cdlemb2

Description: Given two atoms not under the fiducial (reference) co-atom W , there is a third. Lemma B in Crawley p. 112. (Contributed by NM, 30-May-2012)

	Ref	Expression
Hypotheses	cdlemb2.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemb2.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemb2.a	$⊢ A = Atoms (K)$
	cdlemb2.h	$⊢ H = LHyp (K)$
Assertion	cdlemb2	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to \exists r \in A (\neg r \underset{˙}{\leq} W \land \neg r \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$

Proof

Step	Hyp	Ref	Expression
1		cdlemb2.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemb2.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemb2.a	$⊢ A = Atoms (K)$
4		cdlemb2.h	$⊢ H = LHyp (K)$
5		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to K \in HL$
6		simp2ll	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to P \in A$
7		simp2rl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to Q \in A$
8		simp1r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to W \in H$
9		eqid	$⊢ {Base}_{K} = {Base}_{K}$
10	9 4	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
11	8 10	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to W \in {Base}_{K}$
12		simp3	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to P \neq Q$
13		eqid	$⊢ 1. (K) = 1. (K)$
14		eqid	$⊢ ⋖_{K} = ⋖_{K}$
15	13 14 4	lhp1cvr	$⊢ (K \in HL \land W \in H) \to W ⋖_{K} 1. (K)$
16	15	3ad2ant1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to W ⋖_{K} 1. (K)$
17		simp2lr	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to \neg P \underset{˙}{\leq} W$
18		simp2rr	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to \neg Q \underset{˙}{\leq} W$
19	9 1 2 13 14 3	cdlemb	$⊢ ((K \in HL \land P \in A \land Q \in A) \land (W \in {Base}_{K} \land P \neq Q) \land (W ⋖_{K} 1. (K) \land \neg P \underset{˙}{\leq} W \land \neg Q \underset{˙}{\leq} W)) \to \exists r \in A (\neg r \underset{˙}{\leq} W \land \neg r \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
20	5 6 7 11 12 16 17 18 19	syl323anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to \exists r \in A (\neg r \underset{˙}{\leq} W \land \neg r \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$

Metamath Proof Explorer

Theorem cdlemb2

Proof