Theorem cdleme0dN

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 13-Jun-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme0.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme0.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme0c.3	$$⊢V=\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme0dN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge P\ne R\right)\right)\to V\in A$$

Proof

Step	Hyp	Ref	Expression
1		cdleme0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme0.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme0.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme0c.3	$$⊢V=\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W$$
8	1 2 3 4 5 7	lhpat2	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge P\ne R\right)\right)\to V\in A$$