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Theorem pclssN 39896
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 3990 . . . . . 6 (𝑋𝑌 → (𝑌𝑦𝑋𝑦))
213ad2ant2 1135 . . . . 5 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑌𝑦𝑋𝑦))
32adantr 480 . . . 4 (((𝐾𝑉𝑋𝑌𝑌𝐴) ∧ 𝑦 ∈ (PSubSp‘𝐾)) → (𝑌𝑦𝑋𝑦))
43ss2rabdv 4076 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
5 intss 4969 . . 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 1137 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → 𝐾𝑉)
8 sstr 3992 . . . 4 ((𝑋𝑌𝑌𝐴) → 𝑋𝐴)
983adant1 1131 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → 𝑋𝐴)
10 pclss.a . . . 4 𝐴 = (Atoms‘𝐾)
11 eqid 2737 . . . 4 (PSubSp‘𝐾) = (PSubSp‘𝐾)
12 pclss.c . . . 4 𝑈 = (PCl‘𝐾)
1310, 11, 12pclvalN 39892 . . 3 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
147, 9, 13syl2anc 584 . 2 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
1510, 11, 12pclvalN 39892 . . 3 ((𝐾𝑉𝑌𝐴) → (𝑈𝑌) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦})
16153adant2 1132 . 2 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑌) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦})
176, 14, 163sstr4d 4039 1 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) ⊆ (𝑈𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  wss 3951   cint 4946  cfv 6561  Atomscatm 39264  PSubSpcpsubsp 39498  PClcpclN 39889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-psubsp 39505  df-pclN 39890
This theorem is referenced by:  pclbtwnN  39899  pclunN  39900  pclfinN  39902  pclss2polN  39923  pclfinclN  39952
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