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Theorem linepmap 34562
Description: A line described with a projective map. (Contributed by NM, 3-Feb-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
linepmap (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑀‘(𝑃 𝑄)) ∈ 𝑁)

Proof of Theorem linepmap
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 simpl1 1062 . . 3 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝐾 ∈ Lat)
2 simpl2 1063 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝑃𝐴)
3 eqid 2621 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
4 isline2.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4atbase 34077 . . . . 5 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
62, 5syl 17 . . . 4 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝑃 ∈ (Base‘𝐾))
7 simpl3 1064 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝑄𝐴)
83, 4atbase 34077 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
97, 8syl 17 . . . 4 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝑄 ∈ (Base‘𝐾))
10 isline2.j . . . . 5 = (join‘𝐾)
113, 10latjcl 16975 . . . 4 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
121, 6, 9, 11syl3anc 1323 . . 3 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 𝑄) ∈ (Base‘𝐾))
13 eqid 2621 . . . 4 (le‘𝐾) = (le‘𝐾)
14 isline2.m . . . 4 𝑀 = (pmap‘𝐾)
153, 13, 4, 14pmapval 34544 . . 3 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑀‘(𝑃 𝑄)) = {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)})
161, 12, 15syl2anc 692 . 2 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑀‘(𝑃 𝑄)) = {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)})
17 eqid 2621 . . 3 {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)} = {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)}
18 isline2.n . . . 4 𝑁 = (Lines‘𝐾)
1913, 10, 4, 18islinei 34527 . . 3 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑃𝑄 ∧ {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)} = {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)})) → {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)} ∈ 𝑁)
2017, 19mpanr2 719 . 2 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → {𝑟𝐴𝑟(le‘𝐾)(𝑃 𝑄)} ∈ 𝑁)
2116, 20eqeltrd 2698 1 (((𝐾 ∈ Lat ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑀‘(𝑃 𝑄)) ∈ 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  {crab 2911   class class class wbr 4615  cfv 5849  (class class class)co 6607  Basecbs 15784  lecple 15872  joincjn 16868  Latclat 16969  Atomscatm 34051  Linesclines 34281  pmapcpmap 34284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-lub 16898  df-glb 16899  df-join 16900  df-meet 16901  df-lat 16970  df-ats 34055  df-lines 34288  df-pmap 34291
This theorem is referenced by:  cdleme3h  35023  cdleme7ga  35036
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