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 ) )