dihmeetlem10N

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

	Ref	Expression
Hypotheses	dihmeetlem9.b	$⊢ B = {Base}_{K}$
	dihmeetlem9.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	dihmeetlem9.h	$⊢ H = LHyp (K)$
	dihmeetlem9.j	$⊢ \underset{˙}{\lor} = join (K)$
	dihmeetlem9.m	$⊢ \underset{˙}{\land} = meet (K)$
	dihmeetlem9.a	$⊢ A = Atoms (K)$
	dihmeetlem9.u	$⊢ U = DVecH (K) (W)$
	dihmeetlem9.s	$⊢ \underset{˙}{\oplus} = LSSum (U)$
	dihmeetlem9.i	$⊢ I = DIsoH (K) (W)$
Assertion	dihmeetlem10N	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} p) = I (X) \cap I (Y \underset{˙}{\lor} p)$

Proof

Step	Hyp	Ref	Expression
1		dihmeetlem9.b	$⊢ B = {Base}_{K}$
2		dihmeetlem9.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		dihmeetlem9.h	$⊢ H = LHyp (K)$
4		dihmeetlem9.j	$⊢ \underset{˙}{\lor} = join (K)$
5		dihmeetlem9.m	$⊢ \underset{˙}{\land} = meet (K)$
6		dihmeetlem9.a	$⊢ A = Atoms (K)$
7		dihmeetlem9.u	$⊢ U = DVecH (K) (W)$
8		dihmeetlem9.s	$⊢ \underset{˙}{\oplus} = LSSum (U)$
9		dihmeetlem9.i	$⊢ I = DIsoH (K) (W)$
10		simpl1l	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to K \in HL$
11		simpl2	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to X \in B$
12		simpl3	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to Y \in B$
13		simprll	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to p \in A$
14		simprr	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to p \underset{˙}{\leq} X$
15	1 2 4 5 6	dihmeetlem5	$⊢ ((K \in HL \land X \in B \land Y \in B) \land (p \in A \land p \underset{˙}{\leq} X)) \to X \underset{˙}{\land} (Y \underset{˙}{\lor} p) = (X \underset{˙}{\land} Y) \underset{˙}{\lor} p$
16	10 11 12 13 14 15	syl32anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to X \underset{˙}{\land} (Y \underset{˙}{\lor} p) = (X \underset{˙}{\land} Y) \underset{˙}{\lor} p$
17	16	fveq2d	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to I (X \underset{˙}{\land} (Y \underset{˙}{\lor} p)) = I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} p)$
18		simpl1	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to (K \in HL \land W \in H)$
19	10	hllatd	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to K \in Lat$
20	1 6	atbase	$⊢ p \in A \to p \in B$
21	13 20	syl	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to p \in B$
22	1 4	latjcl	$⊢ (K \in Lat \land Y \in B \land p \in B) \to Y \underset{˙}{\lor} p \in B$
23	19 12 21 22	syl3anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to Y \underset{˙}{\lor} p \in B$
24	1 2 3 4 5 6	dihmeetlem6	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to \neg (X \underset{˙}{\land} (Y \underset{˙}{\lor} p)) \underset{˙}{\leq} W$
25	1 2 5 3 9	dihmeetcN	$⊢ ((K \in HL \land W \in H) \land (X \in B \land Y \underset{˙}{\lor} p \in B) \land \neg (X \underset{˙}{\land} (Y \underset{˙}{\lor} p)) \underset{˙}{\leq} W) \to I (X \underset{˙}{\land} (Y \underset{˙}{\lor} p)) = I (X) \cap I (Y \underset{˙}{\lor} p)$
26	18 11 23 24 25	syl121anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to I (X \underset{˙}{\land} (Y \underset{˙}{\lor} p)) = I (X) \cap I (Y \underset{˙}{\lor} p)$
27	17 26	eqtr3d	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} p) = I (X) \cap I (Y \underset{˙}{\lor} p)$

Metamath Proof Explorer

Theorem dihmeetlem10N

Proof