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Metamath Proof Explorer

Theorem latlej1

Description: A join's first argument is less than or equal to the join. ( chub1 analog.) (Contributed by NM, 17-Sep-2011)

Ref Expression
Hypotheses latlej.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latlej.l ⊢ ≤ = ( le ‘ 𝐾 )
latlej.j ⊢ ∨ = ( join ‘ 𝐾 )
Assertion latlej1 ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ≤ ( 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Lat )
5 simp2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
6 simp3 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 )
7 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
8 1 3 7 4 5 6 latcl2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ ∧ ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ( meet ‘ 𝐾 ) ) )
9 8 simpld ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
10 1 2 3 4 5 6 9 lejoin1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) )