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Theorem pclssN 34198
 Description: Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclss.a 𝐴 = (Atoms‘𝐾)
pclss.c 𝑈 = (PCl‘𝐾)
Assertion
Ref Expression
pclssN ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) ⊆ (𝑈𝑌))

Proof of Theorem pclssN
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sstr2 3575 . . . . . 6 (𝑋𝑌 → (𝑌𝑦𝑋𝑦))
213ad2ant2 1076 . . . . 5 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑌𝑦𝑋𝑦))
32adantr 480 . . . 4 (((𝐾𝑉𝑋𝑌𝑌𝐴) ∧ 𝑦 ∈ (PSubSp‘𝐾)) → (𝑌𝑦𝑋𝑦))
43ss2rabdv 3646 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
5 intss 4433 . . 3 ({𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦} → {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦})
64, 5syl 17 . 2 ((𝐾𝑉𝑋𝑌𝑌𝐴) → {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦})
7 simp1 1054 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → 𝐾𝑉)
8 sstr 3576 . . . 4 ((𝑋𝑌𝑌𝐴) → 𝑋𝐴)
983adant1 1072 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → 𝑋𝐴)
10 pclss.a . . . 4 𝐴 = (Atoms‘𝐾)
11 eqid 2610 . . . 4 (PSubSp‘𝐾) = (PSubSp‘𝐾)
12 pclss.c . . . 4 𝑈 = (PCl‘𝐾)
1310, 11, 12pclvalN 34194 . . 3 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
147, 9, 13syl2anc 691 . 2 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
1510, 11, 12pclvalN 34194 . . 3 ((𝐾𝑉𝑌𝐴) → (𝑈𝑌) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦})
16153adant2 1073 . 2 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑌) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦})
176, 14, 163sstr4d 3611 1 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) ⊆ (𝑈𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900   ⊆ wss 3540  ∩ cint 4410  ‘cfv 5804  Atomscatm 33568  PSubSpcpsubsp 33800  PClcpclN 34191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-psubsp 33807  df-pclN 34192 This theorem is referenced by:  pclbtwnN  34201  pclunN  34202  pclfinN  34204  pclss2polN  34225  pclfinclN  34254
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