Theorem pmapssat

Description: The projective map of a Hilbert lattice is a set of atoms. (Contributed by NM, 14-Jan-2012)

	Ref	Expression
Hypotheses	pmapssat.b	$$⊢B={\mathrm{Base}}_K$$
	pmapssat.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	pmapssat.m	$$⊢M=\mathrm{pmap}\left(K\right)$$
Assertion	pmapssat	$$⊢\left(K\in C\wedge X\in B\right)\to M\left(X\right)\subseteq A$$

Proof

Step	Hyp	Ref	Expression
1		pmapssat.b	$$⊢B={\mathrm{Base}}_K$$
2		pmapssat.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
3		pmapssat.m	$$⊢M=\mathrm{pmap}\left(K\right)$$
4		eqid	$$⊢{\le }_K={\le }_K$$
5	1 4 2 3	pmapval	$$⊢\left(K\in C\wedge X\in B\right)\to M\left(X\right)=\left\{p\in A|p{\le }_{K}X\right\}$$
6		ssrab2	$$⊢\left\{p\in A|p{\le }_{K}X\right\}\subseteq A$$
7	5 6	eqsstrdi	$$⊢\left(K\in C\wedge X\in B\right)\to M\left(X\right)\subseteq A$$