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Theorem 2polssN 38381
Description: A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2polss.a 𝐴 = (Atomsβ€˜πΎ)
2polss.p βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
2polssN ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ 𝑋 βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))

Proof of Theorem 2polssN
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 hlclat 37823 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
21ad3antrrr 729 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ 𝐾 ∈ CLat)
3 simpr 486 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ 𝑝 ∈ 𝑋)
4 simpllr 775 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ 𝑋 βŠ† 𝐴)
5 eqid 2737 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
6 2polss.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
75, 6atssbase 37755 . . . . . 6 𝐴 βŠ† (Baseβ€˜πΎ)
84, 7sstrdi 3957 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ 𝑋 βŠ† (Baseβ€˜πΎ))
9 eqid 2737 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
10 eqid 2737 . . . . . 6 (lubβ€˜πΎ) = (lubβ€˜πΎ)
115, 9, 10lubel 18404 . . . . 5 ((𝐾 ∈ CLat ∧ 𝑝 ∈ 𝑋 ∧ 𝑋 βŠ† (Baseβ€˜πΎ)) β†’ 𝑝(leβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘‹))
122, 3, 8, 11syl3anc 1372 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ 𝑝(leβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘‹))
1312ex 414 . . 3 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝐴) β†’ (𝑝 ∈ 𝑋 β†’ 𝑝(leβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘‹)))
1413ss2rabdv 4034 . 2 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ {𝑝 ∈ 𝐴 ∣ 𝑝 ∈ 𝑋} βŠ† {𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘‹)})
15 sseqin2 4176 . . . . 5 (𝑋 βŠ† 𝐴 ↔ (𝐴 ∩ 𝑋) = 𝑋)
1615biimpi 215 . . . 4 (𝑋 βŠ† 𝐴 β†’ (𝐴 ∩ 𝑋) = 𝑋)
1716adantl 483 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (𝐴 ∩ 𝑋) = 𝑋)
18 dfin5 3919 . . 3 (𝐴 ∩ 𝑋) = {𝑝 ∈ 𝐴 ∣ 𝑝 ∈ 𝑋}
1917, 18eqtr3di 2792 . 2 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ∈ 𝑋})
20 eqid 2737 . . . 4 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
21 2polss.p . . . 4 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
2210, 6, 20, 212polvalN 38380 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)))
23 sstr 3953 . . . . . 6 ((𝑋 βŠ† 𝐴 ∧ 𝐴 βŠ† (Baseβ€˜πΎ)) β†’ 𝑋 βŠ† (Baseβ€˜πΎ))
247, 23mpan2 690 . . . . 5 (𝑋 βŠ† 𝐴 β†’ 𝑋 βŠ† (Baseβ€˜πΎ))
255, 10clatlubcl 18393 . . . . 5 ((𝐾 ∈ CLat ∧ 𝑋 βŠ† (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ))
261, 24, 25syl2an 597 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ))
275, 9, 6, 20pmapval 38223 . . . 4 ((𝐾 ∈ HL ∧ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ)) β†’ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) = {𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘‹)})
2826, 27syldan 592 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) = {𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘‹)})
2922, 28eqtrd 2777 . 2 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = {𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘‹)})
3014, 19, 293sstr4d 3992 1 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ 𝑋 βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3408   ∩ cin 3910   βŠ† wss 3911   class class class wbr 5106  β€˜cfv 6497  Basecbs 17084  lecple 17141  lubclub 18199  CLatccla 18388  Atomscatm 37728  HLchlt 37815  pmapcpmap 37963  βŠ₯𝑃cpolN 38368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-p1 18316  df-lat 18322  df-clat 18389  df-oposet 37641  df-ol 37643  df-oml 37644  df-covers 37731  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816  df-pmap 37970  df-polarityN 38369
This theorem is referenced by:  polcon2N  38385  pclss2polN  38387  sspmaplubN  38391  paddunN  38393  pnonsingN  38399  osumcllem1N  38422  osumcllem11N  38432  pexmidN  38435
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