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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 ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝑉 ↔ ∃ 𝑦 ∈ 𝑃 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 ⊢ ( 𝐾 ∈ 𝐴 → ( 𝑋 ∈ 𝑉 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑋 ) ) )
6 5 baibd ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝑉 ↔ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑋 ) )