dalemyeb

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012)

	Ref	Expression
Hypotheses	dalema.ph	$⊢ φ \leftrightarrow (((K \in HL \land C \in {Base}_{K}) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \land (Y \in O \land Z \in O) \land ((\neg C \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg C \underset{˙}{\leq} (Q \underset{˙}{\lor} R) \land \neg C \underset{˙}{\leq} (R \underset{˙}{\lor} P)) \land (\neg C \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg C \underset{˙}{\leq} (T \underset{˙}{\lor} U) \land \neg C \underset{˙}{\leq} (U \underset{˙}{\lor} S)) \land (C \underset{˙}{\leq} (P \underset{˙}{\lor} S) \land C \underset{˙}{\leq} (Q \underset{˙}{\lor} T) \land C \underset{˙}{\leq} (R \underset{˙}{\lor} U))))$
	dalemyeb.o	$⊢ O = LPlanes (K)$
Assertion	dalemyeb	$⊢ φ \to Y \in {Base}_{K}$

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		dalemyeb.o	$⊢ O = LPlanes (K)$
3	1	dalemyeo	$⊢ φ \to Y \in O$
4		eqid	$⊢ {Base}_{K} = {Base}_{K}$
5	4 2	lplnbase	$⊢ Y \in O \to Y \in {Base}_{K}$
6	3 5	syl	$⊢ φ \to Y \in {Base}_{K}$

Metamath Proof Explorer