Metamath Proof Explorer

Theorem latleeqm1

Description: "Less than or equal to" in terms of meet. (Contributed by NM, 7-Nov-2011)

Ref Expression
Hypotheses latmle.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latmle.l ⊢ ≤ = ( le ‘ 𝐾 )
latmle.m ⊢ ∧ = ( meet ‘ 𝐾 )
Assertion latleeqm1 ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 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 1 2 latref ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ≤ 𝑋 )
5 4 3adant3 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ≤ 𝑋 )
6 5 biantrurd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 ≤ 𝑋 ∧ 𝑋 ≤ 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 latlem12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌 ) ↔ 𝑋 ≤ ( 𝑋 ∧ 𝑌 ) ) )
11 7 8 8 9 10 syl13anc ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌 ) ↔ 𝑋 ≤ ( 𝑋 ∧ 𝑌 ) ) )
12 6 11 bitrd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ 𝑋 ≤ ( 𝑋 ∧ 𝑌 ) ) )
13 1 2 3 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 )
14 13 biantrurd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ ( 𝑋 ∧ 𝑌 ) ↔ ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ 𝑋 ≤ ( 𝑋 ∧ 𝑌 ) ) ) )
15 12 14 bitrd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ 𝑋 ≤ ( 𝑋 ∧ 𝑌 ) ) ) )
16 latpos ⊢ ( 𝐾 ∈ Lat → 𝐾 ∈ Poset )
17 16 3ad2ant1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Poset )
18 1 3 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
19 1 2 posasymb ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ 𝑋 ≤ ( 𝑋 ∧ 𝑌 ) ) ↔ ( 𝑋 ∧ 𝑌 ) = 𝑋 ) )
20 17 18 8 19 syl3anc ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ 𝑋 ≤ ( 𝑋 ∧ 𝑌 ) ) ↔ ( 𝑋 ∧ 𝑌 ) = 𝑋 ) )
21 15 20 bitrd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 ∧ 𝑌 ) = 𝑋 ) )