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Metamath Proof Explorer

Theorem cdleme17a

Description: Part of proof of Lemma E in Crawley p. 114, first part of 4th paragraph. F , G , and C represent f(s), f_s(p), and s_1 respectively. We show, in their notation, f_s(p)=(p \/ q) /\ (q \/ s_1). (Contributed by NM, 11-Oct-2012)

		Ref	Expression
	Hypotheses	cdleme17.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdleme17.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdleme17.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdleme17.a	$⊢ A = Atoms (K)$
		cdleme17.h	$⊢ H = LHyp (K)$
		cdleme17.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
		cdleme17.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
		cdleme17.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
		cdleme17.c	$⊢ C = (P \underset{˙}{\lor} S) \underset{˙}{\land} W$
	Assertion	cdleme17a	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Q \underset{˙}{\lor} C)$

Proof

Step	Hyp	Ref	Expression
1		cdleme17.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme17.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme17.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme17.a	$⊢ A = Atoms (K)$
5		cdleme17.h	$⊢ H = LHyp (K)$
6		cdleme17.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme17.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
8		cdleme17.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
9		cdleme17.c	$⊢ C = (P \underset{˙}{\lor} S) \underset{˙}{\land} W$
10	1 2 3 4 5 6 7 8 9	cdleme7a	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} C)$
11	1 2 3 4 5 6 7 9	cdleme9	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F \underset{˙}{\lor} C = Q \underset{˙}{\lor} C$
12	11	oveq2d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} C) = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Q \underset{˙}{\lor} C)$
13	10 12	eqtrid	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Q \underset{˙}{\lor} C)$