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Metamath Proof Explorer

Theorem latmlem1

Description: Add meet to both sides of a lattice ordering. (Contributed by NM, 10-Nov-2011)

Ref Expression
Hypotheses latmle.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latmle.l ⊢ ≤ = ( le ‘ 𝐾 )
latmle.m ⊢ ∧ = ( meet ‘ 𝐾 )
Assertion latmlem1 ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑋 ∧ 𝑍 ) ≤ ( 𝑌 ∧ 𝑍 ) ) )

Proof

Step Hyp Ref Expression
1 latmle.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latmle.l ⊢ ≤ = ( le ‘ 𝐾 )
3 latmle.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 1 2 3 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑍 ) ≤ 𝑋 )
5 4 3adant3r2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∧ 𝑍 ) ≤ 𝑋 )
6 simpl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ Lat )
7 1 3 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑍 ) ∈ 𝐵 )
8 7 3adant3r2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∧ 𝑍 ) ∈ 𝐵 )
9 simpr1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
10 simpr2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
11 1 2 lattr ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑋 ∧ 𝑍 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝑋 ∧ 𝑍 ) ≤ 𝑋 ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ∧ 𝑍 ) ≤ 𝑌 ) )
12 6 8 9 10 11 syl13anc ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( ( 𝑋 ∧ 𝑍 ) ≤ 𝑋 ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ∧ 𝑍 ) ≤ 𝑌 ) )
13 5 12 mpand ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑋 ∧ 𝑍 ) ≤ 𝑌 ) )
14 1 2 3 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑍 ) ≤ 𝑍 )
15 14 3adant3r2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∧ 𝑍 ) ≤ 𝑍 )
16 13 15 jctird ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 → ( ( 𝑋 ∧ 𝑍 ) ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑍 ) ≤ 𝑍 ) ) )
17 simpr3 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
18 8 10 17 3jca ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑍 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) )
19 1 2 3 latlem12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑋 ∧ 𝑍 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( ( 𝑋 ∧ 𝑍 ) ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑍 ) ≤ 𝑍 ) ↔ ( 𝑋 ∧ 𝑍 ) ≤ ( 𝑌 ∧ 𝑍 ) ) )
20 18 19 syldan ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( ( 𝑋 ∧ 𝑍 ) ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑍 ) ≤ 𝑍 ) ↔ ( 𝑋 ∧ 𝑍 ) ≤ ( 𝑌 ∧ 𝑍 ) ) )
21 16 20 sylibd ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑋 ∧ 𝑍 ) ≤ ( 𝑌 ∧ 𝑍 ) ) )