Theorem pclcmpatN

Description: The set of projective subspaces is compactly atomistic: if an atom is in the projective subspace closure of a set of atoms, it also belongs to the projective subspace closure of a finite subset of that set. Analogous to Lemma 3.3.10 of PtakPulmannova p. 74. (Contributed by NM, 10-Sep-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pclfin.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	pclfin.c	⊢ 𝑈 = ( PCl ‘ 𝐾 )
Assertion	pclcmpatN	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑃 ∈ ( 𝑈 ‘ 𝑋 ) ) → ∃ 𝑦 ∈ Fin ( 𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pclfin.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2		pclfin.c	⊢ 𝑈 = ( PCl ‘ 𝐾 )
3	1 2	pclfinN	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑈 ‘ 𝑋 ) = ∪ 𝑦 ∈ ( Fin ∩ 𝒫 𝑋 ) ( 𝑈 ‘ 𝑦 ) )
4	3	eleq2d	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑃 ∈ ( 𝑈 ‘ 𝑋 ) ↔ 𝑃 ∈ ∪ 𝑦 ∈ ( Fin ∩ 𝒫 𝑋 ) ( 𝑈 ‘ 𝑦 ) ) )
5		eliun	⊢ ( 𝑃 ∈ ∪ 𝑦 ∈ ( Fin ∩ 𝒫 𝑋 ) ( 𝑈 ‘ 𝑦 ) ↔ ∃ 𝑦 ∈ ( Fin ∩ 𝒫 𝑋 ) 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) )
6	4 5	bitrdi	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑃 ∈ ( 𝑈 ‘ 𝑋 ) ↔ ∃ 𝑦 ∈ ( Fin ∩ 𝒫 𝑋 ) 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) ) )
7		elin	⊢ ( 𝑦 ∈ ( Fin ∩ 𝒫 𝑋 ) ↔ ( 𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋 ) )
8		elpwi	⊢ ( 𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋 )
9	8	anim2i	⊢ ( ( 𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋 ) → ( 𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋 ) )
10	7 9	sylbi	⊢ ( 𝑦 ∈ ( Fin ∩ 𝒫 𝑋 ) → ( 𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋 ) )
11	10	anim1i	⊢ ( ( 𝑦 ∈ ( Fin ∩ 𝒫 𝑋 ) ∧ 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) ) → ( ( 𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋 ) ∧ 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) ) )
12		anass	⊢ ( ( ( 𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝑋 ) ∧ 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) ) ↔ ( 𝑦 ∈ Fin ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) ) ) )
13	11 12	sylib	⊢ ( ( 𝑦 ∈ ( Fin ∩ 𝒫 𝑋 ) ∧ 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) ) → ( 𝑦 ∈ Fin ∧ ( 𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) ) ) )
14	13	reximi2	⊢ ( ∃ 𝑦 ∈ ( Fin ∩ 𝒫 𝑋 ) 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) → ∃ 𝑦 ∈ Fin ( 𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) ) )
15	6 14	syl6bi	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑃 ∈ ( 𝑈 ‘ 𝑋 ) → ∃ 𝑦 ∈ Fin ( 𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) ) ) )
16	15	3impia	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑃 ∈ ( 𝑈 ‘ 𝑋 ) ) → ∃ 𝑦 ∈ Fin ( 𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ ( 𝑈 ‘ 𝑦 ) ) )