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Theorem pclbtwnN 39900
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 772 . 2 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 ⊆ (𝑈𝑌))
2 simpll 766 . . . 4 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝐾𝑉)
3 simprl 770 . . . 4 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑌𝑋)
4 eqid 2736 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pclid.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
64, 5psubssat 39757 . . . . 5 ((𝐾𝑉𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
76adantr 480 . . . 4 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 ⊆ (Atoms‘𝐾))
8 pclid.c . . . . 5 𝑈 = (PCl‘𝐾)
94, 8pclssN 39897 . . . 4 ((𝐾𝑉𝑌𝑋𝑋 ⊆ (Atoms‘𝐾)) → (𝑈𝑌) ⊆ (𝑈𝑋))
102, 3, 7, 9syl3anc 1372 . . 3 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑌) ⊆ (𝑈𝑋))
115, 8pclidN 39899 . . . 4 ((𝐾𝑉𝑋𝑆) → (𝑈𝑋) = 𝑋)
1211adantr 480 . . 3 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑋) = 𝑋)
1310, 12sseqtrd 4019 . 2 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑌) ⊆ 𝑋)
141, 13eqssd 4000 1 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 = (𝑈𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wss 3950  cfv 6560  Atomscatm 39265  PSubSpcpsubsp 39499  PClcpclN 39890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-psubsp 39506  df-pclN 39891
This theorem is referenced by:  pclfinN  39903
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