Theorem cdleme9a

Description: Part of proof of Lemma E in Crawley p. 113. C represents s_1, which we prove is an atom. (Contributed by NM, 10-Jun-2012)

	Ref	Expression
Hypotheses	cdleme8.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme8.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme8.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme8.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme8.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme8.4	$$⊢C=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme9a	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge P\ne S\right)\right)\to C\in A$$

Proof

Step	Hyp	Ref	Expression
1		cdleme8.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme8.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme8.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme8.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme8.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme8.4	$$⊢C=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
7	1 2 3 4 5 6	lhpat2	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge P\ne S\right)\right)\to C\in A$$