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Theorem pclbtwnN 38674
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 2733 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pclid.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
64, 5psubssat 38531 . . . . 5 ((𝐾𝑉𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
76adantr 482 . . . 4 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 ⊆ (Atoms‘𝐾))
8 pclid.c . . . . 5 𝑈 = (PCl‘𝐾)
94, 8pclssN 38671 . . . 4 ((𝐾𝑉𝑌𝑋𝑋 ⊆ (Atoms‘𝐾)) → (𝑈𝑌) ⊆ (𝑈𝑋))
102, 3, 7, 9syl3anc 1372 . . 3 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑌) ⊆ (𝑈𝑋))
115, 8pclidN 38673 . . . 4 ((𝐾𝑉𝑋𝑆) → (𝑈𝑋) = 𝑋)
1211adantr 482 . . 3 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑋) = 𝑋)
1310, 12sseqtrd 4020 . 2 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑌) ⊆ 𝑋)
141, 13eqssd 3997 1 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 = (𝑈𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wss 3946  cfv 6535  Atomscatm 38039  PSubSpcpsubsp 38273  PClcpclN 38664
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 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-psubsp 38280  df-pclN 38665
This theorem is referenced by:  pclfinN  38677
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