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Metamath Proof Explorer

Theorem cdleme10tN

Description: Part of proof of Lemma E in Crawley p. 113, 2nd paragraph on p. 114. Y represents t_2. In their notation, we prove t \/ t_2 = t \/ r. (Contributed by NM, 8-Oct-2012) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	cdleme10t.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdleme10t.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdleme10t.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdleme10t.a	$⊢ A = Atoms (K)$
		cdleme10t.h	$⊢ H = LHyp (K)$
		cdleme10t.y	$⊢ Y = (R \underset{˙}{\lor} T) \underset{˙}{\land} W$
	Assertion	cdleme10tN	$⊢ ((K \in HL \land W \in H) \land R \in A \land (T \in A \land \neg T \underset{˙}{\leq} W)) \to T \underset{˙}{\lor} Y = T \underset{˙}{\lor} R$

Proof

Step	Hyp	Ref	Expression
1		cdleme10t.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme10t.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme10t.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme10t.a	$⊢ A = Atoms (K)$
5		cdleme10t.h	$⊢ H = LHyp (K)$
6		cdleme10t.y	$⊢ Y = (R \underset{˙}{\lor} T) \underset{˙}{\land} W$
7	1 2 3 4 5 6	cdleme10	$⊢ ((K \in HL \land W \in H) \land R \in A \land (T \in A \land \neg T \underset{˙}{\leq} W)) \to T \underset{˙}{\lor} Y = T \underset{˙}{\lor} R$