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Theorem linepsubclN 39126
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 38537 . . . 4 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
2 eqid 2731 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
3 eqid 2731 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
4 linepsubcl.n . . . . 5 𝑁 = (Linesβ€˜πΎ)
5 eqid 2731 . . . . 5 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
62, 3, 4, 5isline2 38949 . . . 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 480 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ))) β†’ 𝐾 ∈ Lat)
9 eqid 2731 . . . . . . . . . 10 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
109, 3atbase 38463 . . . . . . . . 9 (𝑝 ∈ (Atomsβ€˜πΎ) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
1110ad2antrl 725 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ))) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
129, 3atbase 38463 . . . . . . . . 9 (π‘ž ∈ (Atomsβ€˜πΎ) β†’ π‘ž ∈ (Baseβ€˜πΎ))
1312ad2antll 726 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘ž ∈ (Atomsβ€˜πΎ))) β†’ π‘ž ∈ (Baseβ€˜πΎ))
149, 2latjcl 18397 . . . . . . . 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 1370 . . . . . . 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 39121 . . . . . . 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 590 . . . . . 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 2827 . . . . . 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 490 . . . 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 3210 . . 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 406 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) β†’ 𝑋 ∈ 𝐢)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆƒwrex 3069  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  joincjn 18269  Latclat 18389  Atomscatm 38437  HLchlt 38524  Linesclines 38669  pmapcpmap 38672  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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
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 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 7368  df-ov 7415  df-oprab 7416  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-lines 38676  df-pmap 38679  df-polarityN 39078  df-psubclN 39110
This theorem is referenced by: (None)
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