atmod4i1

Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-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	atmod4i1	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} Y) \to (X \underset{˙}{\land} Y) \underset{˙}{\lor} P = (X \underset{˙}{\lor} P) \underset{˙}{\land} Y$

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		hllat	$⊢ K \in HL \to K \in Lat$
7	6	3ad2ant1	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} Y) \to K \in Lat$
8		simp22	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} Y) \to X \in B$
9		simp23	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} Y) \to Y \in B$
10	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land Y \in B) \to X \underset{˙}{\land} Y \in B$
11	7 8 9 10	syl3anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} Y) \to X \underset{˙}{\land} Y \in B$
12		simp21	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} Y) \to P \in A$
13	1 5	atbase	$⊢ P \in A \to P \in B$
14	12 13	syl	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} Y) \to P \in B$
15	1 3	latjcom	$⊢ (K \in Lat \land X \underset{˙}{\land} Y \in B \land P \in B) \to (X \underset{˙}{\land} Y) \underset{˙}{\lor} P = P \underset{˙}{\lor} (X \underset{˙}{\land} Y)$
16	7 11 14 15	syl3anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} Y) \to (X \underset{˙}{\land} Y) \underset{˙}{\lor} P = P \underset{˙}{\lor} (X \underset{˙}{\land} Y)$
17	1 2 3 4 5	atmod1i1	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} Y) \to P \underset{˙}{\lor} (X \underset{˙}{\land} Y) = (P \underset{˙}{\lor} X) \underset{˙}{\land} Y$
18	1 3	latjcom	$⊢ (K \in Lat \land P \in B \land X \in B) \to P \underset{˙}{\lor} X = X \underset{˙}{\lor} P$
19	7 14 8 18	syl3anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} Y) \to P \underset{˙}{\lor} X = X \underset{˙}{\lor} P$
20	19	oveq1d	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} Y) \to (P \underset{˙}{\lor} X) \underset{˙}{\land} Y = (X \underset{˙}{\lor} P) \underset{˙}{\land} Y$
21	16 17 20	3eqtrd	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land P \underset{˙}{\leq} Y) \to (X \underset{˙}{\land} Y) \underset{˙}{\lor} P = (X \underset{˙}{\lor} P) \underset{˙}{\land} Y$

Metamath Proof Explorer

Theorem atmod4i1

Proof