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Theorem pclbtwnN 33999
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 791 . 2 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 ⊆ (𝑈𝑌))
2 simpll 785 . . . 4 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝐾𝑉)
3 simprl 789 . . . 4 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑌𝑋)
4 eqid 2604 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pclid.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
64, 5psubssat 33856 . . . . 5 ((𝐾𝑉𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
76adantr 479 . . . 4 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 ⊆ (Atoms‘𝐾))
8 pclid.c . . . . 5 𝑈 = (PCl‘𝐾)
94, 8pclssN 33996 . . . 4 ((𝐾𝑉𝑌𝑋𝑋 ⊆ (Atoms‘𝐾)) → (𝑈𝑌) ⊆ (𝑈𝑋))
102, 3, 7, 9syl3anc 1317 . . 3 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑌) ⊆ (𝑈𝑋))
115, 8pclidN 33998 . . . 4 ((𝐾𝑉𝑋𝑆) → (𝑈𝑋) = 𝑋)
1211adantr 479 . . 3 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑋) = 𝑋)
1310, 12sseqtrd 3598 . 2 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → (𝑈𝑌) ⊆ 𝑋)
141, 13eqssd 3579 1 (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 = (𝑈𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  wss 3534  cfv 5785  Atomscatm 33366  PSubSpcpsubsp 33598  PClcpclN 33989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-ov 6525  df-psubsp 33605  df-pclN 33990
This theorem is referenced by:  pclfinN  34002
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