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Theorem isline2 39757
Description: Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012.)
Hypotheses
Ref Expression
isline2.j = (join‘𝐾)
isline2.a 𝐴 = (Atoms‘𝐾)
isline2.n 𝑁 = (Lines‘𝐾)
isline2.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
isline2 (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞)))))
Distinct variable groups:   𝑞,𝑝,𝐴   𝐾,𝑝,𝑞   𝑋,𝑝,𝑞
Allowed substitution hints:   (𝑞,𝑝)   𝑀(𝑞,𝑝)   𝑁(𝑞,𝑝)

Proof of Theorem isline2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . 3 (le‘𝐾) = (le‘𝐾)
2 isline2.j . . 3 = (join‘𝐾)
3 isline2.a . . 3 𝐴 = (Atoms‘𝐾)
4 isline2.n . . 3 𝑁 = (Lines‘𝐾)
51, 2, 3, 4isline 39722 . 2 (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})))
6 simpl 482 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → 𝐾 ∈ Lat)
7 eqid 2735 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
87, 3atbase 39271 . . . . . . . 8 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
98ad2antrl 728 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → 𝑝 ∈ (Base‘𝐾))
107, 3atbase 39271 . . . . . . . 8 (𝑞𝐴𝑞 ∈ (Base‘𝐾))
1110ad2antll 729 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → 𝑞 ∈ (Base‘𝐾))
127, 2latjcl 18497 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝 𝑞) ∈ (Base‘𝐾))
136, 9, 11, 12syl3anc 1370 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → (𝑝 𝑞) ∈ (Base‘𝐾))
14 isline2.m . . . . . . 7 𝑀 = (pmap‘𝐾)
157, 1, 3, 14pmapval 39740 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑝 𝑞) ∈ (Base‘𝐾)) → (𝑀‘(𝑝 𝑞)) = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})
1613, 15syldan 591 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → (𝑀‘(𝑝 𝑞)) = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})
1716eqeq2d 2746 . . . 4 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → (𝑋 = (𝑀‘(𝑝 𝑞)) ↔ 𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)}))
1817anbi2d 630 . . 3 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → ((𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞))) ↔ (𝑝𝑞𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})))
19182rexbidva 3218 . 2 (𝐾 ∈ Lat → (∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞))) ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})))
205, 19bitr4d 282 1 (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wrex 3068  {crab 3433   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  Latclat 18489  Atomscatm 39245  Linesclines 39477  pmapcpmap 39480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-lat 18490  df-ats 39249  df-lines 39484  df-pmap 39487
This theorem is referenced by:  isline3  39759  lncvrelatN  39764  linepsubclN  39934
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