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Theorem paddatclN 38906
Description: The projective sum of a closed subspace and an atom is a closed projective subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
paddatcl.a 𝐴 = (Atomsβ€˜πΎ)
paddatcl.p + = (+π‘ƒβ€˜πΎ)
paddatcl.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
paddatclN ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ (𝑋 + {𝑄}) ∈ 𝐢)

Proof of Theorem paddatclN
StepHypRef Expression
1 hlclat 38314 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
213ad2ant1 1133 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝐾 ∈ CLat)
3 paddatcl.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
4 paddatcl.c . . . . . . . 8 𝐢 = (PSubClβ€˜πΎ)
53, 4psubclssatN 38898 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† 𝐴)
6 eqid 2732 . . . . . . . 8 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
76, 3atssbase 38246 . . . . . . 7 𝐴 βŠ† (Baseβ€˜πΎ)
85, 7sstrdi 3994 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† (Baseβ€˜πΎ))
983adant3 1132 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝑋 βŠ† (Baseβ€˜πΎ))
10 eqid 2732 . . . . . 6 (lubβ€˜πΎ) = (lubβ€˜πΎ)
116, 10clatlubcl 18458 . . . . 5 ((𝐾 ∈ CLat ∧ 𝑋 βŠ† (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ))
122, 9, 11syl2anc 584 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ))
13 eqid 2732 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
14 eqid 2732 . . . . 5 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
15 paddatcl.p . . . . 5 + = (+π‘ƒβ€˜πΎ)
166, 13, 3, 14, 15pmapjat1 38810 . . . 4 ((𝐾 ∈ HL ∧ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)𝑄)) = (((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) + ((pmapβ€˜πΎ)β€˜π‘„)))
1712, 16syld3an2 1411 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)𝑄)) = (((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) + ((pmapβ€˜πΎ)β€˜π‘„)))
1810, 14, 4pmapidclN 38899 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) = 𝑋)
19183adant3 1132 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) = 𝑋)
203, 14pmapat 38720 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜π‘„) = {𝑄})
21203adant2 1131 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜π‘„) = {𝑄})
2219, 21oveq12d 7429 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ (((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) + ((pmapβ€˜πΎ)β€˜π‘„)) = (𝑋 + {𝑄}))
2317, 22eqtr2d 2773 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ (𝑋 + {𝑄}) = ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)𝑄)))
24 simp1 1136 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝐾 ∈ HL)
25 hllat 38319 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
26253ad2ant1 1133 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝐾 ∈ Lat)
276, 3atbase 38245 . . . . 5 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
28273ad2ant3 1135 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
296, 13latjcl 18394 . . . 4 ((𝐾 ∈ Lat ∧ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (((lubβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)𝑄) ∈ (Baseβ€˜πΎ))
3026, 12, 28, 29syl3anc 1371 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ (((lubβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)𝑄) ∈ (Baseβ€˜πΎ))
316, 14, 4pmapsubclN 38903 . . 3 ((𝐾 ∈ HL ∧ (((lubβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)𝑄) ∈ (Baseβ€˜πΎ)) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)𝑄)) ∈ 𝐢)
3224, 30, 31syl2anc 584 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)𝑄)) ∈ 𝐢)
3323, 32eqeltrd 2833 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ (𝑋 + {𝑄}) ∈ 𝐢)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  {csn 4628  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  lubclub 18264  joincjn 18266  Latclat 18386  CLatccla 18453  Atomscatm 38219  HLchlt 38306  pmapcpmap 38454  +𝑃cpadd 38752  PSubClcpscN 38891
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-proset 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18387  df-clat 18454  df-oposet 38132  df-ol 38134  df-oml 38135  df-covers 38222  df-ats 38223  df-atl 38254  df-cvlat 38278  df-hlat 38307  df-pmap 38461  df-padd 38753  df-polarityN 38860  df-psubclN 38892
This theorem is referenced by:  pclfinclN  38907  osumcllem9N  38921  pexmidlem6N  38932
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