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 = ( Base ‘ 𝐾 )
	pmapssat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	pmapssat.m	⊢ 𝑀 = ( pmap ‘ 𝐾 )
Assertion	pmapssat	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) ⊆ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		pmapssat.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		pmapssat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		pmapssat.m	⊢ 𝑀 = ( pmap ‘ 𝐾 )
4		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
5	1 4 2 3	pmapval	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) = { 𝑝 ∈ 𝐴 ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } )
6		ssrab2	⊢ { 𝑝 ∈ 𝐴 ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } ⊆ 𝐴
7	5 6	eqsstrdi	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) ⊆ 𝐴 )