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Metamath Proof Explorer

Theorem cdleme16g

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, Eq. (1). F and G represent f(s) and f(t) respectively. We show, in their notation, (s \/ t) /\ w=(f(s) \/ f(t)) /\ w. (Contributed by NM, 11-Oct-2012)

		Ref	Expression
	Hypotheses	cdleme12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdleme12.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdleme12.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdleme12.a	$⊢ A = Atoms (K)$
		cdleme12.h	$⊢ H = LHyp (K)$
		cdleme12.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
		cdleme12.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
		cdleme12.g	$⊢ G = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
	Assertion	cdleme16g	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} T) \underset{˙}{\land} W = (F \underset{˙}{\lor} G) \underset{˙}{\land} W$

Proof

Step	Hyp	Ref	Expression
1		cdleme12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme12.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme12.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme12.a	$⊢ A = Atoms (K)$
5		cdleme12.h	$⊢ H = LHyp (K)$
6		cdleme12.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme12.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
8		cdleme12.g	$⊢ G = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
9	1 2 3 4 5 6 7 8	cdleme16e	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} T) \underset{˙}{\land} (F \underset{˙}{\lor} G) = (S \underset{˙}{\lor} T) \underset{˙}{\land} W$
10	1 2 3 4 5 6 7 8	cdleme16f	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} T) \underset{˙}{\land} (F \underset{˙}{\lor} G) = (F \underset{˙}{\lor} G) \underset{˙}{\land} W$
11	9 10	eqtr3d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} T) \underset{˙}{\land} W = (F \underset{˙}{\lor} G) \underset{˙}{\land} W$