Metamath Proof Explorer


Theorem lvolneatN

Description: No lattice volume is an atom. (Contributed by NM, 15-Jul-2012) (New usage is discouraged.)

Ref Expression
Hypotheses lvolneat.a 𝐴 = ( Atoms ‘ 𝐾 )
lvolneat.v 𝑉 = ( LVols ‘ 𝐾 )
Assertion lvolneatN ( ( 𝐾 ∈ HL ∧ 𝑋𝑉 ) → ¬ 𝑋𝐴 )

Proof

Step Hyp Ref Expression
1 lvolneat.a 𝐴 = ( Atoms ‘ 𝐾 )
2 lvolneat.v 𝑉 = ( LVols ‘ 𝐾 )
3 hllat ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
4 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
5 4 2 lvolbase ( 𝑋𝑉𝑋 ∈ ( 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 ∧ 𝑋𝑉 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
9 6 1 2 lvolnleat ( ( 𝐾 ∈ HL ∧ 𝑋𝑉𝑋𝐴 ) → ¬ 𝑋 ( le ‘ 𝐾 ) 𝑋 )
10 9 3expia ( ( 𝐾 ∈ HL ∧ 𝑋𝑉 ) → ( 𝑋𝐴 → ¬ 𝑋 ( le ‘ 𝐾 ) 𝑋 ) )
11 8 10 mt2d ( ( 𝐾 ∈ HL ∧ 𝑋𝑉 ) → ¬ 𝑋𝐴 )