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Theorem pclssN 38753
Description: Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclss.a 𝐴 = (Atomsβ€˜πΎ)
pclss.c π‘ˆ = (PClβ€˜πΎ)
Assertion
Ref Expression
pclssN ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) βŠ† (π‘ˆβ€˜π‘Œ))
Proof of Theorem pclssN
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sstr2 3988 . . . . . 6 (𝑋 βŠ† π‘Œ β†’ (π‘Œ βŠ† 𝑦 β†’ 𝑋 βŠ† 𝑦))
213ad2ant2 1134 . . . . 5 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ (π‘Œ βŠ† 𝑦 β†’ 𝑋 βŠ† 𝑦))
32adantr 481 . . . 4 (((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) ∧ 𝑦 ∈ (PSubSpβ€˜πΎ)) β†’ (π‘Œ βŠ† 𝑦 β†’ 𝑋 βŠ† 𝑦))
43ss2rabdv 4072 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ {𝑦 ∈ (PSubSpβ€˜πΎ) ∣ π‘Œ βŠ† 𝑦} βŠ† {𝑦 ∈ (PSubSpβ€˜πΎ) ∣ 𝑋 βŠ† 𝑦})
5 intss 4972 . . 3 ({𝑦 ∈ (PSubSpβ€˜πΎ) ∣ π‘Œ βŠ† 𝑦} βŠ† {𝑦 ∈ (PSubSpβ€˜πΎ) ∣ 𝑋 βŠ† 𝑦} β†’ ∩ {𝑦 ∈ (PSubSpβ€˜πΎ) ∣ 𝑋 βŠ† 𝑦} βŠ† ∩ {𝑦 ∈ (PSubSpβ€˜πΎ) ∣ π‘Œ βŠ† 𝑦})
64, 5syl 17 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ ∩ {𝑦 ∈ (PSubSpβ€˜πΎ) ∣ 𝑋 βŠ† 𝑦} βŠ† ∩ {𝑦 ∈ (PSubSpβ€˜πΎ) ∣ π‘Œ βŠ† 𝑦})
7 simp1 1136 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ 𝐾 ∈ 𝑉)
8 sstr 3989 . . . 4 ((𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ 𝑋 βŠ† 𝐴)
983adant1 1130 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ 𝑋 βŠ† 𝐴)
10 pclss.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
11 eqid 2732 . . . 4 (PSubSpβ€˜πΎ) = (PSubSpβ€˜πΎ)
12 pclss.c . . . 4 π‘ˆ = (PClβ€˜πΎ)
1310, 11, 12pclvalN 38749 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑦 ∈ (PSubSpβ€˜πΎ) ∣ 𝑋 βŠ† 𝑦})
147, 9, 13syl2anc 584 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑦 ∈ (PSubSpβ€˜πΎ) ∣ 𝑋 βŠ† 𝑦})
1510, 11, 12pclvalN 38749 . . 3 ((𝐾 ∈ 𝑉 ∧ π‘Œ βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘Œ) = ∩ {𝑦 ∈ (PSubSpβ€˜πΎ) ∣ π‘Œ βŠ† 𝑦})
16153adant2 1131 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘Œ) = ∩ {𝑦 ∈ (PSubSpβ€˜πΎ) ∣ π‘Œ βŠ† 𝑦})
176, 14, 163sstr4d 4028 1 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) βŠ† (π‘ˆβ€˜π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3432   βŠ† wss 3947  βˆ© cint 4949  β€˜cfv 6540  Atomscatm 38121  PSubSpcpsubsp 38355  PClcpclN 38746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-psubsp 38362  df-pclN 38747
This theorem is referenced by:  pclbtwnN  38756  pclunN  38757  pclfinN  38759  pclss2polN  38780  pclfinclN  38809
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