Theorem psubssat

Description: A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011)

	Ref	Expression
Hypotheses	atpsub.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	atpsub.s	⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
Assertion	psubssat	⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ⊆ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		atpsub.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2		atpsub.s	⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
3		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
4		eqid	⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
5	3 4 1 2	ispsubsp	⊢ ( 𝐾 ∈ 𝐵 → ( 𝑋 ∈ 𝑆 ↔ ( 𝑋 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝑋 ∀ 𝑞 ∈ 𝑋 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ( le ‘ 𝐾 ) ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) → 𝑟 ∈ 𝑋 ) ) ) )
6	5	simprbda	⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ⊆ 𝐴 )