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Theorem paddatclN 38820
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 38228 . . . . . 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 38812 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† 𝐴)
6 eqid 2733 . . . . . . . 8 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
76, 3atssbase 38160 . . . . . . 7 𝐴 βŠ† (Baseβ€˜πΎ)
85, 7sstrdi 3995 . . . . . 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 2733 . . . . . 6 (lubβ€˜πΎ) = (lubβ€˜πΎ)
116, 10clatlubcl 18456 . . . . 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 2733 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
14 eqid 2733 . . . . 5 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
15 paddatcl.p . . . . 5 + = (+π‘ƒβ€˜πΎ)
166, 13, 3, 14, 15pmapjat1 38724 . . . 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 38813 . . . . 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 38634 . . . . 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 7427 . . 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 2774 . 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 38233 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
26253ad2ant1 1134 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝐾 ∈ Lat)
276, 3atbase 38159 . . . . 5 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
28273ad2ant3 1136 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ 𝑄 ∈ 𝐴) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
296, 13latjcl 18392 . . . 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 38817 . . 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 2834 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 3949  {csn 4629  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lubclub 18262  joincjn 18264  Latclat 18384  CLatccla 18451  Atomscatm 38133  HLchlt 38220  pmapcpmap 38368  +𝑃cpadd 38666  PSubClcpscN 38805
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  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 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-pmap 38375  df-padd 38667  df-polarityN 38774  df-psubclN 38806
This theorem is referenced by:  pclfinclN  38821  osumcllem9N  38835  pexmidlem6N  38846
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