atmod2i2

Description: Version of modular law pmod2iN that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-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	atmod2i2	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to (X \underset{˙}{\land} P) \underset{˙}{\lor} Y = X \underset{˙}{\land} (P \underset{˙}{\lor} Y)$

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		hllat	$⊢ K \in HL \to K \in Lat$
7	6	3ad2ant1	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to K \in Lat$
8		simp21	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to P \in A$
9	1 5	atbase	$⊢ P \in A \to P \in B$
10	8 9	syl	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to P \in B$
11		simp23	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to Y \in B$
12	1 3	latjcom	$⊢ (K \in Lat \land P \in B \land Y \in B) \to P \underset{˙}{\lor} Y = Y \underset{˙}{\lor} P$
13	7 10 11 12	syl3anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to P \underset{˙}{\lor} Y = Y \underset{˙}{\lor} P$
14	13	oveq1d	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to (P \underset{˙}{\lor} Y) \underset{˙}{\land} X = (Y \underset{˙}{\lor} P) \underset{˙}{\land} X$
15		simp22	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to X \in B$
16	1 3	latjcl	$⊢ (K \in Lat \land P \in B \land Y \in B) \to P \underset{˙}{\lor} Y \in B$
17	7 10 11 16	syl3anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to P \underset{˙}{\lor} Y \in B$
18	1 4	latmcom	$⊢ (K \in Lat \land X \in B \land P \underset{˙}{\lor} Y \in B) \to X \underset{˙}{\land} (P \underset{˙}{\lor} Y) = (P \underset{˙}{\lor} Y) \underset{˙}{\land} X$
19	7 15 17 18	syl3anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to X \underset{˙}{\land} (P \underset{˙}{\lor} Y) = (P \underset{˙}{\lor} Y) \underset{˙}{\land} X$
20		simp1	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to K \in HL$
21		simp3	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to Y \underset{˙}{\leq} X$
22	1 2 3 4 5	atmod1i2	$⊢ (K \in HL \land (P \in A \land Y \in B \land X \in B) \land Y \underset{˙}{\leq} X) \to Y \underset{˙}{\lor} (P \underset{˙}{\land} X) = (Y \underset{˙}{\lor} P) \underset{˙}{\land} X$
23	20 8 11 15 21 22	syl131anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to Y \underset{˙}{\lor} (P \underset{˙}{\land} X) = (Y \underset{˙}{\lor} P) \underset{˙}{\land} X$
24	14 19 23	3eqtr4d	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to X \underset{˙}{\land} (P \underset{˙}{\lor} Y) = Y \underset{˙}{\lor} (P \underset{˙}{\land} X)$
25	1 4	latmcl	$⊢ (K \in Lat \land P \in B \land X \in B) \to P \underset{˙}{\land} X \in B$
26	7 10 15 25	syl3anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to P \underset{˙}{\land} X \in B$
27	1 3	latjcom	$⊢ (K \in Lat \land Y \in B \land P \underset{˙}{\land} X \in B) \to Y \underset{˙}{\lor} (P \underset{˙}{\land} X) = (P \underset{˙}{\land} X) \underset{˙}{\lor} Y$
28	7 11 26 27	syl3anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to Y \underset{˙}{\lor} (P \underset{˙}{\land} X) = (P \underset{˙}{\land} X) \underset{˙}{\lor} Y$
29	1 4	latmcom	$⊢ (K \in Lat \land P \in B \land X \in B) \to P \underset{˙}{\land} X = X \underset{˙}{\land} P$
30	7 10 15 29	syl3anc	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to P \underset{˙}{\land} X = X \underset{˙}{\land} P$
31	30	oveq1d	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to (P \underset{˙}{\land} X) \underset{˙}{\lor} Y = (X \underset{˙}{\land} P) \underset{˙}{\lor} Y$
32	24 28 31	3eqtrrd	$⊢ (K \in HL \land (P \in A \land X \in B \land Y \in B) \land Y \underset{˙}{\leq} X) \to (X \underset{˙}{\land} P) \underset{˙}{\lor} Y = X \underset{˙}{\land} (P \underset{˙}{\lor} Y)$

Metamath Proof Explorer

Theorem atmod2i2

Proof