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Theorem paddatclN 39124
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 38532 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
213ad2ant1 1132 . . . . 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 39116 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† 𝐴)
6 eqid 2731 . . . . . . . 8 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
76, 3atssbase 38464 . . . . . . 7 𝐴 βŠ† (Baseβ€˜πΎ)
85, 7sstrdi 3995 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† (Baseβ€˜πΎ))
983adant3 1131 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝑋 βŠ† (Baseβ€˜πΎ))
10 eqid 2731 . . . . . 6 (lubβ€˜πΎ) = (lubβ€˜πΎ)
116, 10clatlubcl 18461 . . . . 5 ((𝐾 ∈ CLat ∧ 𝑋 βŠ† (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ))
122, 9, 11syl2anc 583 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ))
13 eqid 2731 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
14 eqid 2731 . . . . 5 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
15 paddatcl.p . . . . 5 + = (+π‘ƒβ€˜πΎ)
166, 13, 3, 14, 15pmapjat1 39028 . . . 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 1410 . . 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 39117 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) = 𝑋)
19183adant3 1131 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) = 𝑋)
203, 14pmapat 38938 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜π‘„) = {𝑄})
21203adant2 1130 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜π‘„) = {𝑄})
2219, 21oveq12d 7430 . . 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 2772 . 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 1135 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝐾 ∈ HL)
25 hllat 38537 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
26253ad2ant1 1132 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝐾 ∈ Lat)
276, 3atbase 38463 . . . . 5 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
28273ad2ant3 1134 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
296, 13latjcl 18397 . . . 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 1370 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ (((lubβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)𝑄) ∈ (Baseβ€˜πΎ))
316, 14, 4pmapsubclN 39121 . . 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 583 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)𝑄)) ∈ 𝐢)
3323, 32eqeltrd 2832 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ (𝑋 + {𝑄}) ∈ 𝐢)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   βŠ† wss 3949  {csn 4629  β€˜cfv 6544  (class class class)co 7412  Basecbs 17149  lubclub 18267  joincjn 18269  Latclat 18389  CLatccla 18456  Atomscatm 38437  HLchlt 38524  pmapcpmap 38672  +𝑃cpadd 38970  PSubClcpscN 39109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-pmap 38679  df-padd 38971  df-polarityN 39078  df-psubclN 39110
This theorem is referenced by:  pclfinclN  39125  osumcllem9N  39139  pexmidlem6N  39150
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