lhpat

Description: Create an atom under a co-atom. Part of proof of Lemma B in Crawley p. 112. (Contributed by NM, 23-May-2012)

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

Proof

Step	Hyp	Ref	Expression
1		lhpat.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		lhpat.j	$⊢ \underset{˙}{\lor} = join (K)$
3		lhpat.m	$⊢ \underset{˙}{\land} = meet (K)$
4		lhpat.a	$⊢ A = Atoms (K)$
5		lhpat.h	$⊢ H = LHyp (K)$
6		simp1l	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to K \in HL$
7		simp2l	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to P \in A$
8		simp3l	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to Q \in A$
9		simp1r	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to W \in H$
10		eqid	$⊢ {Base}_{K} = {Base}_{K}$
11	10 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
12	9 11	syl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to W \in {Base}_{K}$
13		simp3r	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to P \neq Q$
14		eqid	$⊢ 1. (K) = 1. (K)$
15		eqid	$⊢ ⋖_{K} = ⋖_{K}$
16	14 15 5	lhp1cvr	$⊢ (K \in HL \land W \in H) \to W ⋖_{K} 1. (K)$
17	16	3ad2ant1	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to W ⋖_{K} 1. (K)$
18		simp2r	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to \neg P \underset{˙}{\leq} W$
19	10 1 2 3 14 15 4	1cvrat	$⊢ (K \in HL \land (P \in A \land Q \in A \land W \in {Base}_{K}) \land (P \neq Q \land W ⋖_{K} 1. (K) \land \neg P \underset{˙}{\leq} W)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} W \in A$
20	6 7 8 12 13 17 18 19	syl133anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} W \in A$

Metamath Proof Explorer

Theorem lhpat

Proof