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Mirrors > Home > MPE Home > Th. List > latmcl | Structured version Visualization version GIF version |
Description: Closure of meet operation in a lattice. (incom 4139 analog.) (Contributed by NM, 14-Sep-2011.) |
Ref | Expression |
---|---|
latmcl.b | ⊢ 𝐵 = (Base‘𝐾) |
latmcl.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
latmcl | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latmcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2739 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | latmcl.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
4 | 1, 2, 3 | latlem 18136 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(join‘𝐾)𝑌) ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵)) |
5 | 4 | simprd 495 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵) |
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