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Metamath Proof Explorer

Theorem lvolnelpln

Description: No lattice volume is a lattice plane. (Contributed by NM, 19-Jun-2012)

Ref Expression
Hypotheses lvolnelpln.p ⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
lvolnelpln.v ⊢ 𝑉 = ( LVols ‘ 𝐾 )
Assertion lvolnelpln ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ) → ¬ 𝑋 ∈ 𝑃 )

Proof

Step Hyp Ref Expression
1 lvolnelpln.p ⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
2 lvolnelpln.v ⊢ 𝑉 = ( LVols ‘ 𝐾 )
3 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
4 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
5 4 2 lvolbase ⊢ ( 𝑋 ∈ 𝑉 → 𝑋 ∈ ( Base ‘ 𝐾 ) )
6 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
7 4 6 latref ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ ( Base ‘ 𝐾 ) ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
8 3 5 7 syl2an ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
9 6 1 2 lvolnlelpln ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑃 ) → ¬ 𝑋 ( le ‘ 𝐾 ) 𝑋 )
10 9 3expia ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 ∈ 𝑃 → ¬ 𝑋 ( le ‘ 𝐾 ) 𝑋 ) )
11 8 10 mt2d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ) → ¬ 𝑋 ∈ 𝑃 )