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Theorem pclcmpatN 37538
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 37537 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑈𝑋) = 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈𝑦))
43eleq2d 2818 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑃 ∈ (𝑈𝑋) ↔ 𝑃 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈𝑦)))
5 eliun 4885 . . . 4 (𝑃 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈𝑦) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈𝑦))
64, 5bitrdi 290 . . 3 ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑃 ∈ (𝑈𝑋) ↔ ∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈𝑦)))
7 elin 3859 . . . . . . 7 (𝑦 ∈ (Fin ∩ 𝒫 𝑋) ↔ (𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋))
8 elpwi 4497 . . . . . . . 8 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
98anim2i 620 . . . . . . 7 ((𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦𝑋))
107, 9sylbi 220 . . . . . 6 (𝑦 ∈ (Fin ∩ 𝒫 𝑋) → (𝑦 ∈ Fin ∧ 𝑦𝑋))
1110anim1i 618 . . . . 5 ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (𝑈𝑦)) → ((𝑦 ∈ Fin ∧ 𝑦𝑋) ∧ 𝑃 ∈ (𝑈𝑦)))
12 anass 472 . . . . 5 (((𝑦 ∈ Fin ∧ 𝑦𝑋) ∧ 𝑃 ∈ (𝑈𝑦)) ↔ (𝑦 ∈ Fin ∧ (𝑦𝑋𝑃 ∈ (𝑈𝑦))))
1311, 12sylib 221 . . . 4 ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (𝑈𝑦)) → (𝑦 ∈ Fin ∧ (𝑦𝑋𝑃 ∈ (𝑈𝑦))))
1413reximi2 3158 . . 3 (∃𝑦 ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (𝑈𝑦) → ∃𝑦 ∈ Fin (𝑦𝑋𝑃 ∈ (𝑈𝑦)))
156, 14syl6bi 256 . 2 ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑃 ∈ (𝑈𝑋) → ∃𝑦 ∈ Fin (𝑦𝑋𝑃 ∈ (𝑈𝑦))))
16153impia 1118 1 ((𝐾 ∈ AtLat ∧ 𝑋𝐴𝑃 ∈ (𝑈𝑋)) → ∃𝑦 ∈ Fin (𝑦𝑋𝑃 ∈ (𝑈𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wrex 3054  cin 3842  wss 3843  𝒫 cpw 4488   ciun 4881  cfv 6339  Fincfn 8555  Atomscatm 36900  AtLatcal 36901  PClcpclN 37524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-om 7600  df-1o 8131  df-en 8556  df-fin 8559  df-proset 17654  df-poset 17672  df-plt 17684  df-lub 17700  df-glb 17701  df-join 17702  df-meet 17703  df-p0 17765  df-lat 17772  df-covers 36903  df-ats 36904  df-atl 36935  df-psubsp 37140  df-pclN 37525
This theorem is referenced by: (None)
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