Theorem islvol

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	islvol	⊢ ( 𝐾 ∈ 𝐴 → ( 𝑋 ∈ 𝑉 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 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	lvolset	⊢ ( 𝐾 ∈ 𝐴 → 𝑉 = { 𝑥 ∈ 𝐵 ∣ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑥 } )
6	5	eleq2d	⊢ ( 𝐾 ∈ 𝐴 → ( 𝑋 ∈ 𝑉 ↔ 𝑋 ∈ { 𝑥 ∈ 𝐵 ∣ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑥 } ) )
7		breq2	⊢ ( 𝑥 = 𝑋 → ( 𝑦 𝐶 𝑥 ↔ 𝑦 𝐶 𝑋 ) )
8	7	rexbidv	⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑥 ↔ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑋 ) )
9	8	elrab	⊢ ( 𝑋 ∈ { 𝑥 ∈ 𝐵 ∣ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑥 } ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑋 ) )
10	6 9	bitrdi	⊢ ( 𝐾 ∈ 𝐴 → ( 𝑋 ∈ 𝑉 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑋 ) ) )