2lplnm2N

Description: The meet of two different lattice planes in a lattice volume is a lattice line. (Contributed by NM, 12-Jul-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	2lplnm2.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	2lplnm2.m	$⊢ \underset{˙}{\land} = meet (K)$
	2lplnm2.a	$⊢ N = LLines (K)$
	2lplnm2.p	$⊢ P = LPlanes (K)$
	2lplnm2.v	$⊢ V = LVols (K)$
Assertion	2lplnm2N	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to X \underset{˙}{\land} Y \in N$

Proof

Step	Hyp	Ref	Expression
1		2lplnm2.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		2lplnm2.m	$⊢ \underset{˙}{\land} = meet (K)$
3		2lplnm2.a	$⊢ N = LLines (K)$
4		2lplnm2.p	$⊢ P = LPlanes (K)$
5		2lplnm2.v	$⊢ V = LVols (K)$
6		simp22	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to Y \in P$
7		simp1	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to K \in HL$
8		hllat	$⊢ K \in HL \to K \in Lat$
9	8	3ad2ant1	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to K \in Lat$
10		simp21	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to X \in P$
11		eqid	$⊢ {Base}_{K} = {Base}_{K}$
12	11 4	lplnbase	$⊢ X \in P \to X \in {Base}_{K}$
13	10 12	syl	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to X \in {Base}_{K}$
14	11 4	lplnbase	$⊢ Y \in P \to Y \in {Base}_{K}$
15	6 14	syl	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to Y \in {Base}_{K}$
16	11 2	latmcl	$⊢ (K \in Lat \land X \in {Base}_{K} \land Y \in {Base}_{K}) \to X \underset{˙}{\land} Y \in {Base}_{K}$
17	9 13 15 16	syl3anc	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to X \underset{˙}{\land} Y \in {Base}_{K}$
18		eqid	$⊢ join (K) = join (K)$
19	1 18 4 5	2lplnj	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to X join (K) Y = W$
20		simp23	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to W \in V$
21	19 20	eqeltrd	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to X join (K) Y \in V$
22	11 1 18	latlej1	$⊢ (K \in Lat \land X \in {Base}_{K} \land Y \in {Base}_{K}) \to X \underset{˙}{\leq} (X join (K) Y)$
23	9 13 15 22	syl3anc	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to X \underset{˙}{\leq} (X join (K) Y)$
24		eqid	$⊢ ⋖_{K} = ⋖_{K}$
25	1 24 4 5	lplncvrlvol2	$⊢ ((K \in HL \land X \in P \land X join (K) Y \in V) \land X \underset{˙}{\leq} (X join (K) Y)) \to X ⋖_{K} (X join (K) Y)$
26	7 10 21 23 25	syl31anc	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to X ⋖_{K} (X join (K) Y)$
27	11 18 2 24	cvrexch	$⊢ (K \in HL \land X \in {Base}_{K} \land Y \in {Base}_{K}) \to ((X \underset{˙}{\land} Y) ⋖_{K} Y \leftrightarrow X ⋖_{K} (X join (K) Y))$
28	7 13 15 27	syl3anc	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to ((X \underset{˙}{\land} Y) ⋖_{K} Y \leftrightarrow X ⋖_{K} (X join (K) Y))$
29	26 28	mpbird	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to (X \underset{˙}{\land} Y) ⋖_{K} Y$
30	11 24 3 4	llncvrlpln	$⊢ ((K \in HL \land X \underset{˙}{\land} Y \in {Base}_{K} \land Y \in {Base}_{K}) \land (X \underset{˙}{\land} Y) ⋖_{K} Y) \to (X \underset{˙}{\land} Y \in N \leftrightarrow Y \in P)$
31	7 17 15 29 30	syl31anc	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to (X \underset{˙}{\land} Y \in N \leftrightarrow Y \in P)$
32	6 31	mpbird	$⊢ (K \in HL \land (X \in P \land Y \in P \land W \in V) \land (X \underset{˙}{\leq} W \land Y \underset{˙}{\leq} W \land X \neq Y)) \to X \underset{˙}{\land} Y \in N$

Metamath Proof Explorer

Theorem 2lplnm2N

Proof