lhpelim

Description: Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013)

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

Proof

Step	Hyp	Ref	Expression
1		lhpelim.b	$⊢ B = {Base}_{K}$
2		lhpelim.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		lhpelim.j	$⊢ \underset{˙}{\lor} = join (K)$
4		lhpelim.m	$⊢ \underset{˙}{\land} = meet (K)$
5		lhpelim.a	$⊢ A = Atoms (K)$
6		lhpelim.h	$⊢ H = LHyp (K)$
7		eqid	$⊢ 0. (K) = 0. (K)$
8	2 4 7 5 6	lhpmat	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to P \underset{˙}{\land} W = 0. (K)$
9	8	3adant3	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to P \underset{˙}{\land} W = 0. (K)$
10	9	oveq1d	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to (P \underset{˙}{\land} W) \underset{˙}{\lor} (X \underset{˙}{\land} W) = 0. (K) \underset{˙}{\lor} (X \underset{˙}{\land} W)$
11		simp1l	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to K \in HL$
12		simp2l	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to P \in A$
13	11	hllatd	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to K \in Lat$
14		simp3	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to X \in B$
15		simp1r	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to W \in H$
16	1 6	lhpbase	$⊢ W \in H \to W \in B$
17	15 16	syl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to W \in B$
18	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
19	13 14 17 18	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to X \underset{˙}{\land} W \in B$
20	1 2 4	latmle2	$⊢ (K \in Lat \land X \in B \land W \in B) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} W$
21	13 14 17 20	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} W$
22	1 2 3 4 5	atmod4i2	$⊢ (K \in HL \land (P \in A \land X \underset{˙}{\land} W \in B \land W \in B) \land (X \underset{˙}{\land} W) \underset{˙}{\leq} W) \to (P \underset{˙}{\land} W) \underset{˙}{\lor} (X \underset{˙}{\land} W) = (P \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\land} W$
23	11 12 19 17 21 22	syl131anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to (P \underset{˙}{\land} W) \underset{˙}{\lor} (X \underset{˙}{\land} W) = (P \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\land} W$
24		hlol	$⊢ K \in HL \to K \in OL$
25	11 24	syl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to K \in OL$
26	1 3 7	olj02	$⊢ (K \in OL \land X \underset{˙}{\land} W \in B) \to 0. (K) \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \underset{˙}{\land} W$
27	25 19 26	syl2anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to 0. (K) \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \underset{˙}{\land} W$
28	10 23 27	3eqtr3d	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land X \in B) \to (P \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\land} W = X \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem lhpelim

Proof