Metamath Proof Explorer


Theorem latmle1

Description: A meet is less than or equal to its first argument. (Contributed by NM, 21-Oct-2011)

Ref Expression
Hypotheses latmle.b 𝐵 = ( Base ‘ 𝐾 )
latmle.l = ( le ‘ 𝐾 )
latmle.m = ( meet ‘ 𝐾 )
Assertion latmle1 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝑌 ) 𝑋 )

Proof

Step Hyp Ref Expression
1 latmle.b 𝐵 = ( Base ‘ 𝐾 )
2 latmle.l = ( le ‘ 𝐾 )
3 latmle.m = ( meet ‘ 𝐾 )
4 simp1 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → 𝐾 ∈ Lat )
5 simp2 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → 𝑋𝐵 )
6 simp3 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → 𝑌𝐵 )
7 eqid ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
8 1 7 3 4 5 6 latcl2 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ( join ‘ 𝐾 ) ∧ ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ) )
9 8 simprd ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom )
10 1 2 3 4 5 6 9 lemeet1 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝑌 ) 𝑋 )