atmod3i1

Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012) (Revised by Mario Carneiro, 10-May-2013)

	Ref	Expression
Hypotheses	atmod.b	$⊢ B = {Base}_{K}$
	atmod.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	atmod.j	$⊢ \underset{˙}{\lor} = join (K)$
	atmod.m	$⊢ \underset{˙}{\land} = meet (K)$
	atmod.a	$⊢ A = Atoms (K)$
Assertion	atmod3i1	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to P \underset{˙}{\lor} (X \underset{˙}{\land} Y) = X \underset{˙}{\land} (P \underset{˙}{\lor} Y)$

Proof

Step	Hyp	Ref	Expression
1		atmod.b	$⊢ B = {Base}_{K}$
2		atmod.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		atmod.j	$⊢ \underset{˙}{\lor} = join (K)$
4		atmod.m	$⊢ \underset{˙}{\land} = meet (K)$
5		atmod.a	$⊢ A = Atoms (K)$
6		simp1	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to K \in HL$
7		simp21	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to P \in A$
8		simp23	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to Y \in B$
9		simp22	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to X \in B$
10		simp3	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to P \underset{˙}{\leq} X$
11	1 2 3 4 5	atmod1i1	$⊢ (K \in HL \land (P \in A \land Y \in B \land X \in B) \land P \underset{˙}{\leq} X) \to P \underset{˙}{\lor} (Y \underset{˙}{\land} X) = (P \underset{˙}{\lor} Y) \underset{˙}{\land} X$
12	6 7 8 9 10 11	syl131anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to P \underset{˙}{\lor} (Y \underset{˙}{\land} X) = (P \underset{˙}{\lor} Y) \underset{˙}{\land} X$
13		hllat	$⊢ K \in HL \to K \in Lat$
14	13	3ad2ant1	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to K \in Lat$
15	1 4	latmcom	$⊢ (K \in Lat \land X \in B \land Y \in B) \to X \underset{˙}{\land} Y = Y \underset{˙}{\land} X$
16	14 9 8 15	syl3anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to X \underset{˙}{\land} Y = Y \underset{˙}{\land} X$
17	16	oveq2d	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to P \underset{˙}{\lor} (X \underset{˙}{\land} Y) = P \underset{˙}{\lor} (Y \underset{˙}{\land} X)$
18	1 5	atbase	$⊢ P \in A \to P \in B$
19	7 18	syl	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to P \in B$
20	1 3	latjcl	$⊢ (K \in Lat \land P \in B \land Y \in B) \to P \underset{˙}{\lor} Y \in B$
21	14 19 8 20	syl3anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to P \underset{˙}{\lor} Y \in B$
22	1 4	latmcom	$⊢ (K \in Lat \land X \in B \land P \underset{˙}{\lor} Y \in B) \to X \underset{˙}{\land} (P \underset{˙}{\lor} Y) = (P \underset{˙}{\lor} Y) \underset{˙}{\land} X$
23	14 9 21 22	syl3anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to X \underset{˙}{\land} (P \underset{˙}{\lor} Y) = (P \underset{˙}{\lor} Y) \underset{˙}{\land} X$
24	12 17 23	3eqtr4d	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} X) \to P \underset{˙}{\lor} (X \underset{˙}{\land} Y) = X \underset{˙}{\land} (P \underset{˙}{\lor} Y)$

Metamath Proof Explorer

Theorem atmod3i1

Proof