This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cdleme35g

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013)

		Ref	Expression
	Hypotheses	cdleme35.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdleme35.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdleme35.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdleme35.a	$⊢ A = Atoms (K)$
		cdleme35.h	$⊢ H = LHyp (K)$
		cdleme35.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
		cdleme35.f	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
	Assertion	cdleme35g	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (F \underset{˙}{\lor} U) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} F) \underset{˙}{\land} W)) = R$

Proof

Step	Hyp	Ref	Expression
1		cdleme35.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme35.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme35.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme35.a	$⊢ A = Atoms (K)$
5		cdleme35.h	$⊢ H = LHyp (K)$
6		cdleme35.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme35.f	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
8	1 2 3 4 5 6 7	cdleme35a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F \underset{˙}{\lor} U = R \underset{˙}{\lor} U$
9	1 2 3 4 5 6 7	cdleme35e	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} ((Q \underset{˙}{\lor} F) \underset{˙}{\land} W) = P \underset{˙}{\lor} R$
10	8 9	oveq12d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (F \underset{˙}{\lor} U) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} F) \underset{˙}{\land} W)) = (R \underset{˙}{\lor} U) \underset{˙}{\land} (P \underset{˙}{\lor} R)$
11	1 2 3 4 5 6 7	cdleme35f	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (R \underset{˙}{\lor} U) \underset{˙}{\land} (P \underset{˙}{\lor} R) = R$
12	10 11	eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (F \underset{˙}{\lor} U) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} F) \underset{˙}{\land} W)) = R$