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 ) )