dihmeetlem12N

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	dihmeetlem12N	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I (X \underset{˙}{\land} Y) \underset{˙}{\oplus} (I (p) \cap I (Y)) = I (X) \cap I (Y)$

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		simpl1	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
11		simpl2	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to X \in B$
12		simpl3	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Y \in B$
13		simpr1	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (p \in A \land \neg p \underset{˙}{\leq} W)$
14		simpr2	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to p \underset{˙}{\leq} X$
15		simpr3	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (X \underset{˙}{\land} Y) \underset{˙}{\leq} W$
16	1 2 3 4 5 6 7 8 9	dihmeetlem8N	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (p \in A \land \neg p \underset{˙}{\leq} W) \land (p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} p) = I (p) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$
17	10 11 12 13 14 15 16	syl312anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} p) = I (p) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$
18	17	ineq1d	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} p) \cap I (Y) = (I (p) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)) \cap I (Y)$
19	1 2 3 4 5 6 7 8 9	dihmeetlem11N	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} p) \cap I (Y) = I (X) \cap I (Y)$
20	19	3adantr3	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} p) \cap I (Y) = I (X) \cap I (Y)$
21		simpr1l	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to p \in A$
22	1 2 3 4 5 6 7 8 9	dihmeetlem9N	$⊢ ((K \in HL \land W \in H) \land (X \in B \land Y \in B) \land p \in A) \to (I (p) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)) \cap I (Y) = I (X \underset{˙}{\land} Y) \underset{˙}{\oplus} (I (p) \cap I (Y))$
23	10 11 12 21 22	syl121anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (I (p) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)) \cap I (Y) = I (X \underset{˙}{\land} Y) \underset{˙}{\oplus} (I (p) \cap I (Y))$
24	18 20 23	3eqtr3rd	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((p \in A \land \neg p \underset{˙}{\leq} W) \land p \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I (X \underset{˙}{\land} Y) \underset{˙}{\oplus} (I (p) \cap I (Y)) = I (X) \cap I (Y)$

Metamath Proof Explorer

Theorem dihmeetlem12N

Proof