This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lejoin1

Description: A join's first argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011)

Ref Expression
Hypotheses joinval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
joinval2.l ⊢ ≤ = ( le ‘ 𝐾 )
joinval2.j ⊢ ∨ = ( join ‘ 𝐾 )
joinval2.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
joinval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
joinval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
joinlem.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
Assertion lejoin1 ( 𝜑 → 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) )

Proof

Step Hyp Ref Expression
1 joinval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 joinval2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 joinval2.j ⊢ ∨ = ( join ‘ 𝐾 )
4 joinval2.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
5 joinval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 joinval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 joinlem.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
8 1 2 3 4 5 6 7 joinlem ⊢ ( 𝜑 → ( ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ) ) )
9 8 simplld ⊢ ( 𝜑 → 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) )