Theorem elprjspnss

Description: A point of a projective space is a subset of the corresponding free module (an equivalence class of nonzero vectors of that module). (Contributed by SN, 23-Sep-2026)

	Ref	Expression
Hypotheses	prjspnssbas.p	⊢ 𝑃 = ( 𝑁 ℙ𝕣𝕠𝕛n 𝐾 )
	prjspnssbas.w	⊢ 𝑊 = ( 𝐾 freeLMod ( 0 ... 𝑁 ) )
	prjspnssbas.b	⊢ 𝐵 = ( ( Base ‘ 𝑊 ) ∖ { ( 0g ‘ 𝑊 ) } )
	prjspnssbas.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
	prjspnssbas.k	⊢ ( 𝜑 → 𝐾 ∈ DivRing )
	elprjspnss.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
Assertion	elprjspnss	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		prjspnssbas.p	⊢ 𝑃 = ( 𝑁 ℙ𝕣𝕠𝕛n 𝐾 )
2		prjspnssbas.w	⊢ 𝑊 = ( 𝐾 freeLMod ( 0 ... 𝑁 ) )
3		prjspnssbas.b	⊢ 𝐵 = ( ( Base ‘ 𝑊 ) ∖ { ( 0g ‘ 𝑊 ) } )
4		prjspnssbas.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
5		prjspnssbas.k	⊢ ( 𝜑 → 𝐾 ∈ DivRing )
6		elprjspnss.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
7	1 2 3 4 5	prjspnssbas	⊢ ( 𝜑 → 𝑃 ⊆ 𝒫 𝐵 )
8	7 6	sseldd	⊢ ( 𝜑 → 𝐴 ∈ 𝒫 𝐵 )
9	8	elpwid	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )