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Theorem pclcmpatN 37915
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.)
Hypotheses
Ref Expression
pclfin.a 𝐴 = (Atoms‘𝐾)
pclfin.c 𝑈 = (PCl‘𝐾)
Assertion
Ref Expression
pclcmpatN ((𝐾 ∈ AtLat ∧ 𝑋𝐴𝑃 ∈ (𝑈𝑋)) → ∃𝑦 ∈ Fin (𝑦𝑋𝑃 ∈ (𝑈𝑦)))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑈   𝑦,𝐾   𝑦,𝑋   𝑦,𝑃

Proof of Theorem pclcmpatN
StepHypRef Expression
1 pclfin.a . . . . . 6 𝐴 = (Atoms‘𝐾)
2 pclfin.c . . . . . 6 𝑈 = (PCl‘𝐾)
31, 2pclfinN 37914 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑈𝑋) = 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈𝑦))
43eleq2d 2824 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑃 ∈ (𝑈𝑋) ↔ 𝑃 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈𝑦)))
5 eliun 4928 . . . 4 (𝑃 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈𝑦) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈𝑦))
64, 5bitrdi 287 . . 3 ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑃 ∈ (𝑈𝑋) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈𝑦)))
7 elin 3903 . . . . . . 7 (𝑦 ∈ (Fin ∩ 𝒫 𝑋) ↔ (𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋))
8 elpwi 4542 . . . . . . . 8 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
98anim2i 617 . . . . . . 7 ((𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦𝑋))
107, 9sylbi 216 . . . . . 6 (𝑦 ∈ (Fin ∩ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦𝑋))
1110anim1i 615 . . . . 5 ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (𝑈𝑦)) → ((𝑦 ∈ Fin ∧ 𝑦𝑋) ∧ 𝑃 ∈ (𝑈𝑦)))
12 anass 469 . . . . 5 (((𝑦 ∈ Fin ∧ 𝑦𝑋) ∧ 𝑃 ∈ (𝑈𝑦)) ↔ (𝑦 ∈ Fin ∧ (𝑦𝑋𝑃 ∈ (𝑈𝑦))))
1311, 12sylib 217 . . . 4 ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (𝑈𝑦)) → (𝑦 ∈ Fin ∧ (𝑦𝑋𝑃 ∈ (𝑈𝑦))))
1413reximi2 3175 . . 3 (∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈𝑦) → ∃𝑦 ∈ Fin (𝑦𝑋𝑃 ∈ (𝑈𝑦)))
156, 14syl6bi 252 . 2 ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑃 ∈ (𝑈𝑋) → ∃𝑦 ∈ Fin (𝑦𝑋𝑃 ∈ (𝑈𝑦))))
16153impia 1116 1 ((𝐾 ∈ AtLat ∧ 𝑋𝐴𝑃 ∈ (𝑈𝑋)) → ∃𝑦 ∈ Fin (𝑦𝑋𝑃 ∈ (𝑈𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wrex 3065  cin 3886  wss 3887  𝒫 cpw 4533   ciun 4924  cfv 6433  Fincfn 8733  Atomscatm 37277  AtLatcal 37278  PClcpclN 37901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-om 7713  df-1o 8297  df-en 8734  df-fin 8737  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-lat 18150  df-covers 37280  df-ats 37281  df-atl 37312  df-psubsp 37517  df-pclN 37902
This theorem is referenced by: (None)
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