Theorem islvol4

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

	Ref	Expression
Hypotheses	lvolset.b	$$⊢B={\mathrm{Base}}_K$$
	lvolset.c	$$⊢C={⋖}_K$$
	lvolset.p	$$⊢P=\mathrm{LPlanes}\left(K\right)$$
	lvolset.v	$$⊢V=\mathrm{LVols}\left(K\right)$$
Assertion	islvol4	$$⊢\left(K\in A\wedge X\in B\right)\to \left(X\in V\leftrightarrow \exists y\in PyCX\right)$$

Proof

Step	Hyp	Ref	Expression
1		lvolset.b	$$⊢B={\mathrm{Base}}_K$$
2		lvolset.c	$$⊢C={⋖}_K$$
3		lvolset.p	$$⊢P=\mathrm{LPlanes}\left(K\right)$$
4		lvolset.v	$$⊢V=\mathrm{LVols}\left(K\right)$$
5	1 2 3 4	islvol	$$⊢K\in A\to \left(X\in V\leftrightarrow \left(X\in B\wedge \exists y\in PyCX\right)\right)$$
6	5	baibd	$$⊢\left(K\in A\wedge X\in B\right)\to \left(X\in V\leftrightarrow \exists y\in PyCX\right)$$