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Metamath Proof Explorer

Theorem dalem63

Description: Lemma for dath . Combine the cases where Y and Z are different planes with the case where Y and Z are the same plane. (Contributed by NM, 11-Aug-2012)

		Ref	Expression
	Hypotheses	dalem62.ph	$⊢ φ \leftrightarrow (((K \in HL \land C \in {Base}_{K}) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \land (Y \in O \land Z \in O) \land ((\neg C \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg C \underset{˙}{\leq} (Q \underset{˙}{\lor} R) \land \neg C \underset{˙}{\leq} (R \underset{˙}{\lor} P)) \land (\neg C \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg C \underset{˙}{\leq} (T \underset{˙}{\lor} U) \land \neg C \underset{˙}{\leq} (U \underset{˙}{\lor} S)) \land (C \underset{˙}{\leq} (P \underset{˙}{\lor} S) \land C \underset{˙}{\leq} (Q \underset{˙}{\lor} T) \land C \underset{˙}{\leq} (R \underset{˙}{\lor} U))))$
		dalem62.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		dalem62.j	$⊢ \underset{˙}{\lor} = join (K)$
		dalem62.a	$⊢ A = Atoms (K)$
		dalem62.m	$⊢ \underset{˙}{\land} = meet (K)$
		dalem62.o	$⊢ O = LPlanes (K)$
		dalem62.y	$⊢ Y = (P \underset{˙}{\lor} Q) \underset{˙}{\lor} R$
		dalem62.z	$⊢ Z = (S \underset{˙}{\lor} T) \underset{˙}{\lor} U$
		dalem62.d	$⊢ D = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (S \underset{˙}{\lor} T)$
		dalem62.e	$⊢ E = (Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)$
		dalem62.f	$⊢ F = (R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)$
	Assertion	dalem63	$⊢ φ \to F \underset{˙}{\leq} (D \underset{˙}{\lor} E)$

Proof

Step	Hyp	Ref	Expression
1		dalem62.ph	$⊢ φ \leftrightarrow (((K \in HL \land C \in {Base}_{K}) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \land (Y \in O \land Z \in O) \land ((\neg C \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg C \underset{˙}{\leq} (Q \underset{˙}{\lor} R) \land \neg C \underset{˙}{\leq} (R \underset{˙}{\lor} P)) \land (\neg C \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg C \underset{˙}{\leq} (T \underset{˙}{\lor} U) \land \neg C \underset{˙}{\leq} (U \underset{˙}{\lor} S)) \land (C \underset{˙}{\leq} (P \underset{˙}{\lor} S) \land C \underset{˙}{\leq} (Q \underset{˙}{\lor} T) \land C \underset{˙}{\leq} (R \underset{˙}{\lor} U))))$
2		dalem62.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		dalem62.j	$⊢ \underset{˙}{\lor} = join (K)$
4		dalem62.a	$⊢ A = Atoms (K)$
5		dalem62.m	$⊢ \underset{˙}{\land} = meet (K)$
6		dalem62.o	$⊢ O = LPlanes (K)$
7		dalem62.y	$⊢ Y = (P \underset{˙}{\lor} Q) \underset{˙}{\lor} R$
8		dalem62.z	$⊢ Z = (S \underset{˙}{\lor} T) \underset{˙}{\lor} U$
9		dalem62.d	$⊢ D = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (S \underset{˙}{\lor} T)$
10		dalem62.e	$⊢ E = (Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)$
11		dalem62.f	$⊢ F = (R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)$
12	1 2 3 4 5 6 7 8 9 10 11	dalem62	$⊢ (φ \land Y = Z) \to F \underset{˙}{\leq} (D \underset{˙}{\lor} E)$
13	1 2 3 4 5 6 7 8 9 10 11	dalem16	$⊢ (φ \land Y \neq Z) \to F \underset{˙}{\leq} (D \underset{˙}{\lor} E)$
14	12 13	pm2.61dane	$⊢ φ \to F \underset{˙}{\leq} (D \underset{˙}{\lor} E)$