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Theorem pclssN 36897
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 3978 . . . . . 6 (𝑋𝑌 → (𝑌𝑦𝑋𝑦))
213ad2ant2 1128 . . . . 5 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑌𝑦𝑋𝑦))
32adantr 481 . . . 4 (((𝐾𝑉𝑋𝑌𝑌𝐴) ∧ 𝑦 ∈ (PSubSp‘𝐾)) → (𝑌𝑦𝑋𝑦))
43ss2rabdv 4056 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦} ⊆ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
5 intss 4895 . . 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 1130 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → 𝐾𝑉)
8 sstr 3979 . . . 4 ((𝑋𝑌𝑌𝐴) → 𝑋𝐴)
983adant1 1124 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝐴) → 𝑋𝐴)
10 pclss.a . . . 4 𝐴 = (Atoms‘𝐾)
11 eqid 2826 . . . 4 (PSubSp‘𝐾) = (PSubSp‘𝐾)
12 pclss.c . . . 4 𝑈 = (PCl‘𝐾)
1310, 11, 12pclvalN 36893 . . 3 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
147, 9, 13syl2anc 584 . 2 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
1510, 11, 12pclvalN 36893 . . 3 ((𝐾𝑉𝑌𝐴) → (𝑈𝑌) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦})
16153adant2 1125 . 2 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑌) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑌𝑦})
176, 14, 163sstr4d 4018 1 ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) ⊆ (𝑈𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2107  {crab 3147  wss 3940   cint 4874  cfv 6352  Atomscatm 36266  PSubSpcpsubsp 36499  PClcpclN 36890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-psubsp 36506  df-pclN 36891
This theorem is referenced by:  pclbtwnN  36900  pclunN  36901  pclfinN  36903  pclss2polN  36924  pclfinclN  36953
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