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Theorem linepsubclN 38417
Description: A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
linepsubcl.n 𝑁 = (Linesβ€˜πΎ)
linepsubcl.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
linepsubclN ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) β†’ 𝑋 ∈ 𝐢)

Proof of Theorem linepsubclN
Dummy variables π‘ž 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hllat 37828 . . . 4 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
2 eqid 2737 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
3 eqid 2737 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
4 linepsubcl.n . . . . 5 𝑁 = (Linesβ€˜πΎ)
5 eqid 2737 . . . . 5 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
62, 3, 4, 5isline2 38240 . . . 4 (𝐾 ∈ Lat β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ 𝑋 = ((pmapβ€˜πΎ)β€˜(𝑝(joinβ€˜πΎ)π‘ž)))))
71, 6syl 17 . . 3 (𝐾 ∈ HL β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ 𝑋 = ((pmapβ€˜πΎ)β€˜(𝑝(joinβ€˜πΎ)π‘ž)))))
81adantr 482 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ))) β†’ 𝐾 ∈ Lat)
9 eqid 2737 . . . . . . . . . 10 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
109, 3atbase 37754 . . . . . . . . 9 (𝑝 ∈ (Atomsβ€˜πΎ) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
1110ad2antrl 727 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ))) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
129, 3atbase 37754 . . . . . . . . 9 (π‘ž ∈ (Atomsβ€˜πΎ) β†’ π‘ž ∈ (Baseβ€˜πΎ))
1312ad2antll 728 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ))) β†’ π‘ž ∈ (Baseβ€˜πΎ))
149, 2latjcl 18329 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Baseβ€˜πΎ) ∧ π‘ž ∈ (Baseβ€˜πΎ)) β†’ (𝑝(joinβ€˜πΎ)π‘ž) ∈ (Baseβ€˜πΎ))
158, 11, 13, 14syl3anc 1372 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ))) β†’ (𝑝(joinβ€˜πΎ)π‘ž) ∈ (Baseβ€˜πΎ))
16 linepsubcl.c . . . . . . . 8 𝐢 = (PSubClβ€˜πΎ)
179, 5, 16pmapsubclN 38412 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑝(joinβ€˜πΎ)π‘ž) ∈ (Baseβ€˜πΎ)) β†’ ((pmapβ€˜πΎ)β€˜(𝑝(joinβ€˜πΎ)π‘ž)) ∈ 𝐢)
1815, 17syldan 592 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ))) β†’ ((pmapβ€˜πΎ)β€˜(𝑝(joinβ€˜πΎ)π‘ž)) ∈ 𝐢)
19 eleq1a 2833 . . . . . 6 (((pmapβ€˜πΎ)β€˜(𝑝(joinβ€˜πΎ)π‘ž)) ∈ 𝐢 β†’ (𝑋 = ((pmapβ€˜πΎ)β€˜(𝑝(joinβ€˜πΎ)π‘ž)) β†’ 𝑋 ∈ 𝐢))
2018, 19syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ))) β†’ (𝑋 = ((pmapβ€˜πΎ)β€˜(𝑝(joinβ€˜πΎ)π‘ž)) β†’ 𝑋 ∈ 𝐢))
2120adantld 492 . . . 4 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ))) β†’ ((𝑝 β‰  π‘ž ∧ 𝑋 = ((pmapβ€˜πΎ)β€˜(𝑝(joinβ€˜πΎ)π‘ž))) β†’ 𝑋 ∈ 𝐢))
2221rexlimdvva 3206 . . 3 (𝐾 ∈ HL β†’ (βˆƒπ‘ ∈ (Atomsβ€˜πΎ)βˆƒπ‘ž ∈ (Atomsβ€˜πΎ)(𝑝 β‰  π‘ž ∧ 𝑋 = ((pmapβ€˜πΎ)β€˜(𝑝(joinβ€˜πΎ)π‘ž))) β†’ 𝑋 ∈ 𝐢))
237, 22sylbid 239 . 2 (𝐾 ∈ HL β†’ (𝑋 ∈ 𝑁 β†’ 𝑋 ∈ 𝐢))
2423imp 408 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) β†’ 𝑋 ∈ 𝐢)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074  β€˜cfv 6497  (class class class)co 7358  Basecbs 17084  joincjn 18201  Latclat 18321  Atomscatm 37728  HLchlt 37815  Linesclines 37960  pmapcpmap 37963  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-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-lines 37967  df-pmap 37970  df-polarityN 38369  df-psubclN 38401
This theorem is referenced by: (None)
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