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Metamath Proof Explorer

Theorem latjle12

Description: A join is less than or equal to a third value iff each argument is less than or equal to the third value. ( chlub analog.) (Contributed by NM, 17-Sep-2011)

Ref Expression
Hypotheses latlej.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latlej.l ⊢ ≤ = ( le ‘ 𝐾 )
latlej.j ⊢ ∨ = ( join ‘ 𝐾 )
Assertion latjle12 ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍 ) ↔ ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 ) )

Proof

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