Theorem islpln2ah

Description: The predicate "is a lattice plane" for join of atoms. Version of islpln2a expressed with an abbreviation hypothesis. (Contributed by NM, 30-Jul-2012)

	Ref	Expression
Hypotheses	islpln2a.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	islpln2a.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	islpln2a.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	islpln2a.p	$$⊢P=\mathrm{LPlanes}\left(K\right)$$
	islpln2a.y	$$⊢Y=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S$$
Assertion	islpln2ah	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to \left(Y\in P\leftrightarrow \left(Q\ne R\wedge \neg S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		islpln2a.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		islpln2a.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		islpln2a.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		islpln2a.p	$$⊢P=\mathrm{LPlanes}\left(K\right)$$
5		islpln2a.y	$$⊢Y=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S$$
6	5	eleq1i	$$⊢Y\in P\leftrightarrow \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\in P$$
7	1 2 3 4	islpln2a	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\in P\leftrightarrow \left(Q\ne R\wedge \neg S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)$$
8	6 7	syl5bb	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to \left(Y\in P\leftrightarrow \left(Q\ne R\wedge \neg S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)$$