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Theorem paddatclN 38415
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 37823 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
213ad2ant1 1134 . . . . 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 38407 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† 𝐴)
6 eqid 2737 . . . . . . . 8 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
76, 3atssbase 37755 . . . . . . 7 𝐴 βŠ† (Baseβ€˜πΎ)
85, 7sstrdi 3957 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† (Baseβ€˜πΎ))
983adant3 1133 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝑋 βŠ† (Baseβ€˜πΎ))
10 eqid 2737 . . . . . 6 (lubβ€˜πΎ) = (lubβ€˜πΎ)
116, 10clatlubcl 18393 . . . . 5 ((𝐾 ∈ CLat ∧ 𝑋 βŠ† (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ))
122, 9, 11syl2anc 585 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ))
13 eqid 2737 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
14 eqid 2737 . . . . 5 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
15 paddatcl.p . . . . 5 + = (+π‘ƒβ€˜πΎ)
166, 13, 3, 14, 15pmapjat1 38319 . . . 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 1412 . . 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 38408 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) = 𝑋)
19183adant3 1133 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) = 𝑋)
203, 14pmapat 38229 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜π‘„) = {𝑄})
21203adant2 1132 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜π‘„) = {𝑄})
2219, 21oveq12d 7376 . . 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 2778 . 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 1137 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝐾 ∈ HL)
25 hllat 37828 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
26253ad2ant1 1134 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝐾 ∈ Lat)
276, 3atbase 37754 . . . . 5 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
28273ad2ant3 1136 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
296, 13latjcl 18329 . . . 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 1372 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ (((lubβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)𝑄) ∈ (Baseβ€˜πΎ))
316, 14, 4pmapsubclN 38412 . . 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 585 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)𝑄)) ∈ 𝐢)
3323, 32eqeltrd 2838 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ (𝑋 + {𝑄}) ∈ 𝐢)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911  {csn 4587  β€˜cfv 6497  (class class class)co 7358  Basecbs 17084  lubclub 18199  joincjn 18201  Latclat 18321  CLatccla 18388  Atomscatm 37728  HLchlt 37815  pmapcpmap 37963  +𝑃cpadd 38261  PSubClcpscN 38400
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-mpo 7363  df-1st 7922  df-2nd 7923  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-padd 38262  df-polarityN 38369  df-psubclN 38401
This theorem is referenced by:  pclfinclN  38416  osumcllem9N  38430  pexmidlem6N  38441
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