Metamath Proof Explorer


Theorem latlej2

Description: A join's second argument is less than or equal to the join. ( chub2 analog.) (Contributed by NM, 17-Sep-2011)

Ref Expression
Hypotheses latlej.b 𝐵 = ( Base ‘ 𝐾 )
latlej.l = ( le ‘ 𝐾 )
latlej.j = ( join ‘ 𝐾 )
Assertion latlej2 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → 𝑌 ( 𝑋 𝑌 ) )

Proof

Step Hyp Ref Expression
1 latlej.b 𝐵 = ( Base ‘ 𝐾 )
2 latlej.l = ( le ‘ 𝐾 )
3 latlej.j = ( join ‘ 𝐾 )
4 simp1 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → 𝐾 ∈ Lat )
5 simp2 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → 𝑋𝐵 )
6 simp3 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → 𝑌𝐵 )
7 eqid ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
8 1 3 7 4 5 6 latcl2 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ( meet ‘ 𝐾 ) ) )
9 8 simpld ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom )
10 1 2 3 4 5 6 9 lejoin2 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → 𝑌 ( 𝑋 𝑌 ) )