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Theorem pclbtwnN 37193
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 2798 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pclid.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
64, 5psubssat 37050 . . . . 5 ((𝐾𝑉𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
76adantr 484 . . . 4 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 ⊆ (Atoms‘𝐾))
8 pclid.c . . . . 5 𝑈 = (PCl‘𝐾)
94, 8pclssN 37190 . . . 4 ((𝐾𝑉𝑌𝑋𝑋 ⊆ (Atoms‘𝐾)) → (𝑈𝑌) ⊆ (𝑈𝑋))
102, 3, 7, 9syl3anc 1368 . . 3 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑌) ⊆ (𝑈𝑋))
115, 8pclidN 37192 . . . 4 ((𝐾𝑉𝑋𝑆) → (𝑈𝑋) = 𝑋)
1211adantr 484 . . 3 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑋) = 𝑋)
1310, 12sseqtrd 3955 . 2 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑌) ⊆ 𝑋)
141, 13eqssd 3932 1 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 = (𝑈𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  wss 3881  cfv 6324  Atomscatm 36559  PSubSpcpsubsp 36792  PClcpclN 37183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-psubsp 36799  df-pclN 37184
This theorem is referenced by:  pclfinN  37196
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