cdleme18a

Description: Part of proof of Lemma E in Crawley p. 114, 2nd sentence of 4th paragraph. F , G represent f(s), f_s(q) respectively. We show -. f_s(q) <_ w. (Contributed by NM, 12-Oct-2012)

	Ref	Expression
Hypotheses	cdleme18.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme18.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme18.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme18.a	$⊢ A = Atoms (K)$
	cdleme18.h	$⊢ H = LHyp (K)$
	cdleme18.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme18.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
	cdleme18.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((Q \underset{˙}{\lor} S) \underset{˙}{\land} W))$
Assertion	cdleme18a	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg G \underset{˙}{\leq} W$

Proof

Step	Hyp	Ref	Expression
1		cdleme18.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme18.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme18.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme18.a	$⊢ A = Atoms (K)$
5		cdleme18.h	$⊢ H = LHyp (K)$
6		cdleme18.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme18.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
8		cdleme18.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((Q \underset{˙}{\lor} S) \underset{˙}{\land} W))$
9		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (K \in HL \land W \in H)$
10		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
11		simp22	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
12		simp23	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
13		simp3l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \neq Q$
14		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in HL$
15		simp21l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \in A$
16		simp22l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in A$
17	1 2 4	hlatlej2	$⊢ (K \in HL \land P \in A \land Q \in A) \to Q \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
18	14 15 16 17	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
19		simp3r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
20	1 2 3 4 5 6 7 8	cdleme7	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg G \underset{˙}{\leq} W$
21	9 10 11 11 12 13 18 19 20	syl323anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg G \underset{˙}{\leq} W$

Metamath Proof Explorer

Theorem cdleme18a

Proof