This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem clatglbss

Description: Subset law for greatest lower bound. (Contributed by Mario Carneiro, 16-Apr-2014)

Ref Expression
Hypotheses clatglb.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
clatglb.l ⊢ ≤ = ( le ‘ 𝐾 )
clatglb.g ⊢ 𝐺 = ( glb ‘ 𝐾 )
Assertion clatglbss ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) → ( 𝐺 ‘ 𝑇 ) ≤ ( 𝐺 ‘ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 clatglb.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 clatglb.l ⊢ ≤ = ( le ‘ 𝐾 )
3 clatglb.g ⊢ 𝐺 = ( glb ‘ 𝐾 )
4 simpl1 ⊢ ( ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) ∧ 𝑦 ∈ 𝑆 ) → 𝐾 ∈ CLat )
5 simpl2 ⊢ ( ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) ∧ 𝑦 ∈ 𝑆 ) → 𝑇 ⊆ 𝐵 )
6 simp3 ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) → 𝑆 ⊆ 𝑇 )
7 6 sselda ⊢ ( ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) ∧ 𝑦 ∈ 𝑆 ) → 𝑦 ∈ 𝑇 )
8 1 2 3 clatglble ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑦 ∈ 𝑇 ) → ( 𝐺 ‘ 𝑇 ) ≤ 𝑦 )
9 4 5 7 8 syl3anc ⊢ ( ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) ∧ 𝑦 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑇 ) ≤ 𝑦 )
10 9 ralrimiva ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) → ∀ 𝑦 ∈ 𝑆 ( 𝐺 ‘ 𝑇 ) ≤ 𝑦 )
11 simp1 ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) → 𝐾 ∈ CLat )
12 1 3 clatglbcl ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ) → ( 𝐺 ‘ 𝑇 ) ∈ 𝐵 )
13 12 3adant3 ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) → ( 𝐺 ‘ 𝑇 ) ∈ 𝐵 )
14 sstr ⊢ ( ( 𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ 𝐵 ) → 𝑆 ⊆ 𝐵 )
15 14 ancoms ⊢ ( ( 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) → 𝑆 ⊆ 𝐵 )
16 15 3adant1 ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) → 𝑆 ⊆ 𝐵 )
17 1 2 3 clatleglb ⊢ ( ( 𝐾 ∈ CLat ∧ ( 𝐺 ‘ 𝑇 ) ∈ 𝐵 ∧ 𝑆 ⊆ 𝐵 ) → ( ( 𝐺 ‘ 𝑇 ) ≤ ( 𝐺 ‘ 𝑆 ) ↔ ∀ 𝑦 ∈ 𝑆 ( 𝐺 ‘ 𝑇 ) ≤ 𝑦 ) )
18 11 13 16 17 syl3anc ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) → ( ( 𝐺 ‘ 𝑇 ) ≤ ( 𝐺 ‘ 𝑆 ) ↔ ∀ 𝑦 ∈ 𝑆 ( 𝐺 ‘ 𝑇 ) ≤ 𝑦 ) )
19 10 18 mpbird ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇 ) → ( 𝐺 ‘ 𝑇 ) ≤ ( 𝐺 ‘ 𝑆 ) )