Theorem dih0bN

Description: A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dih0b.b	$$⊢B={\mathrm{Base}}_K$$
	dih0b.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	dih0b.o	$$⊢\underset{\dot{}}{0}=\mathrm{0.}\left(K\right)$$
	dih0b.i	$$⊢I=\mathrm{DIsoH}\left(K\right)\left(W\right)$$
	dih0b.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	dih0b.z	$$⊢Z=0_U$$
	dih0b.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	dih0b.x	$$⊢\phi \to X\in B$$
Assertion	dih0bN	$$⊢\phi \to \left(X=\underset{\dot{}}{0}\leftrightarrow I\left(X\right)=\left\{Z\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		dih0b.b	$$⊢B={\mathrm{Base}}_K$$
2		dih0b.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
3		dih0b.o	$$⊢\underset{\dot{}}{0}=\mathrm{0.}\left(K\right)$$
4		dih0b.i	$$⊢I=\mathrm{DIsoH}\left(K\right)\left(W\right)$$
5		dih0b.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
6		dih0b.z	$$⊢Z=0_U$$
7		dih0b.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
8		dih0b.x	$$⊢\phi \to X\in B$$
9	7	simpld	$$⊢\phi \to K\in \mathrm{HL}$$
10		hlop	$$⊢K\in \mathrm{HL}\to K\in \mathrm{OP}$$
11	1 3	op0cl	$$⊢K\in \mathrm{OP}\to \underset{\dot{}}{0}\in B$$
12	9 10 11	3syl	$$⊢\phi \to \underset{\dot{}}{0}\in B$$
13	1 2 4	dih11	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge X\in B\wedge \underset{\dot{}}{0}\in B\right)\to \left(I\left(X\right)=I\left(\underset{\dot{}}{0}\right)\leftrightarrow X=\underset{\dot{}}{0}\right)$$
14	7 8 12 13	syl3anc	$$⊢\phi \to \left(I\left(X\right)=I\left(\underset{\dot{}}{0}\right)\leftrightarrow X=\underset{\dot{}}{0}\right)$$
15	3 2 4 5 6	dih0	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to I\left(\underset{\dot{}}{0}\right)=\left\{Z\right\}$$
16	7 15	syl	$$⊢\phi \to I\left(\underset{\dot{}}{0}\right)=\left\{Z\right\}$$
17	16	eqeq2d	$$⊢\phi \to \left(I\left(X\right)=I\left(\underset{\dot{}}{0}\right)\leftrightarrow I\left(X\right)=\left\{Z\right\}\right)$$
18	14 17	bitr3d	$$⊢\phi \to \left(X=\underset{\dot{}}{0}\leftrightarrow I\left(X\right)=\left\{Z\right\}\right)$$