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Theorem pclcmpatN 39430
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 = (Atomsβ€˜πΎ)
pclfin.c π‘ˆ = (PClβ€˜πΎ)
Assertion
Ref Expression
pclcmpatN ((𝐾 ∈ AtLat ∧ 𝑋 βŠ† 𝐴 ∧ 𝑃 ∈ (π‘ˆβ€˜π‘‹)) β†’ βˆƒπ‘¦ ∈ Fin (𝑦 βŠ† 𝑋 ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦)))
Distinct variable groups:   𝑦,𝐴   𝑦,π‘ˆ   𝑦,𝐾   𝑦,𝑋   𝑦,𝑃

Proof of Theorem pclcmpatN
StepHypRef Expression
1 pclfin.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
2 pclfin.c . . . . . 6 π‘ˆ = (PClβ€˜πΎ)
31, 2pclfinN 39429 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) = βˆͺ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(π‘ˆβ€˜π‘¦))
43eleq2d 2811 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑋 βŠ† 𝐴) β†’ (𝑃 ∈ (π‘ˆβ€˜π‘‹) ↔ 𝑃 ∈ βˆͺ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(π‘ˆβ€˜π‘¦)))
5 eliun 4995 . . . 4 (𝑃 ∈ βˆͺ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(π‘ˆβ€˜π‘¦) ↔ βˆƒπ‘¦ ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (π‘ˆβ€˜π‘¦))
64, 5bitrdi 286 . . 3 ((𝐾 ∈ AtLat ∧ 𝑋 βŠ† 𝐴) β†’ (𝑃 ∈ (π‘ˆβ€˜π‘‹) ↔ βˆƒπ‘¦ ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (π‘ˆβ€˜π‘¦)))
7 elin 3955 . . . . . . 7 (𝑦 ∈ (Fin ∩ 𝒫 𝑋) ↔ (𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋))
8 elpwi 4605 . . . . . . . 8 (𝑦 ∈ 𝒫 𝑋 β†’ 𝑦 βŠ† 𝑋)
98anim2i 615 . . . . . . 7 ((𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋) β†’ (𝑦 ∈ Fin ∧ 𝑦 βŠ† 𝑋))
107, 9sylbi 216 . . . . . 6 (𝑦 ∈ (Fin ∩ 𝒫 𝑋) β†’ (𝑦 ∈ Fin ∧ 𝑦 βŠ† 𝑋))
1110anim1i 613 . . . . 5 ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦)) β†’ ((𝑦 ∈ Fin ∧ 𝑦 βŠ† 𝑋) ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦)))
12 anass 467 . . . . 5 (((𝑦 ∈ Fin ∧ 𝑦 βŠ† 𝑋) ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦)) ↔ (𝑦 ∈ Fin ∧ (𝑦 βŠ† 𝑋 ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦))))
1311, 12sylib 217 . . . 4 ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦)) β†’ (𝑦 ∈ Fin ∧ (𝑦 βŠ† 𝑋 ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦))))
1413reximi2 3069 . . 3 (βˆƒπ‘¦ ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (π‘ˆβ€˜π‘¦) β†’ βˆƒπ‘¦ ∈ Fin (𝑦 βŠ† 𝑋 ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦)))
156, 14biimtrdi 252 . 2 ((𝐾 ∈ AtLat ∧ 𝑋 βŠ† 𝐴) β†’ (𝑃 ∈ (π‘ˆβ€˜π‘‹) β†’ βˆƒπ‘¦ ∈ Fin (𝑦 βŠ† 𝑋 ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦))))
16153impia 1114 1 ((𝐾 ∈ AtLat ∧ 𝑋 βŠ† 𝐴 ∧ 𝑃 ∈ (π‘ˆβ€˜π‘‹)) β†’ βˆƒπ‘¦ ∈ Fin (𝑦 βŠ† 𝑋 ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060   ∩ cin 3938   βŠ† wss 3939  π’« cpw 4598  βˆͺ ciun 4991  β€˜cfv 6543  Fincfn 8962  Atomscatm 38791  AtLatcal 38792  PClcpclN 39416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-om 7869  df-1o 8485  df-en 8963  df-fin 8966  df-proset 18286  df-poset 18304  df-plt 18321  df-lub 18337  df-glb 18338  df-join 18339  df-meet 18340  df-p0 18416  df-lat 18423  df-covers 38794  df-ats 38795  df-atl 38826  df-psubsp 39032  df-pclN 39417
This theorem is referenced by: (None)
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