dihmeetlem6

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014)

	Ref	Expression
Hypotheses	dihmeetlem6.b	$⊢ B = {Base}_{K}$
	dihmeetlem6.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	dihmeetlem6.h	$⊢ H = LHyp (K)$
	dihmeetlem6.j	$⊢ \underset{˙}{\lor} = join (K)$
	dihmeetlem6.m	$⊢ \underset{˙}{\land} = meet (K)$
	dihmeetlem6.a	$⊢ A = Atoms (K)$
Assertion	dihmeetlem6	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to \neg (X \underset{˙}{\land} (Y \underset{˙}{\lor} Q)) \underset{˙}{\leq} W$

Proof

Step	Hyp	Ref	Expression
1		dihmeetlem6.b	$⊢ B = {Base}_{K}$
2		dihmeetlem6.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		dihmeetlem6.h	$⊢ H = LHyp (K)$
4		dihmeetlem6.j	$⊢ \underset{˙}{\lor} = join (K)$
5		dihmeetlem6.m	$⊢ \underset{˙}{\land} = meet (K)$
6		dihmeetlem6.a	$⊢ A = Atoms (K)$
7		simprlr	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to \neg Q \underset{˙}{\leq} W$
8		simpl1l	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to K \in HL$
9	8	hllatd	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to K \in Lat$
10		simpl2	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to X \in B$
11		simpl3	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to Y \in B$
12	1 5	latmcl	$⊢ (K \in Lat \land X \in B \land Y \in B) \to X \underset{˙}{\land} Y \in B$
13	9 10 11 12	syl3anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to X \underset{˙}{\land} Y \in B$
14		simprll	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to Q \in A$
15	1 6	atbase	$⊢ Q \in A \to Q \in B$
16	14 15	syl	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to Q \in B$
17		simpl1r	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to W \in H$
18	1 3	lhpbase	$⊢ W \in H \to W \in B$
19	17 18	syl	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to W \in B$
20	1 2 4	latjle12	$⊢ (K \in Lat \land (X \underset{˙}{\land} Y \in B \land Q \in B \land W \in B)) \to (((X \underset{˙}{\land} Y) \underset{˙}{\leq} W \land Q \underset{˙}{\leq} W) \leftrightarrow ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\leq} W)$
21	9 13 16 19 20	syl13anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to (((X \underset{˙}{\land} Y) \underset{˙}{\leq} W \land Q \underset{˙}{\leq} W) \leftrightarrow ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\leq} W)$
22		simpr	$⊢ ((X \underset{˙}{\land} Y) \underset{˙}{\leq} W \land Q \underset{˙}{\leq} W) \to Q \underset{˙}{\leq} W$
23	21 22	syl6bir	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to (((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\leq} W \to Q \underset{˙}{\leq} W)$
24	7 23	mtod	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to \neg ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\leq} W$
25		simprr	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to Q \underset{˙}{\leq} X$
26	1 2 4 5 6	dihmeetlem5	$⊢ ((K \in HL \land X \in B \land Y \in B) \land (Q \in A \land Q \underset{˙}{\leq} X)) \to X \underset{˙}{\land} (Y \underset{˙}{\lor} Q) = (X \underset{˙}{\land} Y) \underset{˙}{\lor} Q$
27	8 10 11 14 25 26	syl32anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to X \underset{˙}{\land} (Y \underset{˙}{\lor} Q) = (X \underset{˙}{\land} Y) \underset{˙}{\lor} Q$
28	27	breq1d	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to ((X \underset{˙}{\land} (Y \underset{˙}{\lor} Q)) \underset{˙}{\leq} W \leftrightarrow ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\leq} W)$
29	24 28	mtbird	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to \neg (X \underset{˙}{\land} (Y \underset{˙}{\lor} Q)) \underset{˙}{\leq} W$

Metamath Proof Explorer

Theorem dihmeetlem6

Proof