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Metamath Proof Explorer

Theorem cdlemefs29pre00N

Description: FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat . (Contributed by NM, 27-Mar-2013) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	cdlemefs29.b	$⊢ B = {Base}_{K}$
		cdlemefs29.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemefs29.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemefs29.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdlemefs29.a	$⊢ A = Atoms (K)$
		cdlemefs29.h	$⊢ H = LHyp (K)$
	Assertion	cdlemefs29pre00N	$⊢ (((K \in HL \land W \in H) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land s \in A) \to (((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land s \underset{˙}{\lor} (R \underset{˙}{\land} W) = R) \leftrightarrow (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (R \underset{˙}{\land} W) = R))$

Proof

Step	Hyp	Ref	Expression
1		cdlemefs29.b	$⊢ B = {Base}_{K}$
2		cdlemefs29.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemefs29.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemefs29.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemefs29.a	$⊢ A = Atoms (K)$
6		cdlemefs29.h	$⊢ H = LHyp (K)$
7		breq1	$⊢ s = R \to (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
8	1 2 3 4 5 6 7	cdlemefrs29pre00	$⊢ (((K \in HL \land W \in H) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land s \in A) \to (((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land s \underset{˙}{\lor} (R \underset{˙}{\land} W) = R) \leftrightarrow (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (R \underset{˙}{\land} W) = R))$