Theorem dochoc0

Description: The zero subspace is closed. (Contributed by NM, 16-Feb-2015)

	Ref	Expression
Hypotheses	dochoc0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	dochoc0.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	dochoc0.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{ocH}\left(K\right)\left(W\right)$$
	dochoc0.z	$$⊢\underset{\dot{}}{0}=0_U$$
	dochoc0.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
Assertion	dochoc0	$$⊢\phi \to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(\left\{\underset{\dot{}}{0}\right\}\right)\right)=\left\{\underset{\dot{}}{0}\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dochoc0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		dochoc0.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
3		dochoc0.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{ocH}\left(K\right)\left(W\right)$$
4		dochoc0.z	$$⊢\underset{\dot{}}{0}=0_U$$
5		dochoc0.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
6		eqid	$$⊢{\mathrm{Base}}_U={\mathrm{Base}}_U$$
7	1 2 3 6 4	doch0	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to \underset{\dot{}}{\perp }\left(\left\{\underset{\dot{}}{0}\right\}\right)={\mathrm{Base}}_U$$
8	7	fveq2d	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(\left\{\underset{\dot{}}{0}\right\}\right)\right)=\underset{\dot{}}{\perp }\left({\mathrm{Base}}_U\right)$$
9	1 2 3 6 4	doch1	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to \underset{\dot{}}{\perp }\left({\mathrm{Base}}_U\right)=\left\{\underset{\dot{}}{0}\right\}$$
10	8 9	eqtrd	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(\left\{\underset{\dot{}}{0}\right\}\right)\right)=\left\{\underset{\dot{}}{0}\right\}$$
11	5 10	syl	$$⊢\phi \to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(\left\{\underset{\dot{}}{0}\right\}\right)\right)=\left\{\underset{\dot{}}{0}\right\}$$