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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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 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 ∧ 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 ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
8 1 7 3 4 5 6 latcl2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ( join ‘ 𝐾 ) ∧ ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ ) )
9 8 simprd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ )
10 1 2 3 4 5 6 9 lemeet1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 )