Theorem polssatN

Description: The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	polssat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	polssat.p	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
Assertion	polssatN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑋 ) ⊆ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		polssat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2		polssat.p	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
3		eqid	⊢ ( PSubSp ‘ 𝐾 ) = ( PSubSp ‘ 𝐾 )
4	1 3 2	polsubN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑋 ) ∈ ( PSubSp ‘ 𝐾 ) )
5	1 3	psubssat	⊢ ( ( 𝐾 ∈ HL ∧ ( ⊥ ‘ 𝑋 ) ∈ ( PSubSp ‘ 𝐾 ) ) → ( ⊥ ‘ 𝑋 ) ⊆ 𝐴 )
6	4 5	syldan	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑋 ) ⊆ 𝐴 )