Theorem lsatnem0

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

	Ref	Expression
Hypotheses	lsatnem0.o	$$⊢\underset{\dot{}}{0}=0_W$$
	lsatnem0.a	$$⊢A=\mathrm{LSAtoms}\left(W\right)$$
	lsatnem0.w	$$⊢\phi \to W\in \mathrm{LVec}$$
	lsatnem0.q	$$⊢\phi \to Q\in A$$
	lsatnem0.r	$$⊢\phi \to R\in A$$
Assertion	lsatnem0	$$⊢\phi \to \left(Q\ne R\leftrightarrow Q\cap R=\left\{\underset{\dot{}}{0}\right\}\right)$$

Proof

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