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Theorem isline2 39776
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 2737 . . 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 39741 . 2 (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})))
6 simpl 482 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → 𝐾 ∈ Lat)
7 eqid 2737 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
87, 3atbase 39290 . . . . . . . 8 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
98ad2antrl 728 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → 𝑝 ∈ (Base‘𝐾))
107, 3atbase 39290 . . . . . . . 8 (𝑞𝐴𝑞 ∈ (Base‘𝐾))
1110ad2antll 729 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → 𝑞 ∈ (Base‘𝐾))
127, 2latjcl 18484 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝 𝑞) ∈ (Base‘𝐾))
136, 9, 11, 12syl3anc 1373 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → (𝑝 𝑞) ∈ (Base‘𝐾))
14 isline2.m . . . . . . 7 𝑀 = (pmap‘𝐾)
157, 1, 3, 14pmapval 39759 . . . . . 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 2748 . . . 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 3220 . 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 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  Latclat 18476  Atomscatm 39264  Linesclines 39496  pmapcpmap 39499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-lat 18477  df-ats 39268  df-lines 39503  df-pmap 39506
This theorem is referenced by:  isline3  39778  lncvrelatN  39783  linepsubclN  39953
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