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Theorem pclcmpatN 38367
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 38366 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) = βˆͺ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(π‘ˆβ€˜π‘¦))
43eleq2d 2824 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑋 βŠ† 𝐴) β†’ (𝑃 ∈ (π‘ˆβ€˜π‘‹) ↔ 𝑃 ∈ βˆͺ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(π‘ˆβ€˜π‘¦)))
5 eliun 4959 . . . 4 (𝑃 ∈ βˆͺ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(π‘ˆβ€˜π‘¦) ↔ βˆƒπ‘¦ ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (π‘ˆβ€˜π‘¦))
64, 5bitrdi 287 . . 3 ((𝐾 ∈ AtLat ∧ 𝑋 βŠ† 𝐴) β†’ (𝑃 ∈ (π‘ˆβ€˜π‘‹) ↔ βˆƒπ‘¦ ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (π‘ˆβ€˜π‘¦)))
7 elin 3927 . . . . . . 7 (𝑦 ∈ (Fin ∩ 𝒫 𝑋) ↔ (𝑦 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝑋))
8 elpwi 4568 . . . . . . . 8 (𝑦 ∈ 𝒫 𝑋 β†’ 𝑦 βŠ† 𝑋)
98anim2i 618 . . . . . . 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 616 . . . . 5 ((𝑦 ∈ (Fin ∩ 𝒫 𝑋) ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦)) β†’ ((𝑦 ∈ Fin ∧ 𝑦 βŠ† 𝑋) ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦)))
12 anass 470 . . . . 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 3083 . . 3 (βˆƒπ‘¦ ∈ (Fin ∩ 𝒫 𝑋)𝑃 ∈ (π‘ˆβ€˜π‘¦) β†’ βˆƒπ‘¦ ∈ Fin (𝑦 βŠ† 𝑋 ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦)))
156, 14syl6bi 253 . 2 ((𝐾 ∈ AtLat ∧ 𝑋 βŠ† 𝐴) β†’ (𝑃 ∈ (π‘ˆβ€˜π‘‹) β†’ βˆƒπ‘¦ ∈ Fin (𝑦 βŠ† 𝑋 ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦))))
16153impia 1118 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  βˆͺ ciun 4955  β€˜cfv 6497  Fincfn 8884  Atomscatm 37728  AtLatcal 37729  PClcpclN 38353
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-om 7804  df-1o 8413  df-en 8885  df-fin 8888  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-lat 18322  df-covers 37731  df-ats 37732  df-atl 37763  df-psubsp 37969  df-pclN 38354
This theorem is referenced by: (None)
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