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Theorem pclssN 37564
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 3894 . . . . . 6 (𝑋𝑌 → (𝑌𝑦𝑋𝑦))
213ad2ant2 1135 . . . . 5 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑌𝑦𝑋𝑦))
32adantr 484 . . . 4 (((𝐾𝑉𝑋𝑌𝑌𝐴) ∧ 𝑦 ∈ (PSubSp‘𝐾)) → (𝑌𝑦𝑋𝑦))
43ss2rabdv 3975 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
5 intss 4867 . . 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 3895 . . . 4 ((𝑋𝑌𝑌𝐴) → 𝑋𝐴)
983adant1 1131 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → 𝑋𝐴)
10 pclss.a . . . 4 𝐴 = (Atoms‘𝐾)
11 eqid 2739 . . . 4 (PSubSp‘𝐾) = (PSubSp‘𝐾)
12 pclss.c . . . 4 𝑈 = (PCl‘𝐾)
1310, 11, 12pclvalN 37560 . . 3 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
147, 9, 13syl2anc 587 . 2 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
1510, 11, 12pclvalN 37560 . . 3 ((𝐾𝑉𝑌𝐴) → (𝑈𝑌) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦})
16153adant2 1132 . 2 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑌) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦})
176, 14, 163sstr4d 3934 1 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) ⊆ (𝑈𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2114  {crab 3058  wss 3853   cint 4846  cfv 6350  Atomscatm 36933  PSubSpcpsubsp 37166  PClcpclN 37557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7186  df-psubsp 37173  df-pclN 37558
This theorem is referenced by:  pclbtwnN  37567  pclunN  37568  pclfinN  37570  pclss2polN  37591  pclfinclN  37620
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