lsatnem0

Description: The meet of distinct atoms is the zero subspace. ( atnemeq0 analog.) (Contributed by NM, 10-Jan-2015)

	Ref	Expression
Hypotheses	lsatnem0.o	$⊢ \underset{˙}{0} = 0_{W}$
	lsatnem0.a	$⊢ A = LSAtoms (W)$
	lsatnem0.w	$⊢ φ \to W \in LVec$
	lsatnem0.q	$⊢ φ \to Q \in A$
	lsatnem0.r	$⊢ φ \to R \in A$
Assertion	lsatnem0	$⊢ φ \to (Q \neq R \leftrightarrow Q \cap R = \{\underset{˙}{0}\})$

Step	Hyp	Ref	Expression
1		lsatnem0.o	$⊢ \underset{˙}{0} = 0_{W}$
2		lsatnem0.a	$⊢ A = LSAtoms (W)$
3		lsatnem0.w	$⊢ φ \to W \in LVec$
4		lsatnem0.q	$⊢ φ \to Q \in A$
5		lsatnem0.r	$⊢ φ \to R \in A$
6	2 3 5 4	lsatcmp	$⊢ φ \to (R \subseteq Q \leftrightarrow R = Q)$
7		eqcom	$⊢ R = Q \leftrightarrow Q = R$
8	6 7	bitrdi	$⊢ φ \to (R \subseteq Q \leftrightarrow Q = R)$
9	8	necon3bbid	$⊢ φ \to (\neg R \subseteq Q \leftrightarrow Q \neq R)$
10		eqid	$⊢ LSubSp (W) = LSubSp (W)$
11		lveclmod	$⊢ W \in LVec \to W \in LMod$
12	3 11	syl	$⊢ φ \to W \in LMod$
13	10 2 12 4	lsatlssel	$⊢ φ \to Q \in LSubSp (W)$
14	1 10 2 3 13 5	lsatnle	$⊢ φ \to (\neg R \subseteq Q \leftrightarrow Q \cap R = \{\underset{˙}{0}\})$
15	9 14	bitr3d	$⊢ φ \to (Q \neq R \leftrightarrow Q \cap R = \{\underset{˙}{0}\})$

Metamath Proof Explorer