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Metamath Proof Explorer

Theorem cdleme9tN

Description: Part of proof of Lemma E in Crawley p. 113, 2nd paragraph on p. 114. X and F represent t_1 and f(t) respectively. In their notation, we prove f(t) \/ t_1 = q \/ t_1. (Contributed by NM, 8-Oct-2012) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	cdleme9t.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdleme9t.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdleme9t.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdleme9t.a	$⊢ A = Atoms (K)$
		cdleme9t.h	$⊢ H = LHyp (K)$
		cdleme9t.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
		cdleme9t.g	$⊢ F = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
		cdleme9t.x	$⊢ X = (P \underset{˙}{\lor} T) \underset{˙}{\land} W$
	Assertion	cdleme9tN	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F \underset{˙}{\lor} X = Q \underset{˙}{\lor} X$

Proof

Step	Hyp	Ref	Expression
1		cdleme9t.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme9t.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme9t.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme9t.a	$⊢ A = Atoms (K)$
5		cdleme9t.h	$⊢ H = LHyp (K)$
6		cdleme9t.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme9t.g	$⊢ F = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
8		cdleme9t.x	$⊢ X = (P \underset{˙}{\lor} T) \underset{˙}{\land} W$
9	1 2 3 4 5 6 7 8	cdleme9	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F \underset{˙}{\lor} X = Q \underset{˙}{\lor} X$