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Theorem pclcmpatN 34205
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 34204 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑈𝑋) = 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈𝑦))
43eleq2d 2673 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑃 ∈ (𝑈𝑋) ↔ 𝑃 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈𝑦)))
5 eliun 4460 . . . 4 (𝑃 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈𝑦) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈𝑦))
64, 5syl6bb 275 . . 3 ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑃 ∈ (𝑈𝑋) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈𝑦)))
7 elin 3758 . . . . . . 7 (𝑦 ∈ (Fin ∩ 𝒫 𝑋) ↔ (𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋))
8 elpwi 4117 . . . . . . . 8 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
98anim2i 591 . . . . . . 7 ((𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦𝑋))
107, 9sylbi 206 . . . . . 6 (𝑦 ∈ (Fin ∩ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦𝑋))
1110anim1i 590 . . . . 5 ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (𝑈𝑦)) → ((𝑦 ∈ Fin ∧ 𝑦𝑋) ∧ 𝑃 ∈ (𝑈𝑦)))
12 anass 679 . . . . 5 (((𝑦 ∈ Fin ∧ 𝑦𝑋) ∧ 𝑃 ∈ (𝑈𝑦)) ↔ (𝑦 ∈ Fin ∧ (𝑦𝑋𝑃 ∈ (𝑈𝑦))))
1311, 12sylib 207 . . . 4 ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (𝑈𝑦)) → (𝑦 ∈ Fin ∧ (𝑦𝑋𝑃 ∈ (𝑈𝑦))))
1413reximi2 2993 . . 3 (∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈𝑦) → ∃𝑦 ∈ Fin (𝑦𝑋𝑃 ∈ (𝑈𝑦)))
156, 14syl6bi 242 . 2 ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑃 ∈ (𝑈𝑋) → ∃𝑦 ∈ Fin (𝑦𝑋𝑃 ∈ (𝑈𝑦))))
16153impia 1253 1 ((𝐾 ∈ AtLat ∧ 𝑋𝐴𝑃 ∈ (𝑈𝑋)) → ∃𝑦 ∈ Fin (𝑦𝑋𝑃 ∈ (𝑈𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wrex 2897  cin 3539  wss 3540  𝒫 cpw 4108   ciun 4455  cfv 5804  Fincfn 7841  Atomscatm 33568  AtLatcal 33569  PClcpclN 34191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-covers 33571  df-ats 33572  df-atl 33603  df-psubsp 33807  df-pclN 34192
This theorem is referenced by: (None)
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