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Metamath Proof Explorer

Theorem cdleme6

Description: Part of proof of Lemma E in Crawley p. 113. This expresses (r \/ f_s(r)) /\ w = u at the top of p. 114. (Contributed by NM, 7-Jun-2012)

		Ref	Expression
	Hypotheses	cdleme4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdleme4.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdleme4.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdleme4.a	$⊢ A = Atoms (K)$
		cdleme4.h	$⊢ H = LHyp (K)$
		cdleme4.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
		cdleme4.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
		cdleme4.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
	Assertion	cdleme6	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\lor} G) \underset{˙}{\land} W = U$

Proof

Step	Hyp	Ref	Expression
1		cdleme4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme4.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme4.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme4.a	$⊢ A = Atoms (K)$
5		cdleme4.h	$⊢ H = LHyp (K)$
6		cdleme4.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme4.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
8		cdleme4.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
9	1 2 3 4 5 6 7 8	cdleme5	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\lor} G = P \underset{˙}{\lor} Q$
10	9	oveq1d	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\lor} G) \underset{˙}{\land} W = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
11	10 6	eqtr4di	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\lor} G) \underset{˙}{\land} W = U$