ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( 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 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) |