Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  linepsubclN Structured version   Visualization version   GIF version

Theorem linepsubclN 34058
Description: A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
linepsubcl.n 𝑁 = (Lines‘𝐾)
linepsubcl.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
linepsubclN ((𝐾 ∈ HL ∧ 𝑋𝑁) → 𝑋𝐶)

Proof of Theorem linepsubclN
Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hllat 33471 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 eqid 2609 . . . . 5 (join‘𝐾) = (join‘𝐾)
3 eqid 2609 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
4 linepsubcl.n . . . . 5 𝑁 = (Lines‘𝐾)
5 eqid 2609 . . . . 5 (pmap‘𝐾) = (pmap‘𝐾)
62, 3, 4, 5isline2 33881 . . . 4 (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)))))
71, 6syl 17 . . 3 (𝐾 ∈ HL → (𝑋𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)))))
81adantr 479 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝐾 ∈ Lat)
9 eqid 2609 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
109, 3atbase 33397 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
1110ad2antrl 759 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑝 ∈ (Base‘𝐾))
129, 3atbase 33397 . . . . . . . . 9 (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
1312ad2antll 760 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑞 ∈ (Base‘𝐾))
149, 2latjcl 16820 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
158, 11, 13, 14syl3anc 1317 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
16 linepsubcl.c . . . . . . . 8 𝐶 = (PSubCl‘𝐾)
179, 5, 16pmapsubclN 34053 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
1815, 17syldan 485 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
19 eleq1a 2682 . . . . . 6 (((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶 → (𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) → 𝑋𝐶))
2018, 19syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → (𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) → 𝑋𝐶))
2120adantld 481 . . . 4 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋𝐶))
2221rexlimdvva 3019 . . 3 (𝐾 ∈ HL → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋𝐶))
237, 22sylbid 228 . 2 (𝐾 ∈ HL → (𝑋𝑁𝑋𝐶))
2423imp 443 1 ((𝐾 ∈ HL ∧ 𝑋𝑁) → 𝑋𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wrex 2896  cfv 5790  (class class class)co 6527  Basecbs 15641  joincjn 16713  Latclat 16814  Atomscatm 33371  HLchlt 33458  Linesclines 33601  pmapcpmap 33604  PSubClcpscN 34041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-riotaBAD 33060
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-undef 7263  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-p1 16809  df-lat 16815  df-clat 16877  df-oposet 33284  df-ol 33286  df-oml 33287  df-covers 33374  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459  df-lines 33608  df-pmap 33611  df-polarityN 34010  df-psubclN 34042
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator