Metamath Proof Explorer

Theorem latleeqj1

Description: "Less than or equal to" in terms of join. ( chlejb1 analog.) (Contributed by NM, 21-Oct-2011)

Ref Expression
Hypotheses latlej.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latlej.l ⊢ ≤ = ( le ‘ 𝐾 )
latlej.j ⊢ ∨ = ( join ‘ 𝐾 )
Assertion latleeqj1 ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 ∨ 𝑌 ) = 𝑌 ) )

Proof

Step Hyp Ref Expression
1 latlej.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latlej.l ⊢ ≤ = ( le ‘ 𝐾 )
3 latlej.j ⊢ ∨ = ( join ‘ 𝐾 )
4 1 2 latref ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ≤ 𝑌 )
5 4 3adant2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ≤ 𝑌 )
6 5 biantrud ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑌 ) ) )
7 simp1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Lat )
8 simp2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
9 simp3 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 )
10 1 2 3 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑌 ) ↔ ( 𝑋 ∨ 𝑌 ) ≤ 𝑌 ) )
11 7 8 9 9 10 syl13anc ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑌 ) ↔ ( 𝑋 ∨ 𝑌 ) ≤ 𝑌 ) )
12 6 11 bitrd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 ∨ 𝑌 ) ≤ 𝑌 ) )
13 1 2 3 latlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) )
14 13 biantrud ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ∨ 𝑌 ) ≤ 𝑌 ↔ ( ( 𝑋 ∨ 𝑌 ) ≤ 𝑌 ∧ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) ) )
15 12 14 bitrd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( ( 𝑋 ∨ 𝑌 ) ≤ 𝑌 ∧ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) ) )
16 latpos ⊢ ( 𝐾 ∈ Lat → 𝐾 ∈ Poset )
17 16 3ad2ant1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Poset )
18 1 3 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 )
19 1 2 posasymb ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝑋 ∨ 𝑌 ) ≤ 𝑌 ∧ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) ↔ ( 𝑋 ∨ 𝑌 ) = 𝑌 ) )
20 17 18 9 19 syl3anc ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝑋 ∨ 𝑌 ) ≤ 𝑌 ∧ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) ↔ ( 𝑋 ∨ 𝑌 ) = 𝑌 ) )
21 15 20 bitrd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 ∨ 𝑌 ) = 𝑌 ) )