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Theorem pclbtwnN 39880
Description: A projective subspace sandwiched between a set of atoms and the set's projective subspace closure equals the closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclid.s 𝑆 = (PSubSp‘𝐾)
pclid.c 𝑈 = (PCl‘𝐾)
Assertion
Ref Expression
pclbtwnN (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 = (𝑈𝑌))

Proof of Theorem pclbtwnN
StepHypRef Expression
1 simprr 773 . 2 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 ⊆ (𝑈𝑌))
2 simpll 767 . . . 4 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝐾𝑉)
3 simprl 771 . . . 4 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑌𝑋)
4 eqid 2735 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pclid.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
64, 5psubssat 39737 . . . . 5 ((𝐾𝑉𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
76adantr 480 . . . 4 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 ⊆ (Atoms‘𝐾))
8 pclid.c . . . . 5 𝑈 = (PCl‘𝐾)
94, 8pclssN 39877 . . . 4 ((𝐾𝑉𝑌𝑋𝑋 ⊆ (Atoms‘𝐾)) → (𝑈𝑌) ⊆ (𝑈𝑋))
102, 3, 7, 9syl3anc 1370 . . 3 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑌) ⊆ (𝑈𝑋))
115, 8pclidN 39879 . . . 4 ((𝐾𝑉𝑋𝑆) → (𝑈𝑋) = 𝑋)
1211adantr 480 . . 3 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑋) = 𝑋)
1310, 12sseqtrd 4036 . 2 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑌) ⊆ 𝑋)
141, 13eqssd 4013 1 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 = (𝑈𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wss 3963  cfv 6563  Atomscatm 39245  PSubSpcpsubsp 39479  PClcpclN 39870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-psubsp 39486  df-pclN 39871
This theorem is referenced by:  pclfinN  39883
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