This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 biimtrdi ⊢ ( ( 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 ) ) )