dihjatc1

Description: Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change .\/ order of ( X ./\ Y ) .\/ Q here and down? (Contributed by NM, 6-Apr-2014)

	Ref	Expression
Hypotheses	dihjatc1.b	$⊢ B = {Base}_{K}$
	dihjatc1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	dihjatc1.h	$⊢ H = LHyp (K)$
	dihjatc1.j	$⊢ \underset{˙}{\lor} = join (K)$
	dihjatc1.m	$⊢ \underset{˙}{\land} = meet (K)$
	dihjatc1.a	$⊢ A = Atoms (K)$
	dihjatc1.u	$⊢ U = DVecH (K) (W)$
	dihjatc1.s	$⊢ \underset{˙}{\oplus} = LSSum (U)$
	dihjatc1.i	$⊢ I = DIsoH (K) (W)$
Assertion	dihjatc1	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) = I (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$

Proof

Step	Hyp	Ref	Expression
1		dihjatc1.b	$⊢ B = {Base}_{K}$
2		dihjatc1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		dihjatc1.h	$⊢ H = LHyp (K)$
4		dihjatc1.j	$⊢ \underset{˙}{\lor} = join (K)$
5		dihjatc1.m	$⊢ \underset{˙}{\land} = meet (K)$
6		dihjatc1.a	$⊢ A = Atoms (K)$
7		dihjatc1.u	$⊢ U = DVecH (K) (W)$
8		dihjatc1.s	$⊢ \underset{˙}{\oplus} = LSSum (U)$
9		dihjatc1.i	$⊢ I = DIsoH (K) (W)$
10		simp11	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
11		simp11l	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to K \in HL$
12	11	hllatd	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to K \in Lat$
13		simp12	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to X \in B$
14		simp13	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Y \in B$
15	1 5	latmcl	$⊢ (K \in Lat \land X \in B \land Y \in B) \to X \underset{˙}{\land} Y \in B$
16	12 13 14 15	syl3anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to X \underset{˙}{\land} Y \in B$
17		simp2l	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Q \in A$
18	1 6	atbase	$⊢ Q \in A \to Q \in B$
19	17 18	syl	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Q \in B$
20	1 4	latjcl	$⊢ (K \in Lat \land X \underset{˙}{\land} Y \in B \land Q \in B) \to (X \underset{˙}{\land} Y) \underset{˙}{\lor} Q \in B$
21	12 16 19 20	syl3anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (X \underset{˙}{\land} Y) \underset{˙}{\lor} Q \in B$
22		simp2	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
23		simp3l	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Q \underset{˙}{\leq} X$
24	1 2 3 4 5 6	dihmeetlem6	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} X)) \to \neg (X \underset{˙}{\land} (Y \underset{˙}{\lor} Q)) \underset{˙}{\leq} W$
25	10 13 14 22 23 24	syl32anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to \neg (X \underset{˙}{\land} (Y \underset{˙}{\lor} Q)) \underset{˙}{\leq} W$
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	11 13 14 17 23 26	syl32anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to X \underset{˙}{\land} (Y \underset{˙}{\lor} Q) = (X \underset{˙}{\land} Y) \underset{˙}{\lor} Q$
28	27	breq1d	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to ((X \underset{˙}{\land} (Y \underset{˙}{\lor} Q)) \underset{˙}{\leq} W \leftrightarrow ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\leq} W)$
29	25 28	mtbid	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to \neg ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\leq} W$
30	1 2 4	latlej2	$⊢ (K \in Lat \land X \underset{˙}{\land} Y \in B \land Q \in B) \to Q \underset{˙}{\leq} ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q)$
31	12 16 19 30	syl3anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Q \underset{˙}{\leq} ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q)$
32	1 2 4 5 6 3 9 7 8	dihvalcq2	$⊢ ((K \in HL \land W \in H) \land ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q \in B \land \neg ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\leq} W) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land Q \underset{˙}{\leq} ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q))) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) = I (Q) \underset{˙}{\oplus} I (((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
33	10 21 29 22 31 32	syl122anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) = I (Q) \underset{˙}{\oplus} I (((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
34		eqid	$⊢ 0. (K) = 0. (K)$
35	2 5 34 6 3	lhpmat	$⊢ ((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to Q \underset{˙}{\land} W = 0. (K)$
36	10 22 35	syl2anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to Q \underset{˙}{\land} W = 0. (K)$
37	36	oveq2d	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (X \underset{˙}{\land} Y) \underset{˙}{\lor} (Q \underset{˙}{\land} W) = (X \underset{˙}{\land} Y) \underset{˙}{\lor} 0. (K)$
38		simp11r	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to W \in H$
39	1 3	lhpbase	$⊢ W \in H \to W \in B$
40	38 39	syl	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to W \in B$
41		simp3r	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (X \underset{˙}{\land} Y) \underset{˙}{\leq} W$
42	1 2 4 5 6	atmod1i2	$⊢ (K \in HL \land (Q \in A \land X \underset{˙}{\land} Y \in B \land W \in B) \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W) \to (X \underset{˙}{\land} Y) \underset{˙}{\lor} (Q \underset{˙}{\land} W) = ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\land} W$
43	11 17 16 40 41 42	syl131anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (X \underset{˙}{\land} Y) \underset{˙}{\lor} (Q \underset{˙}{\land} W) = ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\land} W$
44		hlol	$⊢ K \in HL \to K \in OL$
45	11 44	syl	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to K \in OL$
46	1 4 34	olj01	$⊢ (K \in OL \land X \underset{˙}{\land} Y \in B) \to (X \underset{˙}{\land} Y) \underset{˙}{\lor} 0. (K) = X \underset{˙}{\land} Y$
47	45 16 46	syl2anc	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to (X \underset{˙}{\land} Y) \underset{˙}{\lor} 0. (K) = X \underset{˙}{\land} Y$
48	37 43 47	3eqtr3d	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\land} W = X \underset{˙}{\land} Y$
49	48	fveq2d	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I (((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\land} W) = I (X \underset{˙}{\land} Y)$
50	49	oveq2d	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I (Q) \underset{˙}{\oplus} I (((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) \underset{˙}{\land} W) = I (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$
51	33 50	eqtrd	$⊢ (((K \in HL \land W \in H) \land X \in B \land Y \in B) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (Q \underset{˙}{\leq} X \land (X \underset{˙}{\land} Y) \underset{˙}{\leq} W)) \to I ((X \underset{˙}{\land} Y) \underset{˙}{\lor} Q) = I (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} Y)$

Metamath Proof Explorer

Theorem dihjatc1

Proof