Metamath Proof Explorer


Theorem islvol4

Description: The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012)

Ref Expression
Hypotheses lvolset.b 𝐵 = ( Base ‘ 𝐾 )
lvolset.c 𝐶 = ( ⋖ ‘ 𝐾 )
lvolset.p 𝑃 = ( LPlanes ‘ 𝐾 )
lvolset.v 𝑉 = ( LVols ‘ 𝐾 )
Assertion islvol4 ( ( 𝐾𝐴𝑋𝐵 ) → ( 𝑋𝑉 ↔ ∃ 𝑦𝑃 𝑦 𝐶 𝑋 ) )

Proof

Step Hyp Ref Expression
1 lvolset.b 𝐵 = ( Base ‘ 𝐾 )
2 lvolset.c 𝐶 = ( ⋖ ‘ 𝐾 )
3 lvolset.p 𝑃 = ( LPlanes ‘ 𝐾 )
4 lvolset.v 𝑉 = ( LVols ‘ 𝐾 )
5 1 2 3 4 islvol ( 𝐾𝐴 → ( 𝑋𝑉 ↔ ( 𝑋𝐵 ∧ ∃ 𝑦𝑃 𝑦 𝐶 𝑋 ) ) )
6 5 baibd ( ( 𝐾𝐴𝑋𝐵 ) → ( 𝑋𝑉 ↔ ∃ 𝑦𝑃 𝑦 𝐶 𝑋 ) )