Theorem pointpsubN

Description: A point (singleton of an atom) is a projective subspace. Remark below Definition 15.1 of MaedaMaeda p. 61. (Contributed by NM, 13-Oct-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pointpsub.p	⊢ 𝑃 = ( Points ‘ 𝐾 )
	pointpsub.s	⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
Assertion	pointpsubN	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝑃 ) → 𝑋 ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		pointpsub.p	⊢ 𝑃 = ( Points ‘ 𝐾 )
2		pointpsub.s	⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
3		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
4	3 1	ispointN	⊢ ( 𝐾 ∈ AtLat → ( 𝑋 ∈ 𝑃 ↔ ∃ 𝑞 ∈ ( Atoms ‘ 𝐾 ) 𝑋 = { 𝑞 } ) )
5	3 2	snatpsubN	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) → { 𝑞 } ∈ 𝑆 )
6	5	ex	⊢ ( 𝐾 ∈ AtLat → ( 𝑞 ∈ ( Atoms ‘ 𝐾 ) → { 𝑞 } ∈ 𝑆 ) )
7		eleq1a	⊢ ( { 𝑞 } ∈ 𝑆 → ( 𝑋 = { 𝑞 } → 𝑋 ∈ 𝑆 ) )
8	6 7	syl6	⊢ ( 𝐾 ∈ AtLat → ( 𝑞 ∈ ( Atoms ‘ 𝐾 ) → ( 𝑋 = { 𝑞 } → 𝑋 ∈ 𝑆 ) ) )
9	8	rexlimdv	⊢ ( 𝐾 ∈ AtLat → ( ∃ 𝑞 ∈ ( Atoms ‘ 𝐾 ) 𝑋 = { 𝑞 } → 𝑋 ∈ 𝑆 ) )
10	4 9	sylbid	⊢ ( 𝐾 ∈ AtLat → ( 𝑋 ∈ 𝑃 → 𝑋 ∈ 𝑆 ) )
11	10	imp	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝑃 ) → 𝑋 ∈ 𝑆 )