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Theorem pclssN 35701
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 3751 . . . . . 6 (𝑋𝑌 → (𝑌𝑦𝑋𝑦))
213ad2ant2 1129 . . . . 5 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑌𝑦𝑋𝑦))
32adantr 472 . . . 4 (((𝐾𝑉𝑋𝑌𝑌𝐴) ∧ 𝑦 ∈ (PSubSp‘𝐾)) → (𝑌𝑦𝑋𝑦))
43ss2rabdv 3824 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
5 intss 4650 . . 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 1131 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → 𝐾𝑉)
8 sstr 3752 . . . 4 ((𝑋𝑌𝑌𝐴) → 𝑋𝐴)
983adant1 1125 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → 𝑋𝐴)
10 pclss.a . . . 4 𝐴 = (Atoms‘𝐾)
11 eqid 2760 . . . 4 (PSubSp‘𝐾) = (PSubSp‘𝐾)
12 pclss.c . . . 4 𝑈 = (PCl‘𝐾)
1310, 11, 12pclvalN 35697 . . 3 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
147, 9, 13syl2anc 696 . 2 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
1510, 11, 12pclvalN 35697 . . 3 ((𝐾𝑉𝑌𝐴) → (𝑈𝑌) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦})
16153adant2 1126 . 2 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑌) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦})
176, 14, 163sstr4d 3789 1 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) ⊆ (𝑈𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wcel 2139  {crab 3054  wss 3715   cint 4627  cfv 6049  Atomscatm 35071  PSubSpcpsubsp 35303  PClcpclN 35694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-psubsp 35310  df-pclN 35695
This theorem is referenced by:  pclbtwnN  35704  pclunN  35705  pclfinN  35707  pclss2polN  35728  pclfinclN  35757
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