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Theorem psubssat 39259
Description: A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
atpsub.a 𝐴 = (Atomsβ€˜πΎ)
atpsub.s 𝑆 = (PSubSpβ€˜πΎ)
Assertion
Ref Expression
psubssat ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ 𝑋 βŠ† 𝐴)

Proof of Theorem psubssat
Dummy variables π‘ž 𝑝 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2728 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
2 eqid 2728 . . 3 (joinβ€˜πΎ) = (joinβ€˜πΎ)
3 atpsub.a . . 3 𝐴 = (Atomsβ€˜πΎ)
4 atpsub.s . . 3 𝑆 = (PSubSpβ€˜πΎ)
51, 2, 3, 4ispsubsp 39250 . 2 (𝐾 ∈ 𝐡 β†’ (𝑋 ∈ 𝑆 ↔ (𝑋 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑋 βˆ€π‘ž ∈ 𝑋 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ 𝑋))))
65simprbda 497 1 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ 𝑋 βŠ† 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058   βŠ† wss 3949   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  lecple 17247  joincjn 18310  Atomscatm 38767  PSubSpcpsubsp 39001
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 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-psubsp 39008
This theorem is referenced by:  psubatN  39260  paddidm  39346  paddclN  39347  paddss  39350  pmodlem1  39351  pmod1i  39353  pmodl42N  39356  elpcliN  39398  pclidN  39401  pclbtwnN  39402  pclunN  39403  pclun2N  39404  pclfinN  39405  polssatN  39413  psubclsubN  39445
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