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Theorem psubssat 39138
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 2726 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
2 eqid 2726 . . 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 39129 . 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 498 1 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ 𝑋 βŠ† 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   βŠ† wss 3943   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  lecple 17213  joincjn 18276  Atomscatm 38646  PSubSpcpsubsp 38880
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-psubsp 38887
This theorem is referenced by:  psubatN  39139  paddidm  39225  paddclN  39226  paddss  39229  pmodlem1  39230  pmod1i  39232  pmodl42N  39235  elpcliN  39277  pclidN  39280  pclbtwnN  39281  pclunN  39282  pclun2N  39283  pclfinN  39284  polssatN  39292  psubclsubN  39324
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