Theorem atcvr0

Description: An atom covers zero. ( atcv0 analog.) (Contributed by NM, 4-Nov-2011)

	Ref	Expression
Hypotheses	atomcvr0.z	$$⊢\underset{\dot{}}{0}=\mathrm{0.}\left(K\right)$$
	atomcvr0.c	$$⊢C={⋖}_K$$
	atomcvr0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	atcvr0	$$⊢\left(K\in D\wedge P\in A\right)\to \underset{\dot{}}{0}CP$$

Proof

Step	Hyp	Ref	Expression
1		atomcvr0.z	$$⊢\underset{\dot{}}{0}=\mathrm{0.}\left(K\right)$$
2		atomcvr0.c	$$⊢C={⋖}_K$$
3		atomcvr0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
5	4 1 2 3	isat	$$⊢K\in D\to \left(P\in A\leftrightarrow \left(P\in {\mathrm{Base}}_K\wedge \underset{\dot{}}{0}CP\right)\right)$$
6	5	simplbda	$$⊢\left(K\in D\wedge P\in A\right)\to \underset{\dot{}}{0}CP$$