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Metamath Proof Explorer

Theorem dalemeea

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 11-Aug-2012)

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

Proof

Step	Hyp	Ref	Expression
1		dalema.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		dalemc.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		dalemc.j	$⊢ \underset{˙}{\lor} = join (K)$
4		dalemc.a	$⊢ A = Atoms (K)$
5		dalemeea.m	$⊢ \underset{˙}{\land} = meet (K)$
6		dalemeea.o	$⊢ O = LPlanes (K)$
7		dalemeea.y	$⊢ Y = (P \underset{˙}{\lor} Q) \underset{˙}{\lor} R$
8		dalemeea.z	$⊢ Z = (S \underset{˙}{\lor} T) \underset{˙}{\lor} U$
9		dalemeea.e	$⊢ E = (Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)$
10	1 2 3 4 7 8	dalemrot	$⊢ φ \to (((K \in HL \land C \in {Base}_{K}) \land (Q \in A \land R \in A \land P \in A) \land (T \in A \land U \in A \land S \in A)) \land ((Q \underset{˙}{\lor} R) \underset{˙}{\lor} P \in O \land (T \underset{˙}{\lor} U) \underset{˙}{\lor} S \in O) \land ((\neg C \underset{˙}{\leq} (Q \underset{˙}{\lor} R) \land \neg C \underset{˙}{\leq} (R \underset{˙}{\lor} P) \land \neg C \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg C \underset{˙}{\leq} (T \underset{˙}{\lor} U) \land \neg C \underset{˙}{\leq} (U \underset{˙}{\lor} S) \land \neg C \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land (C \underset{˙}{\leq} (Q \underset{˙}{\lor} T) \land C \underset{˙}{\leq} (R \underset{˙}{\lor} U) \land C \underset{˙}{\leq} (P \underset{˙}{\lor} S))))$
11		biid	$⊢ (((K \in HL \land C \in {Base}_{K}) \land (Q \in A \land R \in A \land P \in A) \land (T \in A \land U \in A \land S \in A)) \land ((Q \underset{˙}{\lor} R) \underset{˙}{\lor} P \in O \land (T \underset{˙}{\lor} U) \underset{˙}{\lor} S \in O) \land ((\neg C \underset{˙}{\leq} (Q \underset{˙}{\lor} R) \land \neg C \underset{˙}{\leq} (R \underset{˙}{\lor} P) \land \neg C \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg C \underset{˙}{\leq} (T \underset{˙}{\lor} U) \land \neg C \underset{˙}{\leq} (U \underset{˙}{\lor} S) \land \neg C \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land (C \underset{˙}{\leq} (Q \underset{˙}{\lor} T) \land C \underset{˙}{\leq} (R \underset{˙}{\lor} U) \land C \underset{˙}{\leq} (P \underset{˙}{\lor} S)))) \leftrightarrow (((K \in HL \land C \in {Base}_{K}) \land (Q \in A \land R \in A \land P \in A) \land (T \in A \land U \in A \land S \in A)) \land ((Q \underset{˙}{\lor} R) \underset{˙}{\lor} P \in O \land (T \underset{˙}{\lor} U) \underset{˙}{\lor} S \in O) \land ((\neg C \underset{˙}{\leq} (Q \underset{˙}{\lor} R) \land \neg C \underset{˙}{\leq} (R \underset{˙}{\lor} P) \land \neg C \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg C \underset{˙}{\leq} (T \underset{˙}{\lor} U) \land \neg C \underset{˙}{\leq} (U \underset{˙}{\lor} S) \land \neg C \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land (C \underset{˙}{\leq} (Q \underset{˙}{\lor} T) \land C \underset{˙}{\leq} (R \underset{˙}{\lor} U) \land C \underset{˙}{\leq} (P \underset{˙}{\lor} S))))$
12		eqid	$⊢ (Q \underset{˙}{\lor} R) \underset{˙}{\lor} P = (Q \underset{˙}{\lor} R) \underset{˙}{\lor} P$
13		eqid	$⊢ (T \underset{˙}{\lor} U) \underset{˙}{\lor} S = (T \underset{˙}{\lor} U) \underset{˙}{\lor} S$
14	11 2 3 4 5 6 12 13 9	dalemdea	$⊢ (((K \in HL \land C \in {Base}_{K}) \land (Q \in A \land R \in A \land P \in A) \land (T \in A \land U \in A \land S \in A)) \land ((Q \underset{˙}{\lor} R) \underset{˙}{\lor} P \in O \land (T \underset{˙}{\lor} U) \underset{˙}{\lor} S \in O) \land ((\neg C \underset{˙}{\leq} (Q \underset{˙}{\lor} R) \land \neg C \underset{˙}{\leq} (R \underset{˙}{\lor} P) \land \neg C \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg C \underset{˙}{\leq} (T \underset{˙}{\lor} U) \land \neg C \underset{˙}{\leq} (U \underset{˙}{\lor} S) \land \neg C \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land (C \underset{˙}{\leq} (Q \underset{˙}{\lor} T) \land C \underset{˙}{\leq} (R \underset{˙}{\lor} U) \land C \underset{˙}{\leq} (P \underset{˙}{\lor} S)))) \to E \in A$
15	10 14	syl	$⊢ φ \to E \in A$