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Metamath Proof Explorer

Theorem latlem12

Description: An element is less than or equal to a meet iff the element is less than or equal to each argument of the meet. (Contributed by NM, 21-Oct-2011)

Ref Expression
Hypotheses latmle.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latmle.l ⊢ ≤ = ( le ‘ 𝐾 )
latmle.m ⊢ ∧ = ( meet ‘ 𝐾 )
Assertion latlem12 ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑍 ) ↔ 𝑋 ≤ ( 𝑌 ∧ 𝑍 ) ) )

Proof

Step Hyp Ref Expression
1 latmle.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latmle.l ⊢ ≤ = ( le ‘ 𝐾 )
3 latmle.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 latpos ⊢ ( 𝐾 ∈ Lat → 𝐾 ∈ Poset )
5 4 adantr ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ Poset )
6 simpr2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
7 simpr3 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
8 simpr1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
9 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
10 simpl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ Lat )
11 1 9 3 10 6 7 latcl2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ⟨ 𝑌 , 𝑍 ⟩ ∈ dom ( join ‘ 𝐾 ) ∧ ⟨ 𝑌 , 𝑍 ⟩ ∈ dom ∧ ) )
12 11 simprd ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ⟨ 𝑌 , 𝑍 ⟩ ∈ dom ∧ )
13 1 2 3 5 6 7 8 12 meetle ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑍 ) ↔ 𝑋 ≤ ( 𝑌 ∧ 𝑍 ) ) )