Theorem lhpat2

Description: Create an atom under a co-atom. Part of proof of Lemma B in Crawley p. 112. (Contributed by NM, 21-Nov-2012)

	Ref	Expression
Hypotheses	lhpat.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	lhpat.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	lhpat.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	lhpat.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	lhpat.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	lhpat2.r	$$⊢R=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
Assertion	lhpat2	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge P\ne Q\right)\right)\to R\in A$$

Proof

Step	Hyp	Ref	Expression
1		lhpat.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		lhpat.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		lhpat.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		lhpat.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		lhpat.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		lhpat2.r	$$⊢R=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7	1 2 3 4 5	lhpat	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge P\ne Q\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W\in A$$
8	6 7	eqeltrid	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge P\ne Q\right)\right)\to R\in A$$