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Metamath Proof Explorer

Theorem clatlem

Description: Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011)

Ref Expression
Hypotheses clatlem.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
clatlem.u ⊢ 𝑈 = ( lub ‘ 𝐾 )
clatlem.g ⊢ 𝐺 = ( glb ‘ 𝐾 )
Assertion clatlem ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( ( 𝑈 ‘ 𝑆 ) ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ∈ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 clatlem.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 clatlem.u ⊢ 𝑈 = ( lub ‘ 𝐾 )
3 clatlem.g ⊢ 𝐺 = ( glb ‘ 𝐾 )
4 simpl ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → 𝐾 ∈ CLat )
5 1 fvexi ⊢ 𝐵 ∈ V
6 5 elpw2 ⊢ ( 𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵 )
7 6 biimpri ⊢ ( 𝑆 ⊆ 𝐵 → 𝑆 ∈ 𝒫 𝐵 )
8 7 adantl ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → 𝑆 ∈ 𝒫 𝐵 )
9 1 2 3 isclat ⊢ ( 𝐾 ∈ CLat ↔ ( 𝐾 ∈ Poset ∧ ( dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵 ) ) )
10 9 biimpi ⊢ ( 𝐾 ∈ CLat → ( 𝐾 ∈ Poset ∧ ( dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵 ) ) )
11 10 adantr ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( 𝐾 ∈ Poset ∧ ( dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵 ) ) )
12 11 simprld ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → dom 𝑈 = 𝒫 𝐵 )
13 8 12 eleqtrrd ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → 𝑆 ∈ dom 𝑈 )
14 1 2 4 13 lubcl ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( 𝑈 ‘ 𝑆 ) ∈ 𝐵 )
15 11 simprrd ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → dom 𝐺 = 𝒫 𝐵 )
16 8 15 eleqtrrd ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → 𝑆 ∈ dom 𝐺 )
17 1 3 4 16 glbcl ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( 𝐺 ‘ 𝑆 ) ∈ 𝐵 )
18 14 17 jca ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( ( 𝑈 ‘ 𝑆 ) ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ∈ 𝐵 ) )