cdleme3fa

Description: Part of proof of Lemma E in Crawley p. 113. See cdleme3 . (Contributed by NM, 6-Oct-2012)

	Ref	Expression
Hypotheses	cdleme1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme1.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme1.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme1.a	$⊢ A = Atoms (K)$
	cdleme1.h	$⊢ H = LHyp (K)$
	cdleme1.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme1.f	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
Assertion	cdleme3fa	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \in A$

Step	Hyp	Ref	Expression
1		cdleme1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme1.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme1.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme1.a	$⊢ A = Atoms (K)$
5		cdleme1.h	$⊢ H = LHyp (K)$
6		cdleme1.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme1.f	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
8		eqid	$⊢ (P \underset{˙}{\lor} R) \underset{˙}{\land} W = (P \underset{˙}{\lor} R) \underset{˙}{\land} W$
9	1 2 3 4 5 6 7 8	cdleme3h	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \in A$

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